A complex geometric perspective on a,c anomalies
Brian R Williams




Boston University, Department of Mathematics and Statistics
bwill22@bu.edu

A complex geometric perspective on the a,c anomalies
    Brian R. Williams
    April 29, 2024
====================================================






In four-dimensional conformal field theory, the numbers a and c are defined as coefficients of particular terms in the operator product expansion (OPE) of the energy-momentum tensor.
They can be understood as conformal anomalies appearing in the trace of the energy-momentum tensor in a background gravitational field as in

    T_μ^μ = -a/16 π^2(Euler)  + c/16 π^2(Weyl)^2

where (Weyl)^2 is the square of the Weyl curvature and (Euler) is the Euler density.
Generally, conformal (or Weyl) anomalies exist in any even dimension; a complete classification has appeared in <cit.>.
The quantities a,c are informative invariants of a four-dimensional conformal field theory.
For example, reminiscent of Zamolodchikov's c-theorem in two-dimensions <cit.> is the famous “a-theorem” in four-dimensions proposed in <cit.> and proved in <cit.>.
This states that the function a is monotonic along (unitary) RG flow implying, in particular, that the IR value of a is strictly less than its UV value.

The numbers a,c are also well-defined for superconformal field theories in four dimensions, but turn out to be much simpler to characterize.
This is because the superconformal symmetry necessarily mixes spacetime symmetry with supersymmetry and hence a,c admit descriptions in terms of supersymmetric invariants of the theory.
Concretely, as we will review below, the a,c anomalies can be understood in terms of anomalies to R-symmetry.

The R-symmetry is also used to twist a supersymmetric theory <cit.>.
Typically, a supersymmetric theory is defined on some geometric class of spin manifolds.
The R-symmetry allows one to re-label fields of a theory so as to make sense of the theory in other geometric contexts.
So, for a topological twist, one can use R-symmetry to define the theory on an arbitrary smooth manifold.
For a holomorphic twist <cit.>, one can use R-symmetry to define the theory on complex manifolds.
R-symmetry is not the only thing needed to twist a supersymmetric theory; additionally, one must choose the data of a square-zero supercharge.
Fields and operators in the twisted theory can be described in terms of the cohomology with respect to the chosen supercharge.
The twisting process, hence, often provides a significant simplification of the original supersymmetric model.
The holomorphic twist is especially relevant for this paper since we consider four-dimensional supersymmetric theories which do not, generally, admit topological twists.

In this paper, we give an interpretation of the a,c anomalies in terms of complex geometric invariants of the holomorphic twist of a four-dimensional supersymmetric quantum field theory.
The main result is that a,c can be understood, in the twist, in terms of anomalies to holomorphic diffeomorphism symmetry.
Since twisting mixes R-symmetry with spacetime symmetry, we find that while this interpretation is not surprising, it does provide a satisfying complex geometric perspective for the coefficients which is also computationally effective via the Riemann–Roch theorem.

As a consequence of the holomorphic formulation of the a,c anomalies one obtains a classification of complex two-dimensional analogs of the Virasoro algebra in conformal field theory.
The higher dimensional affine Kac–Moody Lie algebra was proposed by Faonte, Hennion, and Kapranov in <cit.> and was connected to symmetries of holomorphic quantum field theory in <cit.>.
The key point is that it is absolutely necessary, in higher dimensions, to work in a derived setting.
Indeed, the n-dimensional Kac–Moody algebras are all obtained as central extensions of the dg Lie algebra of derived sections of the structure sheaf on the punctured n-disk with values in an ordinary Lie algebra g.
Similarly, one can contemplate central extensions of the the dg Lie algebra of derived sections of the tangent sheaf of the punctured n-disk.
In complex dimension two, a,c label all possible central extensions of Γ(D^2 - 0, ).
For an approach to characterizing central extensions of the derived tangent sheaf which uses factorization homology we refer to <cit.>.
While we don't pursue any algebraic structures in this paper, we refer to recent work of Bomans and Wu <cit.> for a characterization of the two-dimensional Virasoro algebra from the point of view of the holomorphic twist of four-dimensional supersymmetery.
We will return to the relationship between the a,c anomalies and the higher dimensional Virasoro algebra from the point of view of factorization algebras in future work.

The key technical aspect of this paper is a formulation of the supersymmetric theory, its twist, and symmetries within the Batalin–Vilkovisky (BV) formalism.
The universal symmetry of a holomorphic field theory is introduced in section <ref>.
In section <ref>, we provide a background of anomalies in the BV formalism, specifically in the context of symmetries which largely follows Costello and Gwilliam's treatment <cit.>. 
We turn to holomorphic theories in section <ref>, and interpret the anomaly to quantizing holomorphic diffeomorphism symmetry as the Chern class of a line bundle over the universal complex manifold; thus affording the Riemann–Roch theorem as an effective method of computation.
In section <ref> we turn to twists of supersymmetric theories and prove the precise relationship between anomalies to holomorphic diffeomorphism with a,c.
Finally, in section <ref> we elaborate on an example involving the twist of supersymmetric quantum chromodynamics (QCD).



 §.§ Acknowledgements


I thank Kevin Costello and Owen Gwilliam for their development of the quantum Noether theorem within the Batalin–Vilkovisky formalism as presented in their book <cit.> which we rely heavily on throughout this paper.
I also am grateful to Ingmar Saberi for his collaboration on twists of four-dimensional =1 superconformal theories which was the catalyst for this paper.
I thank Jingxiang Wu for feedback on a draft of this paper. 
Finally, I thank Boston University and the Center for Advanced Studies (CAS) at the Ludwig Maximilian University (LMU) in Munich, Germany for hospitality during the preparation of this work. 







§ A LOCAL MODEL FOR HOLOMORPHIC VECTOR FIELDS



Let X be a complex manifold.
The -operator associated to a holomorphic vector bundle V defines its Dolbeault complex Ω^0,(X , V).
In degree q, the space Ω^0,q(X, V) denotes the space of smooth sections of the vector bundle ^q _X^* ⊗ V and the -operator extends to a differential Ω^0,q(X,V) →Ω^0,q+1(X,V), ^2 = 0.


By Dolbeault's theorem, the complex provides a resolution for the sheaf of holomorphic sections of the bundle V.


A holomorphic vector field is a holomorphic section of the holomorphic tangent bundle _X.
The sheaf of holomorphic vector fields has the natural structure of a sheaf of Lie algebras with bracket the Lie bracket of vector fields.
Since this bracket involves only holomorphic differential operators, it extends to a bracket on the Dolbeault complex of _X.
In local holomorphic coordinates the Lie bracket is
[f_I,i (z,) _I _z_i , g_J,j (z,) _J _z_j] = f_I,i _z_i g_J,j _I ∧_J _z_j - g_J,j _z_j f_I,i _J ∧_I _z_i.
This bracket endows Ω^0,(X, _X) with the structure of a dg Lie algebra where the differential is .
This leads to the following definition; for a recollection of terminology on local complexes and local Lie algebras we refer to <cit.>.


Let _X be the local dg Lie algebra whose underlying complex of vector bundles is
Ω^0,(X, _X) .
The differential is the -operator associated to the holomorphic bundle _X.
The bracket is the Lie bracket of holomorphic vector fields extended to the Dolbeault complex as described above.

When X = ^n we will abusively refer to this local dg Lie algebra simply as .


By Dolbeault's theorem, for U ⊂ X a Stein open set there is a quasi-isomorphism _X(U) ≃Vect^hol(U), the Lie algebra of holomorphic vector fields on U.

There is also the following relationship between  and formal vector fields, which will be used subsequently.
Let E → M denote a -graded vector bundle on M.
We consider the pro vector bundle of ∞-jets which we will denote by j^∞ E, see <cit.> or <cit.> for instance.
The sheaf of smooth sections of this pro vector bundle carries the natural structure of a D_M-module.
If = Γ(L) is a local L_∞ algebra then j^∞ L is a bundle of L_∞ algebras.


The map
j_0^∞(^n) →vect(n) [[, ]] 
induces a quasi-isomorphism of dg Lie algebras j_0^∞ (^n) ≃vect(n).




§ ANOMALIES AND DESCENT



In this section we rely on the quantum Batalin–Vilkovisky formalism as developed in the series of books <cit.>.



 §.§ Anomalies as local functionals



Let E be a vector bundle on a smooth manifold M.
The C^∞_M-module of local functionals on M is
(E) Dens_X ⊗_D_M ∏_n > 0 Hom_C^∞_M (jet (E) , C^∞_M) ,
where D_M is the algebra of differential operators and jet(E) is the ∞-jet bundle of E.
Concretely, if M is equipped with a volume form, a section of (E) is a linear combination of objects of the form

    (ϕ_1,…,ϕ_n) ↦[ (D_1ϕ_1 ⋯  D_nϕ_n)   _M]

where ϕ_i denote sections of E, where D_i are differential operators → C^∞(M), and where [-] means equivalence classes up to total derivative.

Suppose that (,ω, S) is a classical BV theory on a smooth manifold M.
This means, in particular:


  * = Γ(M,E) is sections of a -graded vector bundle E → M.

  * ω E ⊗ E →Dens_M [-1] is a density valued, non-degenerate, pairing of degree -1.

  * S ∈() is a local functional of cohomological degree zero.


The BV anti-bracket induced from ω equips the complex ()[-1] with the structure of a graded Lie algebra.
The classical master equation is the condition that S be a Maurer–Cartan element {S,S} = 0.
Equivalently, we can decompose {S,-} = Q + {I,-} where S(Φ) = ω(ϕ,Qϕ) + I(ϕ).
Then the classical master equation is Q I + 1/2{I,I} = 0.
When this equation holds, the operator {S,-} is square-zero and endows
((), {S,-}) ,
with the structure of a dg Lie algebra. 

