Instantons and multibananas: Relating elliptic genus and cohomological Donaldson-Thomas theory
Oliver Leigh



Instantons and multibananas]Instantons and multibananas: Relating elliptic genus and cohomological Donaldson-Thomas theory

oliverle@math.uio.no
Dept. of Mathematics, The University of Oslo






In this article the cohomological Donaldson-Thomas theory of local multibanana threefolds is computed using Descombes' hyperbolic localisation formula. 
The resulting expression is precisely given by the elliptic genus of the moduli space of framed instanton sheaves. 
As part of the proof, an infinite-wedge-space trace formula is computed for the elliptic genus of the moduli space of framed instanton sheaves.


The author was supported by the Research Council of Norway grant number 302277 - “Orthogonal gauge duality and non-commutative geometry”.

    
[
    Oliver Leigh
    April 29, 2024
==================










§ INTRODUCTION





 §.§ Cohomological Donaldson-Thomas invariants


      Numerical Donaldson-Thomas (DT) invariants are a mathematical count of BPS states; in particular supersymmetric bound states in type II string compactifications. 
    They were originally developed to study compact Calabi-Yau threefolds, for which they contain important deformation-invariant information, and have since been studied in much more generality <cit.>, <cit.>.

    Let Y be a (not necessarily proper) Calabi-Yau 3-fold over , and consider a curve class β∈ H_2(Y,) and an integer n∈. The numerical DT invariant 𝖣𝖳_β,n(Y) is computed by considering the Hilbert scheme
    
    ^_β,n(Y) 
            := 
            { Z⊂ Y   |   [Z]=βχ(Ø_Z) = n }.

    The invariant 𝖣𝖳_β,n(Y) is given by a weighted Euler characteristic of ^_β,n(Y) <cit.> or, equivalently if Y is proper, a virtual fundamental class of ^_β,n(Y) <cit.>. 

    In recent years, the research group of Joyce has developed a rigorous general theory of  cohomological DT invariants in <cit.>, based on ideas due to Kontsevich and Soibelman in <cit.>, and Dimca and Szendrői in  <cit.> (see also <cit.>, and for an excellent introduction to the field <cit.>).
    Cohomological DT invariants provide more nuanced information, at the expense of deformation invariance (see <cit.> for a discussion), and they are valued in the abelian category of monodromic mixed Hodge structures, = (). 

    The key result needed to define cohomological DT invariants in the present article is:

    
        
        Let ϕ:^∙→ L^∙_^_β,n(Y) be the natural symmetric obstruction theory of Thomas <cit.>. 
        For a given choice of square root (^∙)^1/2 there is a monodromic mixed hodge module
        𝒟𝒯^∙_Y,β,n∈(^_β,n(Y)), which is unique up to canonical isomorphism, satisfying:
        
            
  * If ^_β,n(Y) is locally modelled by the critical locus of a regular function f : U →  on a smooth -scheme U, then 
            𝒟𝒯^∙_Y,β,n is locally modelled by 
            ℋ𝒱^∙_U,f, the monodromic mixed Hodge module lift of the perverse sheaf of vanishing cycles of (U,f).
        
    

    
        The results of <cit.> apply to more general spaces than those given in Theorem <ref>. 
        We have restricted to the case at hand in order to simplify the exposition. 
    

    
        Nekrasov and Okounkov show in <cit.> (see also <cit.>) that a choice of square root (^∙)^1/2 for Theorem <ref> always exists. 
    

    We call 𝒟𝒯^∙_Y,β,n the DT mixed hodge module.
    As discussed in <cit.>, the hypercohomology of a monodromic mixed Hodge module also has the structure of a monodromic mixed Hodge structure.
    Hence, using the monodromic mixed Hodge module from Theorem <ref> one can define cohomological DT invariants as follows:

    
        The cohomological Donaldson-Thomas invariants are defined by 
        
    𝔻𝕋_β,n(Y)
                :=
                ^∙(𝒟𝒯^∙_Y,β,n)
                ∈ K()

        where ^∙(-) is compactly supported hypercohomology. 
    

    
        As discussed in <cit.> and <cit.>, the numerical DT invariant can be recovered from the cohomological DT invariant by 
        
    𝖣𝖳_β,n(X)
                =
                ∑_k∈
                (-1)^k
                ^k(𝒟𝒯^∙_Y,β,n).

    

    Similar to the case of numerical DT invariants, one can assemble the cohomological DT invariants into a partition function
    
    Z^𝔻𝕋(Y)
            :=
            ∑_β∈ H_2(Y,)∑_n∈ 𝔻𝕋_β,n(Y)   Q^β q^n,

    where Z^𝔻𝕋(Y) is an element of K()[q^-1] q, Q. 
    Here an effective basis β_1,…,β_n of H_2(Y,) has been chosen i.e. if β is effective then β=∑ d_iβ_i for d_i ≥0, and the convention Q^β = ∏ Q_i^d_iβ_i has been used. 

    The above partition function is very hard to compute. Indeed, even for the easier numerical DT version, there is no complete computation for any proper Calabi-Yau threefold with trivial first Betti number. The case is worse for cohomological DT theory, but the recent hyperbolic localisation formula of <cit.> (discussed more in Section <ref>) has provided a powerful tool for calculations. 
    In the present article, we use this tool to compute the cohomological DT theory of local multibanana Calabi-Yau threefolds.

    Most previous calculations in this area have been from the view-point of motivic DT invariants, introduced in <cit.>. These take values in a Grothendieck ring of varieties, but can be converted to cohomological DT theory as described in <cit.> and <cit.>.
    From this point of view, the partition function Z^𝔻𝕋(Y) has been computed for local curves in <cit.> (see <cit.> for a review), and the invariants for the one-loop
    quiver with potential were calculated in 
    <cit.>. 
    The theory relating to some local surfaces has also been computed in <cit.>. 






 §.§ Local multibanana threefolds



Local multibanana threefolds are examples of non-singular toric varieties without a finite type fan. Their name is derived from the shape of a curve configuration which is discussed below and depicted in Figure <ref>.

Following Kanazawa and Lau <cit.>, for i,j∈, one can construct the local multibanana threefold of type (i,j)  as a quotient of a toric threefold with a non-finite-type fan. 
We can construct this fan by first considering the lines 𝖫_f,m:={ p∈^3  |  f(p)=m,z=1} and the tiling of { (x,y,1)   |  x,y∈} given by the collection:

    {𝖫_x,l}_l∈∪{𝖫_y,m}_m∈∪{𝖫_x-y,n}_n∈.

This tiling also defines a non-finite-type fan F by taking the all cones over the proper faces of the tiling. This is depicted in Figure <ref> below. 



Now, take the threefold  defined by F and observe that  has a free i⊕ j-action coming from translations of F. 
We define the local multibanana threefold of type (i,j) by

    X_(i,j) 
        :=
        /i⊕ j.

The focus of this article is X_(r,1), and for convenience, we restrict to this space from now on. X_(r,1) has a natural ^2 action and, as explained in <cit.>, the ^2-fixed curves are unions of (-1,-1)-rational curves corresponding to 2-dimensional sub-cones. Following Bryan <cit.>, we call these the banana curves and label them by their sub-cones

    A_k 
           ∼ℝ_≥0Conv({(0,k,1),(0,k + 1,1)}),
        
    
        B_k
           ∼ℝ_≥0Conv({(k+1,k,1),(k,k + 1,1)}),
        
    
        C_k
           ∼ℝ_≥0Conv({(k,0,1),(k+1,0,1)}).


As pointed out in <cit.>, the banana curves have the relations 

    A_k + B_k = A_k + B_k+1
        B_k + C_K = B_k+1 + C_K+1

in H_2(X_(r,1),), and hence the banana curves form the  r+2 classes

    A_0,…,A_r-1,
        
        B = B_0,…,B_r-1,
        
        C = C_0,…,C_r-1.

Moreover, by <cit.> these classes give an effective basis for H_2(X_(r,1),).

The banana curves are either disjoint or meet as the coordinate axis in a copy of ^3. This is depicted in the web diagram of Figure <ref> and the banana diagrams in Figure <ref>. We call the union of the banana curves a banana configuration.
Interestingly, the mirror symmetry of Banana configurations (called perverse curves) was previously studied by Ruddat in <cit.>. 











 §.§ Enumerative geometry of local multibanana threefolds



The enumerative geometry of banana threefolds was first studied by Bryan in <cit.>. He considered an elliptically-fibred proper Calabi-Yau threefold X_𝖻𝖺𝗇 containing subschemes locally isomorphic to X_(1,1), then computes the numerical DT partition function for the banana-curve sublattice. 
We give a cohomological generalisation of the local version of Bryan's formula in Corollary <ref>. 

