title: The Exact Low Energy Effective action of $\mathcal{N}=2$ SYM via Seiberg-Witten Theory and Localisation
authors: Robert Pryor
text: 
   
   
  
  
  
  
  
  
  
  
  shapes,positioning,intersections,quotes,decorations.markings
  
  
  
  
  
  
  
  
  "
  
  
  
   §.§ 
  [runin]
    1em
    
  
  
    §.§.§ 
  [runin]
    1em 
  
  
  
  
  
  remark[2][Remark]
  
    #1#2.                                 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  The Exact Low Energy Action of =2 SYM via Seiberg-Witten Theory and Localisation
  
      Robert Pryor
  
  School of Mathematics and Statisitcs
  
  University of Melbourne
  
  Submitted for the degree of Master of Science
  
      May 2020
  ===============================================================================================================================
  
  
  In the mid 1990's Seiberg and Witten determined the exact low energy effective action of =2 supersymmetric Yang-Mills theory with gauge group SU(2). Later, in the early 2000's Nekrasov calculated this action directly using localisation techniques. This work serves as an introduction to the area, developing both approaches and reconciling their results.
  
  
  
  
  
  roman
  
  
  
  CHAPTER: ACKNOWLEDGEMENTS
  
  
  I would like to thank my supervisor, the late Professor Omar Foda, for his encouragement, insight and patience. I consider myself very lucky to have had the opportunity to work with someone who cared so much for his field and the people in it. Without his guidance I would have remained unaware of the fascinating world of supersymmetry.
  
  I would also like to thank Doctor Thomas Quella for his support and assistance in the final stage of preparing my thesis. In particular I appreciate both his readiness to review my thesis and his offer to put me in contact with others willing to do the same. Finishing this thesis would have been much more difficult without his help.
   
  I would also like to thank Doctor Johanna Knapp and Doctor David Ridout for taking the time to help review my thesis. Their support, insight and willingness to help was invaluable. Their constructive criticism has undoubtedly led to a better finished work than would have otherwise been possible.
  
  Finally, I would like to thank my partner, Gaia, for supporting my choice to pursue mathematics, and without whom none of this would have been possible.
  
  
  
  
  
  
  arabic
  
  
  
  CHAPTER: INTRODUCTION
  
  tocchapterIntroduction
  This thesis aims to introduce the fundamentals of supersymmetric quantum field theory and in particular four dimensional =2 supersymmetric Yang-Mills theory (=2 SYM). In particular we will find that the low energy effective action of =2 SYM is completely determined by a certain holomorphic function called the prepotential. Seiberg-Witten theory will be used to calculate the prepotential for gauge group SU(2) <cit.>. The prepotential will then be calculated again using the localisation techniques of Nekrasov, this time for gauge group SU(N) <cit.>.
  
  
  § THE SEIBERG-WITTEN APPROACH.
  
  The Seiberg-Witten approach proceeds by identifying and imposing certain consistency conditions on the form of the prepotential. More precisely, we examine the moduli space of vacua,  of =2 SYM and the metric on this space. It turns out that the prepotential can be obtained from this metric <cit.>.
  
  The consistency conditions are obtained by considering certain singularities located in regions of weak and strong coupling. The weak coupling singularity can be analysed directly, while the strong coupling singularities can be understood using Seiberg-Witten duality. This duality maps strongly coupled regions of =2 SYM to weakly coupled regions and provides a dual set of coordinates on  which are valid where the original set are not <cit.>.
  
  Examining the monodromy properties of the coordinates about the singularities of  proves to be sufficient to obtain the prepotential <cit.>. We discuss two different ways to do this.
  
  The first method proceeds by identifying the coordinates on  with solutions to a certain differential equation which is fixed uniquely by the monodromy properties.
  
  The second method is the original approach of Seiberg and Witten <cit.>. It involves identifying points on  with certain elliptic curves. The period integrals of these curves are then naturally identified with the coordinates on . Both these approaches yield the SU(2) prepotential in terms of hypergeometric functions.
  
  
  § THE NEKRASOV APPROACH.
  
  The Nekrasov approach uses localisation techniques to construct the instanton partition function, Z_inst explicitly. The prepotential can then be extracted from Z_inst by applying localisation techniques a second time <cit.>.
  
  The instanton partition function is defined in terms of a path integral which we will refer to as the partition function path integral. To obtain Z_inst explicitly, we localise this path integral to the instanton moduli space. It turns out that the k-instanton moduli space is finite dimensional and can be produced explicitly by the so-called ADHM construction <cit.>. This procedure yields Z_inst as a power series in the vacuum expectation values of certain scalar fields, where each term is given by a certain contour integral.
  
  The instanton partition function can be related to the prepotential by considering =2 SYM in a certain deformed spacetime known as the Ω-background <cit.>. Upon localisation, this procedure allows us to obtain Z_inst in the Ω-background in terms of a two-parameter generalisation of the prepotential. Equating the two results in the undeformed limit of the Ω-background then yields the prepotential explicitly as a power series.
  
  
  § RECONCILIATION.
  
  The two approaches discussed above give the prepotential in somewhat different forms. The first in terms of the period integrals of certain elliptic curves, and the second as a power series in certain vacuum expectation values. We aim to reconcile these two results.
  
  The power series form is obtained from the Seiberg-Witten approach by simply expanding the period integrals and performing some power series manipulations <cit.>.
  
  The Seiberg-Witten solution is obtained from the localisation approach by studying the many instanton limit. This limit recovers the Seiberg-Witten differential and thus the elliptic curve approach <cit.>. 
  
  
  § REFORMULATION IN TERMS OF YOUNG DIAGRAMS.
  
  The singularities of the contour integrals encountered when calculating Z_inst via localisation are in one to one correspondence with objects called coloured partitions <cit.>. Reformulating the instanton partition function in terms of these objects gives another interpretation of many of the quantities previously encountered.
  
  
  CHAPTER: THE BASICS OF SUPERSYMMETRY
  
  This section provides a brief introduction to supersymmetric quantum field theory (SUSY QFT). We introduce the SUSY Poincaré algebra and its representations, the superfield formalism and the construction of supersymmetric Lagrangians. In particular the Lagrangian of =2 SYM will be obtained. For a more comprehensive introduction to the topic see <cit.>. For an introduction to ordinary QFT see for example <cit.>.
  
  
  § ORDINARY QUANTUM FIELD THEORY
  
  A quantum field theory can be specified by an action. An action, S is a scalar functional of the fields in the theory. The action is related to the partition function, Z by the path integral approach. The path integral approach consists of taking an often ill-defined integral over an in general uncountably infinite dimensional function space consisting of the fields in the theory:
  
      Z=∫𝒟Xe^iS[X]
  
  This can be viewed as a generalisation of the definition of the ordinary partition function as a sum over states to an infinite dimensional quantum situation.
  
  1.1
  Throughout this thesis we set ħ=1 as in (<ref>).
  
  
  For now we are mostly concerned with fields defined on flat 4-dimensional Minkowski spacetime, ℝ^1,3. The symmetry group of Minkowski space is known as the Poincaré group, and the corresponding Lie algebra is known as the Poincaré algebra. Physically, the Poincaré group is generated by four translations, three rotations and three boosts. The infinitesimal rotations (J_i), and the infinitesimal boosts (K_i), generate a Lie subalgebra of the the Poincaré algebra known as the Lorentz algebra. Its Lie bracket is as follows:
  
      [J_i,J_j]=iϵ_ijkJ_k,   
      [K_i,K_j]=-iϵ_ijkJ_k,   
      [J_i,K_j]=iϵ_ijkK_j ,   
      i,j,k∈{1,2,3}
  
  Complexifying the Lorentz algebra by defining J_k^±=1/2(J_k± iK_k), leads to the following Lie bracket:
  
      [J_i^±,J_j^±]=iϵ_ijkJ_k^±,   
      [J_i^±,J_j^∓]=0
  
  So the complexified Lorentz algebra is isomorphic to the direct sum of two copies of the simple complex Lie algebra 𝔰𝔩(2), or equivalently to the non-simple Lie algebra 𝔰𝔬(4). Often the distinction between the complexified Lorentz algebra and the Lorentz algebra itself will be ignored.
  
  To form the full Poincaré algebra, we must add the four linearly independent translations P_μ. Defining J_0i=K_i, J_ij=ϵ_ijkJ_k, and J_μν=-J_νμ, the resulting commutation relations are <cit.>:
  
      [J_μν,J_ρσ]
         =
      ig_νρJ_μσ-ig_μρJ_νσ-ig_νσJ_μρ+ig_μσM_νρ
      
      [J_μν,P_ρ]
         =
      -ig_ρμP_ν+ig_ρνP_μ
      
      [P_μ,P_ν]
         =
      0
  
  where g_μν=diag(+1,-1,-1,-1) is the Minkowski metric.
  
  One then identifies elementary particles with irreducible representations of the Poincaré algebra.
  
  A representation of the Poincaré algebra on the space of fields is given by the following differential operators:
  
      P_μ   =
      i_μ
      
      J_μν   =
      ix_μ_ν-ix_ν_μ+S_μν
  
  where S_μν is a spin operator.
  
  
  § SUPERSYMMETRY AND THE SUPER POINCARÉ ALGEBRA
  
  Quantum field theories are very hard to analyse in anything more than the perturbative regime, however it is known that they exhibit many interesting non-perturbative phenomena <cit.>. In general we expect a system with more symmetries to be easier to analyse. To this end it is interesting to consider the effect of adding additional symmetries to a given QFT.
  
  The Coleman-Mandula theorem states that the spacetime Poincaré symmetry of a QFT cannot interact non-trivially with any internal symmetry  <cit.>. This theorem is however posited upon a Lie algebraic symmetry structure. If we allow instead a Lie superalgebraic symmetry structure, a non-trivial interaction can be obtained. The idea of supersymmetry is to allow such a symmetry structure, it is achieved by introducing extra fermionic generators known as supercharges.
  
  The more supercharges we incorporate, the less realistic but easier to analyse a theory becomes. The number of supercharges is denoted by . Here we will only be concerned with the cases 𝒩=1, and 𝒩=2.
  
  Upon introducing fermionic supercharges Q_α^A, and Q̅_A,α̇ with spinor indices α=1,2 and α̇=1̇,2̇, as well as supercharge indices A=1,2,…,𝒩, it can be shown that we must have the following commutation and anticommutation relations <cit.>:
  
      [P_μ,Q_α^A]   =0
       ,                  
      [P_μ,Q̅_A,α̇]=0
      
      
      [J_μν,Q_α^A]   =i(σ_μν)_α^βQ_β^A ,           
      [J_μν,Q̅^α̇_A]=i(σ̅_μν)^α̇_β̇Q̅^β̇_A
      {Q_α^A,Q_β^B}   =ϵ_αβZ^AB𝒵 ,     {Q̅_A,α̇,Q̅_N,β̇}=ϵ_α̇β̇Z^*_ABZ
      
      {Q_α^A,Q̅_B,β̇}   =2(σ^μ)_αβ̇P_μδ_A^B
  
  In the above, Z^AB is an 𝒩×𝒩 antisymmetric matrix and 𝒵 is a central element called the central charge. The various σ matrices and indices are defined and explained in Appendix A. The resulting Lie superalgebra is called the SUSY Poincaré algebra, or for short just the SUSY algebra.
  1.2
  Since Z^AB is antisymmetric, if =1, then Z^AB=0. Thus for =1 "unextended", SUSY the central charge is absent.
  
  
  
  § PARTICLES AND IRREDUCIBLE REPRESENTATIONS
  
  In ordinary QFT, elementary particles are identified with irreducible representations of the Poincaré algebra, so in analogy we seek irreducible representations of the SUSY Poincaré algebra. The usual situation is that irreducible representations of the SUSY algebra restrict to reducible representations of the Poincaré algebra and thus correspond to several particles rather than to just one. These sets of particles which transform together irreducibly under the SUSY Poincaré algebra are called supermultiplets. To build supersymmetric theories, we seek to classify these supermultiplets and the corresponding representations.
  
  1.3
  We often abuse terminology and identify supermultiplets with the corresponding representations.
  
  
  For our purposes we only need to classify the massless irreducible representations of spin ≤ 1, since these are the only multiplets involved in pure Yang-Mills theory. The classification is actually rather straightforward but to avoid getting off topic we simply quote the results. More details can be found in <cit.>. 
  1.4
  Although SYM is built from massless supermultiplets, not all particles will be massless. The Higgs mechanism will still be able to generate masses, but we will not need to work with inherently massive supermultiplets.
  
  
  
   §.§ =1 Supermultiplets.
  
  For =1 there are only two supermultiplets of spin not exceeding 1, namely the =1 chiral and vector multiplets.
  
  The =1 chiral multiplet corresponds to a Weyl spinor and a complex scalar in an arbitrary representation of the gauge group. For definitions regarding spinors, see Appendix A.
  
  The =1 vector multiplet corresponds to a massless vector particle (that is to say a massless fundamental representation of the Lorentz group), and a Weyl spinor both necessarily in the adjoint representation of the gauge group.
  
  
   §.§ =2 Supermultiplets.
  
  For =2 there are again only two relevant supermultiplets, namely the =2 vector (or chiral), multiplet, and the =2 hypermultiplet.
  
  The =2 vector multiplet consists of a massless vector particle, a complex scalar and two Weyl spinors, all necessarily in the adjoint representation of the gauge group. The =2 hypermultiplet corresponds to two scalars and two Weyl spinors in an arbitrary representation of the gauge group.
  
  
  § SUPERSPACE, SUPERFIELDS AND SUPERMULTIPLETS
  
  To build supersymmetric field theories we will require representations of the SUSY algebra on spaces of fields. To this end we introduce the notions of superspaces and superfields.
  
  
   §.§ Superspace
  
  Supersymmetric theories are naturally defined on so-called supermanifolds. A supermanifold is a generalisation of a manifold to include fermionic (i.e. Grassmann valued), coordinates. Here the word superspace will refer to the coordinate space of a supermanifold on which a theory of interest is defined.
  
  Rather than delving deeply into the theory of supermanifolds, for the case at hand we simply take our superspace to be a space having d bosonic spacetime coordinates x^μ,  fermionic "left-handed" Grassmann coordinates θ_A^α,  "right-handed" Grassmann coordinates θ̅^A,α̇, and finally one more bosonic coordinate z. Since each Grassmann coordinate has two components, our superspace has d+1 bosonic coordinates and 4 real fermionic coordinates, its superdimension is then denoted (d+1,4).
  
  1.5
  Since Grassmann numbers anticommute, any product involving more than two instances of the same Grassmann coordinate will vanish.
  
  
  A superfield is simply a function defined on superspace. The space of superfields will play the part of our representation space. To this end we write down a set of differential operators which give a (typically reducible), representation of the SUSY algebra on superfields:
  
      𝒵   =
      i/ z
      
      Q_α^A   =
      /θ_A^α
      +
      iσ_αβ̇^μθ̅^A,β̇_μ
      +
      i/2ϵ_αβZ^ABθ^β_B/ z
      Q̅_A,α̇   =
      /θ̅^A,α̇
      +
      iθ_A^βσ_βα̇^μ_μ
      +
      i/2ϵ_α̇β̇Z^*_ABθ̅^B,β̇/ z
  
  where the rest of the representation is given by (<ref>).
  
  
   §.§ SUSY covariant derivatives.
  
  Our first goal is to write =1 and =2 supersymmetric actions; actions which transform trivially under the SUSY algebra. To do so, the irreducible representations of the SUSY algebra must be realised on superspace. To this end it will be helpful to introduce the SUSY covariant derivatives:
  
      _μ   =i∂_μ,     _z=i/ z
      _α^A   =
      /θ_A^α
      -
      iσ_αβ̇^μθ̅^A,β̇_μ
      -
      i/2ϵ_αβZ^ABθ^β_B/ z
      _A,α̇   =
      /θ̅^A,α̇
      -
      iθ^β_Aσ_βα̇^μ_μ
      -
      i/2ϵ_α̇β̇A_AB^*θ̅^B,β̇/ z
  
  The SUSY covariant derivatives are derived by transforming an ordinary derivative with respect to a supercoordinate, then shifting by the non-covariant part of the transformation <cit.>.
  
  It is straightforward to show that the SUSY covariant derivatives anticommute with the supercharges.
  
  An important property of the SUSY covariant derivative is that the quantity y^μ=x^μ-iθ_Aσ^μθ̅^A is covariantly constant in the θ̅^A,α̇ and z directions:
  
      _A,α̇y^μ
      =
      0
      -
      i
      (
      /θ̅^A,α̇(θ_B^γσ^μ_γγ̇θ̅^B,γ̇)
      +
      θ_A^βσ_βα̇^ν_νx^μ)
      =
      -θ_A^γσ^μ_γα̇
      +
      θ_A^βσ^μ_βα̇
      =
      0
  
  This constraint will allow us to explicitly determine some supermultiplets. In preparation, note that for the =1 case Remarks 1.2 and 1.5 imply that a general scalar superfield can be expanded as a finite power series in the Grassmann coordinates:
  
      F(x,θ,θ̅)
      =
      f(x)+θψ(x)+θ̅χ̅(x)+θ^2m(x)+θ̅^2n(x)+θσ^μθ̅A_μ(x)+θ^2θ̅λ̅(x)+θ̅^2θρ(x)+θ^2θ̅^2D(x)
  
  The spinor indices on θ and θ̅ have been suppressed in the above expression.
  
  
   §.§ The =1 chiral multiplet.
  
  To construct the =1 chiral multiplet, consider a scalar function on superspace Φ(x,θ,θ̅), as in (<ref>). This provides an irreducible scalar representation of the Lorentz algebra. However, since the covariant derivative anticommutes with the supercharges, Schur's Lemma says that the representation of the SUSY algebra generated by Φ is reducible. To get an irreducible representation, we fix the constraint 𝒟̅_α̇Φ(x,θ,θ̅)=0. Using that y is covariantly constant this constraint is easy to solve:
  
      Φ(y,θ)=H(y)+√(2)θψ(y)+θ^2f(y)
  
  In the above expression H is a complex scalar field, ψ is a Weyl spinor, and f is an auxiliary scalar field which will be integrated out. The field content of Φ is thus a complex scalar and a Weyl spinor. This is exactly the field content of the chiral scalar multiplet, so we conclude that Φ yields an irreducible representation of this multiplet. Note that Φ can in general be in any representation of the gauge group.
  
  
   §.§ The =1 vector multiplet.
  
  To construct the =1 vector multiplet, we once again start with a general scalar function (<ref>) on =1 superspace and this time impose the reality condition V(x,θ,θ̅)=V(x,θ,θ̅)^†. 
  This leads to the following conditions on the component fields:
  
      f=f^* ,   ψ=χ ,   
      m=n^* ,   
      A_μ^†=A_μ ,   λ=ρ ,   
      D=D^*
  
  We now plug these results back into V(x,θ,θ̅). Upon rescaling and shifting some of the resulting terms for future convenience, we end up with the following expansion:
  
      V(x,θ,θ̅)
      =   
      f
      +
      iθχ
      -
      iθ̅χ̅
      +
      θσ^μθ̅A_μ
      +
      i/2θ^2(M+iN)
      -i/2θ̅^2(M-iN)
      +
      iθ^2θ̅(λ̅+i/2σ̅^μ_μχ)
      
         -
      iθ̅^2θ(λ-i/2σ^μ_μχ̅)
      +
      1/2θ^2θ̅^2(D-1/2^2f)
  
  where -1/2N:=(m), and 1/2M:=(m).
  
  We claim that the corresponding representation is still reducible. To see why, note that V contains 8 bosonic and 8 fermionic components. The classification of irreducible representations of the SUSY algebra (Section 1.3), says that this is impossible for an irreducible representation. We thus seek to gauge out some components. This is done by making the following transformation:
  
      e^2V↦ e^-iΛ^†e^2Ve^iΛ
  
  where Λ(y,θ) is a chiral superfield. Under such a transformation A_μ(x)↦ A_μ(x)-∇_μα(x), where α(x) is the scalar component of Λ, and ∇_μ:=∂_μ-i[A_μ(x),·] is the gauge covariant derivative in the adjoint representation. So indeed this is the supersymmetric generalisation of a gauge transformation. Demanding that this gauge transformation is a symmetry of the action sets χ,f,M, and N to zero. This is known as the Wess-Zumino gauge. It results in the following superfield:
  
      V_WZ(x,θ,θ̅)=θσ^μθ̅A_μ(x)-iθ̅^2θλ(x)+iθ^2θ̅λ̅(x)+1/2θ^2θ̅^2D(x)
  
  Since D is an auxiliary field which will be integrated out, V_WZ has the correct 6 degrees of freedom. We thus identify it with the abstract =1 vector multiplet. Note also that in this expression each component is necessarily in the adjoint representation of the gauge group G, so V=V_aT^a, where the T^a are a basis for Lie(G).
  1.6
  The field V_WZ has the useful property that each term is of degree at least one in θ and θ̅, so that  V_WZ^n=0, for n≥ 3. In particular, e^V_WZ=1+V_WZ+1/2V_WZ^2. The subscript WZ will be suppressed from now on.
  
  According to the classification theorem of Section 1.3, we have now constructed superspace representations of all =1 supermultiplets of spin not exceeding 1.
  
  
  § RENORMALIZABLE SUPERSYMMETRIC ACTIONS
  
  In this section we will obtain the most general =1 supersymmetric Lagrangian for a single chiral multiplet and for a single vector multiplet. Minimal coupling of these theories will then lead to =2 SYM.
  
  
   §.§ A single =1 chiral multiplet.
  
  Let F and W be gauge invariant superfields and in addition suppose W is chiral. It can be shown that the following Lagrangian is SUSY and gauge invariant  <cit.>:
  
      =∫ d^2θ d^2θ̅F(x,θ,θ̅)+∫ d^2θ W(Φ)+∫ d^2θ̅ W(Φ)^†
  
  The superfield W is known as a superpotential. For a renormalizable theory W must be a polynomial of degree less than 3 in H, and F must be of the form Φ^†KΦ for a Hermitian matrix K. We can thus take:
  
      _=1, matter   =
      Tr(
      ∫ d^2θ d^2θ̅Φ^†Φ
      +
      (
      ∫ d^2θ W(Φ)+h.c.
      )
      )
      
         =
      Tr(
      |∂_μH|^2
      -
      iψσ^μ∂_μψ̅
      +
      (
      ∫ d^2θ W(Φ)+h.c.
      )
      )
  
  where the trace is over the gauge group indices which have been suppressed.
  
  One recognizes the usual kinetic terms for a complex scalar and a spinor as well as possibly some interaction governed by W(Φ), so this is indeed a supersymmetric theory of matter.
  
  
   §.§ A single =1 vector multiplet.
  
  To construct SUSY invariant Lagrangians using the =1 vector superfield we define a spinorial superfield, the SUSY field strength:
  
      W_α(x,θ,θ̅)=-1/4_α̇^α̇e^-2V_WZ_αe^2V_WZ
  
  1.7
  Since ^3=0, we have that W_α=0 and so W_α is a chiral superfield.
  
  Recalling the definition of field strength F_μν=∂_μA_ν-∂_νA_μ-i[A_μ,A_ν], and expanding W_α in terms of the component fields yields the following expression:
  
      W_α(y,θ)
      =
      -iλ_α(y)
      +
      θ_αD(y)
      -
      i(σ^μν)_α^βθ_βF_μν(y)
      -
      θ^βθ_βσ^μ_αβ̇∇_μλ̅^β̇(y)
  
  Rescaling the fields by a real coupling constant g and defining the complexified coupling constant τ=Θ/2π+4π i/g^2, the most general renormalizable SUSY and gauge invariant 
  Lagrangian for a single vector superfield is (up to normalization) <cit.>:
  
      _=1, gauge
      =
      1/32πτ∫ d^2θTr W^αW_α
      =
      Tr(-1/4F_μνF^μν-iλσ^μ∇_μλ̅+1/2D^2)+Θ/32π^2g^2Tr F_μν(⋆F)^μν
  
  where (⋆F)^μν:=1/2ϵ^μνρσF_ρσ is the dual field strength tensor.
  
  Note that in the above expression, the trace is over the gauge group indices, which we have suppressed.
  
  The quantity Θ is known as the the instanton angle. It is a real parameter which multiplies the topological part of the action. This part of the action corresponds to instanton configurations, field configurations which obey the classical equations of motion and give a finite non-zero contribution to the action <cit.>. Instantons will be very important later when we discuss localisation.
  
  
   §.§ Minimal coupling.
  
  With Lagrangians for both =1 multiplets at hand, we consider minimal coupling of these theories. Minimal coupling amounts to simply putting the chiral multiplet in some representation of the gauge group (not necessarily the adjoint), and taking Φ^†Φ→Φ^†e^2g  VΦ, which swaps ordinary derivatives for gauge covariant derivatives and introduces the minimal interaction terms necessary for SUSY invariance <cit.>:
  
      _=1, coupled   =
      Tr(
      ∫ d^2θ d^2θ̅Φ^†e^2gV_WZΦ
      +
      ∫ d^2θ W(Φ)
      +
      ∫ d^2θ W(Φ)^†)
      
         =
      Tr(
      |∇_μH|^2
      -
      iψ̅σ̅^μ∇_μψ
      +
      f^†f
      -
      gH^†[D,H]
      -
      i√(2)gH^†{λ,ψ}
         
      +
      i√(2)gψ̅[λ̅,H]
      +
      ∫ d^2θ W(Φ)
      +
      ∫ d^2θ W(Φ)^†)
      
         =
      Tr(
      |∇_μH|^2-iψσ^μ∇_μψ̅
      +
      i√(2)gH^†λψ
      -
      i√(2)gψ̅λ̅ψ
      +
      ∫ d^2θ W(Φ)
      +
      ∫ d^2θ W(Φ)^†)
  
  Here the last equality is up to a total derivative and has been obtained using the equations of motion for f and D. Note that Lagrangians which differ by a total derivative give the same action provided the fields decay sufficiently quickly at spatial infinity.
  
