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Vertex Operators and Moduli Spaces of Sheaves

The Nekrasov partition function in supersymmetric quantum gauge theory is mathematically formulated as an equivariant integral over certain moduli spaces of sheaves on a complex surface. In ``Seiberg-Witten Theory and Random Partitions'', Nekrasov and Okounkov studied these integrals using the representation theory of ``vertex operators'' and the infinite wedge representation. Many of these operators arise naturally from correspondences on the moduli spaces, such as Nakajima's Heisenberg operators, and Grojnowski's vertex operators. In this paper, we build a new vertex operator out of the Chern class of a vector bundle on a pair of moduli spaces. This operator has the advantage that it connects to the partition function by definition. It also incorporates the canonical class of the surface, whereas many other studies assume that the class vanishes. When the moduli space is the Hilbert scheme, we present an explicit expression in the Nakajima operators, and the resulting combinatorial identities. We then apply the vertex operator to the above integrals. In agreeable cases, the commutation properties of the vertex operator result in modularity properties of the partition function and related correlation functions. We present examples in which the integrals are completely determined by their modularity, and their first few values.


Introduction
The original motivation for this paper is a collection of (equivariant) integrals on the moduli space of framed torsion-free sheaves on P 2 .
These functions are the Nekrasov partition functions, and the related correlation functions, defined below. In [27], Okounkov and Nekrasov showed that they may be studied using the representation theory of the infinite wedge representation and vertex operators. The vertex operators are auxilliary in the sense that they are not directly related to the partition functions; they are tools for performing the integration.
The vertex operator is the central object in this paper. In fact, it is as important to us as the original integrals. We present a geometric (in fact, K-theoretic) construction of the vertex operator, with the original intent of generalizing the above methods to other surfaces. It's geometric definition is more general, and is therefore a better logical starting point for this paper than its representation-theoretic definition. Therefore, we begin by constructing the vertex operator, and present it's applications to the partition and correlation functions second.

Hilbert scheme structures
Given a smooth quasi-projective surface S, let S [n] = Hilb n (S) denote the Hilbert scheme of n points on S. Assume that if S is not projective, it carries an action of a complex torus T with compact fixed loci, inducing such an action on S [n] . The point here is that integration make sense on S [n] , either in the usual sense for S projective, or in the sense of localization for S equivariant. Let T m S [n] denote the tangent bundle to S [n] together with an action of C × by scaling the fibers. We consider the generating functions so the Euler class term serves to fill out the remaining degrees necessary to give a top-dimensional cohomology class with the suitable Chern class of the tangent bundle. Given that k 1 , ..., k N are fixed, filling out the remaining terms with the tangent bundle is necessary for a generating function with more than one nonzero term if S does not have a torus action.
Combining the correlation functions in a generating functions is crucial. On S [n] for fixed n, operations such as the cup product are difficult to understand. However, when the Hilbert schemes ranging over all n are considered together, the cohomologies inherit richer structures. That is the strategy we employ for studying the generating function F : each coefficient [q n ]F (q) is difficult to understand in a meaningful way. The generating function, on the other hand, admits a construction by canonical operators on Ideally, commutation relations between such operators lead to properties of F as a function of the generating variable q.
The first of these structures are the Nakajima operators. For each cohomology class c ∈ H * (S, C), Nakajima defines an operator α k (c) : H n → H n−k , by constructing a family of canonical correspondences defined below. They collectively satisfy the relation [α k (c), α l (c )] = kδ k,−l c, c S .
In addition to the commutation relation above, they have the advantage that H is an irreducible representation of the Heisenberg algebra they generate.
Independently, Grojnowski constructed a different set of correspondences on the Moduli spaces of torsion-free sheaves [10] on S. Let M c 1 ,ch 2 be the space of those sheaves F with rank 1, c 1 (F ) = c 1 , ch 2 (F ) = ch 2 . Then M 0,n coincides with the Hilbert scheme, and more generally, we have an identification M c 1 ,ch 2 ∼ = S [n] where n = c 1 , c 1 /2 − ch 2 (see Nakajima [26], chapter 9). In this formulation, the bi-graded vector space inherits the extra structure of a vertex algebra. We do not discuss vertex algebras here, but remark that this structure gives operators for each cohomology class x ∈ V . C((z)) is the ring of formal Laurent series in z. Each operator admits an expression in the Nakajima operators, and isomorphisms Q m : V c,n → V c+m,n using the identification with the Hilbert scheme. Conversely, the Nakajima operators may be reconstructed from the vertex operators, so in some sense the two approaches are equivalent. However, the connection between the vertex operator Y (x, z) and our vertex operator is still mysterious. On one hand, the input into our operator is the first Chern class of a line bundle L on S. If S = C 2 with the torus action (2), and we specialize x = 1 ∈ H 0 (M c 1 (L),c 2 1 /2 ), the two operators actually agree up to the operator Q. On the other hand for general surfaces (for instance, any surface with nontrivial canonical bundle K), our operator differs from the vertex operators Y (x, z) by a c 1 (K) term. This causes asymmetric commutation relations with the Heisenberg operators α k that are not present in actual vertex algebras.