Bryan's results were extended by the author in <cit.>, by also considering the section of the elliptic fibration. 
In forthcoming work, Bryan and Pietromonaco consider related quotient spaces called banana nano-manifolds in <cit.>.
In another direction, Katz's genus 0 Gopakumar-Vafa partition function of X_𝖻𝖺𝗇 for the banana-curve sublattice was partially computed in <cit.>. This calculation was extended to X_(r,1) and X_(2,2) in <cit.>. 

From the discussion in Section <ref> we know A_0,…,A_r-1,B,C is an effective basis for H_2(X_(r,1),), 
and denote the corresponding formal variables by Q_A_0,…,Q_A_r-1,Q_B,Q_C. 
We can also define a restricted partition function Z^𝔻𝕋_c(X_(r,1)) by 

    Z^𝔻𝕋(X_(r,1))
        =
        ∑_c ≥ 0 
        Z^𝔻𝕋_c(X_(r,1))
         
        Q_C^c.

The main result of this article is the following explicit formula for this partition function. 





    
    Let ^1/2 be the monodromic mixed hodge module of →; z ↦ z^2 as detailed in <cit.>, and let ^-1/2 be its formal inverse.  
    Define
    
    Θ_p,y(t) :=
            ∏_n>0(1 - y p^n-1 t)(1 - p^n-1 t)(1 - y^-1 p^n t^-1)(1 - p^n t^-1 ),

    the generalised MacMahon function
    M(u; v, w) 
        :=
        ∏_k,l > 0 ( 1 - u  v^k  w^l)^-1,
    and for a 2D partition Y with (i,j)∈ Y the arm and leg lengths a_Y(i,j) := Y_i - j and l_Y(i,j) := (Y^T)_j - i.

    Then, for any choice of orientation, we have that Z^𝔻𝕋_c(X_(r,1)) is equal to the expression
    
    _0^𝔻𝕋·
            Q_B^(1-r)cγ_0,…,γ_r-1
     s.t. ∑γ_i = c(-u^-1/2v^1/2)^rc∏_i,j=0^r-1 ∏_x ∈γ_i Θ_𝚎_r-1,-u^1/2 v^-1/2 Q_B( 𝚎_i𝚎_j u^-a_γ_i(x)-1 v^-l_γ_j(x))
            
       ∏_z ∈γ_j Θ_𝚎_r-1,-u^1/2 v^-1/2 Q_B( 𝚎_i𝚎_j u^a_γ_j(z) v^l_γ_i(z)+1)

    where 
    u:= - q  ^1/2, v:= - q  ^-1/2 and
    𝚎_i : = ∏_j=0^i Q_A_i Q_B, 
    and _0^𝔻𝕋 is given by the expression
    
    M(v^-1;u,v)^r·∏_0≤ i<j≤ r-1
                M(𝚎_i 𝚎_j^-1v^-1; u,v)
            /
                M(𝚎_j 𝚎_i^-1u^-1; u,v)
            M( -Q_B 𝚎_j 𝚎_i^-1(uv)^-1/2; u,v)/M( -Q_B^-1𝚎_i 𝚎_j^-1 (uv)^-1/2; u,v)
        ·∏_N>01/1-e_r-1^N∏_i,j=0^r-1M(𝚎_r-1^N-1𝚎_i 𝚎_j^-1u^-1; u,v)/M(-Q_B 𝚎_r-1^N-1𝚎_i 𝚎_j^-1 (uv)^-1/2; u,v)M(𝚎_r-1^N𝚎_i 𝚎_j^-1v^-1; u,v)/M(-Q_B^-1𝚎_r-1^N𝚎_i 𝚎_j^-1 (uv)^-1/2; u,v).






    Another interesting refinement of DT invariants is given by the K-theoretic DT invariants of <cit.> (see also <cit.>).
    In the case when Y is proper, these are predicted to be the χ_y-genus of the cohomological DT invariants. 
    However, in the case where Y is non-proper, there is a discrepancy. 
    An example is when Y=^3 and β=0 as computed motivically by <cit.> and K-theoretically by <cit.>. This relationship is discussed in <cit.>. 

    Another discrepancy is for X_(r,1). K-theoretic DT invariants use a refinement parameter arising from the equivariant weight of the canonical bundle. However, since ^2 is a Calabi-Yau torus on X_(r,1), this refinement parameter is 1, and no extra information is obtained in this case. 








 §.§ Elliptic genus of the moduli space of framed instanton sheaves


Theorem <ref> has a striking relationship with the elliptic genus of framed instanton sheaves, which we will discuss in this section. 
We first recall the definition and some key properties of the moduli space of framed instanton sheaves on ^2. 

[<cit.>]
    
    The moduli space of frame instanton sheaves on ^2 (r,N) is the nonsingular scheme of dimension 2rN parameterising pairs
    
    (  E, Φ: E|_ℓ_∞∼⟶Ø_ℓ_∞^⊕ r )

    where E is a rank r torsion free sheaf on ^2 with <c_2(E),[^2]> = n and that is locally free in a neighborhood of ℓ_∞:={[0:z_1:z_2]},
    and Φ is an isomorphism. It has the following properties:
    
        
  * (r,N) is acted upon by a 2+r-dimensional torus, ^2+r, with associated characters t_1,t_2,e_0,…, e_r-1. 
        
  * The fixed points (r,N)^^2+r are isolated and are in bijection to the set of r-tuples of Young diagrams Y=(Y_1,…,Y_r) satisfying ∑_i=1^rY_i = N.
        
  * For a fixed point Y, the equivariant K-theory class of the restricted tangent bundle T_(r,N)|_Y is given by 
        
    ∑_a,b = 0^r-1 e_b e_a^-1·(∑_s∈ Y_a(t_1^-l_Y_b(s)t_2^a_Y_a(s)+1) + ∑_t∈ Y_b(t_1^l_Y_a(t)+1 t_2^-a_Y_b(t) ) ).

    




Recall that elliptic genus can be defined by taking the Chern roots L_i of a tangent bundle and considering the multiplicative class

    _u,y(L) 
         := 
        c_1(L) θ_1(u, y (-L))/θ_1(u, (-L)) = c_1(L)/y^1/2·Θ_u,y(c_1(-L)).

Now, combining Definition-Theorem <ref> with Theorem <ref> we have the following corollary.


    
    Considering ^1/2 as a formal variable, we have the relationship
    
    Z^𝔻𝕋_c(X_(r,1))/Q_B^c ·_0^𝔻𝕋
            =
            Ell_𝚎_r-1,-^1/2Q_B ((r,c); 𝚎_0, …, 𝚎_r-1, ^1/2q^-1, ^1/2 q  )

    where 
    𝚎_i : = ∏_j=0^i Q_A_i Q_B
    and _0^𝔻𝕋 is given in Theorem <ref>.




    As part of the proof of Theorem <ref> and Corollary <ref> we obtain a natural expression for
    
    Ell_u,y((r,N); e_0, …, e_r-1, t_1, t_2  )

    as a trace of vertex operators of Okounkov <cit.> (see also <cit.>). 
    The expression is given in Corollary <ref>. 
    It is hoped that this will be useful in extending the χ_y Vafa-Witten Blowup formula of <cit.> to elliptic genus. 



The relationship between DT theory of banana threefolds and elliptic genus was first observed by Bryan in <cit.>, 
where it was observed that the numerical Donaldson-Thomas partition function for the local banana threefold (r=1) was determined by the equivariant elliptic genus of the Hilbert scheme of points on ^2, where ^* acts diagonally. 

Theorem <ref> extends this result in two ways:

    
  * For r=1 the cohomological DT partition function is determined by the equivariant elliptic genus of the Hilbert scheme of points on ^2 for the larger (^*)^2 torus. 
    
  * For r>1 the cohomological DT partition function is determined by the equivariant elliptic genus of the moduli space of framed instanton sheaves on ^2. 



In the r=1 case we can follow Bryan <cit.> and use Waelder's analogue of the DMVV formula <cit.> to obtain an infinite product formula. 