  
   §.§ =2 SYM.
  
  According to the classification theorem of Section 1.3, the =2 chiral multiplet has the same field content as the combination of an =1 vector and an =1 chiral scalar multiplet. Thus we might hope that some linear combination of _=1,coupled and _=1,gauge will have =2 supersymmetry. This turns out to be the case, yielding the Lagrangian of  =2 Yang Mills theory.
  
  Up to normalisation, the unique linear combination possessing =2 supersymmetry is: 
  
      _=2, YM:=1/g^2_=1, coupled+_=1, gauge
  
  with W(Φ)=0, and Φ in the adjoint representation <cit.>. To make it more clear that _=2,YM is in fact an =2 SUSY invariant Lagrangian, we introduce the =2 chiral superfield in the superspace formalism.
  
  Similarly to the case of the chiral =1 superfield, the =2 chiral superfield is defined by the condition that _A,α̇Ψ(x,θ,θ̅,z)=0. It can be shown that this condition ensures Ψ is in fact independent of z, then expanding as a truncated power series and regrouping terms we find that Ψ can be written in terms of a pair of =1 chiral multiplets Φ and 𝒢, as well as the supersymmetric field strength:
  
      Ψ(y,θ)=Φ(y,θ_1)+i√(2)θ_2W(y,θ_1)+θ_2^2𝒢(y,θ_1)
  
  where Φ and 𝒢 are related by 
  𝒢(y,θ)
  =
  -1/2∫ d^2θ̅Φ^†(y-2iθσθ̅,θ̅)e^2V(y,θ,θ̅)
  <cit.>.
  
  With this notation we can then rewrite _YM up to normalisation as:
  
      _=2, YM=τ∫ d^2θ_1d^2θ_2TrΨ^2
  
  Since Ψ^2 is a chiral superfield, the above expression is clearly =2 SUSY and gauge invariant. The expressions (<ref>) and (<ref>) can be reconciled by expanding out (<ref>) explicitly and carrying out one of the θ integrations in (<ref>) <cit.>.
  1.8
  Unlike the =1 case, for =2 no non-trivial superpotential is permitted. This is due to the uniqueness of (<ref>).
  
  
  
  § =2 SYM FROM DIMENSIONAL REDUCTION
  
  There is another way to obtain the =2 SYM action which will be utilised later when we discuss Lorentz deformation and the Ω-background. This approach consists of reducing an =1 theory in six dimensions to an =2 theory in four dimensions.
  
  Consider =1 SYM in six dimensions. We compactify spacetime as ℝ^3,1×𝕋^2 by taking the compactified coordinates to be x^4 and x^5 with radii of compactification R_4 and R_5 respectively.
  
  Up to normalisation, the =1 d=6 SYM action is <cit.>:
  
      S_=1, d=6
      =
      1/g^2∫ d^4xTr(
      -1/4F_IJF^IJ
      +
      i/2Ψ̅_AΓ^I∇_IΨ^A)
  
  where Ψ^A is a six dimensional Weyl spinor and the Γ^I:=γ_6^I are certain 8× 8 matrices defined in appendix A.
  
  To compactify the theory we assume that the radii R_4,5 are so small that all fields are independent of the corresponding coordinates. This allows the field strength tensor to be simplified as follows:
  
      F_μ 4
      =
      ∂_μA_4-∂_4A_μ-i[A_μ,A_4]
      =
      ∂_μA_4-i[A_μ,A_4]
      =
      ∇_μA_4
  
  And likewise F_μ 5=∇_μA_5.
  
  We suggestively define the following complex scalar field: H=1/√(2)(A_4+iA_5). This definition allows  another component of F_IJ to be simplified:
  
      F_45
      =
      ∂_4A_5-∂_5A_4-i[A_4,A_5]
      =
      0-i[1/√(2)(H+H^†),1/i√(2)(H-H^†)]
      =
      -1/2[H+H^†,H-H^†]
      =
      [H,H^†]
  
  The gauge kinetic term of the Lagrangian can now be written as follows:
  
      -1/4F_IJF^IJ   =
      -1/4(
      F_μνF^μν
      +
      2F_μ 4F^μ 4
      +
      2F_μ 5F^μ 5
      +
      2F_45F^45
      +
      0
      )
      
         =
      -1/4
      F_μνF^μν
      -
      1/2
      (
      -∇_μ(A_4)∇^μ(A_4)
      -
      ∇_μ(A_5)∇^μ(A_5)
      )
      -
      1/2[H,H^†]^2
         =
      -1/4
      F_μνF^μν
      +
      ∇_μH∇^μH^†
      -
      1/2[H,H^†]^2
  
  Upon identifying H with the scalar field of =2, d=4 SYM we see that this term is exactly the bosonic part of the =2, d=4 SYM action!
  
  The spinorial part of S_=2, d=6 must also be considered. Note that since the fields are assumed independent of x^4 and x^5 we can take the corresponding components of the spinors to vanish and write
  Ψ^A
  =
  (ψ^A_α,χ^A_α,0,0)^T.
  Then due to some general properties of spinors in six dimensions (see for example Appendix A of <cit.>), it turns out that χ^A,α̇=ϵ^ABϵ^α̇β̇ψ_B,β̇, allowing the action to be expressed solely in terms of the Weyl spinors ψ^A and their conjugates:
  
      i/2Ψ̅_AΓ^I∇_IΨ^A   =
      i/2Ψ̅^AΓ^μ∇_μΨ^A
      +
      i/2Ψ̅^AΓ^4∇_4Ψ^A
      +
      i/2Ψ̅^AΓ^5∇_5Ψ^A
         =
      i/2
      (0,0,ψ_A,ψ̅_A)
      [                      0_4 [    0  σ^μ; σ̅^μ    0 ]; [    0  σ^μ; σ̅^μ    0 ]                      0_4;                          ][  ∇_μψ^A; ∇_μψ̅^A;     0_2;     0_2 ]
             
      +
      i/2Ψ̅^AΓ^4∇_4Ψ^A
      +
      i/2Ψ̅^AΓ^5∇_5Ψ^A
         =
      i/2
      (
      ψ_Aσ^μ∇_μψ̅^A
      +
      ψ̅_Aσ^μ∇_μψ^A
      )
      +
      1/2
      (
      -
      ψ_A[A_4,ψ^A]
      +
      ψ̅^A[A_4,ψ̅^A]
      )
      
               
      -
      1/2
      (
      ψ_A[A_5,ψ^A]
      +
      ψ̅_A[A_5,ψ̅^A]
      )
      
         =
      i/2
      (
      ψ_Aσ^μ∇_μψ̅^A
      +
      ψ̅_Aσ^μ∇_μψ^A
      )
      -
      i/√(2)ψ_A[H^†,ψ^A]
      +
      i/√(2)ψ̅_A[H,ψ̅^A]
      
         =
      iψ^Aσ^μ∇_μψ̅_A
      -
      i/√(2)ψ_A[H^†,ψ^A]
      +
      i/√(2)ψ̅^A[H,ψ̅_A]
  
  where we have used that in six dimensions the Dirac adjoint swaps components.
  
  Compactification has thus reproduced the spinorial part of the =2 SYM action, so indeed compactifying two directions of =1, d=6 SYM results in =2, d=4 SYM.
  
  
  § NON-RENORMALIZABLE SUPERSYMMETRIC ACTIONS
  
  So far, the form of our SUSY actions have been constrained by renormalizability. We now discuss what happens when this constraint is dropped. Renormalizability is not an issue in the low energy regime, so non-renormalizable actions may be used as so-called effective theories. In this section the construction of such actions for the =2 case will be briefly discussed.
  
  It was previously mentioned that without regard for renormalizability the most general =1 supersymmetric Lagrangian is given by (<ref>). On the other hand, without regard for renormalizability the most general gauge field Lagrangian is:
  
      _NR, gauge=1/16g^2∫ d^2θ f_ab(Φ)W^aαW^b_α+h.c.
  
  where f_ab depends on Φ only and is thus holomorphic, and a,b are gauge group Lie algebra indices <cit.>.
  
  As in the renormalizable case, to obtain the most general non-renormalizable =2 Lagrangian from (<ref>) we must include gauge fields. Schematically this can be achieved by adding the most general kinetic terms for gauge fields as well as converting all derivatives to gauge covariant ones. In practice this can be accomplished in two steps. First we swap Φ^† for ϕ^†e^2gV in the argument of F. Secondly we add an appropriate linear combination of _NR, gauge.
  
  Since the matter and gauge Lagrangians must be related in an =2 invariant theory, the functions F and f_ab must be related for the resulting theory to have =2 supersymmetry. One can show that the correct relation is given by taking w=0, and setting <cit.>:
  
      16π/4g^2f_ab   =:
      -i∂^2/∂ϕ^2
      16π/4g^2F
         =:
      -i/2ϕ^†∂/∂ϕ+h.c.
  
  The holomorphic quantity  is called the =2 prepotential.
  
  This action can be rewritten conveniently in =2 superspace language as <cit.>:
  
      S_eff
      =
      1/8π^2i∫ d^4xd^2θ(Ψ)
  
  Or in =1 language:
  
      S_eff
      =
      1/16π∫ d^4x[1/2∫ d^2θ_ab(Φ)W^aαW^b_α+∫ d^2θ d^2θ̅(Φ^†e^V)^a_a(Φ)]
  
  Upon comparison with (<ref>), we see that in the renormalizable case ∝Ψ^2.
  1.9
  From (<ref>) it is clear that  completely determines the low energy effective action of =2 SYM. Amazingly  can be calculated exactly. Doing so in two different ways is the major goal of this thesis.
  
  
  
  CHAPTER: SEIBERG-WITTEN THEORY
  
  Seiberg-Witten theory provides a way to calculate the =2 prepotential exactly. We present here in detail the original Seiberg-Witten approach for pure SYM with gauge group SU(2) <cit.>. Generalisations to include matter and different gauge groups are well known <cit.>. Introductions to this area include <cit.>.
  
  
  § THE MODULI SPACE OF VACUA
   
  The first important object to introduce is , the moduli space of vacua. Points in  correspond to gauge inequivalent vacua of =2 SYM, that is to gauge inequivalent Poicaré invariant field configurations which minimise the action.
  
  It is well known from ordinary QFT that Lorentz invariance implies all non-scalar fields and all spacetime derivatives must have a vanishing vaccum expectation value (VEV), however a scalar field can have a non-zero VEV. We now discuss the vacuum configurations of =2 SYM. From (<ref>), the full =2 SYM action can be expanded out to give <cit.>:
  
      S_=2 YM   =
      ∫ d^4x
      Tr(-1/4F_μνF^μν-iλσ^μ∇_μλ̅-iψσ^μ∇_μψ̅+|∇ H|^2+Θ/32π^2g^2F_μν(⋆F)^μν
               
      +
      1/2D^2+f^†f+i√(2)gH^†{λ,ψ}-i√(2)g{ψ̅,λ̅}H+gD[H,H^†])
  
  The corresponding scalar potential is thus V=-Tr(1/2D^2+f^†f+gD[H,H^†]). Since f and D are auxiliary fields we can easily integrate them out by solving their equations of motion. The Euler-Lagrange equations give the following equations of motion:
  
      D+g[H,H^†]=f=f^†=0
  
  So the scalar potential is:
  
      V
      =
      1/2g^2Tr([H,H^†])^2
  
  
  
   §.§ Parametrisation of .
  
  By definition, a vacuum minimises the action, and thus the scalar potential V. Clearly V≥ 0, so any minimum H_0 has V(H_0)≥ 0. In fact, unbroken SUSY requires V_0=0, so the possible vacua are parametrised by the solutions of the equation [H,H^†]=0  <cit.>. The moduli space of vacua  is then this space considered up to gauge transformations. It turns out that the prepotential is closely related to the metric on this space.
  
  For pure SYM, all fields are necessarily in the adjoint representation of the gauge group. This means that the fields are Lie(G) valued functions, and thus for the case of SU(2) can be expanded in terms of the Pauli matrices: H(x)=∑_j=1^3(a_j(x)+ib_j(x))τ_j. We assume without loss of generality that not all of the a_j(x) vanish.
  
  In the adjoint representation the gauge group acts as G×Lie(G)→Lie(G), ϕ↦ gϕ g^-1. Such a gauge transformation can be used to set a_1(x)=a_2(x)=0, then [H,H^†]=0 enforces that b_1(x)=b_2(x)=0. So without loss of generality we can write H(x)=1/2a(x)τ_3, with a(x):=a_3(x)+ib_3(x). Let a denote the VEV of a(x) with respect to a particular vacuum. Then a is a parameter labelling the different vacua of the theory.
  2.1
  The condition [H,H^†]=0 says that H is an element of a Cartan subalgebra of Lie(G). This observation proves useful for the case of gauge group SU(N) <cit.>.
  
  Noting that
  [  0 -1;  1  0 ]∈ SU(2)
  sends H→ -H and thus a→ -a, we see that a and -a are also gauge equivalent. Thus gauge inequivalent vacua can be labelled by the gauge invariant parameter Tr H^2 which is given by 1/2a^2 in the vacuum. In general, we define u:=Tr H^2, and H=:1/2aτ_3, then classically u=1/2a^2. The parameter u then labels gauge inequivalent vacua in the full quantum theory and is thus a coordinate on ℳ.
  
  
   §.§ Gauge symmetry breaking and the effective theory.
  
  For H≠ 0, the SU(2) gauge symmetry of =2 SYM is broken, causing the a=1,2 components of the fields to develop masses. This is the well-known Higgs mechanism of ordinary QFT, and can be seen by writing
  H(x)=H'(x)+H_0
  =
  (
  0,
  0,
  1/2(a(x)+a)
  )^T, where H_0 is the VEV of H(x), then expanding the |∇ H|^2 term of (<ref>).
  
  In fact, the gauge symmetry breaks to U(1), so at low energies the theory is described by an =2 theory with gauge group U(1). To see this, note that the vacuum vector is only invariant under the U(1) subgroup of SU(2):
  
      H_0↦
      U H_0U^†
      =
      1/2a
      [   α -β̅;   β  α̅ ][  1  0;  0 -1 ][ α̅ β̅; -β  α ]
      =
      1/2a
      [    |α|^2-|β|^2           2αβ̅;           2α̅β -(|α|^2-|β|^2) ]
  
  which is equal to H_0 if and only if β=0 and |α|^2=1, that is if U∈ U(1)⊂ SU(2).
  
  We have shown that at low energies the theory has a U(1) gauge symmetry and each field has only a single gauge group index. So from the general non-renormalizable =2 action (<ref>) we have that the effective Lagrangian has the form:
  
      S_eff=1/16π∫ d^4x[1/2∫ d^2θ”(Φ)W^αW_α+∫ d^2θ d^2θ̅Φ^†'(Φ)]
  
  for some holomorphic function .
  2.2
  In the above expression, the e^V term has been replaced by 1. To see why, recall that e^V=1+V+1/2V^2, but since V is adjoint valued and U(1) is abelian, only the 1 term remains in a U(1) theory.
  
  A less abstract interpretation of the function  is given as follows. If we expand the effective action term by term we see that ” plays the role of a metric in field space:
  
      S_eff∝∫ d^4x[”(Φ)(|∂_μH|^2-iψσ^μ∂_μψ̅-1/4F_μν(F^μν-i(⋆F)^μν)-iλσ^μ∂λ̅+…)]
  
  By passing to the moduli space (effectively replacing fields by their VEVs), it is clear that the metric on  is given by ds^2=(”(a))dada̅=τ(a)dada̅, where τ(a):=”(a) is the complexified effective coupling. So if we can determine the metric on  we have in principle determined .
  
  
  § SEIBERG-WITTEN DUALITY
  
  An obvious consistency condition is that the metric on  must be positive definite: τ(a)>0 for all a. However this cannot be the case on all of , since  is holomorphic so (τ)=(”) is harmonic. Thus (τ) cannot have a minimum on ℂ, and so we cannot have (τ)>0 everywhere. We conclude that the description of  in terms of τ cannot be valid everywhere, that is when (τ) approaches zero we must switch to a different set of coordinates, a_D and τ_D. These dual quantities are provided by Seiberg-Witten duality <cit.>.
  
  
   §.§ The duality transformation.
  
  Following Seiberg and Witten, we define a dual superfield Φ_D and a dual prepotential _D as the Legendre transform of Φ and :
  
      Φ_D:='(Φ)  _D'(Φ_D)=:-Φ
  
  The form of the second term of the effective action action (<ref>) is easily seen to be invariant under this transformation since:
  
      Φ^†'(Φ)=-((Φ_D^†'_D(Φ_D))^†)=Φ_D^†'_D(Φ_D)
  
  It is less easy to show that the form of the first term is also invariant. To do so we perform a change of variables in the path integral defining the effective partition function and show that this leads to an action of the same form but in the dual variables.
  
  As a preliminary step, note that due to U(1) symmetry the field strength is simply F_μν=_μA_ν-∂_νA_μ, so it obeys the Bianchi Identity: _λF_μν+_μF_νλ+_νF_λμ=0. Upon contracting with ϵ^μνρσ it is clear that 1/2ϵ^μνρσ_νF_ρσ=0. This identity is equivalent to the reality condition D_αW^α=0 <cit.>.
  
  To show invariance of the first term of (<ref>) under duality we change variables in the path integral and enforce the reality condition with a real Lagrange multiplier superfield, V_D. The relevant part of the path integral is
  Z_eff
  =
  ∫𝒟 V exp(i/32π∫ d^2θ”(Φ)W^αW_α)
  . 
  For the Lagrange multiplier we add i/64π∫ d^2θ d^2θ̅V_DD_αW^α to S_eff, yielding the following equality:
  
      Z_eff   =
      ∫𝒟 V exp(i/32π∫ d^2θ”(Φ)W^αW_α)
      
         =
      ∫𝒟 W 𝒟 V_Dexp{i/32π∫ d^4x∫ d^2θ(”(Φ)W^αW_α+1/2d^2θ̅V_DD_αW^α)}
  
  Integrating by parts following <cit.> and defining (W_D)_α=-1/4^2e^-2V_D_αe^2V_D yields:
  
      ∫ d^2θ d^2θ̅^2V_DD_αW^α
      =
      -∫ d^2θ d^2θ̅^2D_αV_DW^α
      =
      ∫ d^2θD̅^2(D_αV_DW^α)
      =
      -4∫ d^2θ (W_D)_αW^α
  
  where we have used that D̅W_α=0, since up to a total spacetime derivative:
  
      D̅^2f(x,θ,θ̅)
         =
      ϵ^α̇β̇/θ̅^α̇/θ̅^β̇f(x,θ,θ̅)
      +
      _v(
      2iθ^α(σ^ν)_α^α̇ f(x,θ,θ̅)/θ̅^α̇)
      
             +
      _μ_ν(
      1/2θ^2(σ̅^μ)^αβ̇(σ^v)_αβ̇f(x,θ,θ̅)
      )
      
         =
      ϵ^α̇β̇/θ̅^α̇/θ̅^β̇f(x,θ,θ̅)
      
         =
      -∫ d^2θ̅f(x,θ,θ̅)̅
  
  Completing the square allows the W integral to be carried out explicitly:
  
      ∫ d^2θ”(Φ)W^αW_α
      +
         1/2∫ d^2θ d^2θ̅V_DD_αW^α
      =
      ∫ d^2θ(”(Φ)W^αW_α-2(W_D)_αW^α)
      
         =
      ∫ d^2θ{”(Φ)(W^α-(W_D)^α/”(Φ))(W_α-(W_D)_α/”(Φ))-(W_D)^α(W_D)_α/”(Φ)}
  
  This results in the following expression for Z_eff:
  
      Z_eff   =
      ∫ W V_Dexp{i/16π∫ d^4x∫ d^2θ”(Φ)(W^α-(W_D)^α/”(Φ))(W_α-(W_D)_α/”(Φ))
      }
             exp{i/16π∫ d^4 x ∫ d^2θ(-1/”(Φ)(W_D)^α(W_D)_α)
      }
         =
      ∫ V_Dexp{i/16π∫ d^4 x ∫ d^2θ(-1/”(Φ)(W_D)^α(W_D)_α)
      }
  
  where we have used that the W integral is Gaussian and thus evaluates to a constant which by appropriate normalisation we can take to be unity.
  
  So indeed the form of this part of the action is invariant up to replacing the effective coupling ”(Φ) with -1/”(Φ)=_D(Φ_D).
  
  In summary we have shown that under the duality transformation (<ref>), S_eff becomes:
  
      1/16π∫ d^4x[1/2∫ d^2θ_D”(Φ_D)W_D^αW_Dα+∫ d^2θ d^2θ̅Φ_D^†_D'(Φ_D)]
  
  That is to say the form of S_eff is duality invariant.
  
  Defining the dual coupling, τ_D=-1/τ we see that the duality transformation maps strongly coupled regions of =2 SYM to weakly coupled regions of =2 SYM and vice versa. It is thus an example of a so called S-duality. Furthermore, as τ→ 0, τ _D →∞, so indeed the dual description should yield extended coordinates on .
  
  
   §.§ The full duality group.
  
  The full group of duality transformations is in fact larger than that derived in the previous section. To see this, we use the dual variables to rewrite S_eff in a more symmetric form:
  
      S_eff
      =
      1/32π∫ d^4xd^2θdΦ_D/dΦW^αW_α
      +
      1/32π i∫ d^4 x d^2θ d^2θ̅
      (
      Φ^†Φ_D
      -
      Φ_D^†Φ
      )
  
  It can now be shown that S_eff is invariant under Φ→Φ, Φ_D→Φ_D+bΦ, where b∈ℤ. This is easy for the second term. For the first term we have:
  
      1/16π∫ d^4xd^2θdΦ_D/dΦW^αW_α→1/32π∫ d^4xd^2θdΦ_D/dΦW^αW_α
      +
      b/32π∫ d^4xd^2θ
      W^αW_α
  
  But
  ∫ d^2θ
  W^αW_α
  =
  -
  F_μν(⋆F)^μν
  so:
  
      b/32π∫ d^4xd^2θ
      W^αW_α
      =
      -2π b1/32π∫ d^4x
      F_μν(⋆F)^μν
      =
      -2π b k∈2πℤ
  
  In the above calculation we have used the fact that k:=1/32π∫ d^4x
  F_μν(⋆F)^μν is the instanton number, and is thus an integer <cit.>. Since Z_eff=e^iS_eff, under this transformation Z_eff→ Z_eff, so indeed this is a symmetry of the theory.
  
  The above transformation can be written as 
  [ Φ_D;   Φ ]→[ 1 b; 0 1 ][ Φ_D;   Φ ]
  , while the original duality transformation (Φ→Φ_D , →_D), can be written as 
  [ Φ_D;   Φ ]→[  0 -1;  1  0 ][ Φ_D;   Φ ]
  .
  Recalling that 
  SL(2;ℤ)
  =
  <
  [ 1 b; 0 1 ]
  ,
  [  0 -1;  1  0 ]>
  , it is clear that the group of duality transformations is at least SL(2;ℤ). In fact it can be shown that this is the full duality group <cit.>.
  
  Seiberg-Witten duality also descends to the level of metrics. Taking expectation values we define a_D=(a)/ a and note H_D=1/2a_Dτ_3. Then da_D=∂ a_D/∂ ada=”(a)da, so ds^2=(da_Dda̅)=i/2(dada̅_D-da_Dda̅). From here it is clear that ds^2 is also SL(2;ℤ) invariant.
  
  
  § THE BPS MASS FORMULA
  
  In this section we introduce the BPS mass formula, named for Bogomolny Prasad and Sommerfield. It relates the masses of particles to their magnetic and electric charges. This will prove useful since Seiberg-Witten duality maps electrically charged states to solitonic magnetic monopoles <cit.>. The BPS mass formula will thus provide a physical interpretation to many of the following arguments. Unfortunately some of the statements required to obtain the BPS mass formula cannot be proven here without going significantly off track. For some additional details see <cit.>.
   
  First recall that the helicity operator is the projection of the spin operator onto the momentum operator, or physically that the helicity of a particle is the component of its spin along its direction of travel. It can be shown that representations of the SUSY algebra can be split into two types; so called short and long multiplets. The long multiplets contain 16 helicity states while the short ones contain 4 <cit.>. Massless states necessarily belong to short multiplets, and since the massive states in our theory obtain their mass via the Higgs mechanism, so do they.
  
  Next, it can be shown that states belong to short multiplets if and only if their mass is given by m^2=2|𝒵|^2, where 𝒵 is the central charge of the SUSY algebra, while states in long multiplets have m^2>2|𝒵|^2 <cit.>. This inequality is known as the BPS bound, and states for which m^2=|𝒵|^2 are called BPS saturated. The BPS mass formula is then obtained by relating the central charge of a state to its electric and magnetic charge.
  
  Finally, a purely electrically charged state has 𝒵=an_e where n_e∈ℤ, so by duality, a purely magnetic state has Z=a_Dn_m with n_m∈ℤ <cit.>. Since the central charge is additive, a general state has 𝒵=an_e+a_Dn_m. In summary, the BPS mass formula is as follows:
  
      m^2=2|𝒵|^2, where 𝒵=an_e+a_Dn_m=(n_m,n_e)(a_D,a)^T.
  
  2.3
  Acting on a and a_D with SL(2;ℤ) is equivalent to transforming the charge vector (n_m, n_e) by right multiplication. For a sanity check note that in this notation, Seiberg-Witten duality indeed maps (1,0) to (0,1).
  