A Prototypical situation
As we pointed out, one desires that the correlation functions (1) admit an expression in terms of some of the natural operators on the Hilbert scheme. In an ideal situation, properties (commutation relations) of the operators lead to functional properties of F in the generating variable q. We show in theorem 2, that the setup with is such a situation. The calculation proceeds as follows. The operator H n → H n given by x → x ∪ ch n (O/I) admits is expressible in the vertex operators mentioned above. Unfortunately, there is no such expression for the operator x → x ∪ e(T m S [n] ). But, if |µ is the element corresponding 1 in H µ , and x, y = x ∪ y, the matrix element is the matrix element Q −µ Y (|µ , 1)x, x . As a result, the correlation function takes the alternative form where d multiplies H n by n. The point is that the operator Y (|µ , 1) has off-diagonal terms (indeed, terms between different Hilbert schemes), though they do not affect the trace. However, we must include them if we are to exploit the commutation relations of Y (|µ , 1). The restriction to m = µt where µ ∈ Z is harmless since the correlation function is a polynomial in m for fixed n. Furthermore, the vertex operator as constructed in this paper is defined for all m ∈ C.
As promised, the expression of F as a trace of canonical operators on H leads to a functional properties of the correlation functions. We prove in theorem 2 that, as a function of q, F (k 1 , ..., k N ; m, q) is a quasimodular form in of a certain weight depending on k j . As a result, one may determine an integral over S [n] by simply determining which quasimodular form represents F , and then extracting the coefficient of q n . This type of calculation is the prototype for all others of interest to us. Essentially every construction in this paper is designed with this in mind.

Outline
With the model scenario above as motivation, we outline the format of this paper. Section 2 summarizes of all necessary background on the Hilbert scheme. This includes the Nakajima operators in both the equivariant and non-equivariant case. It also includes the relationship between the Nakajima operators, and the fixed point basis of equivariant cohomology. We give this description when S = C 2 with the action of the 2-torus T 2 ⊂ GL (2).
Section 3 describes the correlation functions in detail, and motivates a geometric definition of the vertex operator, W , the central object of this paper. This definition has several advantages: 1. W incorporates the canonical class of S, where other studies assume it vanishes. This leads to an asymmetry which is not present in standard vertex operators. (3) is automatically a matrix element of W for any S. This follows directly from definition, not a calculation.

The coefficient
3. m need not be integral.
4. W is defined as a characteristic class of a vector bundle. Therefore it is likely that theorem 1 generalizes to an operator on K-theory.
Section 3 concludes with our main result, which presents W as an expression Γ in the Nakajima operators. This expression justifies the reference to W as a vertex operator. The proof of theorem 1 is the content of section 3.4. Section 4 applies W to the correlation functions for (2). We use the commutation relations among vertex operators to prove the quasimodularity result mentioned above (theorem 2). We give explicit combinatorial expressions for the correlation functions, and connect them with quasimodular forms in simple examples.
Finally, section 5 applies the vertex operator to the moduli space M(2, n) of framed torsion-free sheaves of rank r = 2 on P 2 . We show that W applies the (dual) partition function Z ∨ in the Nekrasov theory. We present a theorem describing Z ∨ as a vector-valued function in the generating variable q satisfying a twisted modularity property (theorem 3).
We will see that the partition function is considerably more complicated in higher rank. In fact, the proof of theorem 3 relies on the principle vertex operator construction for the affine Lie algebra sl 2 C. We do not yet have a geometric explanation for the appearance of sl 2 C, but it seems likely that a generalization of our operator connects with the sl 2 C action described by Licata [17], and the oscillator action described by Baranovsky [2]. We present examples for certain values of the parameter m for which Z ∨ may be classified by regular modular forms.

Cohomology of the Hilbert Scheme
Let S be a smooth quasi-projective surface, possibly together with an action of a complex torus, T = (C × ) d . S [n] is a complex manifold of dimension 2n by Fogarty's theorem. Furthermore, the action of T on S induces an action on S [n] , which has isolated fixed points if S does. Thus we may consider the cohomology groups H * T (S [n] , C). We always consider cohomologies over C in this paper, even though much of what follows makes sense over Q.
It is well understood that it is best to study the cohomologies of S [n] for n ≥ 0 simultaneously instead of fixing n. This is foreshadowed by the fact that the Betti numbers are best expressed as a generating function, called Göttsche's formula [9]: However, the real reason is that the vector space inherits extra structure, the Nakajima operators. It is shown that [Z] has dimension 2n + k + 1, and possesses a welldefined fundamental class

Nakajima's operators
. Given a cohomology class c ∈ H * T (S), the Nakajima operator is defined by where p 3! denotes the Gysin homomorphism for the proper map p 3 : For k > 0, we also define where the dual operator is with respect to the inner product on H given by x, y = S [n] x ∪ y. If S is compact, this takes H to itself. If S is noncompact with a group action, it takes F = H ⊗ R F to itself, where R = H * T (pt) = C[Lie(T ) * ], and F = C(Lie(T ) * ), the fraction field of R.
Remark Notice that this definition depends on the inner-product being defined, which is acceptable since we assumed that integration makes sense on our surface. However, it is possible to define the Nakajima operator for k > 0 if this is not the case by taking c in cohomology with compact support. This would be necessary, for instance, if we considered S = C 2 with no group action.
Nakajima's main result [23] is that Here the bracket is the supercommutator with respect to the grading given by deg(c). If S has no odd cohomology, it reduces to a normal commutator. One can also express this theorem by saying that H is a module over the Heisenberg Lie algebra with generators α k (c). Furthermore, using Göttsche's formula, one can prove that this module is irreducible. Therefore, if |0 represents the element 1 ∈ H 0 (S [0] ) ⊂ H, we obtain a basis of H by successive applications of α −k (c) to |0 .