    
    For r=1 the full partition function is
    
    Z^𝔻𝕋
            =
                ∏_ d_A, d_B d_C ≥ 0k=(k_1,k_2)∈ (1/2)^2 
    (1 - (-Q_A_0)^d_A (-Q_B)^d_B (-Q_C)^d_C v^-k_1 u^k_2)^-b(d,k_1,k_2).

    where 
    d: = (d_A,d_B,d_C), 
    d := 2d_A d_B + 2d_A d_C + 2d_B d_C - d_A^2 - d_B^2 - d_C^2
    and b(a,k_1,k_2) is defined by the expansion as a Laurent series in p,y,t_1^_-1/2,t_2^_1/2 of
    
    ∑_a≥ -1∑_k_1,k_2∈1/2
            b(a,k_1,k_2) 
            Q^a t_1^-k_1 t_2^k_2
            =
            ∑_k∈ Q^k^2(-(t_1t_2)^1/2)^k
            /(
                    ∑_k∈+1/2 Q^2k^2 (-t_1)^-k)
                (
                    ∑_k∈+1/2 Q^2k^2 (-t_2)^k
                )





    It is unclear whether such a product formula exists in the general case.
    On the positive side: The existence of a product formula is equivalent to the existence of the associated Gopakumar-Vafa-invariant formula (see <cit.> for more details). 

    Given the link with elliptic genus of (r,n), one would hope that such a product formula extends to the elliptic genus generating function as well. 
    In the non-framed case, the moduli space of rank 2 stable sheaves on a surface has been shown to have product formula by Göttsche and Kool in <cit.>. 
    A product formula for general rank would be extremely useful in extending the χ_y Vafa-Witten blowup formula of <cit.> to elliptic genus, and similarly extending a forthcoming result of Arbesfeld, Kool and Laarakker <cit.>. 





 §.§ Structure of the article

The article is structured as follows:

    
  - 
    In Section <ref> we compute the partition function in terms of the refined topological vertex. The method is to combine Descombes' hyperbolic localisation formula <cit.> with the K-theoretic computational results of Arbesfeld <cit.>.
    
  - In Section <ref> we compute the formulas in Theorem <ref> and Corollary <ref> using Okounkov's vertex operators <cit.>. 
    We also obtain a trace formula for the elliptic genus of the moduli space of framed instanton sheaves on ^2 (given in Corollary <ref>). 


        


§ COMPUTATION OF COHOMOLOGICAL DT THEORY





 §.§ Descombes' hyperbolic localisation formula



    Cohomological DT invariants are very hard to compute, and the torus localisation techniques used in numerical DT theory <cit.> are insufficient in the cohomological setting. 
    It was suggested by Szendrői <cit.> that the ideas of Braden <cit.> for hyperbolic localisation on intersection cohomological could be applied to cohomological DT theory as well. 
    Hyperbolic localisation for cohomological DT invariants was proved by Descombes in <cit.>, and will be the basis for the results of this article. 

    Hyperbolic localisation applies in the setting of an  algebraic space X with the action of a one dimensional torus ^*. 
    The underlying concept of hyperbolic localisation is the decomposition of Białynicki-Birula <cit.> (extended from smooth varieties by <cit.>).
    In this setting we consider the ^*-fixed space X_0 and also the attracting  variety X^+, which is the subset of points x∈ X such that lim_t→ 0 t· x exists.
    We then consider the hyperbolic localization diagram as in <cit.>, <cit.> and
    <cit.>:
    
    X
                   
                X^+[l,swap,"η",hook']
                [r,"p"]
                   
                X^^*

    Here η is the inclusion and p is the natural morphism. 
    The functor p_!η^* is called the hyperbolic localization functor.

    To apply hyperbolic localisation in the setting of this article, we consider a Calabi-Yau threefold with a ^*-CY-action (i.e. locally the weights (w_1,w_2,w_3) have w_1+w_2+w_3=0), and observe that this action extends to one on _β,n(Y).
    The ^*-action is compatible with the obstruction theory of <cit.> and we obtain a ^*-equivariant perfect obstruction theory ϕ: ^∙→ L^∙__β,n(Y).
    The ^*-fixed locus can then be decomposed into connected components
    
    _β,n(Y)^^* = _λ∈Λ P_λ.

    
    In this article we focus on the case where each P_λ is an isolated and reduced point. 
    In this case
    
    ℋ^0(^∙)^∨≅_P_λ,

    and we define the index at P_λ to be 
    
    _P_λ
            :=
            #{}
            -
            #{}.

    Now, Descombes' hyperbolic localisation formula is given by the following theorem. 

    
        
        Suppose _β,n(Y)^^* is a finite collection of reduced points.
        Then there is an isomorphism of monodromic mixed hodge modules
        
    p_!η^* 𝒟𝒯^∙_Y,β,n≅⊕_λ∈Λ^1/2_λ.

    

    
        In <cit.>, the hyperbolic localisation formula is given in more generality. 
        In particular, the condition on the fixed loci is not needed. 
        In general, each fixed loci would have an associated monodromic mixed hodge module (of the form given in Theorem <ref>) appearing on the right hand side of the formula. 
        In the case of isolated smooth fixed points, we have used that this monodromic mixed hodge module is simply  as described in <cit.>. 
    




 §.§ Analysis of the torus fixed locus


Analysis of the ^2-fixed subschemes of X_(1,1) was carried out by Bryan in <cit.> using toric techniques detailed in <cit.> and <cit.>. 
Essentially, a torus-invariant subscheme is determined by combinatorial
data associated to its web diagram. The data involves a 2D partition for each edge and for each vertex, a 3D partition asymptotic to the the 2D partitions of the incident edges. 

The analysis of the ^2-fixed subschemes of X_(r,1) is essentially the same as for X_(1,1). 
The difference is seen by comparing <cit.> with the web diagram in Figure <ref>. Namely, the ^2-fixed locus of X_(r,1) has 2r vertices and 3r edges. We then arrive at the following lemma.


    
    For a curve class β = ∑ a_i· A_i + b· B + c· C the fixed points of ^n,β(Y) are indexed by the collection Λ of tuples
    
    λ = (α,β,γ; π, η)

    such that 
    
        
  * α = (α_0,…,α_r-1) where α_i is a 2D partition with |α_i| = a_i,
        
  * β = (β_0,…,β_r-1) where β_i is a 2D partition with ∑ |β_i| = b,
        
  * γ = (γ_0,…,γ_r-1) where γ_i is a 2D partition with ∑ |γ_i| = c,
        
  * π = (π_0,…,π_r-1) where π_i is a 3D-partition asymptotic to (α_i,β_i,γ_i),
        
  * η = (η_0,…,η_r-1) where η_i is a 3D-partition asymptotic to (α'_i-1,β'_i,γ'_i),
    
    and 
    
    n = 1/2∑_i( 2|π_i| + 2|η_i| + α_i^2 + α'_i^2 + β_i^2 + β'_i^2 + γ_i^2 + γ'_i^2  ).

    Here for a 2D partition μ we have used μ^2 = ∑μ_i^2 and for a 3D-partition ν we have used the renormalised volume <cit.>:
    
    |ν| := ∑_-0.16em ∈ν(1- # {}).







To apply the hyperbolic localisation formula, we may choose a ^*-subtorus of ^2 given by weights (w_1,w_w) and with character denoted by t. 
The local description of this is given in Figure <ref>. 
The (general) hyperbolic localisation formula is unaffected by the choice of ^*-subtorus and we choose a subtorus such that 

    ^β,n(X_(r,1))^^* 
        =
        ^β,n(X_(r,1))^^2

so that we can apply Theorem <ref>.
(The condition required is that w_1 ≠ n· w_2 for any n∈.) 
We will use the results of <cit.>, where a choice of ^*-subtorus is called a slope. 
Indeed, we assume for later convenience that w_1>>w_2>0 in which case we are using a preferred slope of <cit.>.



    
    Hilb^β,n(X_(r,1)) is equal to its attracting locus, Hilb^β,n(X_(r,1))^+.


    This follows because the web diagram of X_(r,1) contains no half edges, meaning that for [Z]∈^β,n(X_(r,1)) we have that lim_t→ 0 t· [Z] always exists. 




The obstruction theory and the associated ^* weights can be computed from the above combinatorial data following the method of <cit.> (see also, <cit.>, <cit.>). 
In particular from <cit.>, for a 2D partition μ and 3D-partition ν there are Laurent polynomials E_μ(w_1,w_2,w_3) and V_ν(w_1,w_2,w_3) called edge and vertex functions such that 

    T^|_λ
        =
        ∑_i   
        E_α_i(w_1,-w_1-w_2,w_2)
        + E_β_i(w_1,-w_1-w_2,w_2)
        + E_γ_i(w_1,-w_1-w_2,w_2)
        
       
        + V_π_i(w_1,-w_1-w_2,w_2)
        + V_η_i(-w_1,w_1+w_2,-w_2)

where T^ = (^0)^∨ - (^-1)^∨∈ K_0(^β,n(X_(r,1))) is the virtual tangent bundle.
Here we have used the conventions from <cit.> so that the arguments can be read directly off Figure <ref>.