Our goal in the remainder of the section is recall the various versions of symmetry that a theory in the BV formalism can exhibit.
To that end, suppose that g is an arbitrary dg Lie algebra.
Then we can define its Chevalley–Eilenberg complex C^(g) which is a commutative dg algebra.
Furthermore, we can tensor with a shift of local functionals
C^(g) ⊗()[-1]
to obtain a dg Lie algebra.
The subcomplex 

C^_red(g) ⊗()[-1]
is a sub dg Lie algebra.
To write this in a slightly more symmetric way, we recall that S equips [-1] with the structure of a local L_∞ algebra such that {S,-} is precisely the Chevalley–Eilenberg differential.
Thus we can write this sub dg Lie algebra as

C^_red(g) ⊗^([-1])[-1] .


Let  be a local dg Lie algebra (our main example is the local dg Lie algebra _X from the previous section).
The Chevalley–Eilenberg differential equips the C^∞_M-module of local functionals ([1]) on the shift of  with the structure of a cochain complex which we denote ^()
There is a cochain embedding
^()(M) ⊂C_red^((M)) 
from local cochains (by our conventions, local cochains are always reduced) to ordinary Chevalley–Eilenberg cochains of the dg Lie algebra of global sections (M).

Furthermore, the Lie bracket in (<ref>) restricts to a Lie bracket on the subcomplex
^(⊕[-1])[-1] ⊂C^_red(g) ⊗^([-1])[-1] ,
thus equipping the left hand side with the structure of a dg Lie algebra.

The embedding of local dg Lie algebras [-1] →⊕[-1] induces a map of dg Lie algebras
^(⊕[-1])[-1] →^([-1]) [-1] = ()[-1] .
Let InnerAct(,) denote the kernel of this map.
The map of local dg Lie algebras ⊕[-1] → induces an inclusion of dg Lie algebras
^() →^(⊕[-1])[-1] .
This map factors through the dg Lie algebra InnerAct(,) ⊂^(⊕[-1])[-1].
We let 
Act(,) InnerAct(,) ^() 
be the quotient dg Lie algebra.

As cochain complexes there are isomorphisms

    Act(,)    ≃^(⊕[-1])[-1] (^()[-1] ⊕^([-1])[-1] ) 
    InnerAct(,)    ≃^(⊕[-1])[-1] ^([-1])[-1] .

Thus, an element of Act(,) is a local functional on the complex [1] ⊕ which does not depend solely on [1] or .
Likewise, an element of InnerAct(,) is a local functional which does not depend solely on .


Let (,ω, S) be a classical BV theory and  a local Lie algebra.
A classical -background is a Maurer–Cartan element
S^ ∈Act(,)^1 .
Unwrapping the definitions above, this means that ^ is a local functional on [1] ⊕ of cohomological degree zero which satisfies the equation

_̣ S^ + {S, S^} + 1/2 {S^,S^} = 0 ,
where _̣ is the Chevalley–Eilenberg differential for .


The equation (<ref>) is referred to in <cit.> as the -equivariant classical master equation.

In <cit.> the problem of perturbative quantization of classical theories within the BV formalism is put within an obstruction-deformation theoretic framework.
There are two main steps to construct a quantization.
The first is to construct a renormalized, or effective action, {I[L]} which is a family of action functionals depending on a scale parameter L whose L → 0 limit agrees with the classical interaction I (among other conditions).
Additionally, this effective family is required to satisfy the quantum master equation

(Q + ħ_L) e^I[L]/ħ  = 0 ,
where _L is the effective BV Laplacian.

Starting with a classical BV theory (,ω,S = S_kin + I), the problem of quantization order by order in ħ is controlled by the dg Lie algebra ()[-1].
While there are possibly many inequivalent ways to quantize a theory, even at one-loop, there is a scheme-dependent obstruction to the one-loop quantization of a classical BV theory which is measured by a degree +1 cohomology class
Θ_1-loop ∈H^1_loc() = H^2 (()[-1], {S,-}) ,
obtained as the L → 0 limit of the quantum master equation (in other words, it is the failure for an effective action to solve the master equation).
Similarly, if a quantization exists and is fixed to order ħ^n-1 there is an obstruction Θ_n-loop∈ H^1_loc() to extending this to a quantization to order ħ^n.

Now we turn to the question of the equivariant quantization; more specifically, the quantization of -backgrounds as defined above.
Here is the summary of the definition, we refer to <cit.>[Definition 13.2.2.1] for the full definition.


Suppose {I[L]} is an effective family of interactions describing a quantum field theory (satisfies, amongst other conditions, the ordinary quantum master equation (<ref>)).
A quantum -background is a family of effective actions
I^[L] ∈([1] ⊕) ([1]) [[ħ]]
which agrees with I[L] modulo functionals of [1] and
satisfies the following equivariant master equation
_̣I^[L] + Q I^[L] + 1/2 {I^[L],I^[L]}_L + ħ_L I^[L] = 0 
in ([1] ⊕) ([1]) [[ħ]].

Likewise, an inner quantum -background is an effective family in a bigger class of functionals 
I^[L] ∈([1] ⊕) [[ħ]]
satisfying the same quantum master equation.


In the equivariant context the dg Lie algebras Act(,) and InnerAct(,) control different equivariant quantization situations, analogously to how the dg Lie algebra ()[-1] controls the non-equivariant situation.
For simplicity, we consider the problem of quantization to one-loop.
Throughout, we assume that a fixed quantization of the classical BV theory (,ω,Q) is fixed at one-loop (in particular the one-loop obstruction Θ_1-loop vanishes).
The complex which controls quantizations of a classical -background while leaving this quantization of the BV theory fixed is precisely Act(,).



Let S^∈Act(,) be a classical -background and suppose S_1-loop = S + ħ S^(1) is a one-loop quantization of a classical BV theory (,ω, S).
The obstruction to lifting this to a one-loop -background is an element
Θ^, ∈H^2 (Act(,) ) ,
which, in this situation, lifts to an element
Θ^, ∈H^1_loc(⋉[-1]   |   ) .


Similarly, the obstruction to lifting this to a one-loop inner -background is an element
Θ^ ∈H^2 (InnerAct(,) ) 
which, in this situation, lifts to an element 
Θ^ ∈H_loc^1 () = H^2 (^()[-1] ) .




 §.§ Central charges


The local cohomology classes of cohomological degree one deserve a special name which takes the form of the following terminological definition.


The space of central charges for a local Lie algebra  is the vector space 
H_loc^1 () .
In particular, an obstruction to a quantum inner -background determines a central charge for .


The motivation for this definition is the following.
Suppose that our local Lie algebra = Γ(L) is defined on ^d.
Typically, a current is a (d-1)-form valued in Lagrangian densities.
More generally, for  a local Lie algebra, we obtain the space of currents valued in _c by evaluating on the codimension one sphere S^d-1.
Here _c = Γ_c(L) is the cosheaf of compactly supported sections of the underlying vector bundle L.
Some care must be taken in this definition, since strictly it only makes sense to evaluate  on open subsets; thus one should consider the value of  on open sets which are infinitesimal neighborhoods of the (d-1)-sphere.
In nice situations, which we will turn to momentarily, the value of  on a neighborhood of the form (-ϵ,ϵ) × S^d-1 can be written as

Ω^_c (- ϵ, ϵ) ⊗g_S^d-1
where g_S^d-1 is some dg Lie (or more generally L_∞) algebra.
We call g_S^d-1 the Lie algebra of S^d-1-currents associated to .

Now suppose that ϕ∈ C^1_loc() is a local cocycle of cohomological degree +1.
Assume its value on (-ϵ,ϵ) × S^d-1 can be expressed as ∫_-ϵ^ϵ⊠ψ where


  * ∫_-ϵ^ϵΩ^_c (- ϵ, ϵ) →[-1] is the (degree one) integration map along the interval, and

  * ψ is a degree two cocycle for the Lie algebra g_S^d-1.
Automatically, ψ is of cohomological degree two since integration along the interval is of cohomological degree one.

In this situation, ψ determines a central extension 
0 →→g_S^d-1 →g_S^d-1 →0 ,
and we interpret g_S^d-1 as the Lie algebra of centrally extended currents associated to  and ϕ.

We consider an example, studied in <cit.>.
Let g be an ordinary Lie algebra and consider the following local Lie algebra defined on ^n = ^2n
= g ⊗Ω^0,(^n) .
Take the neighborhood ^n - 0 = × S^2n-1 of the sphere.
In this situation, there is a dense embedding of g-valued derived algebraic sections of the structure sheaf into the spherical currents
g ⊗Γ(D^n - 0, ) ↪g_S^2n-1 .
Any invariant and symmetric homogenous polynomial θ∈^n+1(g)^G determines a local cocycle for g⊗Ω^0,(^n) defined by
ϕ(α_0,…,α_n) = ∫_^n θ( α_0 α_1⋯α_n ) .
The corresponding cocycle ψ is precisely the higher dimensional Kac–Moody cocycle <cit.>:
ψ(f_0,…,f_n) = ∮_S^2n-1 θ( α_0 α_1⋯α_n ) 
where f_0,…,f_n ∈g⊗Γ(D^n - 0, )⊂g_S^2n-1.
Thus, every choice of “level” θ defines a central extension of the spherical currents.
In <cit.> we prove that such θ label all such central charges for this local Lie algebra which are GL(n) and translation invariant.

Similarly, one could start with the local Lie algebra  resolving holomorphic vector fields on ^n.
We will explain a characterization of central charges in this case in the next section.
We will not, however, return to the resulting higher dimensional Virasoro algebras such central charges determine.
We refer to <cit.> for results in the complex two-dimensional setting.