Equivariant cohomology of the Hilbert scheme
Suppose now that S is equivariant with isolated fixed points, though not necessarily compact. The action of T on S induces an action on the Hilbert scheme S [n] , also having isolated fixed points. In this case, we have another canonical basis for H, the fixed-point basis for equivariant cohomology. Since T acts on S [n] with isolated fixed points, localization gives an injective map of R-modules, . Tensoring over R with F , this map becomes an isomorphism. Thus, to describe a basis of F = F ⊗ R H, we need to determine the fixed points of S [n] .
We describe the fixed points in the setup A fixed point of (C 2 ) [n] as a manifold is given by a closed point of (C 2 ) [n] as a scheme. Using the functor of points, these are in bijection with ideals of I ⊂ C[x, y] such that dim C C[x, y]/I = n, and I is invariant under T . It is not difficult to see that invariant ideals are just those that are generated by monomials x r y s . Such ideals are indexed by partitions λ = (λ 1 , λ 2 , ...) of n, the correspondence being is the inclusion of the fixed point I λ . If S is not C 2 but has isolated fixed points, we can restrict to the compact torus S 1 × · · · × S 1 ⊂ C × × · · · × C × , and find disjoint equivariant neighborhoods U s of each fixed point s ∈ S. Let U by taking the union of subschemes of S with disjoint support. By the localization theorem [1], this induces an isomorphism This map is easy to describe in the fixed point basis: fixed points of S [n] are indexed by assigning a partition µ s to each fixed point s ∈ S, describing the ideal sheaf I µs inserted at s. The image of this point is just s [I µs ].

Fixed points and Jack polynomials
We now describe the relationship between the fixed-point basis and the Nakajima basis in setup (8). For more details we refer to Wang, Li and Qin [19], though we use the normalizations of Okounkov and Pandharipande [29].
Let α −k = α −k (1), and α µ = i α −µ i for µ a partition. The the elements α µ |0 form a basis of F over F . In fact, as F -vector spaces. The obvious ring structure is unrelated to the cup product ring structure on F, but the Nakajima operators become Notice that, like the fixed-point basis, the Nakajima basis is also indexed by partitions.
The standard way to express the fixed point basis in terms of the Nakajima basis uses the isomorphism where Λ * is the ring of symmetric polynomials in infinitely many variables, and p µ = k p µ k , and p k = i x k i . The connection between F and symmetric polynomials is deeper than the role it plays in this paper (see Haiman [11]); for now, it merely provides a convenient combinatorial description of the change-of-basis matrix.
Under this isomorphism, the inner product x ∪ y takes the form where Here P (θ) is the integral form of the Jack polynomial [20], and we have normalizedJ µ so that J µ , p |µ| 1 = |µ|!. The reason for this normalization is that the element 1 If S is not C 2 , the Nakajima operators can be easily understood in the decomposition (9) over F . Given positive integers m s , n s with m s = m, n s = n, the restriction of Z m,m+k to s U

The Vertex Operator
In this section we consider integrals of Chern classes of tautological bundles on S [n] , where S is a smooth quasi-projective surface, possibly equipped with an action of a complex torus. If S is not projective, neither is S [n] , and we must consider equivariant integrals with respect to a torus action. We present a strategy for studying these integrals that motivates the introduction of the vertex operator, the centerpiece of this paper.

Strategy
Consider the integral Here c is a cohomology class, T m X is the tangent bundle to X taken as an equivariant vector bundle with respect to an action of C × acting trivially on X, but scaling the fibers of T X. e(T m S [n] ) is the equivariant Euler class of this equivariant bundle, and the parameter here is the number m ∈ Lie(C × ), hence the subscript m. If c = 1 and S is as in (2), this is one of Nekrasov's deformed partition functions, reviewed below. As pointed out in the introduction, it is also a tidy way to organize numerical invariants of S [n] : So if S is projective and c has cohomological degree k, the coefficient of [m k ] fills out what is left to give a top degree class. This means the integral makes sense, and yields a number associated to each c.
We summarize the strategy in 3 steps: Strategy 3.1 1. Make a wise choice of an element of H ⊗ H that pulls back to e(T m S [n] ) under ∆ * nn . 2. Using the standard inner product on cohomology a, b = a∪b, this is the same as an operator W : H → H. It has the property that A wise choice of W admits an explicit expression in the Nakajima operators.
3. Use the Nakajima commutation relations to study the trace. Alternatively, compute the trace in the Nakajima basis.
W is the vertex operator.