    
    We define the index associated to the edge and vertex functions via the following procedure. 
    First we recall from <cit.> that 
    
    E_μ(_) = - E_μ(_)^∨
            V_ν(_) = -V_ν(_)^∨

    so we have decompositions
    
    E_μ(_) 
            = ∑_j (u_j - 1/u_j)
            
            V_ν(_) 
            = ∑_j (v_j - 1/v_j).

    Then we write
    
    ∑ u_i = f_-(1t) + a_0 + f_+(t)
            ∑ v_i = g_-(1t) + b_0 + g_+(t)

    for -polynomials f_± and g_± with no constant term and a_0,b_0∈.
    We can then define
    
    (E_μ(_)):= f_+(1) - f_-(1)
            (V_ν(_)):= g_+(1) - g_-(1),

    noting that these are independent of the choice from (<ref>).


The data of _λ is contained in T^|_λ and can be computed via the edge and vertex index functions. 
Recall from <cit.> that _λ = _λ^∨, so we have 
T^|_λ 
    = _λ - _λ^∨.
Thus, extending  additively to (<ref>) gives
(T^|_λ) = _λ.





We now follow <cit.> and define 

    𝖤_λ(w_1,w_2,w_3)
           
        :=
        q^1/2(λ^2 +λ'^2)^1/2(E_λ(w_1,w_2,w_3)),
        
    𝖵_α,β,γ(w_1,w_2,w_3)
           
        :=
        ∑_π∈ P(α,β,γ) 
        q^|λ|^1/2(V_π(w_1,w_2,w_3))

where

    P(α,β,γ)
        :=
        {  }.

(Note: These correspond to Ê_λ,(-1,-1)^σ and V̂_α,β,γ^σ from <cit.>.)

Hence using Theorem <ref> (hyperbolic localisation) and Proposition <ref> we arrive at the following Lemma. 


    
    If β = ∑ a_i· A_i + b· B + c· C then
    
    ∑_n 𝔻𝕋_β,n(X_(r,1))   q^n

    is given by the following expression in K()[q^-1,q:
    
    1.5∑_(α,β,γ)∏_i
            (
                [ 𝖤_α_i(-w_1-w_2,w_2,w_1)
                    ·𝖤_β_i(w_2,w_1,-w_1-w_2)
                    ·𝖤_γ_i(w_1,-w_1-w_2,w_2);                  · 𝖵_α_i,β_i,γ_i(w_1,-w_1-w_2,w_2)
                    ·𝖵_α'_i,β'_i,γ'_i-1(-w_1,w_1+w_2,-w_2) ])
  
    where the sum is over all tuples (α,β,γ) as described in Lemma <ref> without the conditions (iv), (v), and (<ref>).


Following <cit.> we obtain explicit formulas for 𝖤_μ(_) and 𝖵_α,β,γ(_). 
First for convenience we define 

    u := -q ^1/2
        v := -q ^-1/2,

and recall the refined topological vertex of Iqbal, Kozçaz and Vafa <cit.> (modified following <cit.>)

    𝖢_α, β, γ(u,v)
        :=
        
            M(v^-1;u,v) 
            u^-1/2α'^2
            v^-1/2β^2/∏_x∈γ(1 - u^a_γ(x)+1 v^l_γ(x))
        ∑_η(v/u)^1/2|η|
            S_α/η( u^-ρ v^-γ') 
            S_β'/η(u^-γ v^-ρ )

where ρ:= (-n+1/2)_n∈.
Now, applying <cit.>[
    Each row of <cit.> has a typo: t should be interchanged with q. 
    To see this, observe that on page 27 of the unabridged version of <cit.> (https://arxiv.org/abs/1905.04567arXiv:1905.04567v2), the contribution of □∈λ is stated as l(□)- a(□), but is actually a(□)- l(□). 
]
we have

    𝖤_α(w_1,-w_1-w_2,w_2)
           = 
        u^1/2α'^2v^1/2α^2
    𝖤_β(-w_1-w_2,w_2,w_1)
           = 
        u^1/2β'^2v^1/2β^2
    𝖤_γ(w_2,w_1,-w_1-w_2)
           = 
        u^1/2γ^2v^1/2γ'^2
    𝖵_α,β,γ(w_1,-w_1-w_2,w_2)
           = 
        𝖢_α, β, γ(u,v)
        
    𝖵_α',β',γ'(-w_1,w_1+w_2,-w_2)
           = 
        𝖢_α', β', γ'(v,u)


Combining with Lemma <ref> we obtain

    
    If β = ∑ a_i· A_i + b· B + c· C then we have the following equality in K()[q^-1,q:
    
    ∑_n 𝔻𝕋_β,n(X_(r,1))  q^n
            =
            1.5∑_(α,β,γ)∏_i
            (
                [ u^1/2(α'_i^2+β'_i^2+γ_i^2)
                    v^1/2(α_i^2+β_i^2+γ'_i^2);    · 𝖢_α_i, β_i, γ_i(u,v)
                    ·𝖢_α'_i-1, β'_i, γ'_i(v,u) ])
 
    where the sum is over all tuples (α,β,γ) as described in Lemma <ref> without the conditions (iv), (v), and (<ref>).

    
    





§ VERTEX CALCULATIONS





 §.§ Trace expression for cohomological partition function


We begin this section by considering a product factor which arises from the refined topological vertex.
That is, for a 2D partition γ define

    Y_γ
        :=
        u^1/2γ^2v^1/2γ'^2
            M(v^-1;u,v) /∏_x∈γ(1 - u^a_γ(x)+1 v^l_γ(x))
        
            M(u^-1;u,v) /∏_x∈γ'(1 -  u^l_γ'(x) v^a_γ'(x)+1)
        .





    
    Let μ and η be 2D partitions. Then we have the following equation in [u^-1,v^-1] u,v: 
    
    ∑_x∈μ  u^-a_μ(x)-1v^-l_η(x)
            +
            ∑_x∈η  u^a_η(x)v^l_μ(x)+1
       =
            ∑_x∈ℕ^2  u^-a_μ(x)-1v^-l_η(x)
            -
            ∑_x∈ℕ^2  u^-a_∅(x)-1v^-l_∅(x)



    Nakajima and Yoshioka show in the proof of <cit.> that the top line is equal to
    
    ∑_i=1^η'_1∑_j=1^μ_1(
                u^-a_μ(i,j)-1 v^-l_η(i,j) 
                -
                u^-a_∅(i,j)-1 v^-l_∅(i,j))
            +
            ∑_(i,j)∈μ'
            u^η'_1 - iv^j 
            +
            ∑_(i,j)∈η
            u^i-1 v^μ_1-j+1

    Observe that for i > η'_1 we have l_η(i,j) = - j. Hence we have
    
    ∑_i = η'_1+1^∞∑_j=1^μ_1(
                u^-a_μ(i,j)-1  v^-l_η(i,j) 
                -
                u^-a_∅(i,j)-1 v^-l_∅(i,j))
               
            =
            ∑_j=1^μ_1
            v^j∑_i = η'_1+1^∞(
                u^-μ'_j+i-1
                -
                u^i-1)
            
       
            =
            ∑_j=1^μ_1
            v^j ∑_k = 1^μ'_j
                u^η'_1-k.

    The term for j > μ_1, i ≤η'_1 is similarly described, and the term for i > η'_1, j > μ_1 is zero.  
    The desired result now follows immediately.



    
    Using M(k; u, v) := ∏_n,m > 0 ( 1 - k  u^n v^m)^-1
    we have the equality
    
    M(k u^-1; u,v)·∏_n,m>0(1 - k   u^-a_μ(x)-1v^-l_η(x))
            
       
            =
            ∏_x∈μ(1 - k  u^-a_μ(x)-1v^-l_η(x))
            ∏_z∈η(1 - k  u^a_η(x)v^l_μ(x)+1)
            .




Now,  we apply Corollary <ref> after recalling that

    
  * from the definition a_γ(i,j) = l_γ'(j,i) as well as l_γ(i,j) = a_γ'(j,i)
    
  * ∑_x∈μ a_μ(x) = 1/2(μ^2-|μ|)
    and
    ∑_x∈μ (l_μ(x) +1)
    =
    1/2(μ'^2+|μ'|),

to obtain the result:




    Y_γ   
        =
        (-1)^|γ|( v/u)^1/2|γ|
            M(v^-1;u,v) /∏_x∈γ(1 - u^-a_γ(x)-1 v^-l_γ(x))
        
            M(u^-1;u,v) /∏_x∈γ(1 -  u^a_γ(x) v^l_γ(x)+1)
        
       
        =
        (-1)^|γ|( v/u)^1/2|γ|
            M(v^-1;u,v) /∏_x∈γ(1 - u^-a_γ(x)-1 v^-l_γ(x))
        .