 §.§ Diffeomorphism anomalies and Gelfand–Fuks cohomology


We turn our attention to the holomorphic version of diffeomorphism, or gravitational, anomalies.
As recollected, the underlying local Lie algebra describing infinitesimal holomorphic diffeomorphisms is , which resolves the Lie algebra of holomorphic vector fields.

The following result characterizes, in any dimension, the local anomalies to holomorphic reparametrization invariance.


<cit.>

On flat space X = ^n there is a quasi-isomorphism
^(( ^n ) ) ≃C^_red(vect(n)) [2n] .
The space of central charges (on flat space) for  is
H_loc^1 ((^n)) ≃H^2n+1(vect(n)) .


Consider the one-dimensional case n=1.
Then the space of central charges for holomorphic vector fields is H^3(vect_1), which is one-dimensional.
By transgression this space is naturally isomorphic to the cohomology of the holomorphic Witt algebra witt = ((z)) _z, which is the Lie algebra of vector fields on the punctured formal disk.
The unique (up to scale) central charge for the local Lie algebra () corresponds to the usual central charge in conformal field theory.
As a local cocycle for (), a local representative for this central charge is
ϕ(μ, μ') = ∫_J μ∧(J μ') .
Of course, in coordinates μ = f(z,) _z, where f(z,) is a Dolbeault form of type (0,), this recovers the familiar expression for the Virasoro cocycle.

In the notation of the previous section, the space of spherical currents in this example receives a dense embedding of the Lie algebra of algebraic vector fields on the punctured disk [z,z^-1] _z ↪g_S^1.
The corresponding cocycle for [z,z^-1]_z, denoted ψ above, is the standard Virasoro cocycle.

We comment that the result above can be generalized.
In <cit.>, see also <cit.>, it is shown that for any complex manifold X of complex dimension n that there is an isomorphism
H_loc^((X)) ≃H^_dR(X) ⊗H^(vect(n)) [2n] .
When X = ^n we recover the result as stated above.



 §.§ The holomorphic a,c anomalies


In the last section we have seen how the space of central charges for the local Lie algebra of holomorphic vector fields can be expressed in terms of the Gelfand–Fuks cohomology of formal vector fields.
In this section we recall an elegant description of this cohomology, following <cit.> and match with explicit representatives of the corresponding local cocycles in complex dimension two.

Let Gr(n,k) be the complex Grassmannian of n-planes in ^k. 
Denote by Gr(n,∞) the colimit of the natural sequence of topological spaces
{⋆} →Gr(n, n+1) →Gr(n,n+2) →⋯. 
It is a standard fact that Gr(n, ∞) is a model for the classifying space BGL(n) of rank n vector bundles.

From the colimit description, this results in the skeletal filtration of the classifying space:
sk_k BGL(n) = Gr(n,k) .
Let X_n denote the restriction of the universal bundle EGL(n) over the 2n-skeleton:

X_n [r] [d]     E GL(n) [d] 

sk_2n B GL(n) [r]     B GL(n) .









 

There is an isomorphism of graded commutative algebras
H^(vect(n)) ≅H^_dR (X_n) .
The commutative product is the trivial one.


Note that when n = 1 we have sk_2 B U(1) = ^1 ⊂^∞ = B U(1). 
Moreover, the restriction of the universal bundle is the Hopf fibration U(1) → S^3 →^1 so that X_1 ≃ S^3.

It will be useful to say a few words about the proof of this theorem, as we will need a more general version to discuss mixed anomalies shortly.

For a Lie algebra g with a subalgebra h⊂g the relative Lie algebra cohomology is defined to be the cohomology of the subcomplex 
C^(g   |   h) ⊂C^(g)
which consists of h-invariant cochains ϕ∈ C^(g) satisfying ι_X ϕ = 0.
Here ι_X  C^(g) → C^-1 (g) is the contraction with an element X ∈h.
The Hochschild–Serre spectral sequence for a subalgebra h⊂g is a spectral sequence whose E_2 page is given in terms of relative Lie algebra cohomology
E_2 ≃H^(g, h) ⊗H^(h),
and converges to the absolute Lie algebra cohomology H^(g).

In the current context we are interested in the absolute cohomology of vect(n).
The key idea is that the Hochschild–Serre spectral sequence for the subalgebra gl_n ⊂vect(n) consisting of linear vector fields can be interpreted geometrically in terms of classifying spaces.
One can identify the relative Lie algebra cohomology H^(vect(n)   |  gl_n) with the cohomology of the 2n-skeleton sk_2n BGL(n).
This identification can be extended to an isomorphism of spectral sequences between the Serre spectral sequence computing the cohomology of X_n, as a GL(n) space over the 2n-skeleton, and the Hochschild–Serre spectral sequence for the subalgebra gl_n ⊂vect(n) converging to the absolute cohomology of vect(n).

An immediate application of the Serre spectral sequence shows that first nontrivial cohomology above degree zero of X_n is in degree 2n+1 and we have
H^2n+1(vect(n)) ≃H^2n+1(X_n) ≃H^2n+2(BGL(n)) .
In other words, the degree 2n+1 cohomology of formal vector fields is isomorphic to degree 2n+2 polynomials built from universal characteristic classes of rank n vector bundles.

In the previous section we have argued that the space of holomorphic diffeomorphism anomalies in n-dimensions, the space of central charges for , is precisely this first non-trivial cohomology group of X_n.
When n=1 we have that X_1 ≃ S^3 hence the space of central charges for holomorphic vector fields in complex dimension one is one-dimensional, as expected (all being multiples of the universal characteristic class is c_1^2 ∈ H^4 (BGL_1)).
When n = 2 we see that the space of central charges is two-dimensional
H^1_loc((^2)) ≃H^5 (vect_2) ≃H^6(BGL(2)) = ·c_1^3 ⊕·c_1 c_2
spanned by the universal characteristic classes c_1^3 and c_1 c_2.

It will be convenient for our purposes to relabel this cohomology in terms of the degree six components of the universal Chern character:
_1^3 = c_1^3,   _1 _2 = 1/2 (c_1^3 - 2 c_1 c_2) .
Any holomorphic diffeomorphism anomaly represented by a class Θ∈ H^5(vect(n)) can be written as a linear combination of these universal classes

Θ= a^hol ·_1 _2 + c^hol ·_1^3 .
We will see momentarily how the coefficients a^hol,c^hol are related to the standard a,c anomalies in four-dimensional conformal geometry.

We state explicit local models for the central charges _1 _2, _1^3.


The following are explicit degree one local cocycle representatives for the classes _1 _2 and _1^3:
Θ_a^hol = 1/12 ∫_^2 tr ( J μ) tr(J μJ μ) ,
and
Θ_c^hol = 1/6 ∫_^2 tr(J μ) tr(J μ) tr(J μ) 
respectively.


The arguments that follow are elaborated in <cit.>. 

There are two main steps.
The first is to identify explicit representatives in the cohomology of formal vector fields.
The second is in passing from Gelfand–Fuks classes in H^5(vect(2)) to local cohomology classes in H^1_loc((^2)).
Both steps follow the general principle of descent.

The first step is accomplished by considering the double complex associated to the Lie algebra cohomology of vect(2) with coefficients in the formal de Rham complex Ω^(D^2).
Since the de Rham complex is quasi-isomorphic to  in degree zero, the resulting spectral sequence converges to H^(vect(2)).
The first page of the spectral sequence is generated, as an bigraded algebra, by generators a_1,a_2 of bidegree (0,1),(0,3) respectively and by generators (τ_1,τ_2) of bidegree (1,1), (2,2) respectively.
These generators are subject to the relations τ_1^3 = τ_1 τ_2 = τ_2^2 = 0.
In degree five, the following two classes remain at the E_∞ page: a_1 τ_1^2, a_1 τ_2 which are represented by the totally skew symmetric tri-linear functionals on vect(2) with values in Ω^2(D^2):

    a_1 τ_1^2  (X_0,X_1,X_2)    ↦tr(J X_0) tr (J̣ X_1 ∧tr (J̣ X_2) + ⋯
    
    a_1 τ_2  (X_0,X_1,X_2)    ↦tr(J X_0) tr (J̣ X_1 ∧J̣ X_2) + ⋯ .

The ⋯ denote appropriate skew-symmetrization.

The passage from Gelfand–Fuks classes to local cohomology classes uses descent for the -operator.
If ϕ is one of the representatives above, we can extend it to a functional on = Ω^0,(^2, ) with values in densities on ^2 and can then take the local class.
The resulting local functional is of total degree +1 since the functionals are cubic and must have anti-holomorphic Dolbeault degree which sums to two.




 §.§ Mixed anomalies


In this section we generalize the discussion of holomorphic diffeomorphism anomalies on ^n to anomalies for a local Lie algebra built from both infinitesimal holomorphic diffeomorphisms and infinitesimal holomorphic gauge symmetries.
For gauge symmetries of the trivial holomorphic G-bundle, the local Lie algebra is the semi-direct product local Lie algebra
_X ⋉g ⊗Ω^0,_X .
Where the local Lie algebra _X resolving holomorphic vector fields acts by Lie derivative on the Dolbeault complex for the trivial bundle.

Our goal is to describe, just as we did in the case of _X, the space of anomalies for the existence of quantum backgrounds with respect to this local Lie algebra.
Following the discussion of section <ref> the space of such anomalies is the degree one local cohomology
H^1_loc (_X ⋉g ⊗Ω^0,_X ).
We will use the characterization, in particular, to describe diffeomorphism anomalies for holomorphic gauge theory in <ref>.
In analogy with the case that the group G is trivial, we have the following result, which can be proved by similar methods as in theorem <ref>.