  
  
  § THE WEAK AND STRONG COUPLING LIMITS
  
  The weak and strong coupling limits of =2 SYM can now be examined. More precisely, we study the monodromy properties of a(u) and a_D(u) as they encircle certain points of . The points of  for which a(u) and a_D(u) do not return to their original values upon encircling will be referred to as singularities. This analysis will yield several more consistency conditions for the prepotential. Along with Seiberg-Witten duality, these conditions will then  fix  entirely.
  
  
   §.§ The u→∞ limit.
  
  To study the u→∞ limit and the associated monodromy we quote several results without proof. None of these results are particularly out of reach but their derivations would take us quite a long way off topic.
  
  The first result is the tree-level and 1-loop corrections to _class=1/2τ_0Ψ^2, which were determined by Seiberg <cit.>. The result is _pert(Ψ)=i/2πΨ^2log(Ψ^2/Λ^2), where Λ is the dynamically generated energy scale of the theory. In short, this is obtained by using the holomorphicity of  and noting that _pert must be invariant under U(1) gauge transformations.
  
  The next result is the SUSY non-renormalization theorem. In SUSY QFTs it can be shown that due to fermionic/bosonic cancellations certain quantities are either not renormalised or they are not renormalised beyond 1-loop level. This can be done either by exploiting holomorphicity and symmetries, or by the analysis of Feynman diagrams in superspace. For more details and some applications of this fact see <cit.>. In particular, for =2 SUSY the prepotential is not renormalised beyond 1-loop level, and thus _pert is the full perturbative result.
  
  We can now split  into a perturbative and a non-perturbative part: =_pert+_inst. The perturbative part has been obtained without any trouble, it is the non-perturbative instanton contribution _inst which is the tough bit!
  
  In the high energy limit, the full SU(2) theory is known to be asymptotically free. Here the momentum scale p is large and thus so is the mass scale m. The BPS mass formula then implies that this is a region of large a.
  
  As detailed in <cit.>, the effective action is an integral of the full action over heavy modes so the dominant contribution is from regions of large p and thus large a. Asymptotic freedom ensures that in this region the full theory is perturbative, so the integral defining the effective action is dominated by perturbative contributions. This means that _pert→ as u→∞. Furthermore in this limit we must have that u approaches its classical value: u→1/2a^2. 
  
  Since the full prepotential approaches its perturbative value in the u→∞ limit we have that here, (a)→i/2πa^2log(a^2/Λ^2), and as τ(a)=”(a), τ(a)→i/π(log(a^2/Λ^2)+3).
  
  As a sanity check note that in this limit the metric is given by (τ(a))≈2/πlog|a/Λ|, which is indeed positive definite and single valued. Furthermore, since a_D=∂(a)/∂ a we can compute a_D(a)=i/πa(log(a^2/Λ^2)+1).
  
  In summary, as u→∞ we have:
  
      a(u) →√(2u)
         
      a_D(u) →i/π√(2u)(log(2u/Λ^2)+1)
  
  
  The monodromy properties of a and a_D can be examined in this limit by encircling ∞ with a circular anti-clockwise contour on the Riemann sphere. To do so, we take u→ e^2π i tu, t→ 1 and find that a_D→ -a_D+2a. Similarly, we find that a→ -a. The monodromy transformation at infinity is thus given by 
  M_∞
  =
  [ -1  2;  0 -1 ]
  . Since this matrix is non trivial, the point u=∞ is indeed a singularity of  and in particular a branch point. Branch cuts must end somewhere, so  must have at least one more singularity. In fact we will show that it has least three singularities and that these singularities come in pairs.
  
  
   §.§ -Symmetry in general.
  
  To show that the singularities of  come in pairs we must discuss a certain global symmetry of the action known as -symmetry. In general -symmetry refers to a global symmetry of a supersymmetric theory which acts on the supercharges. Symmetries of the action are symmetries of the corresponding classical theory and as such may become anomalous in the quantum regime with only a partial symmetry remaining unbroken. We now discuss the -symmetry of =2 SYM and its breaking pattern as well as the implications for the structure of .
  
  The -symmetry group of =2 SYM is U(1)_× SU(2)_, where the SU(2)_ subgroup rotates the two supercharges, while the U(1)_ subgroup acts on the Grassmann coordinates and fields in the following way:
  H and Φ have charge 2, W,θ, and θ̅ have charge 1, and d^2θ and d^2θ̅ have charge -2 <cit.>. The U(1)_ subgroup turns out to be anomalous, it is broken both perturbatively and non-perturbatively.
  
  For a general simple gauge group G and a general =2 theory, U(1)_ is broken to a discrete ℤ_β, where β is the leading contribution of the β function  <cit.>. In a more abstract context β is the following quantity (see for example Appendix B of <cit.>):
  
      β=ζ(l_adj-∑_ρl_ρ)
  
  where l_ρ is defined in terms of the trace form Tr_ρ and the Killing form Tr_adj as follows: l_adjTr_adj=l_ρTr_ρ, where l_fund is normalised to unity. The factor ζ depends on the group in question (it is 1 for SU(N)), and the sum is over the representations of the matter hypermultiplets included in the theory. Fo the case at hand there are no hypermultiplets and the gauge group is SU(2), so as l_adj=2N+2 for SU(N+1), we have that β=4, and thus U(1)_ is broken to ℤ_4.
  
  
   §.§ -Symmetry for =2 SYM.
  
  The previous discussion is a bit abstract and we haven't proven anything. Luckily for the case at hand, the breaking pattern can be obtained without appealing to general results.
  
  Under U(1)_ we have that _pert transforms as _pert→ e^4iα(i/2πa^2loga^2/Λ^2-a^22α/π), so δ S_eff=2π k4α/π. Thus by the same path integral argument as used in Section 2.2.2, the action is invariant if and only if α=2π n/8 for some n∈ℤ.
  
  We now discuss the transformation properties of _inst. To do so, note that the instanton part of the prepotential can be represented as an infinite sum of the form:
  
      _inst(a,Λ)=∑_k=0^∞Λ^β k_k(a)
  
  for as of yet undetermined coefficients _k∝ a^2-4k. This follows from the renormalization group equations and invariance under the residual -symmetry <cit.>.
  
  The above series transforms under the U(1)_ symmetry as
  _inst→ a^2∑_n=1^∞c_ke^8iα(1-k)(Λ/a)^4k, and so is once invariant if and only if α=2π n/8 for some n∈ℤ. This shows that only the ℤ_8 subgroup of U(1)_ is a symmetry of the quantum theory.
  
  Under the residual ℤ_8 symmetry, H→ e^iπ nH, for n∈ℤ so that for odd n, H^2→ -H^2. This means that for u=Tr H^2≠ 0, ℤ_8 is broken further to ℤ_4, as claimed.
  
  
   §.§ Singularity Counting.
  
  We have shown that for a generic vacuum the residual -symmetry is ℤ_4, while on  itself we have a full ℤ_8 symmetry under which u→ -u. This implies that each singularity of  has a partner under -symmetry, with the only exceptions being the fixed points of this map, namely 0 and ∞.
  
  The only way to have just one additional singularity is if the second is at u=0, and in this case 0 and ∞ must have identical monodromies. So since a^2 was not affected by M_∞ it is not affected by any monodromy, and thus is a valid global coordinate. By the harmonic function argument of Section 2.2 we see that this results in a contradiction, so indeed  must have at least three singularities.
  
  From now on we assume that  has has exactly three singularities. In this case, there must be a pair of non-zero singularities ± u_0 which are interchanged by the global ℤ_8 symmetry. Note that in particular u=0 cannot be a singularity of .
  
  
   §.§ The u→± u_0 limits.
  
  For points of  (i.e. vacua), with enhanced symmetry, the SU(2) gauge symmetry of the full theory does not break all the way to U(1), and thus the effective description as a U(1) gauge theory as per (<ref>) breaks down. Such points are thus singularities of . To detect points of enhanced symmetry, note that at these points the Higgs mechanism partially "turns off", resulting in extra massless particles. We can thus regard singularities of  as being caused by generically massive particles becoming massless.
  
  Seiberg and Witten argue on general grounds that unlike the classical case, the strong coupling singularities cannot be due to massless gauge bosons <cit.>. Instead we assume that since they are the only other generically massive states in the theory, these singularities occur at points of  for which massive dyons become massless. Recall that an (n_m,n_e)-dyon is a soliton of magnetic charge n_m and electric charge n_e.
  
  To begin with assume that a (1,0)-dyon, that is a magnetic monopole, becomes massless. In this case (<ref>) implies that m^2=2|a_D|^2, so this corresponds to a_D=0. We call the point where this occurs u_0.
  
  Near u_0 the theory (in the dual description), consists of a massive but light hypermultiplet corresponding to the magnetic monopole coupled locally to the fundamental chiral multiplet <cit.>, this is exactly =2 SUSY QED for which the β-function is known: μd/dμg_D=g_D^3/8π^2 <cit.>. The energy scale μ is proportional to the mass of our monopole and thus to a_D, so using that Θ=0 for SUSY QED we have that as u→ u_0:
  
      a_Dd/da_Dτ_D=-i/π
  
  Using that τ_D=-da/da_D this ODE can be solved to find that to leading order, a≈ a_0+i/πa_Dlog a_D. Since a is singular near u_0, Seiberg-Witten duality says that a_D should be a good coordinate there, and thus depend linearly on u:
  
      a_D≈ c_0(u-u_0)
      
         
      a≈ a_0+i/π(u-u_0)log(u-u_0)
  
  The constants a_0 and c_0 will be determined later. Taking (u-u_0)→ e^2π i(u-u_0), the corresponding monodromy matrix can be read off:
  M_u_0
  =
  [  1  0; -2  1 ]
  .
  
  To find M_-u_0, we simply note that since  is assumed to have exactly three singularities, a contour about ∞ can be deformed to two contours encircling u_0 and -u_0. This situation is shown in Figure 2.1 and gives the factorisation condition M_∞=M_u_0M_-u_0 (up to a choice of base point, P), which is easily solved to find
  M_-u_0
  =
  [ -1  2; -2  3 ].
  
  
  In summary, the monodromies associated to all three singularities are:
  
      M_∞
      =
      [ -1  2;  0 -1 ],   
      M_u_0
      =
      [  1  0; -2  1 ],   
      M_-u_0
      =
      [ -1  2; -2  3 ]
  
  We would like a physical interpretation for the singularity at -u_0. To this end, note that since mass is a physical observable, the BPS mass formula should be invariant under monodromy. Hence as 𝒵=(n_m,n_e)(a_D,a)^T, the monodromy transformation (a_D,a)^T→ M(a_D,a)^T can instead be interpreted as transforming the charge vector as (n_m,n_e)→ (n_m,n_e)M.
  
  A state of zero mass should be invariant under the corresponding monodromy, so for such a state (n_m,n_e)M=(n_m,n_e), i.e. the charge vector of such a state should be the left eigenvector of the corresponding monodromy matrix. We can thus identify the massless state responsible for -u_0's singular behaviour with the left eigenvector (1,-1) of M_-u_0. The physical interpretation is that this singularity is due to a Dyon with charge (n_e,n_m)=(1,-1) becoming massless.
  
  We finish this section by briefly stating what happens if  is assumed to have more singularities. If  has p singularities it can be shown by a similar argument to the above that they must factorise as M_∞=M_u_1… M_u_p, with 
  M_u_i
  =
  [ 1+2n_m n_e     2n_e^2;    -2n_m^2 1-2n_m n_e ]
  , and (n_m,n_e)∈ℤ^2. It is considered likely that this system has no solutions for p>3 <cit.>.
  
  
  § THE SOLUTION
  
  With the three monodromy matrices (<ref>) at hand, the prepotential can now be determined. We will do so in two ways. For the first method we will identify a and a_D with the solutions of a certain differential equation <cit.>. The second method involves identifying  with a certain Riemann surface for which a and a_D are the periods <cit.>.
  2.4
  From now on we take ± u_0=± 1. This corresponds to a specific choice of Λ and otherwise leaves the discussion unaffected.
  
  
  
   §.§ The differential equation approach.
  
  
  It is a well known fact that functions with non-trivial constant monodromies arise from ODEs with periodic (in the real case), or meromorphic (in the complex case), coefficients and at most regular singular points. For example take the ODE [_uu+V(u)]ψ(u)=0 and fix two linearly independent solutions ψ_1,2(u). If V is meromorphic, then encircling any singularity u_i leaves the ODE invariant so the rotated solutions must be some linear combination of the non-rotated solutions:
  
      [ ψ_1; ψ_2 ]
      (u+e^2π i(u-u_i))=M_i[ ψ_1; ψ_2 ]
      (u)
  
  Where M_i is some monodromy matrix. It is well known that these monodromies are non-trivial and constant if V has only regular singular points.
  
  
    §.§.§ The form of V(u).
  
  We now assume that a and a_D are the solutions to a differential equation of the form [_uu+V(u)]ψ(u)=0. For justification, recall that any ODE with 3 regular singular points can be transformed into some hypergeometric equation. Then since with an appropriate change of variables and choice of V the ODE [_uu+V(u)]ψ(u)=0 becomes an arbitrary hypergeometric equation (as we will see later), the assumption holds.
  
  The known monodromies and the assumption that the ODE has three regular singular points u_i∈{± 1,∞} leads to severe constraints on the form of V. Firstly changing variables to w=1/u we find:
  
      ψ”(w)+2/wψ'(w)-1/w^4V(1/w)ψ(w)=0
  
  So for u=∞ to be a regular singular point, V(1/w) must be (w^2)=(1/u^2). This shows that first order poles (unless they are in a product such as the term 1/(u-1)(u+1)), and regular points lead to essential singularities at infinity and so must be excluded. Poles of degree greater than 2 are also excluded since they correspond to essential singularities at u=± 1. Thus V must be of the form:
  
      V(u)
      =
      -1/4[
      1-λ_1^2/(u+1)^2
      +
      1-λ_2^2/(u-1)^2
      -
      1-λ_1^2-λ_2^2+λ_3^2/(u+1)(u-1)]
      ,   λ_i∈ℂ
  
  The λ_i∈ℂ will eventually be fixed by enforcing the correct monodromy properties.
  
  
    §.§.§ Solving the ODE.
  
  We now solve this ODE and determine the λ_i by transforming it to a hypergeometric equation. To do so,  set ψ(u)=(u+1)^(1-λ_1)/2(u-1)^(1-λ_2)/2f(u+1/2), and take x=(u+1)/2. The ODE then becomes:
  
      x(1-x)f”(x)+[c_3-(c_1+c_2+1)x]f'(x)-c_1c_2f(x)=0
  
  Where c_1=(1-λ_1-λ_2+λ_3)/2, c_2=(1-λ_1-λ_2-λ_3)/2, and c_3=1-λ_1. This is indeed the hypergeometric equation. We pick the following basis of solutions:
  
      f_1(x)
      =
      (-x)^-c_1F(c_1,c_1+1-c_3,c_1+1-c_2;1/x)
      
         
      f_2(x)
      =
      (1-x)^c_3-c_1-c_2
      F(c_3-c_1,c_3-c_2,c_3+1-c_1-c_2;1-x)
  
  The known asymptotic behaviour can now be used to fix the λ_i and match these solutions with a and a_D.
  
  Firstly, as x→∞, V(u)∼ -1-λ_3^2/(2u)^2, resulting in a Cauchy-Euler equation for ψ. This is easy to solve:
  
      ψ(u)∼ 
            Au^(1+λ_3)/2+Bu^(1-λ_3)/2   λ_3≠ 0
      
            Au^1/2+Bu^1/2log(u)    λ_3=0
      ,   
      A,B∈ℂ
  
  Only the λ_3=0 solution can match the known asymptotics (<ref>) as u→∞, so we conclude that λ_3=0.
  
  Next we consider what happens as u→ 1. In this limit the ODE is as follows:
  
      ψ”(u)
      =
      -
      1-λ_2^2/(u-1)^2ψ(u)
      +
      1-λ_1^2-λ_2^2/8(u-1)ψ(u)
      +
      (1)
  
  Recalling that a_D(u)≈ c_0(u-1) as u→ 1, a_D(u) can only be a solution if as u→ 1 we have:
  
      0
      =
      -c_0(1-λ_2^2)/u-1
      +
      c_0/8(1-λ_1^2-λ_2^2)
      +
      (1)
  
  This is only possible for λ_2=1.
  
  Finally, the fact that u→ -u is a symmetry of  implies that V should be an even function. This immediately yields λ_1=1.
  
  In summary, λ_1=λ_2=1 and λ_3=0, so c_1=c_2=-1/2, and c_3=0. Furthermore V(u) is now entirely fixed: V(u)=-1/4(u+1)(u-1).
  
  
    §.§.§ The result.
  
  The solutions ψ_1,2 can still be scaled by constants to get the correct monodromies. It turns out that a(u)=-2iψ_1(u), and a_D(u)=iψ_2(u) are the correct choices as will be shown in Section 2.6. Rewriting the associated hypergeometric functions in integral form gives:
  
      a(u)
         =
      √(2)√(u+1)F(-1/2,1/2,1;2/u+1)
      =
      √(2)/π∫_-1^1dx√(x-u)/√(x^2-1)
      
      a_D(u)
         =
      i/2(u-1)F(1/2,1/2,2;1-u/2)
      =
      √(2)/π∫_1^udx√(x-u)/√(x^2-1)
  
  With a(u) and a_D(u) determined we have now implicitly determined the prepotential .
  
  
   §.§ The elliptic curve approach.
  
  We now present another way to obtain  from the monodromies. This method, due to Seiberg and Witten, relies on some facts from basic differential geometry and the theory of modular curves <cit.>. It is less direct than the differential equations approach but is more readily generalised to theories including matter multiplets <cit.>.
  
  
    §.§.§ Set up.
  
  The information we have is as follows: the metric on  is given by ds^2=(τ)|da|^2 where τ(u)=da_D/du/da/du. This metric is positive definite: (τ)>0. Furthermore,  is the complex u-plane with a ℤ_2-symmetry taking u→ -u and singularities at {±1,∞}. The coordinates (a, a_D) on  are acted on by an SL(2;ℤ)-duality and have monodromies (<ref>) around the singularities of . 
  2.5
  In the language of differential geometry this says that (a,a_D) forms a section of a flat SL(2;ℤ) bundle over  with monodromies M_± 1 and M_∞.
  
  
  
    §.§.§ Identification of .
  
  The first thing to notice is that the three monodromies don't generate all of SL(2;ℤ). In fact they generate the so-called principal congruence subgroup of level 2:
  
      ⟨ M_±1,M_∞⟩
      =
      Γ(2)
      :=
      {
      A∈ SL(2;ℤ)|A≡1_2× 2(mod 2)
      }⊆
      SL(2;ℤ)
  
  There is an action of Γ(2) on the upper half plane, H and in fact ≅ H/Γ(2) <cit.>. Next note that the space H/Γ(2) parametrises the family of complex elliptic curves E_u defined by y^2=(x-1)(x+1)(x-u) so that each point u∈ can be associated to an elliptic curve E_u <cit.>.
  2.6
  For a sanity check, note that the equation defining E_u is invariant under the symmetry group generated by {u→ -u,x→ -x,y→± i y}, which is isomorphic to ℤ_4. Of this symmetry, only the ℤ_2 subgroup acts on u (i.e. on all of ). This is precisely the symmetry structure identified earlier.
  
  2.7
  The curve y^2=(x-1)(x+1)(x-u) is known as the Seiberg-Witten curve for gauge group SU(2).
  
  A given E_u is essentially the surface on which y(x) becomes a single valued function. Since the equation for E_u is quadratic in y, encircling ± 1, u, or ∞ in the x-plane takes y→ -y. For example if x=1 is encircled by a circle of small radius, then writing x=1+δ x and taking (x-1)→ e^2π i(x-1), we have:
  
      y
      =
      √((x-1)(x+1)(x-u))∼√(2δ x(1-u))→√(2e^2π iδ x(1-u))
      =
      -√(2δ x(1-u))
      =
      -y+(δ x)^2
  
  Thus the x-space E_u should be a double cover of the complex plane with points at infinity added. Furthermore it should have square root branch points at ± 1, u, and ∞, which we join pairwise by two cuts. Finally the two sheets are joined along these cuts, that is to say crossing a cut takes us from one sheet to the other. Thus we have that for generic u (that is u not a singularity of ), this Riemann surface is a torus, as shown in Figure 2.2.
  
  
  
    §.§.§ The periods of E_u.
  
  We now seek to relate the family of tori E_u to a and a_D. To do so note that encircling the cut from -1 to 1 corresponds to one of the basic cycles of the torus, while the other basic cycle corresponds to travelling from 1 to u on the first sheet, then back from u to 1 on the second sheet. This is clear from Figure 2.2. We call the first cycle γ_1, and the second γ_2.
  
  Since E_u is generically a torus, the independent cycles γ_1,γ_2 form a basis for the family of first homology groups V_u:=H_1(E_u;ℂ)≅ℤ⊕ℤ. De Rham's theorem then allows us to identify γ_1 and γ_2 with differential forms on E_u. In particular, it says that V_u≅ H_dR^1(E_u;ℂ), that is the cycles on E_u are in one to one correspondence with closed 1-forms modulo exact 1-forms. A basis for the first cohomology group is thus provided by the following 1-forms:
  
      λ_1
      =
      dx/y  λ_2
      =
      xdx/y
  
  The periods of E_u are Ω_i^j=∫_γ_iλ^j. Letting b_i=Ω_i^1 it is a well known fact of differential geometry that for a torus b_1/b_2=τ_u where τ_u is the modular parameter of said torus.
  
  
    §.§.§ Identification of the metric.
  
  In general, τ_u has the fundamental property that τ_u>0. Since this property is shared by the complexified coupling τ(u), we seek to identify these quantities.
  
  Consider a general element λ=a_1(u)λ_1+a_2(u)λ_2∈ H_dR^1(E_u;ℂ). Define a_D=∮_γ_1λ, and a=∮_γ_2λ. We claim that τ(u) has (τ(u))>0 if and only if dλ/du=f(u)λ_1 for some function f.
  
  For one implication, assume that dλ/du=f(u)λ_1 for some function f. In this case
  da_D/du
  =
  ∫_γ_1dλ/du
  =
  f(u)b_1
  , and similarly 
  da/du=f(u)b_2
  so:
  
      τ(u)=da_D/du/da/du=b_1/b_2=τ_u
  
  So we have identified the modular parameter of E_u with the complexified coupling τ(u), and thus also ensured that τ >0.
  
  For the other implication, assume that τ>0 everywhere. Then for each u∈, τ(u) is the modular parameter of some elliptic curve. General considerations show that the family of curves thus determined have the same monodromies and singularities as those determined by τ_u. It can then be shown that the two families coincide and thus τ(u)=τ_u <cit.>.
  
  The condition τ(u)=τ_u implies that da_D/du=(1/b_2da/du)b_1 and da/du=(1/b_1da_D/du)b_2, so 
  
      1/b_2da/du=1/b_1da_D/du:=f(u)
  
  Thus f(u)b_i=∮_γ_idλ/du, and so ∮_λ_i(f(u)λ_1-dλ/du)=0. λ_1 is the unique holomorphic differential on E_u so up to a redefinition of f the integrand vanishes, as required.
  
  It turns out that f is entirely fixed by the asymptotic behaviour of a and a_D near the singularities. In fact we claim that f(u)=-√(2)/4π is the unique choice with the correct properties. To verify that this is the case we need only to show that this choice of f gives the correct monodromy properties. It is then clear that any other choice would have somewhere introduced extra poles or zeros. The monodromy properties will be verified in Section 2.6.
  
  
    §.§.§ Identification of a(u) and a_D(u).
  
  To determine a and a_D explicitly, note that:
  
      dλ/du
      =
      f(u)dx/y
      =
      -
      √(2)/4πdx/√((x-1)(x+1)(x-u))
  
  So upon integration λ=√(2)/2π√(x-u/x^2-1)dx. The constant of integration has been set to zero since constants are entire functions and so would not contribute to a or a_D. Deforming the γ_i to lie along the branch cuts then gives:
  
      a_D(u)
      =
      ∮_γ_1√(2)/2π√(x-u/x^2-1)dx
      =
      √(2)/2π(
      ∫_1^u√(x-u/x^2-1)dx
      -
      ∫_u^1√(x-u/x^2-1)dx
      )
      =
      √(2)/π∫_1^u√(x-u/x^2-1)dx
  
  In the above calculation the negative sign comes from traversing the cut in the opposite direction. Similarly we find that a(u)=√(2)/π∫_-1^1√(x-u/x^2-1)dx.
  
  This is indeed the same result (<ref>) obtained previously from the differential equation approach. All that is left now is to verify that these solutions have the correct monodromy properties.
  
  
  § VERIFICATION OF MONODROMY PROPERTIES
  
  To conclude that the expressions obtained for a(u) and a_D(u) are indeed correct, it must be verified that they have the correct monodromy properties.
  
  
   §.§ The u→∞ limit.
  
  As u→∞, we have:
  
      a(u)
      =
      √(2)/π∫_-1^1√(x-u/x^2-1)dx
      ≈√(2)/π∫_-1^1√(u/1-x^2)dx
      =√(2u)
  
  This result agrees with the assertion that u=1/2a^2 in the semiclassical region and has the correct monodromy. To examine a_D(u) in this limit, set x=uz and note that u→∞ implies z→ 0 for finite x:
  
      a_D(u)
      =
      √(2u)/π∫_1/u^1√(z-1/(z+u^-1)(z-u^-1))dz
      ≈√(2u)/π∫_1/u^1√(z-1/z^2)dz
      ≈
      i√(2u)/π∫_1/u^1dz/z
  
  So as u→∞, a_D(u)≈√(2u)/πi(log(1)-log(1/u))≈√(2u)/πilog(u) in accordance with our previous findings. It is easy to see that this form also has the correct monodromy.
  