The Main theorem
In fact, the vertex operator we construct is slightly more general. It is well-known that the fiber of T S [n] at a point corresponding to an ideal I is given by Here and this is an isomorphism of virtual vector spaces. If S is compact, this is an alternating sum of finite-dimensional vector spaces, and we expect the virtual dimension to be 2n = dim(S [n] ). If S is noncompact with a torus action fixing I, this represents a difference of infinite vector spaces with finite-dimensional torus eigenspaces. Therefore the subtractions are meaningful, and in fact we obtain the tangent space to I as a virtual representation of T , not just a virtual vector space. Given a line bundle L on S, we define an element If L is a trivial bundle with an action of C × , we recover the class of T m S [n] .
over a pair (I, J) of ideal sheaves on S. To be precise, let I, J be the canonical ideal sheaves on S [k] × S and S [l] × S respectively, given by the functor of points. Then define The coordinate z is necessary if we insist on defining H as a direct sum, but later we make use of its matrix elements when z is assigned a numerical value. As pointed out, if L is the trivial bundle with an action of u ∈ C × by scaling, ∆ * nn c 2n (E nn ) = e(∆ * E nn ) = e(T m S [n] ). Thus, c k+l (E kl ) meets the requirements of the cohomology class mentioned in the last subsection. The first step in strategy (3.1) is then complete. The second is the content of our main theorem: where Remark Serre duality on S implies that which is immediately verified for Γ(L, z), using (6).
Remark The theorem for W (L, z) or W (L, z) * applied to |0 is already known. In fact, it is a special case of a formula of Lehn: see [16], theorem 4.6.

A Special case
Before proving theorem, we demonstrate the statement in the setup (8). To do this, we need to calculate the matrix elements (W (L)[I µ ], [I λ ]) in the fixed point basis. W (L) is defined as a characteristic class, so by the Atiyah-Bott localization formula and (17), this is Where e(V ) is the product of the weights λ ∈ Lie(T ) * of a representation V . To compute this coefficient, we need the character of where a µ ( ), l µ ( ) denote the (possibly negative) arm and leg length in µ.
Proof. By localization and Grothendieck-Riemann-Roch for the equivariant map p : where i 0 is the inclusion of 0 ∈ C 2 , check denotes the dual, and Now observe that the terms for which the exponent of z 2 is ≤ 0 occur only in the first sum and correspond to the first sum in (21). The remaining terms may be determined using Serre duality which implies The theorem now takes the purely combinatorial form The expression in (23) is (20) using lemma 1.
Example µ = (2), λ = (1, 1). We haveJ Each term in the inner product in (23) is Taking the inner product (13) gives As a warm up, we can now prove a special case of the theorem: Proof. It is obvious from the definition of Z n,n+k that α −n [ Since only ∅ and [1] are contained in both µ and λ. The cycle Z n,n+1 is nonsingular for all n, so the operator α −1 is easy to understand in the fixed-point basis. We could either use an explicit formula for the tangent bundle to Z n,n+1 [6], or the Pieri rule for Jack polynomials [20]. The elements Γ − ( , z)|0 , Γ + ( , z) * |0 are Chern classes of bundles, so they also have explicit expressions in the fixed-point basis: From localization, Putting these together gives Adding up each term yields which agrees with (23).