On flat space X = ^n there is a quasi-isomorphism
^(( ^n ) ⋉g ⊗Ω^0,(^n) ) ≃C^_red(vect(n) ⋉g ⊗_n) [2n] .
In particular, the space of mixed anomalies can be identified with the following Lie algebra cohomology
H_loc^1 ≃H^2n+1(vect(n) ⋉g ⊗_n) .



Like in the case of pure holomorphic diffeomorphism anomalies, we will identify the space of mixed anomalies with certain characteristic classes.
This will be facilitated by the following result.


Let G be a compact Lie group and g its complexified Lie algebra.
There is a graded ring isomorphism
H^(vect(n) ⋉g ⊗_n   |   gl_n ⊕g) ≃H^≤2n (B GL(n) ×BG) ,
where the right hand side is the cohomology of the 2n-skeleton of the product of classifying spaces.


In this theorem appears the relative Lie algebra cohomology.
In our context we are considering the relative cohomology with respect to the subalgebra

gl_n ⊕g ⊂vect(n) ⋉g ⊗_n 
consisting of linear vector fields and constant g-valued functions.
Using Khoroshkin's result we can proceed similarly as in the case g=0 (see theorem <ref> and the discussion following) to identify the absolute Lie algebra cohomology in terms of classifying spaces.
Indeed, let Y_n be the restriction of the universal bundle over the 2n-skeleton of B GL(n) × BG.
As in the case g=0, there is an isomorphism between the Hochschild–Serre spectral sequence for the subalgebra (<ref>) and the Serre spectral sequence for the GL(n) × G-space Y_n over the 2n-skeleton of BGL(n) × BG.
This leads to the following.


Let G be a compact connected Lie group and g is complexified Lie algebra.
There is an isomorphism of graded rings
H^(vect(n) ⋉g ⊗_n) ≃H^(Y_n) .
In particular, there is an isomorphism
H^2n+1(vect(n) ⋉g ⊗_n) ≃H^2n+2 (BGL(n) ×BG) .


Notice that when G is the trivial group we have Y_n = X_n and this result recovers the  result of Fuks that we recollected in theorem <ref>.
From this theorem we see that the space of mixed anomalies is isomorphic to H^2n+2(BGL(n) × BG).
As in the case of pure holomorphic diffeomorphism anomalies, we can match such cohomology classes with explicit classes in the local cohomology of (X) ⋉g⊗Ω^0,(X).

Let _j ∈ H^2j(BGL(n)), j=1,…,n be the universal Chern characters as before.
Also, let
θ_k ∈^k (g^*)^G ≃H^2k(BG)
denote an invariant polynomial of polynomial degree k.
Then the degree one local functional corresponding to the class
_j ⊗θ_n-j+1 ∈H^2n+2(BGL(n) ×BG),   j=0,…,n,
for example, is proportional to
∫_^n tr(J μ(J μ)^j-1)   θ_n-j+1(A, …, A) .
This can further be extended to the case j=0.
In this case, the local functional corresponding an invariant polynomial θ_n+1 on g of polynomial degree n+1 is
∫_^n θ_n+1(A, A, …, A) . 
This type of local cocycle, which does not involve any holomorphic vector fields, was studied in <cit.> where it is shown how it relates to higher dimensional Kac–Moody algebras <cit.>.



§ HOLOMORPHIC THEORIES AND THEIR QUANTUM ANOMALIES





 §.§ A local index theorem from free field theory


To any elliptic complex (,Q) on a manifold M there exists a free theory in the Batalin–Vilkovisky formalism whose complex of fields (including ghosts, fields, anti-fields, etc.) is `double' the size of :
^* [-1] = ⊕^![-1] ,
see <cit.>.
Here ^! stands is the elliptic complex which is given by sections of the bundle E^* ⊗Dens_M and whose differential is Q^* ⊗𝕀; here E is the bundle whose sections is the underlying graded vector space of the elliptic complex .
A field is a section of ^*[-1] which we denote by the tuple (γ, β).
The BV anti-bracket on observables is determined by the natural (-1)-shifted symplectic structure on the shifted cotangent space—explicitly this utilizes the natural integration pairing between sections of E and sections of E^* ⊗Dens_M.
The action functional is given by
∫_M β(Q γ) ,
so that the equations of motion imply, in particular, that γ is Q-closed.

Some examples of this constructions may be familiar. 
If M is just a smooth manifold, then taking = Ω^(M)[1] to be the shifted smooth de Rham complex of M results in the BV description of abelian topological BF theory.
If M = Σ is a Riemann surface then = Ω^0,(Σ, K^⊗ r) recovers the spin r bosonic ghost system used in string theory.


Next, let  be a local L_∞ algebra on M and suppose that the elliptic complex  is a local -module.
The coupling

S^(α;β,γ) = ∫_M β(α·γ) ,
determines an element S^∈Act(;^* [-1] ).
By the virtue that  is a -module this coupling automatically satisfies the equivariant classical master equation (<ref>).
Thus, any local -module  defines a classical -background for the resulting free BV theory ^*[-1].

Since  is a free theory there is no obstruction to lifting S^ to a quantum -background.
There is, nevertheless, an obstruction to lifting this to an inner quantum -background.
This obstruction admits an elegant description.


Let (,Q) be an elliptic complex which is equipped with a local -action, where  is some local Lie algebra.
Endow the BV theory ^*[-1] with the structure of a classical -background via S^ in equation (<ref>).
The anomaly to the existence of a quantum inner -background for ^*[-1] is the trace of the action of  on ^*[-1].


We specialize to when M = X is a complex n-manifold.
The basic free holomorphic theory we consider is labeled by a holomorphic vector bundle V on X.
The elliptic complex is the Dolbeault complex resolving the sheaf of holomorphic sections of V:
 
_V = Ω^0,(X, V) .
The action functional ∫_X βγ returns the familiar free action of the βγ-ghost system used in string theory.
Notice that in dimension n, however, that the field β is an element
β∈Ω^n,(X,V^*)[n-1],
whose degree zero component is a field of Dolbeault type (n,n-1).

The following will be the most important case for us.
We assume that V → X is a natural holomorphic vector bundle; meaning a bundle built from the tangent bundle by taking duals, tensor products, direct sums, cohomological shifts etc..
This implies that the space of holomorphic sections of V is equipped with the structure of a representation for the Lie algebra of holomorphic vector fields.
The action is by the Lie derivative.
This action extends in a natural way to a local _X-module structure on the elliptic complex _V resolving the sheaf of holomorphic sections of V.

Summarizing, we have the following.


If V → X is a natural holomorphic vector bundle then the corresponding BV theory ^* [-1] _V is equipped with a _X-background defined by the coupling

S^(μ; β,γ) = ∫_X β(L_μγ) ,
where L_μ denotes the Lie derivative.


In this context we can use <ref> to obtain a geometric description of the obstruction to this lifting to an inner _X-background.



 §.§ Riemann–Roch theorem and anomalies


Let  denote the moduli stack of complex n-manifolds.
For the purposes of this note we will not need a precise model for this moduli space, but we will use the following features:


  * There is a universal complex manifold π→ whose fiber over an n-manifold X ∈ is X itself.

  * Any natural bundle (one built from the holomorphic tangent bundle by taking duals, tensor products, direct sums etc.) defines a vector bundle on the universal n-manifold.
For example, there is a universal tangent bundle _univ→ whose restriction to a complex n-manifold X yields _X.

  * A model for the tangent complex of  at a fixed complex n-manifold X is the dg Lie algebra
𝕋_X ≃_X = Ω^0,(X, _X) .


The last item is a formal version of the formalism of Kodaira and Spencer which says that the formal moduli problem for deforming complex structures is controlled by the dg Lie algebra _X.
This means that a formal deformation of a complex n-manifold is a Beltrami differential
μ∈Ω^0,1(X, _X)
which satisfies the Maurer–Cartan equation

μ+ 1/2 [μ,μ] = 0.
The deformed complex n-manifold has Dolbeault complex
(Ω^0,(X) , + μ) .
In other words, a formal deformation has the effect of modifying the  operator as ⇝ + μ.
The condition that this squares to zero is precisely the Maurer–Cartan equation.

Now suppose that V → is a universal holomorphic vector bundle.
We then obtain a family of elliptic complexes _V over  whose fiber over a complex n-manifold X ∈ is the elliptic complex _X,V from (<ref>).
To get a family over the moduli of complex structures  we take the (derived) pushforward along π→ to get the family of elliptic complexes π_* _V over .
The determinant of this virtual bundle is a special case of the determinant, or anomaly, line bundle <cit.> associated to a family of -operators parameterized by a moduli of complex structures.

Now we consider the formal completion of  at X.
By the expectation (3) above, the formal moduli space ^ _X is described by the dg Lie algebra _X:
^ _X ≃B _X .
The resulting family π_* _V over the formal completion corresponds to a _X-representation which is precisely encoded by the _X-background (<ref>).

The consequence of theorem <ref> is that the one-loop anomaly for an inner _X-background is the first Chern class of the virtual bundle π_* _V.


Let V be a natural holomorphic vector bundle and consider the resulting _X-background for the BV theory ^* [-1] _X,V.
When X = ^n, the anomaly to lifting this to an inner _X-background is given by the class
. Td ·(V)|_2n+2 ∈H^2n+2 (BU(n)) ≃H^2n+1 (vect(n)) ↪H^1_loc((X)) .


The consequence of this result in the complex one-dimensional case is hopefully familiar.
The ghost system for the bosonic string is the free theory based off of the complex _Σ, _Σ [1], where _Σ[1] is the holomorphic tangent bundle shifted down by one.
The resulting anomaly in terms of universal Chern characters is
·([1])|_4 = - (1 + 1/2 c_1 + 1/12 c_1^2)(1 + c_1 + 1/2 c_1^2)|_4 = - 13/12 c_1^2 .
The typical normalization for the generator of the Virasoro central charge is 1/24 c_1^2, implying that the central charge of this free system is c = -26 as expected.
To compare, the anomaly associated to the complex free boson CFT is 1/12 c_1^2; thus recovering the fact that the obtained by gauging 13 complex copies of the free boson by holomorphic vector fields is anomaly free.