  
   §.§ The u→± 1 limits.
  
  For u→ 1, we again use that
  a_D(u)
  =
  √(2u)/π∫_1/u^1√(z-1/(z+u^-1)(z-u^-1))dz. Note that as u→ 1 with z∈(u^-1,1), z-1 and z-u^-1 both vanish but z+u^-1→ 2, so:
  
      a_D(u)
      =
      √(2u)/π∫_1/u^1√(z-1/(z+u^-1)(z-u^-1))dz
      ≈√(2)/π∫_1/u^1√(z-1/2(z-u^-1))dz
      =
      1/π∫_1/u^1√(z-1/z-u^-1)dz
  
  This integral can be evaluated:
  
      1/π∫√(z-1/z-u^-1)dz
         =
      1/π∫√(z-1/z-u^-1)dz
      
         =
      -1/π∫1/x^2√(x(u^-1-1)+1)dx
       , where x=1/z-u^-1
         =
      2(1-u^-1)/π∫y^2/(y^2-1)^2dy
       , where y=√(x(u^-1-1)+1)
         =
      1-u^-1/2π∫[
      1/y-1
      -
      1/y+1
      +
      1/(y-1)^2
      +
      1/(y+1)^2]
      dy
      
         =
      1-u^-1/2π[
      -log(y+1/y-1)
      -
      2y/y^2-1]
      
         =
      u^-1-1/π[
      arctanh(√(u^-1-1/z-u^-1+1))
      +
      z-u^-1/u^-1-1√(u^-1-1/z-u^-1+1)]
  
  So as u→ 1:
  
      a_D(u)
      ≈1-u^-1/πlim_z→ u^-1arctanh(i√(1-u^-1/z-u^-1-1))
      =
      i/2(1-u^-1)
      ≈i/2(u-1)
  
  This expression has the correct monodromy and furthermore gives that c_0=i/2.
  
  Finally, we examine a(u) as u→ 1. Unlike the other quantities, a(1) can be computed exactly:
  
      a(1)
      =
      √(2)/π∫_-1^1√(x-1/x^2-1)dx
      =
      √(2)/π∫_-1^1dx/√(x+1)
      =
      4/π
  
  However, to determine the monodromy the leading order non-constant term is required. To obtain it, consider a'(u) and integrate:
  
      a'(u)
      =
      -√(2)/2π∫_-1^1dx/√((x+1)(x-1)(x-u))
  
  Near u=1 this integral develops a factor of 1/(x-1), leading to a logarithmic divergence at x=1:
  
      a'(u)
      ≈
      -1/2π∫_-1^1dx/(x-1)(x-u)
      =
      -1/2πlog(2x-1-u+2√((x-1)(x-u)))|^1_-1
      =
      -1/2πlog(1-u)+(1)
  
  Upon integrating and using the expression for a(1) we have:
  
      a(u)
      =
      -1/2π(u-1)log(1-u)+C+(u)
      =
      4/π-1/2π(u-1)log(1-u)+((u-1)^2log(1-u))
  
  Thus a(u)≈4/π-u-1/2πlog(u-1), which yields the correct monodromy and also gives a_0=4/π.
  
  Rather than analysing the u→ -1 limit, note that the u→ -u symmetry fixes the  u=-1 monodromies. Thus all the monodromies have been verified and we are done!
  
  
  § RE-EXPRESSION AS A POWER SERIES
  
  We have now in principle determined , however it is expressed in a very indirect form. To obtain an explicit form, invert a(u) to get u(a), and substitute to get a_D(a)=/ a. Integrating with respect to a then yields (a). These steps then result in an explicit power series for the prepotential. The result of this process is:
  
      _inst(a)
      =
      -1/2Λ^4/a^2
      -
      5/64Λ^8/a^6
      -
      3/64Λ^12/a^10
      +
      (Λ^16)
  
  Deriving this result is a straightforward but tedious exercise in manipulating power series and so is omitted for brevity. For details see <cit.> and Appendix A of <cit.>.
  
  
  CHAPTER: LOCALISATION
  
  At this point we turn to the second technique for computing the prepotential of =2 SYM; localisation. Localisation involves showing that the partition function path integral only receives contributions from some subspace of the space of fields, that is to say it "localises". This reduces the path integral to a lower dimensional integral. For the case of =2 SYM the resulting integral is finite dimensional.
  
  Localisation is a more direct approach than Seiberg-Witten theory. It gives  as a power series in a from the get-go, and can be more easily generalised to other gauge groups. On the other hand it is much more complicated.
  
  
  § EQUIVARIANT COHOMOLOGY
  
  As a warm-up for supersymmetric localisation we first introduce a "toy model"; the bosonic localisation of finite dimensional integrals with abelian symmetry. From here it is surprisingly only a mild generalisation to obtain localisation formulae for path integrals. The discussion in this section follows <cit.>.
  
  To start, consider a 2l-dimensional boundaryless Riemannian manifold (M,g) with a symmetry group G. We want to reduce integrals over M to integrals over the lower dimensional quotient space M/G.
  
  In general G may not act freely (indeed this is the case for most applications of interest), so this quotient may not be a manifold. In general it is an orbifold, a generalisation of a manifold to include singularities. This complication is dealt with by introducing the notions of equivariant differential forms and equivariant cohomology. These ideas generalise the usual cohomology of manifolds to include singularities.
  
  In this discussion we take G=U(1), although what follows can be generalised to the non-abelian case.
  
  Let V=V^μ∂_μ be a Killing vector on M and assume that it generates the U(1) symmetry. By definition this means that _Vg=0, where _V is the Lie derivative along V. Equivalently we have that ∇_μV_ν+∇_νV_μ=0, where ∇_μV_ν=V_ν,μ-Γ^λ_μνV_λ is the covariant derivative with Levi-Civita connection.
  
  Now let ⋀^nM be the space of differential n-forms on M and let ⋀ M=⊕_n=0^∞⋀^nM be the space of polyforms on M. We define the V-equivariant differential, d_V:⋀ M→⋀ M, d_V=d-ι_V, where d:⋀^nM→⋀^n+1M is the usual exterior differential and ι_V:⋀^nM→⋀^n-1M is contraction in the first slot with the vector V.
  
  3.1
  Remarkably, d_V^2=-_V. This is easy to prove: d^2=0 by definition and ι_V^2=0 by the antisymmetry of forms. The final step is to verify that dι_V+ι_Vd=_V, which is somewhat tedious but straightforward, see for example <cit.>.
  
  
  The space of equivariant polyforms is:
  
      ⋀_VM={α∈⋀ M|_Vα=0}
  
  Remark 3.1 shows that (d_V|_⋀_VM)^2=0, that is to say the equivariant differential acts as a coboundary operator on ⋀_VM. It thus makes sense to define equivariantly closed and equivariantly exact forms analogously to the usual case but with d_V instead of just d. We then define the n'th V-equivariant de Rham cohomology group as the space of equivariantly closed n-forms modulo equivariantly exact n-forms:
  
      H_V^n(M)
      :=
      (d_V|_∧^n_VM)
      /
      (d_V|_∧^n-1_VM)
  
  The point of this construction is that if the U(1) action has no fixed points H_V^n(M)=H_dR^n(M/U(1)), while if it does, H_V^n provides a well defined generalisation of the usual cohomology <cit.>.
  
  Further motivation for introducing equivariant cohomology is provided by the behaviour of polyforms under integration. For a polyform α=α_2l+α_2l-1+…+α_0, we define ∫_Mα=∫_Mα_2l, where the right hand side is the usual integral.
  
  For an equivariantly exact polyform d_Vβ, d_Vβ=(dβ_2l-1)+(dβ_2l-2-ι_Vα_2l)+…+(dα_0-ι_Vα_2)+(-ι_Vα_1) so the top-term is dβ_2l-1 which is exact in the usual sense since d^2=0. Using Stokes' theorem and the fact that M is boundaryless we have that ∫_Md_Vβ=∫_Mdβ_2l-1=0, so for polyforms α and β:
  
      ∫_M(α+d_Vβ)
      =
      ∫_Mα
  
  3.2
  We have shown that the integral is constant on members of an equivariant de Rham cohomology class.
  
  
  
  § BOSONIC LOCALISATION OF ORDINARY INTEGRALS
  
  The zero locus of the Killing vector V is the set of points fixed under the action of V:
  
      M_V
      =
      {
      x∈ M|V|_x=0
      }
  
  Integrals of equivariantly closed polyforms localise to M_V. This will now be proven in two different ways.
  
  
   §.§ An indirect localisation argument.
  
  We claim that an equivariantly closed polyform α is equivariantly exact on M∖ M_V. To prove this, define η to be the 1-form dual to V: η=g(V,·)=V_μdx^μ.
  
  The 1-form η is V-equivariant. To prove this fact let Y be an arbitrary vector field, then:
  
      _V(Y)η
      =
      _Vg(V,Y)
      =
      V(g(V,Y))
      =
      (_Vg(V,·))(Y)+g(V,_VY)
      =
      (_Vg(V,·))(Y)+g(V,[V,Y])
  
  So 
  Rewriting this expression in component form yields:
  
      (_Vη)(Y)
         =
      V^ρ_ρ(g_μνV^μY^ν)-g_μνV^μ(V^ρ_ρY^ν-Y^ρ_ρV^ν)
      
         =
      V^ρ_ρ(g_μνV^μ)Y^ν
      +
      g_μνV^ρV^μ_ρY^ν
      -
      g_μνV^μ(V^ρ_ρY^ν-Y^ρ_ρV^ν)
      
         =
      (
      V^ν_ν(g_μρV^ρ)
      +
      g_νρV^ρ_μV^ν
      )
      Y^μ
         =
      (
      V^ρ_ρg_μν
      +
      g_μρ_νV^ρ
      +
      g_ρν_μV^ρ
      )
      V^νY^μ
         =
      (_Vg)_μνV^νY^μ
         =
      0
  
  Since Y is arbitrary, the above calculation shows that _Vη=0 and so η is equivariant.
  
  It is easy to show that d_Vη=-|V|^2+dη=-|V|^2(1-dη/|V|^2). This expression can be inverted using the geometric series formula:
  
      (d_Vη)^-1
      =
      -1/|V|^2∑_n=0^l(dη/|V|^2)^n
  
  The above series terminates since dη is a 2-form and M=2l.
  
  Equation (<ref>) holds only for |V|≠ 0, which is the case on M∖ M_V. We now claim that (d_Vη)^-1 is equivariantly closed on M∖ M_V. Applying d_V to 1=(d_Vη)(d_Vη)^-1 gives:
  
      0
      =
      (d_V^2η)(d_Vη)^-1
      +
      (d_Vη)d_V(d_Vη)^-1
      =
      0
      -
      (dη-|V|^2)d_V(d_Vη)^-1
  
  Since |V|^2 is a 0-form and dη is a 2-form it must then be that d_V(d_Vη)^-1=0, as claimed.
  
  Finally we define the polyform Θ_V=η(d_Vη)^-1. Applying d_V once gives
  
      d_VΘ_V=(d_Vη)(d_Vη)^-1+η d_V(d_Vη)^-1=1+0=1
  
  Applying it a second time gives _VΘ_V=-d_V^2Θ_V=d_V(1)=0, so Θ_V is equivariant on M∖ M_V.
  
  Let α be an equivariantly closed polyform on M. Then α=(d_VΘ_V)α=d_V(Θ_Vα), so α is equivariantly exact wherever Θ_V is defined, i.e. on M∖ M_V. So since integrals of equivariantly exact forms vanish:
  
      ∫_Mα
      =
      ∫_M∖ M_Vα
      +∫_M_Vα
      =
      ∫_M∖ M_Vd_V(Θ_Vα)
      +∫_M_Vα
      =
      0+
      ∫_M_Vα
      =
      ∫_M_Vα
  
  So indeed the integral has localised to M_V.
  
  
   §.§ A direct localisation argument.
  
  Let α be an equivariantly closed polyform, and define α_t=α e^td_Vβ, where β is a V-equivariant polyform. β is arbitrary at this point, fixing it is called a choice of localisation scheme. Since α_0=α and α_t is a continuous deformation of α, α_t is cohomologous to α, and:
  
      ∫_Mα=∫_Mα_t ,   ∀ t∈ℝ
  
  We are free to take t→∞ as long as this limit exists. This will be the case if the 0-form term of d_Vβ is non-positive with maxima equal to 0. A convenient choice  is β=η, yielding:
  
      ∫_Mα=lim_t→∞∫_Mα e^tdηe^-t|V|^2
  
  To see why this process has localised the integral, note that e^tdη is a polynomial of degree l in t, and e^-t|V|^2 is a Gaussian peaked at M_V. As t→∞, e^-t|V|^2→δ(V), a delta function localised on M_V.
  
  
  § THE BOSONIC ATIYAH-BOTT-BERLINE-VERGNE LOCALISATION FORMULA
  
  The second localisation argument can be extended to derive an explicit formula for the localised integral in the case in which M_V consists of isolated points: M_V={P_k}.
  
  To derive the localisation formula we define local cartesian coordinates (x_i,y_i) on M near some P_k∈ M_V, so the local metric is ds^2≈∑_i=1^l(dx_i^2+dy_i^2). In these coordinates the Killing vector is:
  
      V
      ≈∑_i=1^lω_P_k,i(-y_i/ x_i+x_i/ y_i)
      =∑_i=1^lω_P_k,i/ϕ_i
  
  To verify this, one can evaluate the local connection and show that _Vg=0 locally.
  
  By acting on (x_i,y_i)^T and summing, one can show that the Killing vector generates the following transformation:
  
      e^ω_P,iϕ_i_ϕ_i[ x_i; y_i ]
      =
      [  cos(ω_P,iϕ_i) -sin(ω_P,iϕ_i);  sin(ω_P,iϕ_i)  cos(ω_P,iϕ_i) ][ x_i; y_i ]
      =:
      R(ϕ_i)
      [ x_i; y_i ]
  
  This is an anticlockwise rotation by ϕ_i on the i'th eigenspace. The infinitesimal action of this rotation is specified by  the linear order expansion of the following equation:
  
      L_Vx⃗
      :=
      δx⃗/ϕ_i
      =
      1/ϕ_i(R(ϕ_i)-1)x⃗
      =
      [ -ω_P,iy_i;  ω_P,ix_i ]
      +(ϕ_i)
  
  The above equation is easily solved to yield
  L_V
  =
  [      0 -ω_P,i;  ω_P,i      0 ]
  . 
  
  In local coordinates η and d_Vη are as follows:
  
      η≈∑_i=1^lω_P,ir_i^2dϕ_i,   
      d_Vη≈∑_i=1^l(ω_P,id(t_i^2)∧ dϕ_i-ω_P,i^2r_i^2)
  
  We can now evaluate the contribution to the integral from a neighbourhood N_P of the point P_k:
  
      lim_t→∞∫_N_Pα e^td_Vη   =
      lim_t→∞∫_N_Pα e^t∑_i=1^l(ω_P,id(r_i^2)∧ dϕ_i-ω_P,i^2r_i^2)
         =
      lim_t→∞∫_N_Pα∏_i=1^l e^t(ω_P,id(r_i^2)∧ dϕ_i-ω_P,i^2r_i^2)
         =
      lim_t→∞∫_N_P(α_0+…+α_2l)∏_i=1^l(1+tω_P,id(r_i^2)∧ dϕ_i+…
                                   
      +2/l!(tω_P,id(r_i^2)∧ dϕ_i)^l)e^-tω_P,i^2r_i^2
         =
      lim_t→∞α_0(P)t^l∏_i=1^lω_P,i∫_N_Pd(r_i^2)∧ dϕ_ie^-tω_P,i^2r_i^2
         =
      lim_t→∞α_0(P)t^l∏_i=1^lω_P,i∫_ℝ^2d(r_i^2)∧ dϕ_ie^-tω_P,i^2r_i^2
         =
      lim_t→∞α_0(P)t^l∏_i=1^lω_P,i∫_0^∞d(r_i^2)e^-tω_P,i^2r_i^2∫_0^2πdϕ_i
         =
      lim_t→∞α_0(P)∏_i=1^l2π/ω_P,i
  
  In the fourth equality we have noted that only top (i.e degree 2l), terms contribute to the integral and have kept only the leading order in t term since it is all that survives in the limit. In the fifth equality we have traded an integral over N_P for one over all of ℝ^2 since in the limit the integrand vanishes off N_P.
  
  The above expression can be rewritten so as to make its generalisation to the quantum case more straightforward. Recall the Pfaffian of a 2n× 2n antisymmetric matrix A:
  
      Pf(A)
      =
      1/2^nn!ϵ^i_1… i_2nA_i_ii_2… A_i_2n-1i_2n
  
  Note also that in general Pf(A)^2=(A).
  
  It is easy to verify that:
  
      lim_t→∞∫_N_Pα e^td_Vη
      =
      (2π)^lα_0(P)/Pf(-L_V(P))
  
  So since the integral vanishes off M_V, we have by linearity:
  
      ∫_Mα=(2π)^l∑_x_k∈ M_Vα_0(x_k)/Pf(-L_V(x_k))
  
  This is known as the Atiyah-Bott-Berline-Vergne localisation formula <cit.>. It has reduced an integral over a manifold with abelian symmetry to a sum of contributions from discrete points.
  
  
  § SUPERSYMMETRIC LOCALISATION OF PATH INTEGRALS
  
  The localisation arguments made in the previous section can be generalised to the case of supersymmetric path integrals. In fact, there is a direct correspondence between the various objects introduced in the equivariant case and those necessary in the SUSY case. This correpondence is presented in table 3.1 <cit.>:
  
  
  In Table 3.1, Q is a fermionic supercharge which squares to some bosonic operator B, 𝒪 is a BPS operator (an operator such that Q=0), and  is a space of fields. Finally, the localisation locus M_V is swapped for the so-called "BPS-locus", 𝔉_Q of supersymmetric field configurations.
  
  We now present in very general terms some of the ideas of supersymmetric localisation before specialising to the case of =2 SYM in the next section.
  
  Consider a SUSY gauge theory with supercharge Q and action S. We wish to compute the expectation values of gauge invariant BPS observables:
  
      𝒪_BPS
      =
      ∫_𝔉[ X]𝒪e^iS[X]
  
  3.3
  The partition function is given by the special case _BPS=1.
  
  As in the equivariant case, the expectation value of an operator depends only on its Q-cohomology class:
  
      Q𝒪
      :=
      ∫_𝔉[ X]Q(𝒪)e^iS[X]
      =
      ∫_𝔉[ X](Q(𝒪e^iS[X])+Qe^iS[X])
      =
      ∫_𝔉[ X]Q(𝒪e^iS[X])
      =
      0
  
  In the above calculation we have used that the action of a SUSY theory is necessarily supersymmetric so that QS[X]=0, a supersymmetric generalisation of Stokes' theorem and the assumption that the fields decay sufficiently quickly at spatial infinity.
  
  It is now clear that 𝒪_BPS+Q𝒪=𝒪_BPS, that is the expectation value of a BPS operator in a supersymmetric theory depends only on the Q-cohomology class of said operator.
  
  
   §.§ SUSY path integrals localise.
  
  We now show that SUSY path integrals of BPS observables localise to the BPS locus. Again this can be done in two ways, by generalising either of the two arguments presented in Section 3.2. Here we present briefly the generalisation of the first method, due to Witten <cit.>. For details on generalising the second method see <cit.>. 
  
  Assume there is a (super)group G, generated by a fermionic charge Q and that G acts freely on the field space 𝔉. For a G-invariant operator  we can reduce the path integral to the space 𝔉/G by introducing G-collective coordinates, yielding =Vol(G)∫_𝔉/G X e^iS[X].
  
  However, since G is generated by a fermionic variable and as ∫ dθ = 0, we have that Vol(G)=0. This is a contradiction since if this is the case, even 1 would vanish meaning expectation values would be non-normalisable.
  
  The way out is to conclude that G does not act freely, instead it has fixed points forming the BPS locus 𝔉_Q of Q-invariant field configurations. G then acts freely on the complement 𝔉∖𝔉_Q, so the path integral vanishes there. We thus see that  localises to 𝔉_Q, as claimed.
  
  
  § TOPOLOGICAL TWIST AND LOCALISATION OF =2 SYM
  
  The goal of this section is to establish some more advanced facts about =2 SYM. In particular, we show that the action can be rewritten as S=S_top+Q̅(V_YM+V'), where Q̅ is a fermionic BRST, operator, V_YM and V' are potential terms, and S_top=Θ/32π^2g^2∫ d^4x F_μν(⋆F)^μν is the topological part of the =2 SYM action . Furthermore we identify the localisation locus of =2 SYM which will be explicitly constructed in the next chapter.
  
  
   §.§ Topological QFTs.
  
  In general a QFT is said to be topological if all correlation functions are independent of the metric; δ/δ g_α _1…_α_p. There are two general classes of QFTs which satisfy this condition <cit.>.
  
  The first type are called Schwarz QFTs. A Schwarz QFT is a QFT in which each _α_i is individually metric independent. Such theories are clearly topological.
  
  The second type are called Witten QFTs. A Witten type QFT has a nilpotent symmetry δ such that δ_α(ϕ_i)=0 where _α_i is a gauge invariant supersymmetric observable. Furthermore T_μν=δ G_μν for some tensor G_μν, where T_μν=δ S/δ g. We further assume that the variation of the _α with respect to the metric is δ-exact, that is to say δ_α(ϕ_i)/δ g_μν=δ_α^μν(ϕ_i) for some functional _α^μν(ϕ_i).
  
  Under the previous assumptions:
  
      δ/δ g_μν_α   =
      δ/δ g_μν∫[ϕ_i]_αe^-S[ϕ_i]
         =
      ∫[ϕ_i]
      (
      (δ^μν_α)e^-S[ϕ_i]
      -
      _αT_μνe^-S[ϕ_i])
      
         =
      ∫[ϕ_i]
      δ(
      (^μν_α)e^-S[ϕ_i]
      -
      _αG_μνe^-S[ϕ_i])
      
         =
      0
  
  where the product rule for δ and the fact that δ_α=δ S=0 have been used. We have also assumed that the measure [ϕ_i] is invariant under δ.
  
  So indeed Witten type QFTs are topological. We will show that =2 SYM is, after a topological twist, a Witten type QFT.
  
  
   §.§ The topological twist.
  
  The topological twist is performed by defining the twisted supercharges:
  
      Q̅=ϵ^Aα̇Q̅_A,α̇,   
      Q_μ=σ̅_μ^AαQ_A,α,   Q̅_μν=σ̅_μν^Aα̇Q̅_A,α̇
  
  Introducing the twisted supercharges corresponds to redefining the spacetime symmetry group of =2 SYM. Initially the spacetime symmetry group is SO(3,1)≅ SU(2)_L× SU(2)_R, and the global symmetry group is SO(3,1)× SU(2)_× U(1)_≅ SU(2)_L× SU(2)_R× SU(2)_× U(1)_, where the proceeding isomorphisms are local (i.e. isomorphisms of the corresponding Lie algebras).
  
  The topological twist corresponds to redefining this spacetime symmetry group to
  (SU(2)_L× SU(2)_)× SU(2)_R, where  denotes the diagonal subgroup <cit.>.
  
  Similarly we define the twisted fields:
  
      ψ^μ=σ̅^μ,Aαψ_A,α, ψ̅=ϵ^Aα̇ψ̅_A,α̇, ψ̅^μν=σ̅^μν_α̇Aψ̅^A,α̇
  
  3.4
  The field ψ̅^μν is anti-self-dual: ψ̅^μν=-i(⋆ψ̅)^μν, as is Q̅_μν.
  
  The =2 SYM action can be rewritten using the twisted fields:
  
      S
         =
      1/g^2∫ d^4x
      Tr(
      -1/4F_μνF^μν
      +
      ∇_μH^†∇^μH
      -
      1/2[H,H^†]^2
      +
      i/2ψ^μ∇_μψ̅
      -
      i/2(∇_μψ_nu-∇_νψ_μ)^-ψ̅^μν
             
      +
      i/2√(2)ψ_μ[H^†,ψ^μ]
      -
      i/2√(2)ψ̅[H,ψ̅]
      -
      i/2√(2)ψ̅^μν[H,ψ̅_μν]
      )
      +
      Θ/32π^2∫ d^4x Tr F_μν(⋆F)^μν
  
  Where in general (X)^±=1/2(X∓ i⋆ X) refers to the (anti)self dual part of the field X.
  
  The action of the twisted supercharges on the fields can also be calculated. This is a tedious but simple calculation, and so is deferred to Appendix B.
  
  We can now establish three important facts.
  
    
    * Q̅ is nilpotent up to a gauge transformation with parameter -2√(2)H. This can be shown by using the action of Q̅.
    
    * S is Q̅-exact up to a topological term:
  
      S
      =
      S_top
      +
      [
      Q̅{τ/16π∫ d^4xTr(
      (F_μν)^-ψ̅^μν
      -
      i√(2)ψ^μ∇_μϕ^†
      +
      iψ̅[ϕ,ϕ^†]
      )
      }]
  
  
    * S is Q̅-cohomologous to S_top. This can be seen by using the equations of motion to show that S_top is Q̅-closed.
  
  
  
   §.§ Topological twist from gauge fixing.
  
  The twisted =2 SYM action can also be obtained from S_top by a gauge fixing procedure known as BV quantisation. This process involves introducing  so-called ghost and antighost fields to eliminate unphysical degrees of freedom. A deep discussion of BV quantisation would lead us seriously off topic, but since we need the gauge fixed action to perform explicit computations, we will quote the important results. For more details on the general procedure as well as the case at hand, see <cit.>.
  
  Essentially BV quantisation leads to the introduction of the fields shown in Table 3.2. The action of Q̅ on these fields is given in Appendix B. Using this action and the equations of motion it can be shown that the BRST operator Q̅ coincides with the twisted supercharge Q̅, and is nilpotent up to a gauge transformation <cit.>.
  