Proof of the main theorem
Before proving theorem 1, we prove two lemmas that reduce our considerations to setup (8).
Lemma 3. If theorem 1 holds in the case where S has a torus action with isolated fixed points, it holds for all smooth quasiprojective surfaces with compact fixed loci (and therefore all smooth, projective varieties).
Proof. We prove the equality of W (L) and Γ(L) in the general case by looking at a suitable class of matrix elements of each. As before, let I n on S [n] × S denote the canonical ideal sheaf provided by the functor of points. Given c ∈ H j (S), let It is proved, for instance in [18] that polynomials in these classes span H * (S [n] ), as a vector space. Therefore, it suffices to check the equality on the matrix elements where p i , q i are some integers. The order in the product here matters up to a change in sign, since the cup product is supercommutative.  (24) is the integral over with each S i a different copy of S. The method of Ellingsüd, Göttsche, and Lehn ([7], Proposition 3.1), shows that this becomes the integral of another polynomialf depending only on f on S × S 1 × · · · × S k+l+Nη+N ξ in the variables This statement is also clearly true when we replace W with Γ in (24). Therefore, to check the equality W (L) = Γ(L), it suffices to verify the equality of the polynomials in either case. Checking the identity for every toric surface is enough to determine the coefficients. If there were two polynomialsf 1 ,f 2 , then Let T act on S with isolated fixed points.
Lemma 4. In this decomposition, (1) follows from the naturality property of characteristic classes.
From these two lemmas, it is sufficient to prove the theorem in the setup (8). However, in the proof, we still appeal to other surfaces. W (L) is defined in terms of a localization integral, which is equal to an honest integral in the case where S is projective. This fact by itself imposes relations on W (L) which lead to an induction step.
Proof. By lemmas 3 and 4, it suffices to prove the theorem for (8). However, we will make use of relations on W (L) coming from the fact that equivariant integrals of top degree classes on compact spaces are just numbers that correspond to normal integrals. Consider the following setup: where the group action on Tot(L) ∼ = P 1 × P 1 × C is (z 1 , z 2 , u) · (x 1 , x 2 ; s) = (z 1 x 1 , z 2 x 2 ; us).
On P 1 × P 1 we have equivariant cycles Also let U ab ≡ C 2 be the T -equivariant chart of P 1 × P 1 by containing the point a, b ∈ {0, ∞} Given partitions µ and ν such that |µ| = k and |ν| = l, define and similar quantities g µν , g [ab] µν replacing W with Γ. Notice that w µν may be computed either as a non-equivariant integral, or using localization. To compute it using localization, we must make a choice of L 0 versus L ∞ for each occurence of L in the expression (26), though the answer is independent of this choice. This corresponds to a de- . Using all three parts of lemma 4 we arrive at an expression relating w µν and w [ab] µν , where the sum is over terms such that . The same relation holds replacing w with g. We can now prove by induction on (µ), (ν) that w [ab] µν . As noted above, the case w µν proves the theorem. Assume that (µ) > 1 or (ν) > 1, and that w µ ν which also holds for g. By induction, we are done.
What remains are the base cases µ = ∅, ν = ∅, and ( (µ), (ν)) = (1, 1). As noted in the above remark, the first two are proved by Lehn's theorem. As for the third, it is in fact enough to demonstrate the claim between any collection of elements (ξ k , η l ) ∈ S [k] × S [l] that have nonzero inner product with α −k |0 and α −l |0 . This is done in lemma 2.

Correlation Functions and Quasimodularity
In this section, we investigate the aforementioned integrals on the Hilbert scheme. We remain true to our strategy, and compute these integrals as traces in the Nakajima basis. We eventually specialize to setup (2) mentioned in the introduction. We then prove that the correlation functions are quasimodular forms, and show how this can be used to compute them.

The Partition function
To begin, suppose that the cohomology class c that we are integrating against is 1. We use the following lemma, Lemma 5. We have the following commutation relations between vertex operators: Some explanation is necessary. First, if x, y are viewed as formal variables, the expression (1 − y x ) a must be taken as a formal series in y x . If we assign values to x and y, the operator Γ ± (L) does not take values in F. However, if we are interested the trace of this operator, we only need its matrix elements. Matrix elements make perfect sense for numerical choices of the variables as long as x > y in the lemma. The proof of the lemma is obvious from the Nakajima commutation relations.
We may now calculate one of our integrals. Let F n be the graded component F ⊗ R H * (S [n] ) in F, and d the operator that multiplies F n by n. Then for |q| < 1,

Correlation functions
Next, we turn to more general correlation functions, in setup (2). This is equivalent to making the specialization t 1 = −t 2 = t in (8). We see that the vertex operator depends only on µ = m/t, so we may further assume t = 1.
Under this specialization, the relationship between the fixed-point basis and the Nakajima basis become much simpler: On the symmetric function side, when t 1 = −t 2 = 1 the Jack polynomials specialize to Schur polynomials. On the vertex operator side, the operator Γ(L, z) simplifies because the canonical bundle now becomes trivial equivariantly (hence the term K disappears). Also, the operator c ∪ on cohomology admits a simple expression in the vertex operators. This also stems from the vanishing of K.
The combinatorics of this situation can be described by that of the charge zero infinite wedge representation. The description of the infinite wedge representation is standard (see, for instance Kaç [13]), but we recall it here. Let V be the vector space with basis v n , n ∈ Z. The infinite wedge space ∞ V is the vector space with basis such that i 1 > i 2 > ..., and such that for n >> 0, i n = n + m − 1. The integer m is called the charge of v I . The charge induces a grading on ∞ V , and the charge m component is labeled Finally let α 0 be the operator that multiplies ∞ m V by m. Each component has a basis indexed by partitions: ∞ V has operators ψ n , ψ * n given by These operators generate a Clifford algebra with commutation relations over which ∞ V forms a representation called the infinite wedge representation. In fact, ∞ V is generated by applying the elements ψ n+1 , ψ * −n , n ≥ 0 to the vacuum vectors |m = v m ∧ v m−1 ∧ · · · . We are now in a position to describe the combinatorics of F C 2 with the above torus action. Let Φ : F → ∞ 0 V be the isomorphism sending where h( ) = a µ ( ) + l µ ( ) + 1, the hook length of . Recall that α k = (−1) k+1 α * −k . Thanks to the hook length normalization above, the inner product transports to v µ , v λ = δ µ,λ .