Let's move to the complex n=2 dimensional situation.
As an example, consider the case where V is the trivial rank one bundle.
Then the universal Riemann–Roch theorem implies that the anomaly is represented by the local cocycle corresponding to the universal class 
|_6 = - 1/24 _1 _2 + 1/48 _1^3 .
In other words a^hol(V = triv) = -1/24, c^hol(V = triv) = 1/48 so that
- 4 π^2 Θ_V = triv = -1/24 ∫_^2 tr ( J μ) tr(J μJ μ) + 1/48 ∫_^2 tr(J μ) tr(J μ) tr(J μ) .


We can generalize this example to an arbitrary natural line bundle which we will write as V = K^λ for λ a rational number, where K is the canonical bundle on ^2.
We record the result here.


There are the following expressions for the holomorphic central charges associated to the free holomorphic theory ^* [-1] _K^λ:
a^hol (K^λ) = 1/24 r   and   c^hol(K^λ) = - 1/48 r^3 ,
where r = 2 λ -1.


This follows from the expression in universal Chern classes
. ·(K^λ)|_6 = 1/12 (λ- 1/2) _1 _2 - 1/6 (λ- 1/2)^3 _1^3 ,
which is straightforward to verify.


Observe that under λ↦ 1- λ one has r ↦ - r.
In particular, a^hol and c^hol satisfy 
a^hol(K^λ) = - a^hol (K^1-λ),   c^hol(K^λ) = - c^hol (K^1-λ) .





 §.§ Computing the anomaly using the holomorphic gauge


Recall that the free part of a holomorphic theory is determined by the  operator associated to some holomorphic vector bundle.
In <cit.> it is shown that any holomorphic theory admits a natural gauge where the gauge fixing operator is Q^GF = ^⋆.
The corresponding renormalization is well-behaved.
On flat space ^n the main results concerning renormalization for holomorphic theories can be summarized as follows.



Suppose (,ω,S = S_free + I) is a holomorphic theory on ^n and let S[L] denote the effective action, at scale L, constructed using the gauge fixing operator Q^GF = ^⋆ whose corresponding propagator is P_ϵ <L. 


  * To first-order in ħ the effective action is finite
I[L] = lim_ϵ→0 W(P_ϵ^L , I) ħ^2
In particular, all one-loop counterterms are identically zero.

  * Suppose the free part of the action is of the form S_free (Φ) = ω(Φ, Φ).
The anomaly to solving the quantum master equation to first-order in ħ can be expressed as the sum over wheels with exactly (d+1)-vertices. 
Explicitly, as a local functional this anomaly is

Θ_1-loop = lim_L →0 lim_ϵ→0 ∑_Γ∈Wheel_n+1, e W_Γ(P_ϵ< L, K_ϵ,I) .
where the sum is over all wheels with (n+1)-vertices and distinguished edges thereof.
Here K_ϵ is the heat kernel associated to the generalized Laplacian built using the gauge fixing operator ^⋆.



Let's return to the context of the previous section.
We consider the free BV theory _V on X = ^n, where V is some natural vector bundle, in the -background S^(ξ;β,γ) from (<ref>).
In this case we can use the trivialization V ≃^n × V_0 to identify the -operator for V with ⊗𝕀 where  is simply the ordinary -operator on ^n.
Correspondingly, we choose the gauge fixing operator is ^⋆⊗𝕀 where ^⋆ is Hodge adjoint of  constructed using the flat hermitian metric on ^n. 
Explicitly, the heat kernel is given by
K_L(z,w) = 1/(2πi L)^n e^-|z-w|^2/4L ⊗𝕀_V ∈_V ⊗ _V [1] 
where 𝕀_V ∈ V_0 ⊗ V_0^* represents the identity morphism V_0 → V_0.
The propagator is 
P_ϵ<L (z,w) = ∫_T = ϵ^L Ṭ   (^⋆⊗𝕀)   K_T(z,w) .


Since ^*[-1]_V is a free BV theory, item (1) of theorem <ref> implies that 
S^ [L] lim_ϵ→0 W(P_ϵ^L , I) ħ^2 
is a one-loop quantum -background.
Further, item (2) implies that the obstruction to lifting this to a one-loop inner -background is
Θ= lim_L →0 lim_ϵ→0 ∑_Γ∈Wheel_n+1, e W_Γ(P_ϵ< L, K_ϵ,I) ,
which automatically determines a degree one cocycle in ^((^n)).

We specialize to complex dimension n=2.
In this case, the anomaly Θ is expressed as a sum over wheels with three vertices.
Since the effective action is GL(2,)-invariant it is straightforward to see that based on symmetry the anomaly local cocycle Θ is cohomologous to a linear combination of the cocycles Θ_a^hol and Θ_c^hol from proposition <ref>.
That is
- 4 π^2 Θ_V = a^hol(V) Θ_a^hol + c^hol(V) Θ_c^hol
for some coefficients a^hol(V),c^hol(V).
Instead of computing these coefficients explicitly using the effective action, which appears to be quite cumbersome, we will refer to the Riemann–Roch theorem as outlined in the previous subsection to fix them.





 §.§ Riemann–Roch theorem for mixed anomalies



So far we have only discussed anomalies for the action of holomorphic vector fields on free theories within the BV formalism.
These anomalies are avatars of gravitational, or conformal, anomalies as portrayed in the physics literature.
For non-free theories, like gauge theories, there are also possible internal anomalies, like gauge anomalies, which must be trivial in order for quantization to exist.

In this section we consider anomalies present when holomorphic gauge theory is put in a -background.
In addition to the holomorphic gravitational anomalies we have been focused on thus far, there will also be pure gauge and mixed anomalies which must be trivialized.
To make the discussion as parallel as possible with the case of pure diffeomorphism anomalies, we consider the case that both diffeomorphisms and gauge symmetries are considered as background fields.
From this we will be able to deduce mixed anomalies for interacting gauge theories.

Let G be a Lie group.
Let _G denote the moduli of pairs (X,P) of a complex n-manifold X together with a principal holomorphic G-bundle P on X.

As in the case of complex n-manifolds, we will not develop a precise model for this moduli space, but we will use the following expected features:


  * There is a universal moduli π_G →_G whose fiber over a pair (X,P) ∈_G is the principal bundle P → X itself.

  * Any natural bundle (one built from the holomorphic tangent bundle by taking duals, tensor products, direct sums etc.) together with a representation of G defines a vector bundle on the universal n-manifold.


  * A model for the tangent complex of  at the pair (X,P_triv), where P_triv is the trivial G-bundle, is the semi-direct product dg Lie algebra
𝕋_X _G ≃= Ω^0,(X, _X) ⋉Ω^0,(X) ⊗g .
This dg Lie algebra is a resolution of the sheaf of holomorphic sections of the sheaf of Lie algebras ^hol_X ⋉^hol⊗g.


The last point says that a formal deformation of a pair (X,P_triv) is a Beltrami differential
μ∈Ω^0,1(X, _X), satisfying (<ref>), as before together with a (0,1) connection 
A ∈Ω^0,1(X) ⊗g
which together satisfy the Maurer–Cartan equation
(+ μ) A + 1/2 [A,A] = 0.
The deformed complex n-manifold has Dolbeault operator + μ and the deformed principal bundle has Dolbeault operator + A.



Let V be a natural holomorphic vector bundle equipped with a G-action and consider the resulting _X ⋉Ω^0,_X ⊗g-background for the BV theory ^* [-1] _X,V.
When X = ^n, the anomaly to lifting this to an inner _X ⋉Ω^0,_X ⊗g-background is given by the class
. Td ·_GL(n) ×G (V)|_2n+2 ∈H^2n+2 (BGL(n) ×BG) ↪H^1_loc((X) ⋉Ω^0,(X) ⊗g) ,
where _GL(n) × G (V) is the GL(n) × G-character of the representation V.


Let's consider a simple example when the complex dimension n=1 and G = GL_1() = ^×.
Suppose that V is the trivial super vector bundle W[] = W ⊕Π W where W is some natural vector bundle on a curve.
In this case, the pure holomorphic gravitational anomaly for the BV theory ^* [-1] _,W[] vanishes.
So, the anomaly corresponds to an element in
H^2(BGL_1) ⊗H^2(B G) ⊂H^4(BGL_1 ×B G), 
which turns out to be proportional to c_1  c_1 where c_1 represents the `gravitational' first Chern class and c_1 is the background gauge G=GL_1() class.
Explicitly, as a local functional this class is represented by
∫_J μTr(A) 
where μ is a background holomorphic vector field and A is a background gauge field.
This anomaly was found independently in <cit.> where it is interpreted as the famous Calabi–Yau anomaly for the two-dimensional B-model.
Indeed, the holomorphic theory we are considering here is a holomorphic twist of the two-dimensional =2 chiral multiplet valued in W.
The GL_1() × GL_1() represents, at the group level, the dilations and R-symmetry which remains inside of the two-dimensional =2 superconformal algebra after twisting.



 §.§ Holomorphic gauge theories


In this section we have only discussed anomalies for background holomorphic diffeomorphism symmetry or background holomorphic gauge symmetry where it suffices to just consider the underlying free theory.
For non-free theories, like gauge theories, there are also possible internal anomalies, like gauge anomalies, which must be trivial in order for quantization to exist.
In this section we explain how holomorphic diffeomorphism symmetry mixes with possible internal gauge symmetries.