  
  Furthermore BV quantisation imposes the following conditions: 
  
      ∇^μA_μ=(F_μν)^-=∇^μψ_μ=0
  
  For later convenience, we  make the following field redefinitions:
  
      ϕ=-2√(2)H
      ,   λ=-2√(2)H^†,   χ_μν=ψ̅_μν,   η=-4ψ̅
  
  The gauge fixed action is then given by S_top+Q̅(V_YM+V'), where <cit.>:
  
      V_YM
      =
      1/g^2∫ d^4x
      Tr[
      1/2χ^μν(
      (F_μν)^-
      +
      1/4H_μν)
      +
      i/8λ∇_μψ^μ
      +
      c̅
      (
      ∇_μA^μ
      +
      b
      )
      ]
      
         
      V'
      =
      -
      i/128g^2∫ d^4x
      Tr(
      η[ϕ, λ]
      )
  
  One can then show that indeed S=S_top+Q̅(V_YM+V').
  
  
   §.§ The localisation locus of =2 SYM.
  
  We are now in a position to determine the localisation locus of =2 SYM. Since the twisted action is Q̅-cohomologous to S_top, it can be deformed by a Q̅-exact term. We choose:
  
      V
      =
      ∫ d^4x
      Tr[
      -χ^μν(
      t(F_μν^--1/4H_μν)
      )
      +
      iλ∇_μψ^μ]
  
  for some parameter t, then deform the twisted action to S_top+Q̅V.
  
  The equation of motion for H^μν will be required to proceed. We now determine the part of the V integrand which involves H^μν:
  
      Q̅(
      -tχ^μν(F_μν)^-
      +1/4χ^μνH_μν)
         =
      -tQ̅(χ^μν)(F_μν)^-
      +1/4Q̅(χ^μνH_μν)
      
         =
      -tH^μν(F_μν)^-
      +
      1/4(H^μν-i{c,χ^μν})H_μν
      -
      iχ^μν[c,H_μν]
      
         =
      -tH^μν(F_μν)^-
      +
      1/4H^μνH_μν
      -
      i/4{c,χ^μν}H_μν
      -
      iχ^μν[c,H_μν]
  
  where these equalities are up to terms not involving H^μν.
  
  This results in the following equation of motion:
  
      H^μν
      =
      2t(F^μν)^-
      +
      i/2{c,χ^μν}
  
  Evaluating the action of Q̅ and substituting in this equation of motion, the deformed action becomes:
  
      S
      =
      S_top
      +
      ∫ d^4x
      Tr(
      -t^2(F_μν)^-(F^μν)^-
      +
      tχ^μν(∇_μψ_ν-∇_νψ_μ)^-
      +
      iη∇^μψ_μ
      +
      iλ∇^μ∇_μϕ)
  
  So as per the usual localisation procedure we have that upon taking t→∞ the action becomes large and negative leading to a vanishing exponential e^S unless (F_μν)^-(F^μν)^-=0. Thus the localisation locus consists of field configurations for which the field tensor satisfies (F^μν)^-=0, or equivalently F^μν=i(⋆F)^μν. This is called the self-dual equation, and such configurations are called self dual.
  
  Satisfying the self-dual equation is in fact a sufficient condition for satisfying the Yang-Mills equation, so such configurations satisfy the classical equations of motion. They are called instanton configurations. The space of such configurations is called the instanton moduli space.
  
  We have now shown that the partition function path integral localises to the instanton moduli space. This space turns out to be the direct sum of certain finite dimensional spaces, thus reducing a path integral to a sum of finite dimensional integrals.
  
  
  CHAPTER: THE ADHM CONSTRUCTION
  
  The subject of this chapter is to explicitly construct the space of self dual field configurations of =2 SYM, thus providing a finite dimensional model of the instanton moduli space. The approach we will take to this problem is known as the ADHM construction, after Atiyah, Drinfeld, Hitchin and Manin <cit.>.
  
  The ADHM construction proceeds by explicitly constructing a family of connections A_μ, one for each k, then showing that the corresponding curvatures F_μν are self dual. The connections constructed in this way then provide a family of instantons indexed by k. To show that this process does in fact yield all possible instantons, we will show that the solutions of the corresponding Dirac equation are in bijection with the constructed configurations.
  
  
  § THE CONSTRUCTION
  
  The ADHM construction is the only part of the localisation approach which is strongly gauge group dependent. For this thesis we will limit ourselves to the SU(N) case, although the generalisation to other gauge groups is quite straightforward <cit.>. Our discussion follows  <cit.> and <cit.>.
  
  Take k∈ℤ_>0 and introduce (N+2k)× 2k matrices =(_s,rI) and =(B_s,rI) with 1≤ s≤ N+2k, 1≤ r≤ k, and I∈{1,2}.
  4.1
  The matrices  and  can be seen as operators ℂ^N+2k→ℂ^k⊗ℂ^2.
  
  Next we introduce a complex structure on ℝ^4 by defining:
  
      x
      =
      (x_IJ)
      =
      σ^μ
      x_μ
      =
      [  x_0-ix_3 -ix_1-x_2; -ix_1+x_2  x_0+ix_3 ] ,   
      x_μ∈ℝ^4
  
  
  Several more definitions are necessary. Firstly, define Δ(x)=+ x which is assumed to have the maximal rank 2k for all x. Next introduce an (N+2k)× N matrix, v(x) which consists of a basis for the null space of Δ^†: Δ^†v(x)=0. We normalise this basis in the following way: v^†v=1_N.
  
  Now define a candidate connection:
  
      A_μ(x)
      =
      iv^†(x)_μv(x)
  
  This connection is in fact Hermitian:
  
      A_μ(x)^†
      =
      -i(_μv^†(x))v(x)
      =
      -i(_μ(v^†(x)v(x))-v(x)^†_μv)
      =
      A_μ(x)-i_μ(1_N)
      =
      A_μ(x)
  
  For A_μ(x) to lead to a self dual curvature we require a further factorisation condition on Δ:
  
      (Δ^†Δ)_rI,sJ=(ℛ^-1(x))_rsδ_IJ
  
  Using this factorisation condition and the assumption that Δ is of maximal rank:
  
      (Δ^†Δ)_rI,sJ
      =
      (ℛ^-1)_rsδ_IJ
      ⇒        ℛΔ^†Δ
      =
      1
      ⇒      ΔℛΔ^†Δ
      =
      Δ
  
  But (1-vv^†)Δ=Δ-v(Δ^†v)^†=Δ, since Δ^† annihilate v, so:
  
      (ΔℛΔ^†-(1-vv^†))Δ=0
  
  By the maximality of the rank of Δ, a right inverse exists and thus:
  
      ΔℛΔ^†=1-vv^†
  
  We can use the above result to show that F_μν is indeed self dual:
  
      F_μν   =
      _μA_ν
      -
      _νA_μ
      -i[A_μ,A_ν]
      
         =
      i((∂_μv^†)∂_νv+v^†(_μv)v^†(∂_νv))-(μ↔ν)
      
         =
      i(∂_μv^†)(1-vv^†)(∂_νv)+(∂_μ1)-(μ↔ν)
      
         =
      i(∂_μv^†)(1-vv^†)(∂_νv)-(μ↔ν)
      
         =
      i(∂_μv^†)ΔℛΔ^†(∂_νv)-(μ↔ν)
      
         =
      i(∂_μ((Δ^†v)^†)-v^†(∂_μΔ))ℛ(∂_ν(Δ^†v)-∂_ν(Δ^†)v)-(μ↔ν)
      
         =
      iv^†(∂_μΔ)ℛ∂_ν(Δ^†)v-(μ↔ν)
  
  So noting that _μΔ(x)=0+σ_ν∂_μ(x^ν)=σ_μ, we have:
  
      F_μν
      =
      iv^†(σ_μσ_ν^†-σ_νσ_μ^†)^†v
      =
      iv^†(σ_μσ_ν^†-σ_νσ_μ^†)^†v
      =
      4iv^†σ_μν^†v
  
  where we have used that =(_rs) and σ_μ=(σ_μ,IJ)
  4.2
  The above condition says that as matrices,  and σ_μ have a block diagonal structure. 
  
  The quantity σ_μν is easily shown to be self-dual, so F_μν=i(⋆F)^μν, as required. We have thus constructed a family of self-dual field configurations indexed by the non-negative integer k.
  
  
  § REFORMULATION AS LINEAR OPERATORS
  
  Before showing that the previous construction is complete, we reformulate things in the language of linear operators. To do so, the factorisation condition (<ref>) is reinterpreted to yield the so-called ADHM constraints. This reformulation will illuminate some residual freedom in the ADHM data which can then be used to determine a canonical form for the matrices  and , thus yielding the ADHM equations.
  
  
   §.§ The ADHM constraints.
  
  Noting that that δ^α̇_β̇δ^β̇_α̇=2, we have from (<ref>):
  
      ^-1(x)
         =
      1/2δ^β̇_α̇Δ^†α̇Δ_β̇
         =
      1/2δ^β̇_α̇(
      ^†,α̇_β̇
      +
      ^†α̇^βx_ββ̇
      +
      x̅^αα̇^†_α_β̇
      +
      x̅^αα̇^†_α^βx_ββ̇)
      
         =
      1/2(
      ^†,γ̇_γ̇
      +
      ^†α̇^αx_αα̇
      +
      x̅^αα̇^†_α_α̇
      +
      x̅^αβ̇^†_α^βx_ββ̇)
  
  From the third line of the above expression we can easily obtain a Taylor series for ^-1(x) which terminates after three terms:
  
      ^-1(x)
         =
      1/2^†,γ̇_γ̇
      +
      1/2x^μ(
      ^†α̇^ασ_μ,αα̇
      +
      σ̅_μ^αα̇^†_α_α̇)
      +
      1/4
      x^μx^ν(
      σ̅_μ^αβ̇^†_α^βσ_ν,ββ̇
      +
      σ̅_ν^αβ̇^†_α^βσ_μββ̇)
      
         =
      1/2^†,γ̇_γ̇
      +
      1/2(
      ^†α̇^αx_αα̇
      +
      x̅^αα̇^†_α_α̇)
      +
      1/4x̅^αβ̇^†_α^β
      x_ββ̇
  
  where (x_αα̇)^†=x_μσ̅^μ,αα̇:=x̅^αα̇=x^†,αα̇.
  
  Using the above Taylor series as well as (<ref>), we have:
  
      ^†,α̇_β̇
      +
      ^†α̇^βx_ββ̇
      +
         x̅^αα̇^†_α_β̇
      +
      x̅^αα̇^†_α^βx_ββ̇
         =
      δ^α̇_β̇(
      1/2^†,γ̇_γ̇
      +
      1/2(
      ^†γ̇^αx_αγ̇
      +
      x̅^αγ̇^†_α_γ̇)
      +
      1/4x̅^αγ̇^†_α^β
      x_βγ̇)
  
  Equating coefficients in the above expression then gives the following three conditions:
  
      ^†_α^β
      =
      1/2δ^β_α^†_γ^γ ,   ^†_α_α̇
      =
      ^†_α̇_α ,   ^†α̇_β̇
      =
      1/2δ^α̇_β̇^†γ̇_γ̇
  
  For example, the first condition is obtained by equating:
  
      x̅^αα̇^†_α^βx_ββ̇
      =
      1/2δ^α̇_β̇x̅^αγ̇^†_α^β
      x_βγ̇
  
  which holds for all x and thus for x=x̅=1_2, so:
  
      δ^αα̇^†_α^βδ_ββ̇
      =
      1/2δ^α̇_β̇δ^αγ̇^†_α^βδ_βγ̇
      =
      1/2δ^α̇_β̇δ^α_β^†_α^β
      =
      1/2δ^α̇_β̇^†_γ^γ
  
  Multiplying through by δ^β_α then yields:
  
      δ^β_αδ^αα̇^†_α^βδ_ββ̇
      =
      1/2δ^α̇_β̇δ^β_α^†_γ^γ
      ⇒       δ^α̇_β̇(
      ^†_α^β)
      =
      δ^α̇_β̇(
      1/2δ^β_α^†_γ^γ)
  
  The result is then immediately obtained by multiplying through with δ^β̇_α̇.
  
  The three position independent conditions (<ref>) are called the ADHM constraints. They are a set of coupled quadratic conditions on  and  which must be obeyed to yield a self-dual connection.
  
  
   §.§ Residual freedom.
  
  The input data for the construction consists of the two conditions defining v and the factorisation condition on Δ (or equivalently the three ADHM constraints). We now want to see what freedom we have with this data. In particular we perform the transformations; Δ↦ UΔ M, v↦ Uv and ↦ M^† M with U unitary and M invertible. It is easy to show that this does not effect any of the ADHM conditions (<ref>).
  
  
   §.§ Canonical forms.
  
  The above set of symmetries allows the freedom to choose a nice canonical form for  and . Firstly since  is assumed to be of rank 2k we can take Δ↦ UΔ M to set 
  =
  (0|')^T
  , with ' 2k× 2k. This is essentially a partial singular value decomposition.
  
  The first ADHM condition says that ^†'∝1, so '^†' is real and symmetric. So from basic linear algebra there exist matrices O and μ with O real orthogonal and μ real diagonal such that '^†=Oμ O^T.
  
  Using the symmetry transformation again with U=1 and M=Oμ^-1/2, '↦' O μ^-1/2. Thus:
  
      '^†'↦μ^-1/2O^T'^†'Oμ^-1/2=μ^-1/2O^†Oμ O^TOμ^-1/2=1
  
  This means that the matrix 
  [ [ 1 0 ];     '^† ]
   is unitary, allowing us to perform the transformation one more time with M=1 and
  U
  =
  [ [ 1 0 ];     '^† ] as follows:
  
      ↦[ [ 0 0 ];    '^†' ]
      =
      [    0; 1_2k ]
      =
      [       0; 1_k⊗1_2 ]
  
  In this form the third ADHM constraint is manifestly obeyed. We then partition  and v according to the form of :
  
      =
      [    S_α̇; X^μ⊗σ_μ ] ,   
      v
      =
      [   T; Q_α ]
  
  Where S_1̇ and S_2̇ are N× k, X^μ is k× k, T is N× N and Q_1 and Q_2 are k× N.
  
  The redundancy in this description can be further reduced by transforming with elements of the unitary subgroups of U(N+2k) and GL(2k) since such transformations preserve the ADHM conditions (<ref>). This amounts to the following component transformations:
  
      S_α̇↦ U_NS_α̇U_k^† ,  
      X^μ↦ U_kX^μU_k^-1 ,  
      T↦ U_NT
       ,  
      Q_α↦
      U_kQ_α ,  ↦ (U_k^†⊗1_2)R(U_k⊗1_2)
  
  4.3
  This residual freedom arising from the transformations (<ref>) defines an action of U(k) on the ADHM fields. For different gauge groups G, the residual transformations define actions of different groups. The resulting group is called the dual group G_D. We have just shown that for G=SU(N), the k-instanton dual group is U(k).
  
  
  
   §.§ The ADHM equations.
  
  The first ADHM constraint is automatically satisfied with  in canonical form, but what about the other two? It is easy to see that the second constraint leads to X^μ=(X^μ)^†, that is the matrix X^μ is necessarily hermitian. The third condition is slightly harder to interpret. First note that for any Pauli matrix τ_i we have upon contraction:
  
      μ^i
      :=
      (^†α̇)(τ_i)_α̇^β̇_β̇=1/2Tr(τ_i)^†γ̇_γ̇=0
  
  Since {1_2,τ_i} is a basis for the set of 2× 2 hermitian matrices and as ^† is hermitian (and 2× 2 if considered to be matrix valued), this condition is enough to conclude that ^†∝1, where the constant of proportionality is obtained by taking the trace. This shows that the third ADHM condition is obeyed if and only if μ^i=0 for i=1,2,3. These three non-linear conditions are a first form of the so-called ADHM equations.
  
  In summary, for a self-dual curvature, X^μ must be hermitian and each μ^i=0 for i=1,2,3.
  
  To bring the ADHM equations into their most well known form, we define the following quantities:
  
      J
      =
      S_1̇,   
      J
      =
      S_2̇,   
      B_1
      =
      X^0-iX^3,   
      B_2
      =
      -iX^1+X^2
  
  The columns of X^μ⊗σ_μ can be reordered using (<ref>) to obtain:
  
      X^μ⊗σ_μ
      =
      [    B_1 -B_2^†;    B_2  B_1^† ]
  
  Then _1̇=(S_1̇,B_1,B_2)^T and _2̇=(S_2̇,-B_2^†,B_1^†)^T, which yields:
  
      μ^1
      =
      J^†I^†+[B_2^†,B_1^†]+IJ+[B_1,B_2]
      
         μ^2
      =
      i
      (
      -J^†I^†-[B_2^†,B_1^†]+IJ+[B_1,B_2]
      )
      
         μ^3
      =
      -II^†+J^†J-[B_1,B_1^†]-[B_2,B_2^†]
  
  Next note that μ^1 and μ^2 are purely hermitian and antihermitian respectively, so defining
  μ^ℝ=-μ^3 and μ^ℂ=1/2(μ^1-iμ^2), we have the final form of the ADHM equations:
  
      μ^ℝ=μ^ℂ=0
  
  The matrices I,J,B_1, and B_2 can now be interpreted as linear operators acting on vector spaces 𝒱≅ℂ^k and 𝒲≅ℂ^N, which obey the ADHM equations:
  
      I:𝒲→𝒱  
      J:𝒱→𝒲  
      B_1,2:𝒱→𝒱
  
  We call the space of such operators (modulo the dual group transformations (<ref>), 𝔪_k, and claim that this is the k-instanton moduli space. The last step to verifying this claim is to show that the ADHM construction does indeed yield all instantons. From a given A_μ yielding a self-dual F_μν we must be able to recover matrices  and  satisfying the ADHM equations and hermiticity properties derived above. This can be done using the relevant massless Dirac equation.
  
  
  § COMPLETENESS OF THE CONSTRUCTION
  
  Consider the massless Dirac equation in the instanton background, that is for A_μ a connection corresponding to a self-dual field strength. We first study its solutions in the formalism of ADHM, then later show how knowledge of said solutions allows inversion of the ADHM construction.
  
  
   §.§ Recovering  and .
  
  The massless Dirac equation is given by γ^μD_μΨ=0, where D_μ=∂_μ+A_μ, Ψ=(ψ^+,ψ^-)^T, and 
  γ^μ
  =
  [    0  σ^μ; σ̅^μ    0 ].
  This gives two equations for the two component Weyl spinors ψ^±:
  
      σ^μD_μψ^+=σ̅^μD_μψ^-=0
  
  It can be shown that there are no solutions to the positive chirality equation, and exactly k linearly independent solutions to the negative chirality one, where k is the instanton number associated with the connection A_μ, <cit.>. We thus rename ψ^- to ψ and arrange these solutions in an N× 2k matrix which turns out to be <cit.>:
  
      ψ^α
      =
      v^†^α
  
  Where , ,  and v are the matrices determined in the previous section <cit.>. In particular  has its canonical form and  is defined by the equation:
  
      2^-1(x)
      =
      ^†,γ̇_γ̇
      +
      (
      ^†α̇^αx_αα̇
      +
      x̅^αα̇^†_α_α̇)
      +
      x̅^αβ̇^†_α^β
      x_ββ̇
      =
      ^†
      +
      2X^μx_μ
      +
      x^21_k
  
  Certain moments of these solutions can then be calculated, as can some relevant asymptotic behaviour. Together this is enough to recover that X^μ is hermitian as well as the ADHM constraints, thus showing that the ADHM construction is complete.
  
  The moments are calculated via the following formula: ψ̅_αψ^α=-1/4_μ^μ <cit.>. This formula and the definition of  give:
  
      ∫ d^4xψ̅_αψ^α
      =
      π^21_k ,   ∫ d^4xψ̅_αψ^αx^μ
      =
      -π^2X^μ
  
  The first equation says that the solutions to the Dirac equation are orthonormal up to a constant, and the second says that part of  can be recovered from these solutions. In particular if we can find a complete orthonormal set of solutions to the massless Dirac equation, we can determine part of the matrix  by calculating the first moment of ψ̅_αψ^α.
  
  To recover S_α̇, we use that <cit.>:
  
      ψ^αx_αα̇→-1/x^2S_α̇ , as 
      x→∞.
  
  Given this asymptotic behaviour as a definition, an orthonormal set of solutions to the Dirac equation gives S_α̇ by simply taking a limit.
  
  
   §.§ Inverting the ADHM construction.
  
  Given a connection A_μ corresponding to a self-dual curvature, the ADHM construction can be inverted as follows: we first solve the corresponding massless Dirac equation to obtain a complete set of orthonormal solutions ψ^α. The corresponding X^μ and S_α̇ are then determined by doing a moment integral and taking a limit respectively.
  
  With these quantities at hand, we define a derived ,  and thus Δ by the canonical forms (<ref>) and (<ref>). The corresponding v is then obtained by simply finding the null space of Δ^†. We have thus recovered all the initial data.
  
  To conclude that the ADHM construction is complete it must also be shown that the derived quantities satisfy the required hermiticty properties as well as the ADHM equations.
  
  The hermiticity of X^μ and the ADHM equations were both derived from the assumption that Δ^†Δ=^-11_2, which is itself derived from the assumption that Δ^†Δ commutes with the quaternions, so it is sufficient to show that this holds for the derived Δ  <cit.>. A lengthy but relatively straightforward calculation involving Green's functions establishes the result. The calculation will be omitted for brevity, but the details can be found in <cit.>.
  
  We have now explicitly constructed the k-instanton moduli space 𝔪_k of =2 SYM with gauge group SU(N). It consists of linear operators I,J,B_1,2 (<ref>) satisfying the ADHM equations (<ref>), modulo some dual group transformations (<ref>). The full instanton moduli space is then:
  
      _inst:=⊕_k=1^∞𝔪_k
  
  
  4.4
  For large k the ADHM equations become very difficult so the integration must be performed without introducing local coordinates on the k-instanton moduli space. This can be achieved using equivariant integration.
  
  
  
  § APPLICATION TO =2 SYM
  
  In section 3.5.4, the path integral of =2 SYM was shown to localise to the instanton moduli space _inst. In section 4.3.2 an explicit construction of _inst was obtained. With these two pieces of information we can now deduce localisation formulae for expectation values in terms of finite dimensional integrals. It is at this point we perform a Wick rotation, taking our spacetime from ℝ^1,3 to ℝ^4 and introducing a factor of i in (<ref>).
  
  For a BRST closed, gauge invariant operator , we have that 
  𝒪
  =
  ∫__inst[ X]𝒪e^-S[X].
  Recalling the localisation argument of Section 3.4, a Q̅-exact term can be added to the action without changing the value of this integral. Since S is Q̅-cohomologous to S_top and recalling the gauge fixing potential V (<ref>), we have:
  
      𝒪
      =
      ∫__inst[ X]𝒪e^-(S_top[X]+Q̅V(x,t))
  
  Taking t→∞ then sets F_μν=i(⋆F)^μν everywhere that the integral is non-vanishing. Substituting this condition into the action leads to following contribution from 𝔪_k:
  
      S|_𝔪_k   =
      (
      S_top
      -
      1/4g^2∫ d^4xTrF_μνF^μν
      +
      0
      )
      |_𝔪_k
         =
      (
      kΘ
      -
      i/4g^2∫ d^4xTrF_μν(⋆F)^μν
      +
      0
      )
      |_𝔪_k
         =
      -Θ k
      -
      i8π^2k/g^2
         =
      i(2π i k τ)
  
  The expectation value of  is then:
  
      𝒪
      =
      ∫__inst[ X]𝒪e^-i(S_top[X]+Q̅V(x,t))
      =
      ∑_k=0^∞
      e^2π ikτ∫_𝔪_k_k
  
  where _k is Q̅-cohomologous to . The expectation value of a general BRST closed and gauge invariant observable has thus been reduced to a sum of finite dimensional integrals over known spaces!
  
  
  CHAPTER: LORENTZ DEFORMATION, THE OMEGA BACKGROUND AND THE PREPOTENTIAL
  
  Using the results of the previous chapter, we can calculate the partition function of =2 SYM with gauge group SU(N). However, to access the prepotential a little more work will be necessary. This will lead to a second localisation argument which gives the partition function in terms of the prepotential. We can then equate the two expressions for the partition function to obtain the prepotential in terms of known quantities.
  
  The idea is to compactify a six dimensional =1 SYM theory in a non-trivial spacetime to give a deformed four dimensional =2 theory. This is called a Lorentz deformation and the resulting spacetime is called the Ω-background. It turns out that the partition function of this theory is easily expressed in terms of the prepotential. We then take the limit in which the deformation of the Ω-background vanishes and demand that the two expressions for the partition function agree, thus yielding the prepotential.
  
  
  § LORENTZ DEFORMATION
  
  This section follows <cit.>, which also contain additional details. To carry out the Lorentz deformation we exploit the dimensional reduction construction of =2 SYM from Section 1.6. This construction used the six dimensional flat metric:
  
      ds^2
      =
      g_μνdx^μdx^ν-(dx^4)^2-(dx^5)^2
  
  To accomplish the Lorentz deformation, let the torus T^2 act on ℝ^3,1 by Lorentz rotations. This is achieved by introducing two independent Lorentz rotations corresponding to the x^4 and x^5 directions, packaged as follows:
  
      V_a^μ
      =
      (Ω_a)^μ_νx^ν,   
      a=4,5.
  
  The deformed metric is:
  
      ds^2
      :=
      g_μν(dx^μ+V_a^μdx^a)(dx^ν+V_b^νdx^b)-(dx^4)^2-(dx^5)^2
      =:
      G_IJdx^Idx^J
  
  The components of the deformed metric and its inverse can easily be determined.
  