Correlation functions and the vertex operator
As pointed out, This representation of F has the advantage that the operator c ∪ takes a simple form. We now make use of this fact, and compute the desired integrals. Specifically, we consider (35) This is an equivariant integral, and such take values in the dual of Lie(T ). We evaluate this quantity at (t = 1, m), yielding a function of q, m for each (k 1 , ..., k N ).
We can give an explicit combinatorial expression for F using the localization formula. The vector bundle O/I at a fixed point I µ has torus character (i,j)∈µ e (j−i)t . Therefore The weights of the C × action on T S [n] are given by {h( ), −h( )}| ∈µ , so the integral may be evaluated by localization: (36) We represent the operator of cup-product against a ch(O/I) in the infinite wedge picture. The trick is to write ch(H 0 (O/I)) = ch(H 0 (O)) − ch(H 0 (I)), and express ch(I) ∪ in the vertex operators. Then Proof. The operator [y 0 ]ψ(xy)ψ * (y) is the same as which converges for |x| > 1. In this range, we get Thus, it suffices to consider the auxilliary functions As pointed out before, for fixed numerical values of x i , y i , this is technically not an operator on ∞ 0 , which is defined as a direct sum. However, the matrix elements of this operator make perfect sense as long as Furthermore, the trace converges as long as q is sufficiently small. In this case, it is enough that |x 1 y 1 | < |1/q|. We now complete the final step of strategy 3.1, and compute G in this range.
Using the boson-fermion correspondence, theorem 1, and the method of subsection 4.2, Here we have repeatedly cycled around the trace as in subsection 4.2. The constraint (39) gives us the conditions we require for the commutation relations in lemma 5. We can repeat this process on Γ + (x 1 y 1 ) −1 , Γ − (y 1 ) −1 , Γ + (y 1 ), with the difference that the Γ + terms must move to the right, whereas the Γ − terms move left. Using the theta function θ(x; q) = (xq; q) ∞ (x −1 ; q) ∞ (q; q) −2 ∞ , we arrive at Here Z(m, q) = G(∅; ∅; m, q), which was calculated in the last subsection. Taking t = 1, we get This turns into the following proposition about F , the original correlation functions: Proposition 2. Let x k = e 2πiz k , y k = e 2πiw k , θ(z k ; q) = θ(x k ; q). Then F (k 1 , ..., k n ; m, q) = r 1 i+1 where −1 < r 1 < · · · < r N < 0, 0 < i < r i+1 − r i , the second F is given recursively by and G is the expression in (40).

Quasimodularity of the correlation Functions
If we know the coefficient of z k in θ(z; q) = θ(x; q) for all k, proposition 2 gives an explicit expression for F (k 1 , ..., k N ; m, q). However, a concise classification for these functions is preferable to a complicated explicit expression. In this subsection, we give such a classification that lands F in a finite-dimensional vector space of functions of q, whose dimension depends on k j . Specifically, we prove that F is a quasimodular form in the variable q.
Before stating the theorem, we classify the coefficients of θ(z; q). Let the classical Eisenstein series, for k ≥ 1. Here B 2k are the Bernoulli numbers, where a k (q) ∈ C[E 2 , E 4 , E 6 ]. Furthermore, if E 2k has weight 2k, then a k (q) has weight 2k − 1.
Remark An element of C[E 2 , E 4 , E 6 ] is called a quasimodular form (for the full modular group Γ). Quasimodular forms were introduced in [14]. We do not develop their theory here, since we only use them as a tool for classifying the coefficients of θ(z; q). They contain the set of Modular forms, but also contain the element E 2 , which has the property Theorem 2. As a function of q, F (k 1 , ..., k N ; m, q)/Z(m, q) is a quasimodular form of weight ≤ 2N + i k i . As a function of m, it is a polynomial of degree 2N + 2 i k i 2 . Proof. If R is a graded algebra, we get a grading on R((z)) by giving z weight −1. For instance, if R = C[E 2 , E 4 , E 6 ], then the power series of θ(z; q) about z = 0 has weight −1 since the coefficient of z k has weight k − 1. On the other hand, θ(z, q) −1 has weight 1. Let Here X = B, C, and ABC is the expression in (40). We start by proving that the power series of (BC) in z j about 0 is an expression of weight 0 in the algebra In particular, we notice that BC itself is a quotient f /g of holomorphic functions in z, w of the same weight. We show that in the case m ∈ Z, taking the integral over w j reduces to a residue calculation, each one raising wt(f ) − wt(g) by 1. To do this, notice that by the theta relation θ(z + τ, τ ) = e −2πi(z−τ ) θ(z, τ ), we find that BC → e −2πimz 1 BC under w 1 → w 1 + τ . Then where is the parallelogram with vertices ir 1 , ir 1 +1, ir 1 +1+τ, ir 1 +τ . If m is an integer, this becomes a simple residue calculation. It is easy to see that taking the residue over w 1 yields a quotient f /g in the remaining variables such that wt(f ) = wt(g) − 1. By induction, it follows that where D is a quotient f /g such that wt(f ) − wt(g) = −N . Since the product has inhomogeneous weight ≤ N and (BC) is holomorphic near z i = 0, it must be inhomogeneous of weight ≤ 0. The remaining term A has weight N , each term (1−e 2πiz i ) −1 appearing in proposition 2 has weight −1, and differentiating raises the weight by 1. Therefore, F (k 1 , ..., k N ; m, q) is quasimodular of weight ≤ 2N +k 1 +...+k N when m is an integer. It remains to show that F is a polynomial in m. This also eliminates the assumption that m is integral, since any polynomial which is 0 at every integer is 0. To do this, check that the coefficient n vanishes to order n in z 1 , ..., z N . In other words, taking fewer than n derivatives total in any of the variables z 1 , ..., z N and evaluating at z j = 0 gives zero identically in y j , τ . The expression B is holomorphic near z i = 0, and has a pole of order 2 at z i = 0, and only even terms in the Laurent expansion. This explains the funny expression for the degree as a polynomial in m.