Now, for an r-tuple of 2D partitions γ = (γ_0,…,γ_r-1), define

    Z_γ :=
           ∑_(α,β)∏_i
         
        Q_A_i^|α_i|  Q_B^|β_i| ∑_η_i(v/u)^1/2|η_i|
            S_α_i/η_i( u^-ρ v^-γ_i') 
            S_β_i'/η_i(u^-γ_i v^-ρ )
        
       ·∑_δ_i(u/v)^1/2|δ_i|
            S_β_i/δ_i( u^-ρ  v^-γ'_i)
            S_α'_i-1/δ_i( u^-γ_i v^-ρ ).
 

So that from Corollary <ref> we have

    Z_c^𝔻𝕋
        =
           
     |γ|= c∑_(α,β)∏_i
        (
            1.5[ u^1/2(α'_i^2+β'_i^2+γ_i^2)
                v^1/2(α_i^2+β_i^2+γ'_i^2);    · 𝖢_α_i, β_i, γ_i(u,v)
                ·𝖢_α'_i-1, β'_i, γ'_i(v,u) ])
        
    
        =
           
     |γ|= c
        Y_γ_0⋯ Y_γ_r-1· Z_γ.
 



We will now determine an operator trace expression for Z_γ.
Defining the change of variables
_A_i := Q_A_i^-1/2 = Q_A_i u^-1/2 v^1/2
and
_B := Q_B^1/2 = Q_B u^1/2 v^-1/2
we have

    Z_γ 
        :=
           ∑_(α,β)∏_i
         _A_i^|α_i| _B^|β_i| ∑_η_i(v/u)^1/2|η_i|
            u^1/2|α_i|
            v^-|α_i|+1/2|β_i|
            S_α_i/η_i( u^-ρ v^-γ_i') 
            S_β_i'/η_i(u^-γ_i v^-ρ )
        
       ·∑_δ_i(u/v)^1/2|δ_i|
            u^-1/2|β_i|
            v^1/2|α_i-1|
            S_β_i/δ_i( u^-ρ  v^-γ'_i)
            S_α'_i-1/δ_i( u^-γ_i v^-ρ )
        
    
        =
           ∑_(α,β)∏_i
         _A_i^|α_i| _B^|β_i| ∑_η_i(v/u)^1/2|η_i|
            S_α_i/η_i( u^-ρ +1/2 v^-γ_i'-1) 
            S_β_i'/η_i(u^-γ_i v^-ρ+1/2 )
        
       ·∑_δ_i(u/v)^1/2|δ_i|
            S_β_i/δ_i( u^-ρ-1/2  v^-γ'_i)
            S_α'_i-1/δ_i( u^-γ_i v^-ρ+1/2 ).




We can rewrite the above expression using vertex operators of <cit.> (see also <cit.>). 
These operators have the properties

    Γ_-(𝐱)  | μ> = ∑_λ≻μ S_λ/μ(𝐱)  
         | λ>,
        Γ_+(𝐱)  | λ> = ∑_μ≺λ S_λ/μ(𝐱) 
         | μ>,


    Γ'_-(𝐱)  | μ> = ∑_λ' ≻μ' S_λ'/μ'(𝐱)  
         | λ>,
        Γ'_+(𝐱)  | λ> = ∑_μ' ≺λ' S_λ'/μ'(𝐱) 
         | μ>,


    Q^H  | μ> 
        =
        Q^|μ| | μ>.

Hence, we have the following expression:

    Z_γ   
        =
        tr(
            ∏_i=0^r-1 _A_i^H Γ'_-(u^-γ_i v^-ρ+1/2) 
            Γ'_+( u^-ρ-1/2 v^-γ'_i)
             _B^H Γ_-(u^-γ_i v^-ρ+1/2)
            Γ_+( u^-ρ+1/2 v^-γ'_i-1))


The weight operator Q^H commutes with the other operators with the following relations (see for example <cit.>):

    Q^H  Γ_-(𝐱) 
        =
        Γ_-(Q𝐱)  Q^H
        Γ_+(𝐱)   Q^H 
        =
        Q^H Γ_+(Q𝐱)


    Q^H  Γ'_-(𝐱) 
        =
        Γ'_-(Q𝐱)  Q^H
        Γ'_+(𝐱)   Q^H 
        =
        Q^H Γ'_+(Q𝐱)

Using these relations to commute the _B^H terms to the left of each of the “i” factors gives that Z_γ is the trace of the following expression:

    ∏_i=0^r-1 (Q_BQ_A_i)^H Γ'_-(u^-γ_i v^-ρ+1/2) 
            Γ'_+( u^-ρ-1/2 v^-γ'_i)
            Γ_-(_Bu^-γ_i v^-ρ+1/2)
            Γ_+( _B^-1u^-ρ+1/2 v^-γ'_i-1)

Now, commuting the (Q_BQ_A_i)^H terms to the left and using the cyclic property of tr(_) 
gives 

    Z_γ
        =
        tr(
         p^H
        ∏_i=0^r-1Γ'_-( 𝚟_i) 
         Γ'_+(  u^-1𝚞_i) 
         Γ_-( _B 𝚟_i)
         Γ_+(  _B^-1 v^-1𝚞_i)
        )

where we have defined
𝚎_i : = ∏_j=0^i (Q_BQ_A_i)^-1, 
p:=(e_r-1)^-1
and the following vectors:

    𝚞_i 
        =
        ( 𝚎_i^-1  u^n v^-(γ'_i)_n)_n>0𝚟_i
        =
        ( 𝚎_i  u^-(γ_i)_n v^n)_n>0.






 §.§ Computing elliptic genus from trace formula

Consider an r-tuple of Young diagrams γ = (γ_0,…,γ_r-1) and define vectors

    =
        ( e_i^-1 t_1^(γ'_i)_n t_2^n)_n>0
        =
        ( e_i  t_1^-n t_2^-(γ_i)_n)_n>0

inspired by those of (<ref>). Also consider the following trace expression inspired by Equation (<ref>):

    W_γ   
        :=
        tr(
         p^H
        ∏_i=0^r-1Γ'_-( )
         Γ'_+(  t_2^-1) 
         Γ_-( -y )
         Γ_+( -y^-1 t_1 )
        )

Note that the right-hand side of Equation <ref> can be obtained by substituting 

    e_i  = 𝚎_i,
        
        t_1 = 1/v,
        
        t_2 = u,
        
        y= -_B.


We will compute a product expression for W_γ by employing the method of <cit.>, namely commuting the Γ^*_± terms and obtaining an expressions with increasingly large powers of p. Then considering this expression mod(q^M) for M→∞. 
The key tools are the commutation relations, which recall from <cit.> as

    [Γ_+(a), Γ'_-(b)] 
          =  [Γ'_+(a), Γ_-(b)]
          =  ∏_n,m>0 (1+ a_n b_m )


    [Γ_+(a), Γ_-(b)] 
          =  [Γ'_+(a), Γ'_-(b)]
          =  ∏_n,m>0 (1- a_n b_m )^-1,

where a = (a_n)_n∈ and b = (b_m)_m∈. 


Now, commuting the 
Γ'_+(  t_2^-1) terms 
with the 
Γ_-( -y ) 
terms, followed by commuting the 
Γ'_+(  t_2^-1) 
 Γ_+( -y^-1 t_1 )
terms to the right we obtain

    W_γ
        =
            ∏_i=0^r-1∏_n,m (1 - y  t_2^-1,n ,m)
        ·tr(
         p^H
        ∏_i=0^r-1Γ'_-(   ) 
        Γ_-(  -y )
        Γ'_+(  t_2^-1) 
        Γ_+( -y^-1 t_1 )
        )
        
    
        =    ∏_i=0^r-1∏_n,m (1 - y  t_2^-1,n ,m)
        ·∏_0≤ i<j≤ r-1∏_n,m(1 - y  t_2^-1,n ,m)(1 - y^-1 t_1 ,n ,m)/(1 - t_2^-1,n ,m)(1 - t_1 ,n ,m)
        ·tr(
         p^H
        ∏_j=0^r-1Γ'_-(   ) 
        Γ_-( -y )
        ∏_i=0^r-1Γ'_+(  t_2^-1) 
        Γ_+( -y^-1 t_1 )
        )


For a 2D partition, Y, with (m,n)∈ Y recall the arm and leg functions a_Y(m,n) := Y_m - n and l_Y(m,n) := (Y^T)_n - m. Hence we have  

    ,n ,m 
        =
        e_i e_j^-1 
        t_1^(γ_j')_n - m
        t_2^-(γ_i)_m+n
        =
        e_i e_j^-1 
        t_1^l_γ_j(m,n)
        t_2^-a_γ_i(m,n)
        =
        e_i e_j^-1 
        t_1^a_γ'_j(n,m)
        t_2^-l_γ'_i(n,m).