      G_μν=g_μν ,             
      G^μν=g^μν-V_a^μV_a^ν
         
      G_a,μ=V_aμ ,             
      G^aμ=V_a^μ
         
      G_ab=-δ_ab+V_a^μV_b,μ ,   
      G^ab=-δ^ab
  
  A more tedious but straightforward computation shows that G:= G_IJ=-1, and that the curvature tensor vanishes if Ω_4 and Ω_5 commute. Since we want a flat metric, this will from now on be assumed to be the case.
  
  To facilitate computations, introduce a vielbein:
  
      ds^2
      =
      g_μνe_I^(μ)e_J^(ν)dx^Idx^J-e_I^(a)e_J^(a)dx^Idx^J
  
  The components can be calculated by expanding the metric:
  
      e_ν^(μ)=δ^μ_ν,   
      e_a^(μ)=V^μ_a,   
      e_μ^(a)=0
      ,        
      e_b^(a)=δ^a_b
         
      e^μ_(ν)=δ^μ_ν,   
      e^a_(μ)=0
      ,      
      e^μ_(a)=-V^μ_a,   
      e^a_(b)=δ^a_b
  
  We now rewrite the =1, d=6 SYM action (<ref>) in this background and once again compactify in the x^4 and x^5 directions. This process will yield =2 SYM in the Ω-background.
  
  We first rewrite the gauge kinetic term in flat vielbein indices:
  
      F_IJF^IJ   =
      (e_I^(N)e_J^(M)F_(N)(M))(e_K^(O)e_L^(P)F_(O)(P))G^IKG^JL
         =
      F_(N)(M)F_(O)(P)G^(N)(O)G^(M)(P)
         =
      F_(I)(J)F^(I)(J)
  
  Using that F_(I)(J)=e^K_(I)e^L_(J)F_KL, the components of the curvature can be obtained:
  
      F_(μ)(ν)=F_μν,   
      F_(a)(μ)=F_aμ-V^ρ_aF_ρμ,   
      F_(a)(b)=V^μ_aV^ν_bF_μν-F_aνV^ν_b-V^μ_aF_μ b+F_ab
         
      F^(μ)(ν)=V^μ_aV^ν_bF^ab+F^μ aV^ν_a+V^μ_aF^aν+F^μν,   
      F^(a)(μ)=F^aμ+V_b^μF^ab,   
      F^(a)(b)=F^ab
  
  Recalling the definition of H and defining V^μ and Ω_μν analogously, the gauge term can be rewritten as:
  
      -1/4F_IJF^IJ
      =
      -
         1/4F_μνF^μν
      +
      (∇_μH+V^νF_νμ)(∇^μH^†+V̅^νF_ν^μ)
      
         -
      1/2(
      [H,H^†]-iV̅^μV^νF_μν-i(V^μ∇_μH^†-V̅^ν∇_νH)
      )^2
  
  This resembles the undeformed expression with some shifts having been introduced. In fact this observation can be strengthened by defining a deformed version of H:
  
      :=
      H-iV^μ∇_μ
      ,  ^†
      :=
      H^†-iV̅^μ∇_μ
  
  Recalling that [∇_μ,∇_ν]=-iF_μν, we have:
  
      [,^†]
      =
      [H,H^†]
      -
      iV̅^μV^νF_μν
      -
      i
      (
      V^μ∇_μH^†
      -
      V̅^μ∇_μH
      )
      -
      i
      (
      HV̅^μ
      -
      H^†V^μ
      )
      ∇_μ
  
  However for commuting Ω_a's the last term vanishes, yielding:
  
      -1/4F_IJF^IJ
      =
      -
      1/4F_μνF^μν
      +
      (∇_μH+V^νF_νμ)(∇^μH^†+V̅^νF_ν^μ)
      -
      1/2
      [,^†]^2
  
  Proceeding similarly for the fermionic term (and adding a spin  operator term to the definition of , as detailed in <cit.>), and rewriting a little it turns out that the only difference between the undeformed action (<ref>) and the deformed action is the swapping of H for  and the addition of the following term:
  
      -
      1/4Ω̅_ρμF^ρμ
      -
      1/4Ω_ρμF^ρμ^†
      -
      1/2√(2)Ω_μνψ̅^Aσ̅^μνψ̅_A
      -
      1/2√(2)Ω̅_μνψ^Aσ^μνψ_A
  
  In the =1 superspace formalism this corresponds to making the coupling constant coordinate dependent:
  
      τ↦τ(x,θ):=τ-1/√(2)(Ω̅_μν)^+θ^μθ^ν
  
  
  
  § LOCALISATION IN THE Ω-BACKGROUND
  
  We wish to use localisation to evaluate the partition function of =2 SYM in the Ω-background. This requires a nilpotent (to define a cohomology), BRST operator with respect to which the Lorentz deformed action is still exact. This is the case with the deformed BRST operator:
  
      Q̅_Ω
      =
      Q̅
      +
      1/2√(2)Ω^μ_νx^νQ_μ
  
  provided we make another shift of the coupling constant <cit.>:
  
      τ(x,θ)
      :=
      τ-1/√(2)(Ω̅_μν)^+θ^μθ^ν
      +
      1/2√(2)Ω̅_μνΩ^μ_ρx^ρx^ν
  
  Since this is a real shift it only changes the topological term of the action.
  
  From the transformation properties, {Q̅,Q_μ}=4i∇_μ, and Q̅^2=Q_μ^2=0. Thus we have that up to a gauge transformation:
  
      Q̅_Ω^2
      =
      1/2√(2)Ω^μ_νQ̅(x^ν)Q_μ
      +
      1/2√(2)Ω^μ_νx^ν{Q̅,Q_μ}
      +
      1/8Ω^μ_νΩ^ρ_σx^νQ_μ(x^σQ_ρ)
      =
      √(2)iΩ^μ_νx^ν∇_μ+C
      =
      √(2)iΩ^μ_νx^ν_μ
  
  where we have used that constants are gauge equivalent to zero, and that the covariant derivative is gauge equivalent to the ordinary derivative.
   
  Recalling the antisymmetry of Ω we have that:
  
      Ω^μ_νx^ν_μf
      =
      -Ω^ν_μx^ν_μf
      =
      -Ω^ν_μ(_μ(x^νf)-δ^ν_μf)
      =
      -_μ(Ω^ν_μx^νf)-Tr(Ω)f
      =
      _μ(-Ω^ν_μx^νf)
  
  So up to gauge transformations and total derivatives, we have that Q̅_Ω^2=0. Thus Q̅_Ω is nilpotent and a valid BRST operator.
  
  Since the action is Q̅_Ω-exact, we can once again localise the path integral defining the partition function. Recall that the action is 
  1/8π∫ d^4x d^4θτTrΨ^2
  , 
  so the path integral localises to the zero modes of Ψ. At low energies the effective action (<ref>) is valid, so <cit.>:
  
      Z_Ω(a)
      =
      ∫_Zero modes Xe^S[X]
      =
      exp{(
      1/8π i∫ d^4x d^4θ(-1/2√(2)a;Λ(x,θ))
      )
      }
  
  where we have used that zero modes correspond to all non-scalar fields vanishing and the Higgs field being given by the constant a as discussed in Section 2.1.
  
  
  § THE PREPOTENTIAL
  
  To access the prepotential, the superspace integral in (<ref>) must be evaluated. To this end we define another operator:
  
      _Ω
      :=
      θ^μ_μ
      +
      1/2√(2)Ω^μ_νx^ν/θ^μ
  
  The key observation is that τ(x,θ) is annihilated by this operator:
  
      -4_Ωτ(x,θ)
      =
      Ω^μ_νx^ν/θ^μ
      (
      (Ω̅_ρσ)^+θ^ρθ^σ
      )
      -
      θ^σ_σ
      (
      Ω̅_μνΩ^μ_ρx^ρx^ν
      )
      =
      0
  
  Recall that the renormalization group equation relates the complex coupling constant to the dynamically generated scale: Λ=e^2π i/β(τ-τ_0). So in the Ω-background, Λ is effectively superspace dependent and _ΩΛ(x,θ)=0. Furthermore _Ω is nilpotent up to total derivatives and gauge transformations <cit.>.
  
  Since a is a constant and Λ(x,θ) is _Ω-closed, we see that the integrand on the right hand side of (<ref>) is _Ω-closed. A slight generalization of the localisation formula derived in Section 3.3 can thus be applied. To do so, we fix a system of coordinates where Ω has the following canonical form:
  
      Ω
      =
      1/√(2)[    0    0    0 -ϵ_1;    0    0 -ϵ_2    0;    0  ϵ_2    0    0;  ϵ_1    0    0    0;      ]
  
  In the above expression, the termsϵ_1 and ϵ_2 measure the deformation of spacetime and will eventually be sent to zero to recover =2 SYM on undeformed spacetime.
  
  The infinitesimal torus action is given by Ω, yielding in the language of Section 3.3, Pf(-L_V(0))=1/2ϵ_1ϵ_2. The only fixed point is x=θ=0, so:
  
      (1/8π i∫ d^4x d^4θ(-1/2√(2)a;Λ(x,θ))
      )
      =
      1/ϵ_1ϵ_2(a,Λ|_0;ϵ)
  
  Where we have used that  is homogeneous of degree 2, and where (a,Λ|_0;ϵ) is the Lorentz deformed prepotential.
  5.1
  By assumption, the Lorentz deformed prepotential is related to the usual prepotential by:
  
      (a,Λ)=lim_ϵ_1,2→ 0(a,Λ|_0;ϵ)
  
  
  The prepotential can now be evaluated! We equate the Lorentz deformed and undeformed partition functions, use the ADHM construction to evaluate the undeformed partition function, then take the ϵ_1,2→ 0 limit:
  
      lim_ϵ_1,2→ 0e^1/ϵ_1ϵ_2(a,Λ|_0;ϵ)
      =
      ∑_k=0^∞e^2π ikτ∫_𝔪_k1
      =
      ∑_k=0^∞e^2π ikτVol(𝔪_k)
  
  For later convenience, define Z_k(a)=Vol(𝔪_k).
  
  
  CHAPTER: EXPLICIT CALCULATION OF THE PARTITION FUNCTION
  
  With the ADHM construction at hand, the integrals (<ref>) over the instanton moduli space can be explicitly evaluated. This process will yield the right hand side of (<ref>), and thus the prepotential. Directly evaluating the prepotential in this way gives a power series which will agree order by order with that derived from Seiberg Witten theory.
  
  
  § TRANSFORMATION PROPERTIES
  
  Recall that the k-instanton moduli space 𝔪_k is the space of linear operators I,J,B_1,2 (<ref>) satisfying the ADHM equations (<ref>), modulo the dual group transformations (<ref>). To deal with the fact that the operators I,J,B_1,2 must satisfy the ADHM equations we introduce two supplementary multiplets which act as Lagrange multiplers:
  
      (χ_ℝ,H_ℝ),   (χ_ℂ,H_ℂ)
  
  The transformation properties of the ADHM fields under the torus action on ℝ^3,1 will be required. They are obtained from the transformation properties of the position vector x^μ. Recalling that Ω=1/√(2)(Ω_4+iΩ_5): 
  
      x^0↦ -ϵ_1x^3  
      x^1↦ -ϵ_2x^2  
      x^2↦ϵ_2x^1  
      x^3↦ϵ_1x^0
  
  yielding for example:
  
      B_2=-iX^1+X^2↦ iϵ_2X^2+ϵ_2X^1=iϵ_2B_2
  
  so that under finite rotations, B_2↦ e^iϵ_2B_2.
  
  For the spinorial quantities I and J, recall that spinors transform under infinitesimal rotations as 
  ψ↦ e^-i/2θψ
  . So since T^2≅ U(1)× U(1) is abelian,
  ψ↦ e^-i/2(ϵ_1+ϵ_2)ψ
  .
  
  The end result is:
  
      B_1↦ e^iϵ_1B_1,   B_2↦ e^iϵ_2B_2
      ,   
      I↦ e^-iϵ_+I,   J↦ e^-iϵ_+J
      ,  μ^ℝ↦μ^ℝ,  μ^ℂ↦ e^iϵμ^ℂ
  
  where ϵ:=ϵ_1+ϵ_2, and ϵ_+:=1/2ϵ.
  
  The BRST transformation properties of the ADHM fields and the supplementary multiplets will also be needed. A detailed derivation would be quite off topic, so we present only a brief overview. More details can be found in <cit.>. The transformation properties are:
  
      Q̅_ΩB_1,2=ψ_1,2,   Q̅_Ωψ_1,2=[ϕ,B_1,2]+iϵ_1,2B_1,2
         Q̅_ΩI=ψ_I,         Q̅_Ωψ_I=ϕ I-Ia-iϵ_+I
      
         Q̅_ΩJ=ψ_J,        Q̅_Ωψ_J=-Jϕ+aJ-iϵ_+J
      
         Q̅_Ωχ_ℝ=H_ℝ,      Q̅_ΩH_ℝ=[ϕ,χ_ℝ]
      
         Q̅_Ωχ_ℂ=H_ℂ,      Q̅_ΩH_ℂ=[ϕ,χ_ℂ]+iϵχ_ℂ
         Q̅_Ωλ = η,         Q̅_Ωη = [λ,ψ]
  
  Where ψ_1,2,I,J are the basis of "1-forms" corresponding to the "coordinates" B_1,2,I,J, on 𝔪_k.
  
  The H and ψ expressions come from considering the weights of Q̅_Ω^2, and that Q̅_Ω squares to the sum of a gauge transformation with parameter a, and a dual group transformation with parameter ϕ. The ϵ terms come from the torus transformations derived above.
  
  
  § THE CALCULATION
  
  In this section we finally reduce the instanton partition function to a sum of contour integrals.
  
  
   §.§ Set up.
  
  The terms Z_k(a;ϵ) of the partition function sum (<ref>) are:
  
      Z_k(a;ϵ)
      =
      ∫_𝔪_k1
      =
      1/Vol(G_D)∫ϕηλ Hχ B_1 B_2 I Jψ e^iQ̅(χ·μ+tχ· H+ψ· V(λ))
  
  where we have converted an integral over the quotient space 𝔪_k≅μ^-1(0)/G_D to an integral over the full space by imposing the constraints defining 𝔪_k via Lagrange multipliers <cit.>.
  
  In the above expression the dotted terms are:
  
      χ·μ
      =
      Tr(
      χ_ℝμ_ℝ+1/2(χ_ℂ^†μ_ℂ+χ_ℂμ_ℂ^†)
      )
      
         χ· H
      =
      Tr(
      χ_ℝH_ℝ+1/2(χ_ℂ^†H_ℂ+χ_ℂH_ℂ^†)
      )
  
  and V(ψ) is the dual group flow:
  
      ψ· V(λ)
      =
      Tr(
      ψ_1[λ,B_1^†]
      +
      ψ_2[λ,B_2^†]
      +
      ψ̅_1[λ,B_1]
      +
      ψ̅_2[λ,B_2]
      +
      ψ_Iλ I
      -
      I^†λψ̅_I
      -
      Jλψ̅_J
      +
      ψ_Jλ J^†)
  
  To make the computation easier, we make a choice of localisation scheme by adding another Q̅-exact term to the action:
  
      Q̅(
      it'Tr
      (
      χ_ℝλ
      )
      -
      1/2t”Tr{∑_s=1^2(B_s^†ψ_s-ψ̅_̅s̅B_s)
      -
      I^†ψ_I
      +
      ψ̅_II
      -
      J^†ψ_J
      +
      ψ̅_JJ
      })
  
  
  The idea is to now take t,t',t”→∞ which will integrate out various fields, eventually resulting in a sum of contour integrals.
  
  
   §.§ The t'→∞ limit.
  
  We first take t'→∞. To see what this does, note that
  Q̅Tr
  (
  χ_ℝλ
  )
  =
  Tr(H_ℝλ-χ_ℝη), so the exponent contains the term 
  -t'Tr(H_ℝλ-χ_ℝη). We can thus choose to first localise H_ℝ to zero, giving a delta function at t'Tr(λ) and allowing us to integrate out λ:
  
      Z_k(a;ϵ)
         =
      1/Vol(G_D)∫λ(…)δ(t'Tr(λ))e^iQ̅(χ·μ+tχ· H+ψ· V(λ))+…
         =
      1/Vol(G_D)∫λ(…)δ(Tr(λ))t'^-λe^iQ̅(χ·μ+tχ· H+ψ· V(λ))+…
  
  Where t'^-λ=(t'λ)^-1 is the Jacobian of the bosonic field redefintion; λ↦ t'λ⇒λ↦(t'λ)^-1λ'.
  
  Now, λ=1/2(adjrep(G_D))=1/2k^2 (the factor of 1/2 is because λ is real), so the integral over λ sets λ=0 and introduces a factor of t'^-1/2k^2.
  
  In this limit the χ_ℝ integral localises to a delta function at t'Tr(η), allowing us to integrate out the fermionic field η. To do so, redefine the field; η↦ t'η. Since η is fermionic, this changes the integration measure by a factor of (t')=t'^1/2k^2 (as is familiar from Grassmann integrals). This leads to cancellation of the determinant factors. 
  
  The partition function now becomes:
  
      Z_k(a;ϵ)
      =
      1/Vol(G_D)∫ϕ H_ℂχ_ℂ B_1 B_2 I Jψ e^iQ̅(χ·μ+tχ· H+ψ· V(λ))+…|_λ=η=H_ℝ=χ_ℝ=0
  
  
  
   §.§ The ϕ integration measure.
  
  We now discuss the ϕ part of the integration measure. This is where the non-trivial part of the resulting integral comes from. Since ϕ∈Lie(G_D), this integral can be reduced from the whole Lie algebra to its maximal torus by using the Weyl integral formula <cit.>.
  
  The maximal torus of the Lie group G_D=U(k) can be parametrised as T={e^iθ_1,…,e^iθ_k|θ_j∈ℝ}, so an element of the corresponding Cartan subalgebra can be given by ϕ={iϕ_1,…,iϕ_k}. The Weyl integral formula then reduces the integration measure:
  
      ϕ↦1/k!∏_i=1^kdϕ_i/2π i∏_i<j≤ k(ϕ_i-ϕ_j)
  
  
  
   §.§ The t→∞ limit.
  
  We now take t→∞ and expand the Q̅(χ· H) term of the exponent to see which terms can be integrated out:
  
      Q̅(χ· H)
      =
      Tr(
      H_ℝ^2+H_ℂH_ℂ^†+χ_ℝ[ϕ,χ_ℝ]+χ_ℂ^†([ϕ,χ_ℂ]+iϵχ_ℂ)
      )
      =
      Tr(
      H_ℂH_ℂ^†+χ_ℂ^†([ϕ,χ_ℂ]+iϵχ_ℂ)
      )
  
  The only part of the integral containing H_ℂ is a Gaussian factor ∫ H_ℂe^itH_ℂH_ℂ^†, which can be evaluated:
  
      ∫ H_ℂ H_ℂ^†e^itH_ℂH_ℂ^†
      =
      e^-Tr(log(t1_k^2))
      =
      t^-k^2
  
  Where we have used that H_ℂ is k^2× k^2 since it is in the adjoint representation of the dual group, U(k). This factor will later be eliminated by integrating out the fermionic χ_ℂ.
  
  6.1
  The cancellation of  and 1/ terms arising from bosonic and fermionic fields is the famous bosonic/fermionic cancellation of supersymmetry.
  
  
  
  The fields χ_ℂ and χ_ℂ^† can now be integrated out. To do so, note that ϕ_ij=√(-1)ϕ_iδ_ij so:
  
      itTr(
      χ_ℂ^†
      (
      [ϕ,χ_ℂ]
      +
      iϵχ_ℂ
      )
      )
         =
      it∑_i,j=1^k
      (χ_ℂ^†)_ij
      (
      [ϕ,χ_ℂ]
      +
      iϵχ_ℂ
      )_ji
         =
      -t
      (
      ∑_i,j=1^k(ϕ_i-ϕ_j)χ^†_jiχ_ij
      +
      ϵ∑_i,j=1^kχ_ji^†χ_ij)
      
         =
      -t
      (
      ∑_i,j=1^k(ϕ_i-ϕ_j)χ̅_̅i̅j̅χ_ij
      +
      ϵ∑_i,j=1^kχ̅_̅i̅j̅χ_ij)
      
         =
      -t
      (
      ∑_i<j≤ k(
      (ϕ_i-ϕ_j+ϵ)|χ_ij|^2
      +
      (ϕ_j-ϕ_i+ϵ)|χ_ji|^2)
      +
      ϵ∑_i=1^k|χ_ii|^2)
  
  Since the corresponding integral is fermionic, integrating out χ_ij and χ_ij^† gives a factor t(ϕ_i-ϕ_j+ϵ) for i≠ j, or a factor tϵ, for i=j. The case i=j occurs k times, so we have:
  
      ∫χ_ℂ^†χ_ℂe^itTr(
      χ_ℂ^†
      (
      [ϕ,χ_ℂ]
      +
      iϵχ_ℂ
      )
      )
      =
      t^k^2ϵ^k∏_i<j≤ k(ϕ_i-ϕ_j+ϵ)(ϕ_j-ϕ_i+ϵ)
      =
      t^k^2ϵ^k∏_i<j≤ k((ϕ_i-ϕ_j)^2+ϵ^2)
  
  
  The current result is as follows:
  
      Z_k(a;ϵ)
      =   ϵ^k/k!Vol(G_D)∫∏_i=1^kdϕ_i/2π i B_1 B_2 I Jψ∏_i<j≤ k(ϕ_i-ϕ_j)((ϕ_i-ϕ_j)^2+ϵ^2)
      
         
      e^
      Q̅(
      -
      1/2t”Tr{∑_s=1^2(B_s^†ψ_s-ψ̅_̅s̅B_s)
      -
      I^†ψ_I
      +
      ψ̅_II
      -
      J^†ψ_J
      +
      ψ̅_JJ
      })
  
  
  
   §.§ The t”→∞ limit.
  
  We now send t”→∞ to integrate out B_1,2,I,J and the corresponding fermionic fields. To do so, first note that the exponent is:
  
      -t”(
      ψ̅_1ψ_1
      +
      ψ̅_2ψ_2
      +
      ψ̅_Iψ_I
      +
      ψ̅_Jψ_J
      +   ∑_s=1^2
      B_s^†([ϕ,B_s]+iϵ_sB_s)
      
         
      -
      I^†(ϕ I-Ia-iϵ_+I)
      -
      J^†(-Jϕ+aJ-iϵ_+J)
      )
  
  The B_s terms above have the exact same form as the χ_ℂ terms discussed previously, so by identical workings and noting that the B_s are complex bosonic fields, we find that the corresponding integrals yield:
  
      t”^-2k^2∏_s=1^21/ϵ_s^k∏_i<j≤ k1/(ϕ_i-ϕ_j)^2-ϵ_s^2
  
  The Gaussian integrals over the ψ_s then eliminate the t”^-2k^2 factor.
  
  The process for I and J is similar. The only difference is that this time rather than a commutator with ϕ, an a appears in the Q̅_Ω action. Noting that a∈Lie(SU(N)) is a gauge parameter at infinity, we can through similar reasoning to the ϕ case write a=(ia_1,…,ia_N). Proceeding as above, the resulting factor (after also integrating out ψ_I,J), is:
  
      ∏_i=1^k∏_j=1^N1/(ϕ_i-a_j)-ϵ_+^2
  
  Only the ϕ_i integrals remain!
  
  
   §.§ The result.
  
  To write the final result nicely, we introduce some special polynomials:
  
      Δ_±(x)=∏_i<j≤ k((ϕ_i±ϕ_j)^2-x^2)
      ,   𝒫(x)=∏_i=1^N(x-a_i)
  
  The k'th contribution to the partition function can now be written as follows:
  
      Z_k(a;ϵ)=∫∏_i=1^kdϕ_i/2π i𝔷_k(a,ϕ;ϵ)
       ,  𝔷_k(a,ϕ;ϵ)
      =
      1/k!ϵ^k/ϵ_1^kϵ_2^kΔ_-(0)Δ_-(ϵ)/Δ_-(ϵ_1)Δ_-(ϵ_2)∏_i=1^k1/𝒫(ϕ_i+ϵ_+)𝒫(ϕ_i-ϵ_+)
  
  In principle, the partition function and prepotential of =2 SYM with gauge group SU(N) have now been determined:
  
      Z(a;ϵ)=Z_pert(a;ϵ)∑_k=0^∞e^2π i k τZ_k(a;ϵ)=lim_ϵ_1,ϵ_2→∞e^1/ϵ_1,ϵ_2(a,Λ) ,   
      Z_k(a;ϵ)=lim_ϵ_1,ϵ_2→∞∫∏_i=1^kdϕ_i/2π i𝔷_k(a,ϕ;ϵ)
  
  6.2
  The factor 1/k!Vol(G_D) has been removed in the above by normalisation.
  
  
  
  § INSTANTON CORRECTIONS FOR GAUGE GROUP SU(2)
  
  The problem of calculating the prepotential has been reduced to calculating certain contour integrals and limits. In this section we consider in detail the SU(2) case. In particular, the k=1 contribution will be calculated exactly and the procedure for calculating higher terms will be clarified.
  
  
   §.§ Calculating Z_1.
  
  The integrals Z_k(a;ϵ) must be calculated explicitly. This is done by the standard complex analytic trick of integrating around a large semicircular contour in the complex plane and taking the radius to infinity. Naively it appears that the ϕ_i integrals pass through the poles of the 𝔷_k(a,ϕ;ϵ), which would would lead to non-convergence. To cure this problem we introduce a small imaginary shift. Schematically; ϵ_s→ϵ_s +i0.
  6.3
  The shift ϵ_s→ϵ_s +i0 can be derived rigorously by considering a modified contour which excises the poles using small semicircular contours. 
  