The idea is the following.
We start with a holomorphic gauge theory which has a background symmetry by holomorphic vector fields.
Before discussing the issue of quantizing this background symmetry, we first need to know that the gauge theory has no internal anomaly.
Once this internal anomaly is trivialized, we can then treat the underlying free theory as a theory with a background symmetry by ⋉Ω^0,⊗g.
We then treat the problem just as we did before with two caveats.
First, since we have already trivialized the pure gauge anomaly, it suffices to only look at anomalies which depend non-trivially on a holomorphic vector field.
Second, the terms in the anomaly which have a nontrivial dependence on the gauge field represent the anomaly to having a quantum -background.
Once these are trivialized, we can then contemplate the anomaly to having a quantum inner -background; this is represented by the terms in the anomaly which are independent of the gauge field.

Let's consider an explicit example.
Let g be a Lie algebra of a compact Lie group G.
The basic holomorphic gauge theory we consider is holomorphic BF theory which can be defined on any complex manifold X.
Globally, holomorphic BF theory is the shifted cotangent bundle to the moduli stack of holomorphic G-bundles on X.
Within the perturbative BV formalism the fields consist of pairs 

    A    ∈Ω^0,(X, g)[1] 
    
    B    ∈Ω^n,(X, g^*)[n-2]

and the action is of standard BF-type: S_BF = ∫_X B F_A.
This theory enjoys a classical backround -symmetry defined by
S^= ∫_X B (L_μA) .


Holomorphic BF theory suffers from an internal gauge anomaly.
As usual, we focus just on the case X = ^n.
Using theorem <ref> one can show the following. 



Holomorphic BF theory on ^n is anomalous at one-loop in perturbation theory.
The local functional representing this anomaly is
- 1/(n+1)!∫_^n Tr_ad (A (A)^n) ,
where the trace denotes the trace is in the adjoint representation of g on itself.
















For example, in complex dimension one n=1, the holomorphic BF theory associated to any semi-simple Lie algebra is anomalous.
In complex dimension two n=2 the anomaly for holomorphic BF theory vanishes for complex semi-simple Lie algebras.


We now turn to thinking of the underlying free theory as a theory with a background symmetry by ⋉Ω^0,⊗g.
Such anomalies to quantization will be labeled by certain elements in
H^2n+2 (BGL(n) ×BG) 
as in the last section.
The thing that distinguishes this discussion from the last section is that we are assuming that the component in H^2n+2(BG), which we just summarized in the previous proposition, vanishes
Explicitly, this is the component -_n+1(_Bg) = - _n+1^G(g_ad).

The following is a consequence of proposition <ref>.

Suppose _n+1(_Bg) = 0.
The anomaly to lifting S^ to a quantum -background is represented by the class

    - . Td·ch^G(g^ad) |_2n+2^⋆ = - ∑_j=1^n_j_n+1-j (_B g[1]) 
    ∈⊕_j H^2j(BGL(n)) ⊗ H^2n+2-2j(BG) ⊂ H^2n+2(BGL(n) × BG)

where the ⋆ indicates that we do not include the component _2n+2.


The reason we have not included the term _2n+2 is that this term represents the anomaly to having an inner -action, which is only well-defined once the above class vanishes.
It is very rare that pure holomorphic BF theory admits a quantum -background; the above anomaly conditions are very restrictive.
We obtain many examples however, once we include matter.
We will detail an explicit example motivated by four-dimensional supersymmetry in the next section, but for now we state the general result.

Suppose that V is a G-equivariant natural holmorphic vector bundle on X.
Examples of such bundles include K_X^⊗λ⊗ U where U is a G-representation.
In particular, on X = ^n, we can consider its GL(n) × G-character
ch^GL(n) ×G (V) ∈H^(GL(n) ×G) .


We consider coupling to holomorphic BF theory for the Lie group G the following holomorphic matter system valued in V.
The fields, in the BV formalism, are
γ∈Ω^0,(X, V)
together with the conjugate field β∈Ω^n,(X, V)[n-1].
The total action functional is 
∫_X B F_A + ∫_X β_A γwhere _A =  + A.
We refer to this as holomorphic BF theory coupled to the βγ system with values in V.
The classical -background is 
S^ = S^= ∫_X B (L_μA)  + ∫_X β(L_μγ) .
The proof of the proposition below combines the results above.



Holomorphic BF theory coupled to the βγ system with values in V on ^n is anomalous at one-loop in perturbation theory.
The invariant representing this anomaly is

^G_n+1 (V) - ^G_n+1(g^ad) ∈H^2n+2 (BG) .


Suppose the invariant (<ref>) vanishes.
The anomaly to lifting S^ to a quantum -background is represented by the class

. Td ·(ch^GL(n) ×G(V) - ch^G(g^ad) ) |_2n+2^⋆∈H^2n+2(BGL(n) ×BG) 
where the ⋆ indicates that we do not include the component in H^2n+2(BGL(n)).

Suppose the invariant (<ref>) vanishes.
Then the -central charge of the system is 
. Td ·(ch^GL(n) ×G(V) - ch^G(g^ad) ) |_2n+2 ∈H^2n+2 (BGL(n)) .




 §.§ Compactification


Suppose we have a holomorphic theory defined on ^n × X where X is some compact complex manifold of dimension m.
Then, we obtain a holomorphic theory on ^n via compactification along ^n × X →^n.
We turn to the problem of characterizing the _^n-anomaly polynomial from the anomaly polynomial in n+m dimensions.

Recall that the space of anomalies in (n+m)-dimensions is
H^1_loc(_^n+m) = H^2n+2m+1(vect(n+m)) = H^2n+2m+2(BU(n+m)) .
Let X be a compact m-dimensional manifold, then we have the composition

H^2n+2m+2 (BU(n+m)) [d,equal] [r]     H^2n+2m+2(BU(n) ×BU(m)) [r]     H^2n+2(BU(n)) [d,equal] 
 
H^1_loc(_^n+m) [rr,dotted,"∫_X"]         H^1_loc(_^n) .

The first map on the top line is induced from the block diagonal U(n) × U(m) → U(n+m) and the second map restricts along the tangent classifier X → BU(m) and then integrates along X.
We denote this composition by ∫_X.


If Θ is the anomaly to a quantum -background on ^n × X then ∫_X Θ is the anomaly to a quantum -background on ^n.


As an example, consider the basic higher dimensional βγ system on ×Σ where Σ is a Riemann surface. 
The fields are 

    γ   ∈Ω^0,(×Σ) 
    β   ∈Ω^2,(×Σ)[1]  .

The compactification along Σ is simply the ordinary βγ system on  with values in the graded vector space H^(Σ, )[More accurately, this is the two-dimensional bosonic βγ system with values in H^0(Σ, ) coupled to the two-dimensional fermionic bc system with values in H^1(Σ, ).], whose fields are

    γ_2d   ∈Ω^0,() ⊗ H^(Σ, ) 
    β_2d   ∈Ω^1,() ⊗ H^(Σ, K) [1]  .

In particular, the the _ anomaly polynomial is simply
χ^hol(Σ,) ·1/12 c_1^2 ∈H^4(BU(1)) ,
where χ^hol(Σ, ) is the holomorphic Euler characteristic.

On the other hand, we have seen that the universal anomaly on ^2 for this higher dimensional βγ system is simply
Td|_6 = - 1/24 ch_1 ch_2 + 1/48 ch_1^3 ∈H^6(BU(2)) .
Now, ∫_Σch = 1 + ch_1 + ∫_Σ c_1(Σ) where c_1(Σ) is the first Chern class of the tangent bundle of Σ, and ∫_Σch_2 = 1/2ch_1^2.
Since χ^hol(Σ, ) = 1/2∫_Σ c_1(Σ) we have
∫_ΣTd|_6 = - 1/24 ·1/2 ∫_Σc_1(Σ) ch_1^2 + 1/48 ·3 ·∫_Σ c_1(Σ) _1^2 = 1/12 χ^hol(Σ, ) _1^2 ∈H^4(BU(1)) . 





§ RELATIONSHIP TO THE A,C ANOMALIES



Most of the work up until now held in arbitrary complex dimension.
Now we specialize to complex dimension two, and specifically to holomorphic quantum field theories obtained from twisting four-dimensional superconformal theories.

As reviewed in the introduction, to a four-dimensional conformal theory one associates two invariants.
These are the coefficients c of (Weyl)^2 and a of (Euler) in the conformal anomaly T_μ^μ.

A four-dimensional superconformal theory is, in part, a theory which has a symmetry by the four-dimensional superconformal algebra. 
There are variants of this algebra depending on the amount of supersymmetry, but in this section we will only need the minimal =1 supersymmetric version.
In particular, being superconformal means that the theory has a symmetry by the so-called R-symmetry group U(1) which commutes with supersymmetry (and conformal transformations).
Crucially, for this section, in such a four-dimensional superconformal theory, the coefficients a,c are related to certain coefficients in the R-symmetry anomaly which we recall momentarily.

This paper concerns holomorphic quantum field theories.
To any four-dimensional supersymmetric theory there is a twist (unique up to R-symmetry, if it exists) which renders the theory holomorphic <cit.>. 
A natural question is: what structure does the holomorphic twist of a superconformal field theory posess?
The answer to this question is one of the main results of <cit.>.


Suppose  is the holomorphic twist of four-dimensional =1 superconformal theory of Yang–Mills type.[By this, we mean a theory consisting of some number of chiral and vector multiplets with both gauge interactions and possibly a superpotential.
In particular this is not just pure gauge theory as there are possibly matter fields as well.]
Then  admits a classical -background.


In other words, the holomorphic twist of a =1 superconformal theory admits a symmetry by holomorphic vector fields.
In this sense, we argue in <cit.> that the Lie algebra of holomorphic vector fields is an enhanced twisted superconformal algebra.

The main result of this section shows how the coefficients a,c of the supersymmetric theory relate to the -equivariant anomaly polynomial of the holomorphically twisted theory.