We use this to consider the factors of the above expression of W_γ. 



    
    We have the equality
    
    ∏_i=0^r-1∏_n,m (1 - y  t_2^-1,n ,m)
            ·∏_0≤ i<j≤ r-1∏_n,m(1 - y  t_2^-1,n ,m)(1 - y^-1 t_1 ,n ,m)/(1 - t_2^-1,n ,m)(1 - t_1 ,n ,m)
       
            =
            y^(1-r) |γ|∏_0≤ i<j≤ r-1M(e_i e_j^-1t_1 ; t_1^-1,t_2)/M( y^-1 e_i e_j^-1t_1 ; t_1^-1,t_2)M( y e_j e_i^-1t_2^-1; t_1^-1,t_2)/M(e_j e_i^-1t_2^-1; t_1^-1,t_2)
          ·∏_i=0^r-1∏_n,m (1 -  t_2^-1,n ,m)
            ·∏_i,j=0^r-1∏_n,m(1 - y  t_2^-1,n ,m)/(1 - t_2^-1,n ,m).



    We refer to Equation (<ref>) so that
    using Corollary <ref> 
    with k = y^-1e_i e_j^-1, then inverting the monomials and reordering the products; before using 
    Corollary <ref> again 
    with k = y e_j e_i^-1t_1 t_2 gives
    
    M( y^-1 e_i e_j^-1t_1; t_1^-1,t_2)/M(e_i e_j^-1t_1; t_1^-1,t_2)·∏_n,m(1 - y^-1 t_1 ,n ,m)/(1 - t_1 ,n ,m)
       
            =
            ∏_x∈γ'_i(1 - y^-1e_i e_j^-1 
            t_1^a_γ'_j(x)+1
            t_2^-l_γ'_i(x))/(1 - e_i e_j^-1 
            t_1^a_γ'_j(x)+1
            t_2^-l_γ'_i(x))∏_x∈γ'_j(1 - y^-1e_i e_j^-1 
            t_1^-a_γ'_j(x)
            t_2^l_γ'_i(x)+1)/(1 - e_i e_j^-1 
            t_1^-a_γ'_j(x)
            t_2^l_γ'_i(x)+1)
       
            =
            y^-|γ'_j|-|γ'_i|∏_x∈γ'_j(1 - y e_j e_i^-1 t_1^a_γ'_j(x)
            t_2^-l_γ'_i(x)-1)/(1 - e_j e_i^-1 t_1^a_γ'_j(x)
            t_2^-l_γ'_i(x)-1)∏_x∈γ'_i(1 - y e_j e_i^-1 t_1^-a_γ'_j(x)-1
            t_2^l_γ'_i(x))/(1 - e_j e_i^-1 t_1^-a_γ'_j(x)-1
            t_2^l_γ'_i(x))
       
            =
            y^-|γ'_j|-|γ'_i|·M( y e_j e_i^-1t_2^-1; t_1^-1,t_2)/M(e_j e_i^-1t_2^-1; t_1^-1,t_2)·∏_n,m(1 - y  t_2^-1,n ,m)/(1 - t_2^-1,n ,m).

    Finally, in the set { (i,j) |  0≤ i<j≤ r-1 } each element of {0,…, r-1} appears r-1 times. Hence, we have
    
    ∑_0≤ i<j≤ r-1( |γ'_j|+|γ'_i|) =  (r-1) ∑_i=0^r-1 |γ_j|

    and the desired result now follows. 




    
    We have the equality
    
    𝖳𝗋
            :=    tr(
            p^H
            ∏_j=0^r-1Γ'_-(   ) 
            Γ_-( -y )
            ∏_i=0^r-1Γ'_+(  t_2^-1) 
            Γ_+( -y^-1 t_1 )
            )
            
    
            =    ∏_N>01/1-p^N∏_i,j=0^r-1∏_n,m>0(1 - p^N y  t_2^-1,n ,m)(1 - p^N y^-1 t_1 ,n ,m)/(1 -p^N t_2^-1,n ,m)(1 - p^N t_1 ,n ,m)



    Using the cyclic property of the trace, we move the
    ∏_i=0^r-1Γ'_+(  t_2^-1) 
        Γ_+( -y^-1 t_1 )
    term to the left and then commute it back to the right using the commutation relations. 
    Repeating this M times gives
    
    𝖳𝗋
            =
               ∏_i,j=0^r-1∏_N=1^M∏_n,m>0(1 - p^N y  t_2^-1,n ,m)(1 - p^N y^-1 t_1 ,n ,m)/(1 -p^N t_2^-1,n ,m)(1 - p^N t_1 ,n ,m)
       ·tr(
            p^H
            ∏_j=0^r-1Γ'_-(   ) 
            Γ_-( -y )
            ∏_i=0^r-1Γ'_+(  p^M t_2^-1) 
            Γ_+( -p^M y^-1 t_1 )
            ).

    Considering this mod(q^M) implies that
    
    𝖳𝗋
            =
               ∏_i,j=0^r-1∏_N>0∏_n,m>0(1 - p^N y  t_2^-1,n ,m)(1 - p^N y^-1 t_1 ,n ,m)/(1 -p^N t_2^-1,n ,m)(1 - p^N t_1 ,n ,m)
       ·tr(
            p^H
            ∏_j=0^r-1Γ'_-(   ) 
            Γ_-( -y )
            ).

    Now, consider the remaining trace factor and commute p^H term to the right and then use the cyclic property to move it back to the left. Repeating this M times gives
    
    tr(
            p^H
            ∏_j=0^r-1Γ'_-(   ) 
            Γ_-( -y )
            )
            =
            tr(
            p^H
            ∏_j=0^r-1Γ'_-(  p^M ) 
            Γ_-( -p^M y )
            )

    Hence, this term is equal to 
    tr(
        p^H
        )
        =
        ∑_μ<μ |  p^H | μ>
        = 
        ∏_N>0(1-p^N)^-1,
    the partition function for the number of 2D partitions. 




    
    We have the equality
    
    ∏_N>0∏_i,j=0^r-1∏_n,m>0(1 - p^N-1 y  t_2^-1,n ,m)(1 - p^N y^-1 t_1 ,n ,m)/(1 -p^N-1 t_2^-1,n ,m)(1 - p^N t_1 ,n ,m)
       
            =
            ∏_N>0∏_i,j=0^r-1M(p^N-1e_i e_j^-1t_2^-1; t_1^-1,t_2)/M(p^N-1 y  e_i e_j^-1 t_2^-1; t_1^-1,t_2)M(p^Ne_i e_j^-1t_1 ; t_1^-1,t_2)/M(p^N y^-1e_i e_j^-1 t_1 ; t_1^-1,t_2)
          ·∏_i,j=0^r-1 ∏_x ∈γ_iΘ_p,y( e_ie_jt_1^l_γ_j(x)t_2^a_γ_i(x)+1)
             ∏_z ∈γ_jΘ_p,y( e_ie_jt_2^a_γ_j(z)t_1^l_γ_i(z)+1)

    where we use the notation 
    
    Θ_p,y(t) 
            :=
            ∏_n>0(1 - y p^n-1 t)(1 - p^n-1 t)(1 - y^-1 p^n t^-1)(1 - p^n t^-1 ).