  For k=1 the integrand is:
  
      𝔷_1(a,ϕ;ϵ)
      =
      ϵ/ϵ_1ϵ_21/(ϕ_1+ϵ_+-a_1)(ϕ_1+ϵ_+-a_2)(ϕ_1-ϵ_+-a_1)(ϕ_1-ϵ_+-a_2)
  
  The contour of integration is shown in Figure 6.1 in which the poles of 𝔷_1(a,ϕ;ϵ) are denoted by black dots. Noting that the 𝔷_k(a,ϕ;ϵ) decay sufficiently fast at infinity, the residue theorem yields:
  
  
      Z_1(a;ϵ)
      =
      ϵ/ϵ_1ϵ_2(
      1/2ϵ_+(a_1-a_2+2ϵ_+)(a_1-a_2)
      +
      1/2ϵ_+(a_2-a_1+2ϵ_+)(a_2-a_1))
      =
      -
      1/2ϵ_1ϵ_21/a^2-ϵ_+^2
  
  where 2a:=a_1-a_2.
  
  
   §.§ Calculating _1.
  
  To calculate the corresponding term _1(a) of the prepotential note that _inst(a)=lim_ϵ_1,2→ 0(ϵ_1ϵ_2Z(a;ϵ)). Setting q:=Λ^2N then gives:
  
      ∑_k=1^∞_k(a)q^k=lim_ϵ_1,2→ 0(ϵ_1ϵ_2∑_k=0^∞q^kZ_k(a;ϵ))
  
  Viewing q as a generating parameter and setting Z_0(a;ϵ)=1, we can calculate _k(a) by taking k derivatives and setting q=0. Explicitly for k=1:
  
      _1(a)
      +
      (q)
      =
      lim_ϵ_1,2→ 0(
      ϵ_1ϵ_2/1+(q)(Z_1(a;ϵ)+(q))
      )
  
  So that upon setting q=0 we find _1(a)=-1/2lim_ϵ_1,2→ 0(1/a^2-ϵ_+^2)=-1/21/a^2.
  
  
   §.§ The instanton series.
  
  The above procedure is easily generalised to larger values of k at the cost of increasingly complicated contour integrals. The general solution will be discussed in Chapter 8. Proceeding in this way, the first three terms are <cit.>:
  
      _inst(a)
      =
      -1/2Λ^4/a^2
      -
      5/64Λ^8/a^6
      -
      3/64Λ^12/a^10
      +
      (Λ^16)
  
  This result is in agreement with the result of Chapter 2.
  
  
  CHAPTER: THE MANY INSTANTON LIMIT
  
  In the previous chapter, the prepotential was determined as a power series with terms indexed by k. In this form its analytic properties are not very clear. To this end we now discuss the large k limit which will lead to two important results.
  
  Firstly, the perturbative part of the prepotential (which has so far been neglected), will be recovered. Secondly, the Seiberg-Witten geometry will emerge. This will verify that the localisation approach is consistent with the Seiberg-Witten approach. In particular it will guarantee that the power series arising from the two methods agree with one another to arbitrarily high order.
  
  
  § THE K→∞ LIMIT
  
  The main contribution to Z(a,Λ;ϵ) comes from the region k∼1/ϵ_1ϵ_2 <cit.>. Since we take ϵ_1,ϵ_2→ 0, it follows that the dominant contribution comes from the region where k→∞.
  
  The large k limit is equivalent to taking some sort of semiclassical limit,  effectively converting a problem in field theory to one in quantum mechanics. Each ϕ_i is interpreted as a particle, converting the many contour integrals to a single quantum mechanical path integral. To achieve this in practice, we must derive the Hamiltonian of the corresponding quantum mechanical system.
  
  
  § THE HAMILTONIAN FROM THE EQUIVARIANT INDEX
  
  The integrand appearing in a quantum mechanical path integral generally has the form e^-1/ϵ_1ϵ_2H, so the integrand 𝔷_k of Z_k should be converted to a similar form. From general considerations, it turns out that the relevant Hamiltonian is given by  <cit.>:
  
      H
      =
      -lim_ϵ_1ϵ_2→ 0∑_αϵ_αlog|w_α/Λ|
  
  where the w_α are the weights of a certain torus action, and the ϵ_α are ± 1 depending on whether the coordinates from which the corresponding weights come are bosonic or fermionic.
  
  
   §.§ Weights from Ind_q.
  
  The weights w_α are obtained from the equivariant index of the Dirac operator:
  
      Ind_q=∑_αϵ_αe^w_α
  
  For the case of SU(N) in the adjoint representation, this index is:
  
      Ind_q   =
      1/(e^iϵ_1-1)(e^iϵ_2-1)(
      N+∑_l≠ m^Ne^ia_l-ia_m)
      
             
      -
      ∑_i=1^k∑_l=1^N(e^iϕ_i-iϵ_+-ia_l+e^-iϕ_i+ia_l-iϵ_+)
      +
      k(1-e^-iϵ_1)(1-e^-iϵ_2)
      
             
      +
      ∑_i≠ j^k(e^iϕ_i-iϕ_j+e^iϕ_i-iϕ_j-iϵ_1-iϵ_2-e^iϕ_i-iϕ_j-iϵ_1-e^iϕ_i-iϕ_j-iϵ_2)
  
  The above result is derived using the equivariant Atiyah-Singer index theorem and known results about Chern classes <cit.>.
  
  The weights can be extracted by using the following integral transform to convert e^w_α to log(w_α):
  
      f(x)
      ↦d/ds|_s=0Λ^s/Γ(s)∫_0^∞dt/tt^sf(itx)
  
  which is closely related to the Mellin transform. In the case at hand we have:
  
      e^iw_α↦d/ds|_s=0Λ^s/Γ(s)∫_0^∞dt/tt^se^-tw_α
      =
      log|
      w_α/Λ|
  
  Approximating this integral in the small ϵ limit then allows the weights w_α to be extracted.
  
  
   §.§ Extracting the weights.
  
  To determine the contribution to H from the first line of (<ref>), define:
  
      γ_ϵ_1ϵ_2(a_l-a_m,Λ)
      :=
      d/ds|_s=0Λ^s/Γ(s)∫_0^∞dt/tt^se^-t(a_l-a_m)/(e^-tϵ_1-1)(e^-tϵ_2-1)
  
  And expand to (1/ϵ_1ϵ_2):
  
      [b]
      γ_ϵ_1ϵ_2(a_l-a_m,Λ)
         =
      1/ϵ_1ϵ_2d/ds|_s=0Λ^s/Γ(s)∫_0^∞t^s-3e^-t(a_l-a_m)dt
      +(1)
      
         =
      1/ϵ_1ϵ_2d/ds|_s=0Λ^s/Γ(s)(a_l-a_m)^2-sΓ(s-2)
      +(1)
      
         =
      1/ϵ_1ϵ_2d/ds|_s=0Λ^s/(s-1)(s-2)(a_l-a_m)^2-s
      +(1)
      
      ⇒γ_ϵ_1ϵ_2(a_l-a_m,Λ)
         =
      1/ϵ_1ϵ_2(a_l-a_m)^2/2(
      log|Λ/a_l-a_m|+3/2)
      +(1)
  
  The first line of (<ref>) thus gives a contribution of ∑_l≠ m 1/2(a_l-a_m)^2(log|a_l-a_m/Λ|-3/2) to the Hamiltonian.
  7.1
  This contribution to H is the SU(N) generalisation of the perturbative part of the prepotential as determined using Seiberg-Witten theory in Section 2.4.1
  
  The contribution to H from each term in the last line of (<ref>) is -ϵ_1ϵ_21/(ϕ_i-ϕ_j)^2+…, as is easily verified using the following identity:
  
      f(0)+f(ϵ_1+ϵ_2)-f(ϵ_1)-f(ϵ_2)=-ϵ_1ϵ_2f”(0)+…
  
  Similarly the term k(1-e^-iϵ_1)(1-e^-iϵ_2) from the second line of (<ref>) gives a contribution -kϵ_1ϵ_2 to H.
  
  The final term of (<ref>) is rather simple as well. Since it is already in an exponential form we can read off the contribution right away:
  
      ∑_i=1^k∑_l=1^N(
      log(
      (ϕ_i-a_l)-ϵ_+/Λ)
      +
      log(
      -(ϕ_i-a_l)-ϵ_+/Λ)
      )
      =
      2∑_l=1^Nlog(
      |𝒫(ϕ_i)/Λ^N|
      )
  
  where for the last equality we have taken ϵ_+→ 0.
  
  The full Hamiltonian has now been determined:
  
      H
      =
      -
      ∑_l≠ m 1/2(a_l-a_m)^2(log|a_l-a_m/Λ|-3/2)
      +
      2ϵ_1ϵ_2∑_i=1^klog(
      |𝒫(ϕ_i)/Λ^N|
      )
      +
      (ϵ_1ϵ_2)^2∑_i≠ j1/(ϕ_i-ϕ_j)^2
  
  
  
   §.§ Particle densities and the profile function.
  
  As k→∞ the number of ϕ_i fields becomes infinite and it makes sense to instead work with particle densities. To this end we introduce the density function:
  
      ρ(x)
      =
      ϵ_1ϵ_2∑_i=1^kδ(x-ϕ_i)
  
  The Hamiltonian can now be rewritten:
  
      H
      =
      -
      ∑_l≠ m 1/2(a_l-a_m)^2(log|a_l-a_m/Λ|-3/2)
      +
      2∑_l=1^N∫ dx ρ(x)
      log(
      |x-a_l/Λ|
      )
      +
      _x≠ ydxdy
      ρ(x)ρ(y)/(x-y)^2
  
  where the bar denotes a principal value integral <cit.>.
  
  We now introduce the so-called profile function f(x):
  
      f(x)
      =
      -2ρ(x)
      +∑_l=1^N|x-a_l|
  
  7.2
  The profile function will later be the key to connecting the partition function to integer partitions and Young diagrams.
  
  Upon integration by parts, one can show that the Hamiltonian can be written in a very nice form:
  
      H[f]
      =
      -1/4∫ dxdy
      f”(x)f”(y)k_Λ(x-y)
  
  where k_Λ(x):=ϵ_1ϵ_2γ_ϵ_1ϵ_2(x,Λ). This expression is tough to dream up, but straightforward to verify.
  
  The deformed partition function in the many instanton limit can now be expressed as follows:
  
      Z(a,Λ;ϵ)
      ∼∫ f
      e^-1/ϵ_1ϵ_2H[f]
  
  
  
  § LAGRANGE MULTIPLIERS AND THE SPACE OF  PROFILE FUNCTIONS
  
  To analyse the path integral (<ref>) we nee to understand the space of profile functions f. The profile function was defined in terms of ρ which is itself dependent on the particles ϕ_i. The first goal of this section is to derive some easier to analyse conditions which also fix f. We will then introduce Lagrange multipliers to enforce these conditions, thus allowing (<ref>) to be replaced with an unconstrained path integral.
  
  For future convenience we make several remarks on the parameters {a_1,…,a_rank(G)}. These parameters are Cartan subalgebra elements of Lie(G), so for the SU(N) case there are N-1 of them which sum to some non-zero matrix. If an extra parameter a_N:=-(a_1+…+a_N-1) is introduced, then only the differences a_1-a_2,…,a_N-1-a_N matter and ∑_l=1^Na_l=0. From now on ths will be assumed to be the case.
  
  Following <cit.>, note that by definition ρ(x) is only supported on the compact set {ϕ_i}_i=1^k, so f(x)∼ N|x| as x→∞. This implies that f satisfies f(+∞)=f(-∞). Also note that f”(x)=2∑_l=1^Nδ(x-a_l)-2ρ”(x), so ∫_-∞^∞ f”(x)dx=2N since the a_l are distinct.
  
  In a similar way, the first two moments of f”(x) can be determined. Explicitly ∫_-∞^∞xf”(x)dx=2∑_l=1^Na_l=0, and ∫_-∞^∞x^2f”(x)dx=2∑_l=1^Na_l^2-4ϵ_1ϵ_2k. In fact, since the a_l are distinct and f”(x) has compact support, there exist intervals [α_l^-,α_l^+] containing a single a_l, so that:
  
      ∫_α_l^-^α_l^+xf”(x)dx=2a_l
  
  It turns out that it is sufficient to only enforce this condition in the path integral (<ref>) <cit.>.
  
  We now introduce the Lagrange multipliers ξ_l and consider the following modified Lagrangian:
  
      L[f,ξ]
      :=
      H[f]+∑_l=1^Nξ_l(
      1/2∫_α_l^-^α_l^+xf”(x)dx-a_l)
      =
      S[f,ξ]-∑_l=1^Nξ_la_l
  
  where S[f,ξ]:=H[f]+1/2∑_l=1^Nξ_l∫_α_l^-^α_l^+xf”(x)dx.
  7.3
  Physically, the Lagrange multipliers ξ_l can be interpreted as dual charges to the a_l.
  
  
  
  § EXTREMISING THE ACTION
  
  One can show that in the ϵ_1,2→ 0 limit the functional L[f,ξ] has a unique extremum <cit.>. The extremising f corresponds to the limiting partition profile shape, and is closely related to the Seiberg-Witten curve. This extremiser will be constructed explicitly.
  
  
   §.§ The surface tension function.
  
  Given the unique extremiser f_*(x) of L[f,ξ], we will also need the stationary points with respect to the ξ_i. Solving L[f_*,ξ]/ξ_i=0 easily gives the following condition for the stationary points:
  
      S[f_*,ξ_l]/ξ_l=a_l
  
  The idea now is to solve this system for the ξ_l in terms of the a_l and substitute back into H to get the value of the Hamiltonian at the extremiser. There is however, an issue. We need N independent equations to solve for the a_l but since ∑_l=1^Na_l=0, at most N-1 of these equations are independent. This issue can be resolved by introducing the surface tension function σ.
  
  Recalling that f(x)∼ N|x| as x→∞, we have that f'(+∞)=N and f'(-∞)=-N. Similarly since ∫_α_l^-^α_l^+f”(x)=2, we have that f'(α_l^+)=2+f'(α_l^-). It can be assumed without loss of generality that α_0^-=-∞, α_N^+=+∞, and α_l+1^-=α_l^+ (while possibly deleting some isolated points such as the ϕ_i). With this in mind it is clear that for x∈ [α_l^-,α_l^+], f'(x)∈[-N+2(l-1),-N+2l]. This observation motivates the following definition of the surface tension function.
  
  The surface tension, σ:[-N,N]→ℝ is defined to be the unique continuous piecewise linear function such that:
  
      σ'(t)=ξ_l, for t∈[-N+2(l-1),-d+2l]
      
         σ(-N)+σ(N)=0
  
  Upon integration the first condition gives several linear functions at various vertical positions. Demanding continuity yields a system of N-1 equations for N unknowns and the final condition gives the N'th equation needed to solve the system. For example, if (ξ_1,ξ_2,ξ_3,ξ_4)=(4,1,-2,-3), these conditions are easily solved to yield:
  
      σ(t)=4t+16, on [-4,-2]
       ,   σ(t)=t+10, on [-2,0]
      
      σ(t)=-2t+10, on [0,2]
       ,   σ(t)=-3t+12, on [2,4]
  
  The second term of the action functional S[f,ξ] can now be rewritten in a nicer way. Working backwards we have:
  
      _-∞^∞σ(f'(x))dx
         =
      _-∞^∞1σ(f'(x))dx
      
         =
      lim_a→∞[
      xσ(f(x))
      ]|_-a^a
      -
      ∫_-∞^∞xσ'(f'(x))f”(x)dx
      
         =
      lim_a→∞(
      a(σ(N)+σ(-N))
      )
      -
      ∑_l=1^N∫_α_l^-^α_l^+σ'(f'(x))xf”(x)dx
      
         =
      -
      ∑_l=1^Nξ_l∫_α_l^-^α_l^+xf”(x)dx
  
  We thus have:
  
      1/2∑_l=1^Nξ_l∫_α_l^-^α_l^+xf”(x)dx
      =
      -1/2_-∞^∞σ(f'(x))dx
  
  This implies that ∑_l=1^Nξ_l=0, and thus the ξ_l are in principle all fixed <cit.>.
  
  We seek to extremise the Lagrange functional L[f,ξ]. To do so we extremise the action functional:
  
      S[f,ξ]
      =
      -1/4∫ dxdy
      f”(x)f”(y)k_Λ(x-y)
      -1/2_-∞^∞σ(f'(x))dx
  
  Since σ is not smooth there will be two cases. One for points of continuity of σ ' and one for its discontinuities.
  
  
   §.§ Variation at points of continuity.
  
  At a point of continuity of σ' we vary S[f,ξ] with respect to f'(x) to obtain the corresponding Euler Lagrange equation:
  
      S[f',ξ]
         →
      -1/4∫ dxdy
      (f”(x)+_yδ f'(x))(f”(y)+_yδ f'(y))k_Λ(x-y)
      -1/2_-∞^∞σ(f'(x)+δ f'(x))dx
      
         =
      -1/4∫ dxdy
      (f”(x)f”(y)+2f”(y)_xδ f'(x))k_Λ(x-y)
      -1/2_-∞^∞σ(f'(x))+σ'(f'(x))δ f'(x)dx
      
         =
      S[f',ξ]
      -1/2∫ dx
      (
      ∫_x(δ f'(x))f”(y)k_Λ(x-y)dy
      +
      σ'(f'(x))δ f'(x)
      )
      
         =
      S[f',ξ]
      +1/2∫ dx
      (
      ∫ f”(y)k_Λ'(x-y)dy
      -
      σ'(f'(x))
      )δ f'(x)
   
  so that:
  
      2δ S[f(x),ξ]/δ f'(x)
      =
      ∫ f”(y)k_Λ'(x-y)dy
      -
      σ'(f'(x))
  
  Recalling the definition of k_Λ, we see that a function f(x) extremises S at a point of continuity if and only if
  ∫ f”(y)(x-y)(log|x-y/Λ|-1)dy
  =
  σ'(f'(x)). For later convenience, define the following integral transform:
  
      [Xf](x)
      =
      ∫ f”(y)(x-y)(log|x-y/Λ|-1)dy
  
  Then f(x) extremises S at a point of continuity if and only if [Xf](x)=σ '(f'(x)).
  
  
   §.§ Variation at the discontinuities.
  
  The discontinuities of σ' occur when σ(f(x))=ξ_l, that is when f'(x)∈{-N+2l|l∈ℤ_[1,N-1]}. At these points the left hand side of the previous calculation is unchanged, but the right hand side does change.
  
  For f'(x)=-N+2l, σ(-N+2l+δ f'(x)) cannot be determined since it depends on the sign of the variation. All that can be said is that in this case: 
  
      σ(-N+2l+δ f'(x))∈σ(-N+2l)+{σ(-N+2l-0)δ f'(x),σ(-N+2l+0)δ f'(x)}
  
  leading to the requirement that for f'(x)=-N+2l, ξ_l>[Xf](x)>ξ_l+1.
  
  
   §.§ Equivalent conditions for a critical point.
  
  In summary, a function f_*(x) is a critical point of S[f,ξ] if and only if the following two conditions are met for l=0,1,…,N:
  
      (i).   [Xf_*](x)=ξ_l, for -N+2(l-1)<f_*'(x)<-N+2l
      
      
         
      (ii).  ξ_l>[Xf_*](x)>ξ_l+1, for f_*'(x)=-N+2l
  
  where we define ξ_0=-∞, ξ_N+1=+∞ and without loss of generality order ξ_0<ξ_1<…<ξ_N+1.
  
  The conditions (<ref>) have a geometric interpretation which allows them to be solved quite easily. To this end, define the following function:
  
      ϕ(x)
      =
      f_*'(x)
      +
      1/iπ[Xf_*]'(x)
  
  We claim that f_*(x) obeying (<ref>) is equivalent to the condition that f_*(x) maps the real line to the boundary of the slitted strip domain Δ:
  
      Δ
      =
      {z||(z)|<N,(z)>0}∖{z|(z)=-N+2l,(z)∈[0,η_l], l=1,2,…,N-1}
  
  for some η_1,…,η_N-1∈ℝ.
  
  
  To see this, first note that in case (i), ϕ(x)=f_*'(x)+0, so -N+2(l-1)<ϕ(x)<-N+2l. Thus ϕ maps the corresponding parts of ℝ to the "gaps" between the slits of Δ.
  
  In case (ii), f_*'(x)=-N+2l, so ϕ(x) is positioned on a slit. Then ϕ(x)=1/π[Xf_*]'(x)≥ 0 (Since the ξ's are in increasing order and ξ_l>[Xf_*](x)>ξ_l+1), and thus ϕ(x)∈{-N+2l+i[0,η_l]|l=1,2,…,N-1} for some η_l∈ℝ_>0. So ϕ maps the corresponding sections of ℝ to the slits of Δ.
  
  We have now shown that indeed ϕ:ℝ→Δ. Each statement above is trivially reversed to show that this is equivalent to f_*(x) obeying the conditions (<ref>). 
  7.4
  Since ξ_0=-∞ and ξ_N+1=+∞, η_0=η_N+1=∞.
  
  
  
  § CONSTRUCTION OF THE EXTREMISING PROFILE FUNCTION AND THE SEIBERG-WITTEN DIFFERENTIAL
  
  Using the geometric interpretation developed in the previous section, the extremiser f_*(x) can now be constructed explicitly. Doing so will also lead to the Seiberg-Witten differential, which will also help to show that this construction is complete.
  
  
   §.§ Construction of f_*(x).
  
  We first construct ϕ(x) by defining a conformal map
  Φ(z;η_1,…,η_N-1):ℍ→Δ and restricting back to the real line. The map Φ is guaranteed to exist by the Riemann Mapping Theorem, where the slitted strip is interpreted along with the point at infinity as a polygon on the Riemann sphere. In principle, this map can be recovered from the Schwarz-Christoffel formula of complex analysis, although the details are somewhat messy.
  
  One can show that up to normalisation and the addition of a constant  the following map is the unique choice <cit.>:
  
      Φ(z)=N+2/iπlog(w)
  
  where P_N(z) is defined to be a monic polynomial of degree N such that all roots of
  
      P_N(z)^2-4Λ^2N=∏_l=1^N(z-α_l^+)(z-α_l^-)
  
  are real, and w is the smaller root of
  
      P_N(z)=Λ^N(w+1/w)
  
  We now have a map from the upper half plane to Δ, but desire a map between the real line and Δ. To this end, ϕ is defined as follows:
  
      ϕ(x):=Φ(x+iϵ)
      ,   
      f_*'(x):=ϕ(x)
  
  It remains to show that the constructed f_*(x) obeys (<ref>) and that for a given set of ξ_l's there exists a set of η_l's such that f_*'(x):=ϕ(x) extremises S[f,ξ]. To do so, we first make the connection to Seiberg-Witten theory.
  
  
   §.§ The Seiberg-Witten differential.
  
  Note that <ref> defines a genus N-1 hyperelliptic curve, 𝒞_u which is in particular a Riemann surface. This curve will be identified with the Seiberg-Witten curve.
  
  We introduce basic cycles a_l and b_l with unit intersection number on 𝒞_u. These cycles are illusrated in Figure 7.2 for the case N=3 . 
  
  
  Using the Schwarz reflection principle, it can be shown that <cit.>:
  
      Φ'(z)
      =
      1/iπ∫_-∞^∞f_*”(x)/x-zdx
      =:
      -2/iπR_f_*(z)
  
  Let a_l be a circular contour encircling a_l and no other a_k for k≠ l, then:
  
      [b]
      2
      ∮_a_l
      zR_f_*(z)dz
      =
      ∮_a_ldz∫_ℝdxzf_*”(x)/z-x
      =
      ∫_ℝdxf”_*(x)
      {[  2π i x , x∈a_l;       0 , x∉a_l;         ].
      =
      π i
      ∫_α_l^-^α_l^+dxxf”_*(x)
      =
      4π i a_l
  
  where the circular contour has been deformed to the interval in which a_l lies.
  
  Noting the following identity:
  
      dw/w
      =
      1/wdw/dzdz
      =
      (d/dzlog(w))dz
      =iπ/2Φ'(z)dz
  
  the arbitrary N Seiberg-Witten differential dS=1/2π izdw/w can be rewritten as  dS=1/2π izR_f_*(z), and thus:
  
      a_l=∮_a_ldS
  
  in agreement with the Seiberg-Witten approach for N=2, and now generalised to arbitrary N.
  
  To complete the connection to Seiberg-Witten theory we also need the dual quantities a_D,l=/ a_l, which will allow us to fix . The a_D,l were originally defined as the Legendre transform of the a_l's. Inverting the Legendre transform, we find that as expected <cit.>:
  
      a_D,l=/ a_l=2π i∮_b_ldS=ξ_l
  
  
  
   §.§ Verification of the extremising property of f_*(x).
  
  The map ϕ is easily shown to have the following properties: ϕ(x)≥ 0, ϕ(x)=0 on the "bands", [α_l^-,α_l^+], ϕ'(z)≥ 0 and ϕ(x)=0 on the "gaps", (α_l^+,α_l+1^-).
  
  Since ϕ(z)=-1/π[Xf_*]'(x), [Xf_*](x) is monotonically decreasing everywhere and is constant on the bands. We thus identify ξ_l with the value of [Xf_*](x) on the l'th band.
  
  Since ϕ(z) is constant on the gaps, on the l'th gap its value is 
  ϕ(α_l^+)
  .
  Here w=-1, so we see that ϕ(z)=-N+2l. 
  
  Taking into account the monotonicity properties, (<ref>) have now been verified and thus f_*(x) does indeed extremise S[f,ξ].
  
  
   §.§ Completeness of the construction.
  