Consider the holomorphic twist of a four-dimensional superconformal theory on ^4 of Yang–Mills type and let a,c be the gravitational anomaly coefficients of this supersymmetric theory.
The anomaly polynomial for -equivariant quantization of the holomorphic twist is 

Θ= 2/3 (a-c) _1 _2 + 1/9 (c - 5/3 a) _1^3 .


In other words, there is the following relation involving the holomorphic anomalies a^hol,c^hol (see equation (<ref>)):

    a^hol   = 2/3 (a-c) 
    
    c^hol   = 1/9(c - 5/3 a)

Alternatively, the expression for a,c in terms of a^hol,c^hol is


a     = - 9/4 (a^hol + 6 c^hol ) 

c     = - 3/4 (5 a^hol + 18 c^hol ) .





 §.§ R-symmetry and twisting data


A four-dimensional superconformal theory is, in particular, a theory with R-symmetry.
The minimal amount of R-symmetry is with respect to the group U(1), which we denote by U(1)_R when speaking of R-symmetry.

Let R be the generator of the U(1)_R-symmetry in the original supersymmetric theory.
We will utilize the following well-known relationship between the a,c coefficients of the anomaly polynomial and the R-symmetry, see <cit.>:


a     = 3/32 (3 Tr R^3 - Tr R ) 

c     = 1/32 (9 Tr R^3 - 5 Tr R) .

For standard multiplets, the R-charges appearing in the above formulas are those of the chiral fermions in the theory.
In terms of the anomaly polynomial for mixed gravitational and R-symmetry (in this case, U(1)_R) the coefficients (Tr R^3), (Tr R) appear as

Θ^=1 = 1/3! ( (Tr R^3) c_1^3 - 1/4 (Tr R) c_1 p_1)
where c_1 is the generator of H^2(BU(1)_R) and p_1 ∈ H^4 (BSO(4)) is the universal first Pontryagin class, which we have normalized with a factor of -1/4 to fit with physics conventions.
Our goal is to relate these coefficients to objects in the holomorphic twist.

Generally speaking, the operation of `twisting' a supersymmetric theory involves the following steps.
Throughout, we fix a supersymmetry algebra together with an R-symmetry group G_R.


  * Find a supercharge Q, which is square zero.
That is, Q is an odd element of the supersymmetry algebra which satisfies [Q,Q] = 0.

  * Find a `grading homomorphism' α U(1) → G_R, where G_R is the R-symmetry group. 
The supercharge Q must be weight one with respect to this homomorphism.

  * Find a `twisting homomorphism'.
This amounts to choosing a Lie group G, together with homomorphisms ι G → Spin(n) and ϕ G → G_R with the property that Q is G-invariant with respect to the induced homomorphism ι×ϕ G → Spin(n) × G_R.


The four-dimensional =1 supersymmetry algebra has an essentially unique supercharge which is square-zero <cit.>.
Any other supercharge is in the same R-symmetry orbit.
We fix such a supercharge Q^hol.
This supercharge is holomorphic, in the sense that the rank of the map [Q^hol,-] is two (half the dimension of spacetime).
For more details on twisting we refer to <cit.>.



Consider a free supersymmetric theory on ^4 built from chiral and vector multiplets with R-symmetry generator R = (r_i) where r_i is the R-charge of the ith multiplet.
Our convention is that for chiral (respectively, vector) multiplets, the scalar (respectively gauge) field has R-charge r_i+1 (respectively, r_i-1).

Then, there exists twisting data such that the holomorphic twist of of this theory is equivalent to the cotangent theory (in the BV sense) to
⊕_i Ω^0,(^2, K^(r_i+1)/2) .


As an example, consider a single free vector multiplet with its standard R-charge r = 1 (so that the gauge field in the multiplet has vanishing R-charge R(A) = 0).
This lemma implies that the holomorphic twist is the cotangent theory to Ω^0,(^2,K) = Ω^2,(^2) which consists of fields 
(B,A) ∈Ω^2,(^2) ⊕Ω^0,(^2) [1],
thus recovering the free limit of holomorphic BF theory.

As another example, the R-charge for a single chiral multiplet which saturates the BPS bound for the superconformal algebra is r = -1/3 (so that the scalar in the multiplet has R-charge R(ϕ)=2/3).
The holomorphic twist of this theory is the theory whose underlying space of BV fields is 
(γ, β) ∈Ω^0,(^2, K^1/3) ⊕Ω^0,(^2, K^2/3)[1] ,
and action is ∫βγ.
This is the two-dimensional βγ system twisted by K^1/3.


To compute the -anomaly it suffices to consider the underlying free theory, hence we can appeal to proposition <ref>.

From (<ref>) we have the following relation between the R-symmetry generator and the gravitational anomaly coefficients:
Tr R = 16 (a-c) ,   Tr R^3 = 16/3 (5/3 a - c) .


From proposition <ref> we know that we can express the holomorphic twist as a βγ-system with coefficients in a sum of line bundles determined by R. 
Using Lemma <ref> we see that
a^hol = 1/24 16 (a-c) = 2/3 (a-c) 
and
c^hol = - 1/48 ·16/3 (5/3 a - c) = 1/9 (c - 5/3 a)
as desired.



There is an alternative way of deducing these holomorphic anomaly coefficients from the anomaly polynomial of the supersymmetric theory.
Indeed, we can start with the R-symmetry anomaly polynomial of the untwisted theory in equation (<ref>) and proceed with the following two steps: first, break SO(4) to MU(2) and second, twist by setting c_1 = -1/2ch_1.
The first step has the effect of setting p_1 ↦ 2 ch_2 (this is the induced homomorphism H^4(BSO(4)) → H^4(BU(2)) and the second step follows from the standard twisting homomorphism for four-dimensional supersymmetry, see <cit.>.









 §.§ Anomalies for basic supersymmetric theories


We unpack the anomaly polynomial for basic four-dimensional supersymmetric gauge theories.
The results, including anomalies for theories with more supersymmetry, are summarized in table <ref>.





  §.§.§ =1 gauge theory


In =1 gauge theory there are vector multiplets and chiral multiplets.
We have described the twist of the superconformal vector and chiral multiplet after the statement of proposition <ref>.
To compute the coefficients a^hol,c^hol we could refer to the formula (<ref>).
Instead, we will compute the coefficients directly from the Riemann–Roch theorem.

In fact, we have already done this.
The anomaly polynomial for the chiral multiplet is obtained by plugging λ = 1/3 into the coefficients of lemma <ref>.
Similarly, the anomaly polynomial for a single vector multiplet is obtained as the negative of the coefficients in proposition <ref> when λ = 0.
The sign arises because this is a gauge theory.

Translating back to the physical coefficients as in equation (<ref>) we obtain the expected values of a,c <cit.>.




  §.§.§ =2 gauge theory


We focus on two multiplets relevant for four-dimensional =2 supersymmetric gauge theory.
The first is the vector multiplet, which is defined from the data of a Lie group G and a principal bundle, which we take to be trivial.
The second is the hypermultiplet, which depends on the choice of a representation R of G.

In the BV formalism, the holomorphic twist of pure gauge theory is holomorphic BF theory for the holomorphic local Lie algebra whose underlying holomorphic vector bundle is 
g ⊗∧^K^1/3 = g ⊕Πg ⊗K^1/3 .
The Lie algebra structure is determined by the Lie bracket on g.
Note that

(g[1] ⊗∧^K^1/3 ) = (- 1/3 _1 + 1/18 _1^2 - 1/162 _1^3 ) G .
Thus
(g[1] ⊗∧^K^1/3 )|_6 = (1/36 _1 _2 - 13/648 _1^3 ) G .
(Alternatively, note that as =1 multiplets a =2 vector is the same as a =1 vector plus a =1 chiral).

The hypermultiplet is the cotangent theory to 
R ⊗∧^K^1/3 ⊗K^1/3 = R ⊗K^1/3 ⊕ΠR ⊗K^2/3,
From this we see that the anomaly polynomial for the hypermultiplet valued in R is
(-1/36 _1 _2 + 1/648 _1^3 ) R .


Translating back to the physical coefficients as in equation (<ref>) we obtain the expected values of a,c <cit.>.






  §.§.§ =4 gauge theory

Consider four-dimensional =4 supersymmetric gauge theory for gauge group G near the trivial G-bundle.
In the BV formalism, the holomorphic vector bundle underlying the holomorphic twist of this theory is
g[1] ⊗∧^(K^1/3 ⊗^3) . 
Notice that in the cohomology of BGL(2) we have the formal expression
(K^λ/3) = 1 - λ/3 _1 + λ^2/2 ·3^2 _1^2 - λ^3/3! ·3^3 _1^3 . 
where λ is any integer.
From this we compute

    ∧^(K^1/3⊗^3)     = 1 - 3  (K^1/3) + 3  (K^2/3) - (K) 
        = 1/27_1^3 .


Thus, we see directly that for =4 supersymmetric Yang–Mills theory we have a^hol = 0 and 
c^hol = -1/2 ·27 G = -G/54 .


Translating back to the physical coefficients as in equation (<ref>) we obtain the expected values of a,c <cit.>.



§ ANOMALY MATCHING IN HOLOMORPHIC QCD



In this section we consider anomalies in the holomorphic twist of four-dimensional =1 supersymmetric quantum chromodynamics (QCD).



 §.§ The holomorphic theory


Four-dimensional =1 supersymmetric Yang–Mills theory depends on a choice of compact Lie group G, with complexified Lie algebra , and a representation  that comprises the supersymmetric matter.







Typically, in supersymmetric QCD one takes G = SU(N_c) and 
= (^N_c)^⊕N_f ⊕(^N_c)^⊕N_f
where ^N_c (respectively, ^N_c) denotes the fundamental (respectively, anti-fundamental)  SU(N_c) representation.
One says that N_c is the number of `colors' and N_f is the number of `flavors'.