    We recall Equation (<ref>) use and Corollary <ref> to give
    
    ∏_i,j=0^r-1∏_N>0M(p^N-1 y  e_i e_j^-1 t_2^-1; t_1^-1,t_2)/M(p^N-1e_i e_j^-1t_2^-1; t_1^-1,t_2)M(p^N y^-1e_i e_j^-1 t_1 ; t_1^-1,t_2)/M(p^Ne_i e_j^-1t_1 ; t_1^-1,t_2)
         ·∏_N>0∏_n,m(1 - p^N-1 y  t_2^-1,n ,m)/(1 -p^N-1 t_2^-1,n ,m)(1 - p^N y^-1 t_1 ,n ,m)/(1 - p^N t_1 ,n ,m)
       
            =
            ∏_i,j=0^r-1∏_N>0∏_x∈γ_i(1 - p^N-1 y  e_i e_j^-1 t_1^l_γ_j(x)
            t_2^-a_γ_i(x)-1)/(1 - p^N-1  e_i e_j^-1 t_1^l_γ_j(x)
            t_2^-a_γ_i(x)-1)∏_z∈γ_j(1 - p^N-1 y  e_i e_j^-1 t_1^-l_γ_i(x)-1
            t_2^a_γ_j(x))/(1 - p^N-1 e_i e_j^-1 t_1^-l_γ_i(x)-1
            t_2^a_γ_j(x))
       ∏_x∈γ_i(1 - p^N y^-1 e_i e_j^-1 t_1^l_γ_j(x)+1
            t_2^-a_γ_i(x))/(1 - p^N  e_i e_j^-1 t_1^l_γ_j(x)+1
            t_2^-a_γ_i(x))∏_z∈γ_j(1 - p^N y^-1 e_i e_j^-1 t_1^-l_γ_i(x)
            t_2^a_γ_j(x)+1)/(1 - p^N e_i e_j^-1 t_1^-l_γ_i(x)
            t_2^a_γ_j(x)+1)
       
            =
            ∏_i,j=0^r-1∏_N>0∏_x∈γ_i(1 - p^N-1 y  e_i e_j^-1 t_1^l_γ_j(x)
            t_2^-a_γ_i(x)-1)/(1 - p^N-1  e_i e_j^-1 t_1^l_γ_j(x)
            t_2^-a_γ_i(x)-1)∏_z∈γ_j(1 - p^N-1 y  e_i e_j^-1 t_1^-l_γ_i(x)-1
            t_2^a_γ_j(x))/(1 - p^N-1 e_i e_j^-1 t_1^-l_γ_i(x)-1
            t_2^a_γ_j(x))
       ∏_x∈γ_j(1 - p^N y^-1 e_j e_i^-1 t_1^l_γ_i(x)+1
            t_2^-a_γ_j(x))/(1 - p^N  e_j e_i^-1 t_1^l_γ_i(x)+1
            t_2^-a_γ_j(x))∏_z∈γ_i(1 - p^N y^-1 e_j e_i^-1 t_1^-l_γ_j(x)
            t_2^a_γ_i(x)+1)/(1 - p^N e_j e_i^-1 t_1^-l_γ_j(x)
            t_2^a_γ_i(x)+1)
       
            =
            ∏_i,j=0^r-1∏_x ∈γ_iΘ_p,y( e_ie_jt_1^l_γ_j(x)t_2^a_γ_i(x)+1)
             ∏_z ∈γ_jΘ_p,y( e_ie_jt_2^a_γ_j(z)t_1^l_γ_i(z)+1).






Hence, combining Lemmas <ref>, <ref> and <ref> with Definition-Theorem <ref>
we arrive at the following formula for the elliptic genus of the moduli space of framed instanton sheaves. 


    
    For an r-tuple γ=(γ_0,…,γ_r-1) of 2D partitions, define the vectors 
    
    =
            ( e_i^-1 t_1^(γ'_i)_n t_2^n)_n>0
            =
            ( e_i  t_1^-n t_2^-(γ_i)_n)_n>0.

    Then we have 
    
    Ell_p,y((r,n); e_0,…,e_r-1,t_1,t_2 )
       
            = 
            1/y^n∏_0≤ i<j≤ r-1M( y^-1 e_i e_j^-1t_1 ; t_1^-1,t_2)/M(e_i e_j^-1t_1 ; t_1^-1,t_2)M(e_j e_i^-1t_2^-1; t_1^-1,t_2)/M( y e_j e_i^-1t_2^-1; t_1^-1,t_2)
          ·∏_N>0
            (1-p^N)
            ∏_i,j=0^r-1M(p^N-1 y  e_i e_j^-1 t_2^-1; t_1^-1,t_2)/M(p^N-1e_i e_j^-1t_2^-1; t_1^-1,t_2)M(p^N y^-1e_i e_j^-1 t_1 ; t_1^-1,t_2)/M(p^Ne_i e_j^-1t_1 ; t_1^-1,t_2)
          ·
     |γ|= ntr(
            p^H
            ∏_i=0^r-1Γ'_-( )
             Γ'_+(  t_2^-1) 
             Γ_-( -y )
             Γ_+( -y^-1 t_1 )
            )/∏_i=0^r-1∏_n,m (1 -  t_1 ,n ,m).







 §.§ Proof of the main theorem and corollaries


We recall from Equations (<ref>) and (<ref>) that Z_γ can be obtained from W_γ by substituting 

    e_i  = 𝚎_i,
        
        p=  = 𝚎_r-1,
        
        t_1 = 1/v,
        
        t_2 = u,
        
        y= -_B.

Moreover, from Equation (<ref>) we have

    Z_c^𝔻𝕋
        =
           
     |γ|= c
        Y_γ_0⋯ Y_γ_r-1· Z_γ.

Hence we obtain the expressions 

    Z_c^𝔻𝕋
        =
            
        M(v^-1;u,v)^r·∏_0≤ i<j≤ r-1M(𝚎_i 𝚎_j^-1v^-1; u,v)/M( -_B^-1𝚎_i 𝚎_j^-1v^-1; u,v)M( -_B 𝚎_j 𝚎_i^-1u^-1; u,v)/M(𝚎_j 𝚎_i^-1u^-1; u,v)
        ·∏_N>01/1-e_r-1^N∏_i,j=0^r-1M(𝚎_r-1^N-1𝚎_i 𝚎_j^-1u^-1; u,v)/M(-_B 𝚎_r-1^N-1𝚎_i 𝚎_j^-1 u^-1; u,v)M(𝚎_r-1^N𝚎_i 𝚎_j^-1v^-1; u,v)/M(-_B^-1𝚎_r-1^N𝚎_i 𝚎_j^-1 v^-1; u,v)
        ·(v/u)^1/2 c_B^c
        
     |γ|= c
        (-_B)^-r|γ|∏_i,j=0^r-1 ∏_x ∈γ_iΘ_𝚎_r-1,-_B( 𝚎_i𝚎_j u^-a_γ_i(x)-1 v^-l_γ_j(x))
        
       ∏_z ∈γ_jΘ_𝚎_r-1,-_B( 𝚎_i𝚎_j u^a_γ_j(z) v^l_γ_i(z)+1)
        
    
        =
            _0^𝔻𝕋·(v/u)^1/2 c_B^c
        ·_𝚎_r-1,-_B( (r,c); 𝚎_0,…𝚎_r-1, v^-1, u ),

where we have defined _0^𝔻𝕋:= Z_0^𝔻𝕋,
proving Theorem <ref> and Corollary <ref>.




 §.§ Product formula for r=1 case

We now consider the r=1 and use the techniques of <cit.>. In this case we have

    _𝚎_0,-_B( (1,c); 𝚎_0, v^-1, u )
        =
        Ell_𝚎_0,-_B(^n(^2); v^-1,u ),

where ^n(^2) is the Hilbert scheme of n points on ^2. 
In this special case we can use Waelder's analogue of the DMVV formula where (^*)^2 acts with weight (-1,1):


    
    Considering the Laurent expansion of
    _p,y(^2; t_1^-1,t_2)
    in p, y, t_1^-1, t_2 
    
    ∑_m≥ 0∑_l,k_1,k_2∈
            c(m,l,k_1,k_2)  p^m y^l t_1^-k_1 t_2^k_2,

    we have the following formula
    
    ∑_n ≥ 0_p,y(^n(^2); t_1^-1,t_2)
             Q^n
            = 
            ∏_m,n≥ 0∏_l,k_1,k_2∈(1 - p^n Q^m y^l t_1^-k_1 t_2^k_2)^-c(nm,l,k_1,k_2).




Now define 
_C := Q_C^-1/2 = Q_C u^-1/2 v^1/2
and consider

    Z^𝔻𝕋/_0^𝔻𝕋
        =
            ∑_c≥ 0Ell_ _A_0_B,-_B(^n(^2); v^-1,u )
         _B^c
         _C^c
        
    
        =
            ∏_n≥ 0∏_m>0∏_l,k_1,k_2∈(1 - (Q_A_0Q_B)^n ( Q_B Q_C )^m (-Q_B (uv)^1/2)^l v^-k_1 u^k_2)^-c(nm,l,k_1,k_2)
    
        =
            ∏_n≥ 0∏_m>0∏_l,k_1,k_2∈(1 - (-Q_A_0)^n (-Q_B)^m+n+l (-Q_C)^m v^-k_1-1/2l u^k_2+1/2l)^-c(nm,l,k_1,k_2)


Now consider the following generalisation of <cit.>. 