  It only remains to show that this construction is complete, that is for generic ξ_l there must exist corresponding slit lengths η_l. To this end we must first recover the ξ_l.
  
  Recalling that [Xf_*]'(x)=-πϕ(x), and integrating along the l'th gap gives:
  
      ξ_l+1-ξ_l
      =
      -π∫_α_l^+^α_l+1^-ϕ(x)dx
      =
      -π[xϕ(x)]|_α_l^+^α_l+1^-
      +
      π∫_α_l^+^α_l+1^-xdϕ(x)
      =
      0
      +
      π∫_α_l^+^α_l+1^-xdϕ(x)
  
  Then ϕ(x)=-1/π[Xf_*]'(x)=i(ϕ(x)-f_*'(x)), so as f_*'(x) is constant on the gaps, dϕ(x)=-idϕ(x):
  
      ξ_l+1-ξ_l
      =
      -iπ∫_α_l^+^α_l+1^-xdϕ(x)
      =
      4iπ∫_α_l^+^α_l+1^-dS
      =
      2iπ∮_b_l-b_l+1dS
  
  Where the last equality is clear upon considering Figure 7.2.
  
  So for a choice of slit lengths we can calculate the Seiberg-Witten differential, integrate along the gaps and (since ∑_l=1ξ_l=0), solve the resulting system of equations to recover the corresponding ξ_l's.
  
  With the ξ_l in hand we define the period map <cit.>:
  
      (η_1,…,η_N-1)
      ↦
      (ξ_1>…>ξ_N)
  
  This map is a continuous map between open sets, and since the extremiser f_* is unique, it is injective. Surjectivity follows from the general fact that a continuous map which maps boundaries to boundaries is surjective. This map is thus invertible, showing that for any choice of dual charges ξ_l, there exists a corresponding set of slit lengths η_l.
  
  
  CHAPTER: CHARGED PARTITIONS AND YOUNG DIAGRAMS
  
  So far the Z_k contour integrals have only been evaluated to low order (as in Chapter 6), or evaluated in the large k limit (as in Chapter 7). However not only can these integrals be evaluated to arbitrary order, doing so provides a connection to integer partitions. Making this connection will also provide a nice interpretation of some of the quantities introduced in the previous chapter. In this chapter we follow <cit.>.
  
  
  § CHARGED AND COLOURED PARTITIONS
  
  To make the connection to integer partitions we must first introduce some notation. A coloured integer partition of k∈ℤ_≥ 0 is an N-tuple k⃗=(k_1,…,k_N), where each k_l is itself an integer partition of some k_l:=|k_l|<k, and ∑_lk_l=k. That is to say k_l=(k_l,1≥ k_l,2≥…≥ k_l,n_l>k_l,n_l+1=0=…), such that |k⃗|:=k=∑_l,ik_l,i.
  
  More visually, a coloured partition consists of N integer partitions. Each one can be drawn as a (possibly empty), Young diagram in the usual way, resulting in N Young diagrams with k boxes shared between them.
  
  A charged partition of k∈ℤ_≥ 0 is a set {k_i'=k_i+a|i∈ℤ_>0} of non-increasing integers such that (k_1≥ k_2≥…≥ k_n>k_n+1=0=…) is a partition of k. The integer a is  called the charge.
  
  For a given partition k, the corresponding dual partition is obtained by swapping the rows and columns of the Young diagram of k. It is denoted by k:=(ν^1≥…≥ν^k_1>0).
  
  
  § THE ARBITRARY K RESIDUE FORMULA
  
  Recall the expression for the integrands of the partition function instanton series (<ref>):
  
      𝔷_k(a,ϕ;ϵ)
      =
      1/k!ϵ^k/ϵ_1^kϵ_2^k∏_i=1^k1/𝒫(Φ_i)𝒫(Φ_i+ϵ)∏_1≤ i<j≤ kΦ_ij^2(Φ_ij^2-ϵ^2)/(Φ_ij^2-ϵ_1^2)(Φ_ij^2-ϵ_2^2)
  
  where we have defined Φ_i=ϕ_i-ϵ_+, and Φ_ij:=Φ_i-Φ_j.
  
  We seek to classify the singularities of 𝔷_k and find their residues, thus allowing the calculation of the corresponding contour integrals.
  
  Each singularity of 𝔷_k is a simple pole and thus must have Φ_IJ≠ 0, otherwise the numerator would vanish. Each such singularity comes from some Φ_i taking on the value Φ_i=a_l+ϵ_1(α-1)+ϵ_2(β-1), for some α,β∈ℤ_≥ 0. This is easy to see for the 𝒫 factors and can be seen recursively for the remaining product by noting that its poles occur at Φ_i=Φ_j±ϵ_1,2, so each successive application of the residue formula replaces one Φ_i with some term of the form a_l+ϵ_1(α-1)+ϵ_2(β-1).
  
  In fact it can be shown that the poles of Φ_i are in one to one correspondence with the points a_l+ϵ_1(α-1)+ϵ_2(β-1), where 0≤α≤ν^l,β, and 0≤β≤ k_l,α, i.e. box (α,β) of the l'th Young tableau  <cit.>. Schematically:
  
      k⃗↔
      a_l+ϵ_1(α-1)+ϵ_2(β-1)
  
  So coloured partitions of k are in one to one correspondence with the poles of 𝔷_k.
  
  The residue corresponding to an arbitrary k⃗ is <cit.>:
  
      1/(ϵ_1ϵ_2)^k∏_l∏_α=1^ν^l,1∏_β=1^k_l,α𝒮(ϵ_1(α-1)+ϵ_2(β-1))/(ϵ(ℓ(s)+1)-ϵ_2h(s))(ϵ_2h(s)-ϵℓ(s))
           ×∏_l<m∏_α=1^ν^l,1∏_β=1^k_m,1(
      (a_lm+ϵ_1(α-ν^m,β)+ϵ_2(1-β))(a_lm+ϵ_1α+ϵ_2(k_l,α+1-β))/(a_lm+ϵ_1α+ϵ_2(1-β))(a_lm+ϵ_1(α-ν^m,β)+ϵ_2(k_l,α+1-β)))^2
  
  where we define:
  
      𝒮(x)
      =
      (
      ∏_m≠ l(x+a_lm)(x+ϵ+a_lm)
      )^-1 ,   ℓ(s)=k_l,α-β ,   
      h(s)=k_l,α+ν^l,β-α-β+1
  
  After some simplification, we then have the following contribution from this coloured partition:
  
      Z_k⃗(a;ϵ_1,ϵ_2)
      :=
      1/ϵ_2^2N|k⃗|∏_(l,i)≠(n,j)Γ(k_li-k_nj+ν(j-i+1)+b_ln)Γ(ν(j-i)+b_ln)/Γ(k_li-k_nj+ν(j-i)+b_ln)Γ(ν(j-i+1)+b_ln)
  
  where b_ln=a_l-a_n/ϵ_2, and ν=-ϵ_1/ϵ_2.
  
  Since clearly Z_k=∑_k⃗,|k⃗|=kZ_k⃗, we can now relabel the partition function sum to a sum over partitions k⃗, rather than instanton number k:
  
      Z(a,Λ;ϵ_1,ϵ_2)
      =
      Z^pert(a,Λ;ϵ_1,ϵ_2)∑_k=0^∞Λ^2NkZ_k(a;ϵ_1,ϵ_2)
      =
      Z^pert(a,Λ;ϵ_1,ϵ_2)∑_k⃗Λ^2N|k⃗|Z_k⃗(a;ϵ_1,ϵ_2)
  
  8.1
  Physically we have reinterpreted the partition function as a sum over an ensemble of random coloured partitions.
  
  
  
  § PARTITIONS AND THE MANY INSTANTON LIMIT
  
  Relating the reformulation of the partition function in terms of Young Diagrams to the many instanton limit exposes a new interpretation of the profile function f, and in particular the extremiser f_*.
  
  
   §.§ Partitions and their profiles.
  
  We draw Young diagrams in the so-called Russian convention, that is rotated 90 degrees and arranged right to left as in Figure 8.1.
  
  
  The profile f_k of a partition k is defined to be the piecewise linear curve forming the upper boundary of the Young Diagram corresponding to k. For example, in Figure 8.1 f_k is  shown in bold.
  
  In the Russian convention the profile has a convenient expression:
  
      f_k(x)
      =
      |x|
      +
      ∑_i=1^∞(
      |x-k_i+i-1|
      -
      |x-k_i+i|
      +
      |x+i|
      -
      |x+i-1|
      )
  
  We also define the profile of a squeezed Young diagram. A squeezed Young diagram is one for which the two axes have been scaled by constants ϵ_1, and ϵ_2. Explicitly:
  
      f_k(x|ϵ_1,ϵ_2)
      =
      |x|
      +
      ∑_i=1^∞(
      |x-ϵ_2k_i+ϵ_1(i-1)|
      -
      |x-ϵ_2k_i+ϵ_1i|
      +
      |x+ϵ_1i|
      -
      |x+ϵ_1(i-1)|
      )
  
  It is straightforward to verify that the profile of a squeezed Young Diagram satisfies the following properties:
  
      f_k'(x|ϵ_1,ϵ_2)=± 1,   
      f_k(x|ϵ_1,ϵ_2)≥ |x|,   
      f_k(x|ϵ_1,ϵ_2)∼ |x|, as  x→∞
  
  The profile of a charged partition of charge a is defined to be f_a;k(x|ϵ_1,ϵ_2):=f_k(x-a|ϵ_1,ϵ_2).
  
  Finally, the profile of a general charged and coloured partition is defined as the sum of the constituent profiles:
  
      f_a;k⃗(x|ϵ_1,ϵ_2)
      =
      ∑_l=1^∞
      f_a_l;k_l(x|ϵ_1,ϵ_2)
  
  Where k⃗ is a coloured partition and a is some vector of charges.
  
  
   §.§ The ϵ_1,2→ 0 limit.
  
  The connection between profiles of partitions and the profile function f(x) is established via the ϵ_1,2→ 0 limit. To understand this limit, note that for a charged partition <cit.>:
  
      |k|
      =
      a^2/2ϵ_1ϵ_2
      -
      1/4ϵ_1ϵ_2∫ dx f”_a;k(x;ϵ_1,ϵ_2)
  
  So for ϵ_1,2→ 0 the size of a typical partition goes as 1/ϵ_1ϵ_2, a very large number! Thus we seek to convert the sum (<ref>) over discrete Young diagrams to an integral over continuous Young diagrams.
  
  A continuous Young diagram is a continuous function f satisfying the following conditions:
  
      |f(x)-f(y)|≤|x-y|,   _ℝdxf'(x)=0,   ∫_ℝdx(f(x)-|x|)<∞,   
      f(x)∼|x|, as x→∞
  
  8.2
  The above conditions are a weaker version of the conditions (<ref>) for the profile of a discrete Young diagram.
  
  Associating profiles with the corresponding partitions gives a measure on the space of continuous Young Diagrams. In the ϵ_1,2→ 0 kimit this measure concentrates to a delta measure at a single function. This function is the limiting profile shape of the random partition, it ends up being the extremised profile function f_* derived earlier <cit.>.
  
  
   §.§ The partition function in terms of Young Diagrams.
  
  It can be shown from the representation theory of the symmetric group and the Plancherel measure on partitions that <cit.>:
  
      Z_f(ϵ_1,ϵ_2,Λ)
      :=
      Z_k⃗(a;ϵ_1,ϵ_2,Λ)
      =
      exp(
      -1/4 dxdy
      f_a,k⃗”(x|ϵ_1,ϵ_2)
      f_a,k⃗”(y|ϵ_1,ϵ_2)
      γ_ϵ_1,ϵ_2(x-y,Λ)
      )
  
  where we have changed notation to associate a partition with its profile.
  
  The partition function can now be reformulated as a sum over Γ_a^discrete, the set of discrete paths of the form f=f_a,k⃗:
  
      Z(a;ϵ_1,ϵ_2,Λ)
      =
      ∑_Γ_a^discrete
      Z_f(ϵ_1,ϵ_2,Λ)
  
  
  Heuristically, taking ϵ_1,2→ 0 turns this sum into an integral over paths of the form
  f(x)
  =
  ∑_l=1^N
  f_l(x-a_l)
  , where f_l is a continuous Young Diagram. We call the set of such paths Γ_a.
  
  It is then clear that Z_f∼ e^1/ϵ_1ϵ_2H[f], and:
  
      Z∼∫_Γ_a fe^1/ϵ_1ϵ_2H[f]
  
  where H is the Hamiltonian as derived in Chapter 7.
  
  We have thus recovered the form of the large k partition function which was assumed earlier on general grounds and also shown that f_* corresponds to the profile of the limiting partition.
  
  
  CHAPTER: CONCLUSION AND FURTHER DIRECTIONS
  
  tocchapterConclusion and Further Directions
  This thesis began with a brief overview of supersymmetric quantum field theory. In particular we introduced the SUSY Poincaré algebra and its representations on superspace. These representations were then used to construct SUSY invariant actions including the action of =2 SYM. Effective SUSY theories were briefly discussed, in particular it was found that the low energy effective action of =2 SYM is completely determined by a holomorphic function  called the prepotential.
  
  Seiberg-Witten theory was used to determine the low energy effective action of =2 SYM. This involved identifying and imposing various consistency conditions on . In particular the monodromies of the coordinates a and a_D about the singularities of  were determined. The monodromy about the weak coupling singularity was able to be analysed directly, while Seiberg-Witten duality allowed us to determine the monodromies about the strong coupling singularities.
  
  The moduli space  was identified as the triply punctured complex plane endowed with a metric possessing known monodromies. From here, we were able to determine the metric and thus . This was done in two ways. Firstly by arguing that the coordinates on  were the solution to a certain ODE and secondly by identifying the metric with the modular parameter of a certain family of elliptic curves.
  
  As a warm up for SUSY localisation we introduced the bosonic localisation of ordinary integrals with abelian symmetry. This lead to the idea of equivariant cohomology and the Atiyah-Bott-Berline-Vergne localisation formula for discrete fixed points. The localisation arguments for ordinary integrals were then generalised to SUSY QFTs and in particular to =2 SYM (via a topological twist). This reduced the partition function path integral of =2 SYM to an integral over the instanton moduli space. A model for the instanton moduli space was provided by the ADHM construction.
  
  With a further modification of =2 SYM to the Ω-background, we managed to reduce the partition function to a sum of contour integrals indexed by the instanton number, k. A second localisation argument then allowed us to determine the prepotential by taking the undeformed limit.
  
  The many instanton limit was used to recover the Seiberg-Witten geometry from the localisation approach. This limit reduced the =2 SYM partition function to a quantum mechanical path integral. Solving the corresponding equations of motion recovered the Seiberg-Witten geometry.
  
  We briefly explored the connection between =2 SYM and integer partitions. In particular the correspondence between coloured partitions and the poles of the =2 SYM partition function contour integrals 
  was discussed. This correspondence allowed the partition function to be re-cast as a sum indexed by coloured integer partitions. Taking the many instanton limit then recovered the quantum mechanical problem discussed earlier, and showed that the profile function corresponded to the profile of the profile of the limiting partition.
  
  A particular direction for further study would be to extend the localisation techniques utilised throughout this thesis to gauge groups other than SU(N) and to theories including matter multiplets as in <cit.>.
  
  A less obvious direction is that of the AGT correspondence, a correspondence between certain four dimensional =2 SU(2) SYM theories and Liouville theory on certain punctured Riemann surfaces <cit.>. 
  
  The methods of topological field theory which we briefly encountered in Chapter 3 have pure mathematical applications. In particular, topological QFTs can be used to calculate certain topological invariants and so are relevant to the study of 3 and 4-manifolds <cit.>. Topologcal QFTs also find use in the field of knot theory <cit.>.
  
  Many of the arguments presented throughout this thesis have analogues in string theory and M-theory <cit.>.
  
  
  
  CHAPTER: DEFINITIONS AND CONVENTIONS
  
  In this appendix we provide definitions of some of the objects encountered throughout this thesis, as well as an overview of the various conventions used.
  
  
  § INDICES
  
  The conventions for various indices are as follows:
  
      ∙Greek 3+1 dimensional spacetime indices μ,ν,ρ∈{0,1,2,3}
         ∙Lower case latin indices i,j,k∈{1,2,3,4}
         ∙Lower case latin gauge group Lie algebra indices a,b,c∈{1,2,…,rank(G)}
         ∙Capital latin supercharge indices A,B,C∈{1,2}
         ∙Capital latin 6+1 dimensional spacetime indices I,J,K∈{0,1,…,6}
         ∙Greek undotted spinor indices α,β∈{1,2}
         ∙Greek dotted spinor indices α̇,β̇∈{1̇,2̇}
  
  
  
  § SPINORS
  
  A Weyl spinor is is an element of a two dimensional irreducible representation of SL(2;ℂ). Throughout this work the word spinor will refer to a Weyl spinor.
  
  So-called "undotted" spinors ψ_α and ψ^α belong to the fundamental and dual representations of SL(2;ℂ) respectively. On the other hand, "dotted" spinors ψ_α̇ and ψ^α̇ belong to the conjugate of the fundamental and dual representations respectively.
  
  An undotted spinor is also called a left-handed spinor. A dotted spinor is also called a right-handed spinor. Objects with more than one dotted or undotted spinor index belong to tensor products of spinor representations.
  
  Roughly speaking, the fundamental and dual representations of SL(2;ℂ) are related via the operations of raising andlowering indices. More precisely we have <cit.>:
  
      ψ^α=ϵ^αβψ_β ,   ψ_α=ϵ_αβψ^β
         ψ^α̇=ϵ^α̇β̇ψ_β̇ ,   ψ_α̇=ϵ_α̇β̇ψ^β̇
  
  where the matrices
  
      ϵ^αβ
      =
      ϵ^α̇β̇
      =[  0  1; -1  0 ] , and   ϵ_αβ
      =
      ϵ_α̇β̇
      =[  0 -1;  1  0 ]
  
  act as a metric on spinor space.
  
  
  § Σ AND Γ MATRICES
  
  The Pauli matrices are:
  
      τ_1
      =
      [ 0 1; 1 0;   ] ,   τ_2
      =
      [  0 -i;  i  0;    ] ,   τ_3
      =
      [  1  0;  0 -1;    ]
  
  The matrices iτ_i form a basis for the Lie algebra 𝔰𝔩(2).
  
  Also of interest are the Minkowski σ-matrices, defined in terms of the Pauli matrices as follows:
  
      (σ^μ)_αα̇=(1_2,-τ_1,-τ_2,-τ_2)_αα̇ ,   
      (σ̅^μ)^α̇α=(1_2,τ_1,τ_2,τ_2)^α̇α
  
  These matrices naturally have one dotted and one undotted spinor index as well as a single spacetime index <cit.>. The spinor indices will sometimes be suppressed.
  
  We also define the generalised σ-matrices:
  
      (σ^μν)^α_β
      =
      1/4(
      σ^μσ̅^ν-σ^νσ̅^μ)^α_β ,   
      (σ̅^μν)^α̇_β̇
      =
      1/4(
      σ̅^μσ^ν-σ̅^νσ^μ)^α̇_β̇
  
  
  The γ-matrices of Section 1.6 are defined as follows:
  
      γ_6^μ
      =
      [     0 γ_4^μ; γ_4^μ     0 ] ,     γ_6^4
      =
      [   0 Γ_4; Γ_4   0 ] ,     γ_6^5
      =
      [    0  1_4; -1_4    0 ]
  
  The above matrices are in turn defined by:
  
      Γ_4
      =
      [ -i1_2     0;     0  i1_2 ] ,     γ_4^μ
      =
      [    0  σ^μ; σ̅^μ    0 ]
  
  
  
  CHAPTER: TRANSFORMATION PROPERTIES
  
  Throughout this thesis the transformation properties of fields under various operators have been utilised. In this appendix we show briefly the general method by which to obtain said transformations and list the results for easy reference.
  
  Firstly note from the discussion of supersymmetry in Chapter 1 that the SUSY Poincaré algebra has generators; P_μ,J_μν,Q^A_α,Q̅^α̇_A,𝒵, so a general element of this algebra can be written as follows:
  
      -ia^μP_μ-i/2ω^μνJ_μν+ζ_A^αQ^A_α+ζ̅^B,β̇Q̅_B,β̇-it𝒵
  
  where a^μ,ω^μν,ζ^α_A,ζ̅^B,β̇, and t are constant parameters. 
  
  Restricting to the SUSY part of the algebra, a general SUSY transformation of a (z-independent), superfield is given by:
  
      δ_ζ,ζ̅F(x,θ,θ̅)
      =
      (ζ_AQ^A+ζ̅^AQ̅_A)F(x,θ,θ̅)
  
  A representation of the SUSY Poincaré algebra on superspace is then given by identifying the generators with the following differential operators:
  
      P_μ=i_μ,   
      J_μν=ix_μ_ν-ix_ν_μ+S_μν,   𝒵
      =
      i/ z
         
      Q_α^A
      =
      /θ_A^α
      +
      iσ_αβ̇^μθ̅^A,β̇_μ
      +
      i/2ϵ_αβZ^ABθ^β_B/ z,   Q̅_A,α̇
      =
      /θ̅^A,α̇
      +
      iθ_A^βσ_βα̇^μ_μ
      +
      i/2ϵ_α̇β̇Z^*_ABθ̅^B,β̇/ z
  
  where S_μν is the spin operator. From here it is easy to derive the coordinate transformations by acting with the general SUSY Poincaré algebra element (<ref>):
  
      δ x^μ
      =
      a^μ
      +
      ω^μνx_ν
      +
      iζ^α_Aσ^μ_αβ̇θ̅^A,β̇
      -
      iθ^α_Bσ^μ_αβ̇ζ̅^B,β̇
         δθ^α_A
      =
      ζ^α_A
      +
      1/2ω^μν(σ_μν)^α_βθ^β_A
         δθ̅^A,α̇
      =
      ζ̅^A,α̇
      +
      1/2ω^μν(σ̅_μν)^α̇_β̇θ̅^A_β̇
         δ z
      =
      t
      +
      i/2ζ^α_Aϵ_αβZ^ABθ_B^β
      +
      i/2ζ̅^A,α̇ϵ_α̇β̇Z^*_ABθ̅^B,β̇
  
  Proceeding similarly one can then deduce the SUSY transformations of the component fields of any superfield. We tabulate the result of this procedure for the =2 chiral multiplet:
  
      δ_ζ,ζ̅H
      =
      √(2)ζ_Aψ^A
         δ_ζ,ζ̅H^†
      =
      √(2)ζ̅^Aψ̅_A
         δ_ζ,ζ̅ψ^A_α
      =
      (σ^μν)_α^βζ^A_βF_μν
      +
      iζ^A_α[H,H^†]
      -
      i√(2)σ^μ_α,β̇ζ̅^A,β̇∇_μH
      
         δ_ζ,ζ̅ψ̅^α̇_A
      =
      (σ̅^μν,α̇)_β̇ζ̅^β̇_AF_μν
      -
      iζ̅^α̇_A[H,H^†]
      -
      i√(2)σ̅^μ,α̇βζ_A,β∇_μH^†
         δ_ζ,ζ̅A_μ
      =
      iζ^Aσ_μψ̅_A-iψ^Aσ_μζ̅_A
  
  These transformations can be used to derive the action of the twisted supersymmetry generators. For example, taking ζ=0 in δ_ζ,ζ̅H^†, we obtain Q̅_A,α̇=√(2)ψ̅_A,α̇, so that Q̅H^†=√(2)ϵ^A,α̇ψ̅_A,α̇=√(2)ψ̅. The result of this procedure is tabulated below:
  
      Q̅H=0
      ,   
      Q_μH=√(2)ψ_μ,   Q̅_μνH=0
      
         Q̅H^†=√(2)ψ̅,   
      Q_μH^†=0
      ,   Q̅_μνH^†=√(2)ψ̅_μν
         Q̅ψ̅=2i[H,H^†]
      ,   
      Q_μψ̅=2i√(2)∇_μH^†,   Q̅_μνψ̅=2(F_μν)^-
         Q̅ψ_ρ=2i√(2)∇_ρH
      ,   
      Q_μψ_ρ=-4(F_μν)^++2ig_μρ[H,H^†]
      ,   Q̅_μνψ_ρ=-2i√(2)(g_μρ∇_νH-g_νρ∇_μH)^-
         Q̅ψ̅_ρτ=-2(F_ρτ)^-,   
      Q_μψ̅_ρτ=-2i√(2)(g_μρ∇_τH^†-g_μτ∇_ρH^†)^-,   
         Q̅_μνψ̅_ρτ=-(g_ρμ(F_τν)^--g_τμ(F_ρν)^-+g_τν(F_ρμ)^--g_ρν(F_τμ)^-)^-+i(g_μρg_ντ-g_μτg_νρ)^-[H,H^†]
      
         Q̅A_ρ=-iψ_ρ,   
      Q_μA_ρ=-ig_μρψ̅-2iψ̅_μρ,   Q̅_μνA_ρ=-i(g_μρψ_ν-g_νρψ_μ)^-
  
  As mentioned in Section 3.5, the process of BV quantisation involved adding certain ghost fields to the theory. The action of Q̅ on these fields is as follows <cit.>:
  
      Q̅b=0
      ,   Q̅c=-i/2{c,c}-ϕ,   Q̅c̅=b
      ,   Q̅ϕ=-i[c,ϕ]
      ,   Q̅η=-i[ϕ,λ]-i{c,η}
         Q̅λ=η-i[c,λ]
      ,   Q̅A_μ=-∇_μc-iψ_μ,   Q̅ψ_μ=-i∇_μϕ-i{c,ψ_μ}
         Q̅H_μν=-i[ϕ,χ_μν]-i[c,H_μν]
      ,   Q̅χ_μν=H_μν-i{c,χ_μν}
  
  
  habbrv
  
  
  