In the =1 supersymmetry algebra there is (up to equivalence) a unique twisting supercharge.


The twist of four-dimensional =1 supersymmetric Yang–Mills theory on ^4 associated to the pair (G,) is equivalent to holomorphic BF theory on ^2 with gauge group G_ coupled to the holomorphic βγ system on ^2 valued in .

If the supersymmetric theory is equipped with a superpotential W, then there is a holomorphic superpotential in the twisted theory, see below.


Following this theorem, we arrive at the holomorphic version of supersymmetric QCD obtained from the previous theorem as the twist of supersymmetric QCD with N_c colors and N_f flavors.
Explicitly, this the holomorphic theory whose fields are:


  * the gauge fields are those of holomorphic BF theory valued in sl(N_c), whose fields are

    A ∈Ω^0,(^2 , sl(N_c))[1]   and   B ∈Ω^2,(^2, sl(N_c)^*);


  * the “quarks” are fields of a holomorphic βγ system valued in 

    V = (^N_c)^N_f,
 
where ^N_c is the fundamental SU(N_c) representation whose fields are

    γ∈Ω^0,(^2, V)   and  β^†∈Ω^2,(^2 , V^*)[1];


  * the “antiquarks” are fields of a holomorphic βγ system valued in the dual representation 

    V^* ≅ (^N_c)^N_f,

where ^N_c is the anti-fundamental SU(N_c) representation whose fields are

    γ^†∈Ω^0,(^2 , V^*)   and  β∈Ω^2,(^2 , V) [1].[The superscript † does not refer to adjoint here.]




The action functional is

    S_QCD = ∫ B F_A + ∫β^†_A γ + ∫β_A γ^†,

where _A =  + A is the covariant -operator.

If, additionally, the supersymmetric theory is equipped with a superpotential W, then there is a term in the holomorphic action
∫^̣2 z   W(γ) = ∫^̣2 z   (1/2 W”(γ^0) γ^0,1 γ^0,1 + W'(γ^0) γ^0,2) .
Notice that this depends on the choice of a holomorphic volume form and does not define a theory with a cohomological -grading.
Both of these drawbacks stem from the fact that the superpotential breaks R-symmetry in the original supersymmetric theory.


In the above description we have described the fields on flat space.
Even on flat space, a -background will depend on which bundle the quark fields transform under.
For example, for a single superconformal chiral multiplet, the field γ is twisted by the line bundle K^1/3_^2.

For holomorphic QCD we will do something more general.
We suppose that γ (and γ^†) transform as sections of K^⊗λ_^2⊗ V (and K^⊗λ_^2⊗ V^*) where λ is some rational number.
From section <ref>, recall that this corresponds to the twist of a chiral multiplet with r = 2 λ - 1.

The -background is 

I^,r = ∫B L_μA + ∫β^†L_μγ+ ∫βL_μγ^†.
With respect to the standard coordinates μ = μ_i (z) _z_i and this coupling reads
I^,r = ∫B μ_i _z_i A + (1-r) ∫β^†μ_i _z_i γ+ (1-r) ∫βμ_i _z_i γ^†+ r ∫_z_i μ_i (β^†γ+ βγ^†) .





















 §.§ There are no gauge anomalies


Consider holomorphic QCD with gauge group SU(N_c) and N_f flavors valued in the fundamental representation.
First we consider the internal anomalies.
For pure sl(N_c) holomorphic gauge theory is no pure gauge anomaly as Tr_adj(X^3) = 0 for any X ∈sl(N_c), see proposition <ref>.
The mixed gauge-matter anomalies vanish since the matter comprises an equal number of fundamental and anti-fundamental representations.



 §.§ Anomaly to a quantum -background



Next we consider the anomaly to having background of holomorphic vector fields  using the generalized coupling I^,r from equation (<ref>).
We find an exact agreement with the anomaly cancelation for the R-symmetry current in supersymmetric QCD.


Holomorphic QCD admits a quantum -background with classical coupling I^,r if and only if
r = - N_c/N_f ,
equivalently, λ = 1/2 (1-N_c/N_f).


This is a consequence of proposition <ref>.
The graded holomorphic vector bundle underlying holomorphic QCD is 
V_QCD = g[1] ⊕K^⊗λ ⊗V ⊕K^⊗λ ⊗V^* .
In the expansion of (<ref>) we see that since the gauge group is simple, the only the terms in anomaly live in
H^2(BGL(2)) ⊗H^4(B G) ,
where here G = SU(N_c).
Explicitly, this is the term

Td_0 _3^GL(2) ×G(V_QCD) + Td_1 _2^G(V_QCD) .


Using the form of V_QCD above, we see that the first term in equation (<ref>) is
2 F _2(K^⊗λ) = - 2λF _1 ch^G_2(fun) .
We have used the relation ch_1(K^⊗λ) = - λ_1 where, as usual, ch_1 is the standard generator of H^2(BGL(2)). 
The factor of two comes from the fact that we have quarks and anti-quarks.

Likewise, the second term in (<ref>) is
1/2 _1 (- _1 ch^G_2(adj) + ·2 F ·_2^G(fun)) .


In total we see that the anomaly to a quantum -background is
_1 (-1/2 _2^G(adj) + 2 (-λ+ 1/2) F _2^G(fun) ).
For any X ∈sl(N_c) we have the relation Tr_adj(X^2) = 2 N_c Tr_fun(X^2).
In other words, for G = SU(N_c), we have ch_2^G(adj) = 2 N_c ch_2^G(fun). 
Using this relation, the result follows.
















 §.§ a,c for QCD


We are now in a position to compute the values of a^hol,c^hol for holomorphic QCD.
As always, this only depends on the underlying holomorphic vector bundle of holomorphic QCD
V_QCD = g[1] ⊕K^⊗λ ⊗V ⊕K^⊗λ ⊗V^* ,
where, as we found in the last section, one has λ = 1/2 (r+1) = 1/2(1 - N/F) in order to guarantee anomaly cancellation for the -background.

According to this decomposition, the a^hol coefficient decomposes as a^hol_QCD = a^hol_gauge + a^hol_quark + a^hol_anti-quark, and similarly for c^hol.
Notice that a^hol_quark = a^hol_anti-quark and c^hol_quark = c^hol_anti-quark as they transform identically under the action of holomorphic vector fields.
We read these coefficients directly.

First
a^hol_gauge = 1/24g = 1/24(N_c^2 -1),
as read off from table <ref>.
From lemma <ref> we see
a^hol_quark = 1/24 (- N_c/N_f ) V = - 1/24 N_c^2 .
Thus in total we see that

a^hol_QCD = - 1/24 (N_c^2 + 1)
Interestingly, this is independent of the number of flavors N_f.

Similarly, from table <ref>, c^hol_gauge = -1/48g = - 1/48 (N^2 - 1),
and from <ref> we see
c^hol_quark = - 1/48 (- N_c/N_f)^3 V = 1/48 N_c^4/N_f^2 .
Thus

c^hol_QCD = 1/48N_f^2(2N_c^4 - N_c^2 N_f^2+N_f^2).


Plugging into equation (<ref>) we recover the expected physical values for these coefficients

    a_QCD    = -3/16 N_f^2(3N_c^4-2N_c^2 N_f^2+N_f^2) 
    
    c_QCD    = - 1/16 N_f^2(9 N_c^4 - 7 N_c^2 N_f^2 + 2 N_f^2 ) .




 §.§ Magnetic anomalies


In <cit.>, Seiberg discovered a duality in four-dimensional supersymmetric gauge theory that posits an equivalence between =1 supersymmetric QCD for gauge group SU(N_c), called the `electric' theory, and another =1 supersymmetric QCD, called the `magnetic' theory.

The magnetic theory has gauge group SU(N_f - N_c) and the matter consists of N_f fundamental chiral multiplets together with N_f^2 `mesons' which transform trivially under the gauge group.
There is also a superpotential which we will not need so do not comment any further on.
Therefore, at the level of the holomorphic twist, the bundle underlying the magnetic dual to holomorphic QCD is thus of the form
V_QCD = g[1] ⊕K^⊗λ ⊗V ⊕K^⊗λ ⊗V^* ⊕K^⊗λ_M ⊗End(^F) ,
where g = sl(N_f-N_c) and the matter is labeled by the representation V = (^N_f - N_c)^N_f.

Since the holomorphic mesons transform trivially under g they do not contribute to any possible gauge anomalies.
In particular, we can compute the anomaly to a quantum -background just as we did in section <ref>.
We find that a quantum -background exists if and only if
λ= N_c/2 N_f,
equivalently r = - N_f - N_c/N_f.

From this, we can proceed as above to calculate the a,c anomalies for the magnetic version of QCD.
As a function of r_M = 2λ_M-1, we find

    a^hol_QCD    = 1/24(-1 + 2 N_f N_c - N_c^2 + N_f^2(r_M - 1) ) 
     c^hol_QCD    = 1/48 N_f^2(N_f^2 + N_f^4 - 6 N_f^3 N_c + 11N_f^2 N_c^2-8 N_f N_c^3+2N_c^4-r_M^3 N_f^4) .



The a,c anomalies match for the holomorphic twists of Seiberg dual theories
a_QCD = a_QCD,   c_QCD = c_QCD
if and only if r_M = 1 - 2N_c/N_f. 


It suffices to prove the statement for the holomorphic coefficients a^hol,c^hol. 
Immediate computation shows that a_QCD^hol from equation (<ref>) matches with a^hol_QCD if and only if r_M = 1 - 2N_c/N_f.
Plugging this value of r_M into c_QCD^hol we find a match with equation (<ref>).

