    
    Consider the expansion of 
    _p,y(^2; t_1,t_2)
    from Theorem <ref> in and define integers b(m,l,k_1,k_2) via the expansions as Laurent series in p,y,t_1^_-1/2,t_2^_1/2 
    
    ∑_m≥ 0∑_l,k_1,k_2∈
            c(m,l,k_1,k_2)  p^m y^l t_1^-k_1 t_2^k_2   
            =
            m≥ 0, l∈
     k_1,k_2∈1/2
            b(m,l,k_1,k_2)  p^m (y/(t_1 t_2)^1/2)^l t_1^-k_1 t_2^k_2.

    Then b(m,l,k_1,k_2) depends only on (4m-l^2, k_1,k_2). Writing
    
    b(m,l,k_1,k_2) = b(4m-l^2, k_1,k_2)

    we have b(a,k_1,k_2) = 0 if a< -1 and the Laurent series in Q,t_1^_-1/2,t_2^_1/2 
    
    ∑_a≥ -1∑_k_1,k_2∈1/2
            b(a,k_1,k_2) 
            Q^a t_1^-k_1 t_2^k_2
            =
            ∑_k∈ Q^k^2(-(t_1t_2)^1/2)^k
            /(
                    ∑_k∈+1/2 Q^2k^2 (-t_1)^-k)
                (
                    ∑_k∈+1/2 Q^2k^2 (-t_2)^k
                )



    The proof is essentially the same as that of <cit.>, but we include it for completeness. 
    By definition we have 
    
    _p,y(^2; t_1^-1,t_2)
            =
            y^-1Θ_p,y(t_1^-1)
            Θ_p,y(t_2)
            =
            θ(p, y t_1)/θ(p, t_1)θ(p, y t_2^-1)/θ(p, t_2^-1),

    and considering the denominator we have an expansion
    
    1/θ(p, t_1)θ(p, t_2^-1)
            =
            q^-1/4t_1^-1/2/1-t_1^-1t_2^1/2/1-t_2∑_i≥ 0δ_i(t_1,t_2) p^i

    for some δ_i(t_1,t_2)∈[t_1^± 1,t_2^± 1]. 
    The numerator can be expanded using the Jacobi triple product formula to give
    
    θ(p, y t_1)
            θ(p, y t_2^-1)
               
            =
            ∑_n,m∈
            p^1/2(n+1/2)^2+1/2(m+1/2)^2
            (-yt_1)^n+1/2
            (-yt_2^-1)^m+1/2
       
            =
            l,b∈
     l≡ b+1 mod 2
            p^1/4(l^2+b^2-1)
            y^l
            (-t_1)^1/2(l+b)
            (-t_2)^-1/2(l-b)
       
            =
            l,b∈
     l≡ b+1 mod 2
            p^1/4(l^2+b^2-1)(y  t_1^1/2/t_2^1/2)^l( -(t_1 t_2)^1/2)^b

    where we have used the change of variables l= m+n+1 and b=n-m. 
    We then obtain the expression for _p,y(^2; t_1^-1,t_2)
    
    q^-1/4t_1^-1/2/1-t_1^-1t_2^1/2/1-t_2i ≥0,  l,b∈
     l≡ b+1 mod 2δ_i(t_1,t_2)
              
            p^i+1/4(l^2+b^2-1)(y  t_1^1/2/t_2^1/2)^l( -(t_1 t_2)^1/2)^b,

    and observe that the terms q^n(y  t_1^_1/2t_2^_-1/2)^l occur when n = i+1/4(l^2+b^2-1).
    Thus they occur when 4n-l^2 = 4i+b^2 -1 ≥ -1, and the coefficient of q^n(y  t_1^_1/2t_2^_-1/2)^l depends only on this value, and is zero when this value is less than -1. 
    
    In summary 
    
    Coeff_p^m (y  t_1^_1/2t_2^_-1/2)^l[ _p,y(^2;t_1^-1,t_2) ]
            =
            ∑_k_1,k_2∈1/2
            b(4m-l^2,k_1,k_2)  t_1^-k_1 t_2^k_2

    where 
    
    ∑_a≥ -1∑_k_1,k_2∈1/2
            b(a,k_1,k_2)  t_1^-k_1 t_2^k_2   
            =
            -
            t_1^-1/2/1-t_1^-1t_2^1/2/1-t_2∑_i≥ 0∑_b∈
            Q^4i+b^2-1( -(t_1 t_2)^1/2)^bδ_i(t_1,t_2)
            
       
            =
            -Q^-1t_1^-1/2/1-t_1^-1t_2^1/2/1-t_2∑_i≥ 0δ_i(t_1,t_2)
            Q^4i∑_b∈
            Q^b^2( -(t_1 t_2)^1/2)^b
       
            =
            θ_1(Q^4,t_1)^-1θ_1(Q^4,t_2^-1)^-1∑_b∈
            Q^b^2( -(t_1 t_2)^1/2)^b




    
    The coefficients b(a,k_1,k_2) for a = -1,0 are given by
    
    b(-1,k_1,k_2) 
               
            =
            {[        -1          ;         0 therwise. ].
            
    
            b(0,k_1,k_2) 
               
            =
            {[ 1  ; 2  ; 0 . ].



    Expanding in t_1^-1 and t_2, the result follows immediately from 
    
    ∑_k_1,k_2 ∈1/2 b(-1,k_1,k_2)
            =  t_1^-1/2 t_2^1/2/(1-t_1^-1)(1-t_2),
        ∑_k_1,k_2 ∈1/2 b(0,k_1,k_2)
            = - 1+t_1^-1t_2/(1-t_1^-1)(1-t_2).





Now applying Lemma <ref> and considering the variables

    d_A = n, 
        
        d_B = m+n+l,
        
        d_C = m

with 
d = (d_A,d_B,d_C) := 2d_A d_B + 2d_A d_C + 2d_B d_C - d_A^2 - d_B^2 - d_C^2 = 4nm-l^2
gives

    Z^𝔻𝕋/_0^𝔻𝕋
        =
            ∏_n≥ 0∏_m>0, l∈∏_k_1,k_2∈1/2(1 - (-Q_A_0)^n (-Q_B)^m+n+l (-Q_C)^m v^-k_1 u^k_2)^-b(4nm-l^2,k_1,k_2)
    
        =
            ∏_d_C>0∏_ d_A, d_B ≥ 0∏_k_1,k_2∈1/2(1 - (-Q_A_0)^d_A (-Q_B)^d_B (-Q_C)^d_C v^-k_1 u^k_2)^-b(d,k_1,k_2).

Note that, a priori we must take the product over all d_B∈, however as pointed out in <cit.> when d_B<0 and d_C>0 we have d < -1 meaning these factors don't contribute. 

Consider the case when d_C=0. In this case we have
d = -(d_B - d_A)^2,
for which b(d,k_1,k_2) is zero unless d∈{-1,0}. Now by Corollary <ref> the terms where d=0 are:

    M(u^-1;u,v)^r·
        M(v^-1;u,v)^r
       
        =
        ∏_ d_A = d_B = 0k_1,k_2∈1/2
    (1 - (-Q_A_0)^d_A (-Q_B)^d_B (-Q_C)^d_C v^-k_1 u^k_2)^-b(d,k_1,k_2).



    ∏_N>0  
            M(Q_A_0^NQ_B^N u^-1; u,v)
            M(Q_A_0^NQ_B^N v^-1; u,v)
        /
            (1-Q_A_0^NQ_B^N)
        
       
        =
        ∏_ d_A = d_B > 0∏_k_1,k_2∈1/2(1 - (-Q_A_0)^d_A (-Q_B)^d_B (-Q_C)^d_C v^-k_1 u^k_2)^-b(d,k_1,k_2).

and the terms where d=-1 are:

    ∏_N>01/M(-Q_A_0^N-1Q_B^N  (uv)^-1/2; u,v) M(-Q_A_0^NQ_B^N-1 (uv)^-1/2; u,v)
       
        =
         d_A, d_B ≥ 0 
    ∏_k_1,k_2∈1/2(1 - (-Q_A_0)^d_A (-Q_B)^d_B (-Q_C)^d_C v^-k_1 u^k_2)^-b(d,k_1,k_2).


This completes the proof of Corollary <ref>. 




§ ACKNOWLEDGEMENTS


The author wishes to thank Noah Arbesfeld, Pierre Descombes, Martijn Kool, Nikolas Kuhn and Georg Oberdieck for very useful conversations which contributed to this article. 



amsalpha