The Moduli of Sheaves
We start with some very brief motivation from gauge theory, then move on to study the partition function for sheaves of rank 2. This material is located in [28].

Instantons and the moduli space
In quantum gauge theory, one seeks to integrate over the space of gauge fields, i.e. rank-r vector bundles on a Riemannian 4-manifold S together with a connection, modulo the action of the group of gauge symmetries. The integrand may be a function of the form exp(−βS[A]), where S is some energy functional of a connection A.
If β is large, one may assume that the minimizers of S dominate, and reduce to an integral over the minimizers of S. One is then led to integrals over finite-dimensional smooth manifolds. The minimizers are called instantons. Specifically, let be the curvature of A, and where * is the Hodge star. The minimizers here are solutions to the Yang-Mills (anti) self-duality equation A rank-r instanton is a U (r) vector bundle E on M , together with a connection satisfying the self-duality or anti self-duality equation. The solutions are grouped by a discrete invariant, charge, The letter c 2 is used because each it represents the second Chern class of the bundle E. This is even so in the noncompact case M = R 4 , because instantons on R 4 extend to instantons on S 4 . The resulting vector bundle on S 4 may not be trivial, and in fact c 2 is the second Chern class of this bundle in H 4 (S 4 ) ∼ = Z. Solutions to the self duality equation have c 2 < 0, while the anti self duality solutions have c 2 > 0. One can switch between c 2 > 0 and c 2 < 0 by reversing the orientation of R 4 , so it is fair to assume instantons are anti-self-dual with c 2 ≥ 0. In Donaldson theory, charge n U (r) instantons on R 4 are identified with a moduli space Where E is a rank-r vector bundle on P 2 with c 2 (E) = n, is a framing, i.e. a particular trivialization of E| P 1 ∞ , and P 1 ∞ is the curve z = 0 in P 2 . This space has the advantage of a complex structure. Notice that c 1 (E) = 0, just by the existence of the framing.
However, it is still not compact. As a first step towards compactification we embed M(r, n) into the larger moduli space M(r, n) = (rank-r torsion-free sheaf F , Φ : . Example Let F be a rank 1 torsion-free sheaf with a framing. We have a canonical map F → F ∨∨ = O to the double dual, which must be an injection. This means F must come from an ideal sheaf on the complement of P 1 ∞ which is C 2 . Thus, M(1, n) = C 2 [n] , the Hilbert scheme on C 2 .
As this example shows, M(r, n) is still not compact. In the next subsection, we will see that we must again turn to equivariant integration.
We now give a rough description of what the integrand e(T m M ) means from the point of view of gauge theory. Suppose that we would like to integrate not just over the space of connections on a bundle E (or the above compactification), but over connections together with a section of E ∨ ⊗ E. In physics, this corresponds to a matter field interacting with the field A. For any fixed field, the literal space of sections is badly behaved from a mathematical point of view. A related but better behaved quantity is which is a virtual vector space that extends to sheaves F in the larger moduli space M(r, n).
The simplest thing we could integrate over this space of "sections" is the element 1 in cohomology. To make sense of such an integral, we regularize by adding a scaling action by u ∈ C × , and consider an equivariant integral. Localization the leads to the quantity which becomes rigorous after adding a group action to M(r, n), and regularizing the infinite-dimensional virtual bundle which is χ(F, F ) over each point F .
To salvage the integral, we write As we note in the next section, the term on the right is isomorphic to the tangent bundle to M(r, n). The term on the left is infinite, but uninteresting. Taking out it's contribution (called Z pert ), the integral that remains is M(r,n) When r = 1, this is the partition function from the last section.

The Nekrasov partition function
As for the Hilbert scheme, a torus action T P 2 induces a torus action on the moduli space. Unfortunately, this action does not have isolated or even compact fixed loci for r > 1. To resolve this problem, we follow [27] and enlarge the group to include the action of the group GL(r) on the choice of framing by composition on the right. In fact, it is enough to include the action of the standard torus T r ∈ GL(r).
We can now define the Nekrasov partition function. Consider the action where the second term acts as described, C × acts trivially, and the first T action is induced from the action (2) on C 2 ⊂ P 2 . The partition function is defined as a product of a factor (called Z pert ) times the series, Z inst (a 1 , ..., a r ; t; m, q) = n q rn M(r,n) e(T m M(r, n)).
We describe the factor below.
The parameters a j here lie in Lie(T r ), and m ∈ C × acts by scaling the tangent bundle as before. To give a combinatorial expression using localization, we must find the character of the tangent bundle to the fixed sheaves.
Example Let I, J be a pair of ideal sheaves such that O/I, O/J are zero dimensional and supported away from P 1 ∞ . Then F = I ⊕ J is a rank 2 torsion-free sheaf on P 2 . The map I, J → O restricted to P 1 ∞ gives an automatic framing. Furthermore, each I, J has a two-step resolution by vector bundles, which shows that ch 2 (F ) = c 2 (F ) = dim C (O/I) + dim C (O/J), and c 1 (F ) = ch 1 (F ) = 0. Claim 1. The fixed points of the action of T × T r on M(r, n) are the sheaves I µ [1] ⊕ · · · ⊕ I µ [r] .

The Dual partition function
It seems natural here to define the vertex operator analogously to the operator W (L, z) on the Hilbert scheme. While there are operators analogous to the Nakajima operators on cohomology, (cf. Licata [17] and Baranovsky [2]), their commutation relations are those of the affine Lie algebra sl 2 C, which are more complicated than those of the Heisenberg algebra. At any rate, it is unclear what should replace theorem 1.
Instead, we study the dual partition function, which is a reduction of Z to the rank 1 case. Among other things, this has the advantage that we may use the vertex operator W (m) as it is. We see that in the analysis of Z ∨ , the representation of sl 2 C on ∞ 0 V is critical. We have no geometric basis for introducing sl 2 C, but its appearance is perhaps not surprising in light of the sl 2 C action above. It seems likely that the original partition function may be studied using a generalization of W (m), and the natural sl 2 C action.
According to the relations above, and again assuming without loss of generality that t = 1, This is a sort of Fourier transform of the original function Z, with weights e ζ·b keeping track of the shift b of the partition µ, often referred to as the charge of µ (not the same the previous notion of charge). The remainder of the paper focuses on an analog of theorem 2 for this function.

Charge and the affine Lie algebra
To employ the vertex operator, we must represent the operator h 0 : where b is the charge of µ from the previous subsection. In fact, this operator can be expressed in terms of a natural representation of the affine lie algebra sl 2 C on ∞ 0 V . The representation can be found in Kaç [13], but we describe it briefly here. Let be basis vectors of sl 2 C. We obtain an action of the Lie algebra , h, f . Other than h 0 , each operator x j induces an operator on ∞ 0 V defined by h 0 , however, leads to an infinite sum. The solution is to subtract the infinite part of the sum: This gives a projective representation of C[t, t −1 ] ⊗ sl 2 C, which extends to an honest representation of sl 2 C by sending K → 1. In this representation, we see that where b(µ) is the charge as defined above. To understand Z ∨ We must understand the interaction between h 0 and the vertex operator.

The Principal vertex operator construction
Strategy (3.1) has allowed us to compute correlations only for operators on F expressed in the Nakajima basis. To understand Z, we must find such an expression for In fact, the entire action of sl 2 C admits such a description, known as the principal vertex operator construction [13]: Proposition 3. (principal vertex operator construction for sl 2 C). Let where Γ odd ± (z) = exp   n>0, n odd z ∓n n α ±n   .
The representation ∞ 0 V of sl 2 C is given in the Nakajima basis by This can be found in [13], chapter 14.

Vector-Valued Modularity
Now that we have enough vertex operators, let us attempt an analogue of theorem 2 for the Nekrasov partition function. We will see that this leads us to a modularity condition for vector-valued operators.

A Representation of a modular subgroup
We describe how f (z, τ ) m behaves under changes of the coordinate τ in terms of an action of a subgroup Γ ⊂ SL(2, Z) on the homology cycles of the Riemann surface R. Given any path c in C 2 avoiding the endpoints of the slits 1 4 + τ 2 + 1 2 Z + τ Z, we obtain a path in R. If we identify any cycle c + τ with e πim c, and make use of the moves shown in figure (2) Since Γ preserves the points 1 4 + τ 2 + 1 2 Z + τ Z via its action on R 2 ∼ = R · τ ⊕ R · 1 ∼ = C, we obtain a representation To write [A · γ ( ) ] as a linear combination of [γ], [γ ], we use the equivalence [γ ( ) + τ ] = e πim [γ ( ) ], and the homotopy equivalence in figure  2.
Making use of the move (2), we see that

An Example
Let us apply the theorem to compute Z ∨ when m is odd. The situation then greatly simplifies, since one can easily check that ρ m (A) is diagonal for A ∈ G. Therefore v k,m (A · τ ) 1 k = −1 (cτ + d) k v k,m (τ ) 1 k for A ∈ G. Since v k,m is periodic in τ , we may also drop the condition b ∼ = 2 (mod 4), and extend the group to all of Γ 1 (4). Since η(2τ ) 2 η(τ ) −4 satisfies the same modularity condition, Z ∨ k (m, q)/(q; q) m 2 −1 ∞ is modular of weight 0. We would like it to be a modular form, but it may have poles at the cusps of X 1 (4). Multiplying by suitable power of the η function takes care of the problem. For instance, we see that Z ∨ 2 (3, q) = −16q + 128q 5 − 320q 9 + 1120q 17 − 1024q 21 + ...