Instantons, Topological Strings, and Enumerative Geometry

Abstract

We review and elaborate on certain aspects of the connections between instanton counting in maximally supersymmetric gauge theories and the computation of enumerative invariants of smooth varieties. We study in detail three instances of gauge theories in six, four, and two dimensions which naturally arise in the context of topological string theory on certain noncompact threefolds. We describe how the instanton counting in these gauge theories is related to the computation of the entropy of supersymmetric black holes and how these results are related to wall-crossing properties of enumerative invariants such as Donaldson-Thomas and Gromov-Witten invariants. Some features of moduli spaces of torsion-free sheaves and the computation of their Euler characteristics are also elucidated.

1. Introduction

Topological theories in physics usually relate BPS quantities to geometrical invariants of the underlying manifolds on which the physical theory is defined. For the purposes of the present article, we will focus on two particular and well-known instances of this. The first is instanton counting in supersymmetric gauge theories in four dimensions, which gives the Seiberg-Witten and Donaldson-Witten invariants of four-manifolds. The second is topological string theory, which is related to the enumerative geometry of Calabi-Yau threefolds and computes, for example, Gromov-Witten invariants, Donaldson-Thomas invariants, Gopakumar-Vafa BPS invariants, and key aspects of Kontsevich's homological mirror symmetry conjecture.

From a physical perspective, these topological models are not simply of academic interest, but they also serve as exactly solvable systems which capture the physical content of certain sectors of more elaborate systems with local propagating degrees of freedom. Such is the case for the models we will consider in this paper, which are obtained as topological twists of a given physical theory. The topologically twisted theories describe the BPS sectors of physical models, and compute nonperturbative effects therein. For example, for certain supersymmetric charged black holes, the microscopic Bekenstein-Hawking-Wald entropy is computed by the Witten index of the relevant supersymmetric gauge theory. This is equivalent to the counting of stable BPS bound states of D-branes in the pertinent geometry, and is related to invariants of threefolds via the OSV conjecture [1].

From a mathematical perspective, we are interested in counting invariants associated to moduli spaces of coherent sheaves on a smooth complex projective variety 𝑋. To define such invariants, we need moduli spaces that are varieties rather than algebraic stacks. The standard method is to choose a polarization on 𝑋 and restrict attention to semistable sheaves. If 𝑋 is a Kähler manifold, then a natural choice of polarization is provided by a fixed Kähler two-form on 𝑋. Geometric invariant theory then constructs a projective variety which is a coarse moduli space for semistable sheaves of fixed Chern character. In this paper we will be interested in the computation of suitably defined Euler characteristics of certain moduli spaces, which are the basic enumerative invariants. We will also compute more sophisticated holomorphic curve counting invariants of a Calabi-Yau threefold 𝑋, which can be defined using virtual cycles of the pertinent moduli spaces and are invariant under deformations of 𝑋. In some instances the two types of invariants coincide.

An alternative approach to constructing moduli varieties is through framed sheaves. Then there is a projective 𝑄𝑢𝑜𝑡 scheme which is a fine moduli space for sheaves with a given framing. A framed sheaf can be regarded as a geometric realization of an instanton in a noncommutative gauge theory on 𝑋 [2–4] which asymptotes to a fixed connection at infinity. The noncommutative gauge theory in question arises as the worldvolume field theory on a suitable arrangement of D-branes in the geometry. In Nekrasov's approach [5], the set of observables that enter in the instanton counting are captured by the infrared dynamics of the topologically twisted gauge theory, and they compute the intersection theory of the (compactified) moduli spaces. The purpose of this paper is to overview the enumeration of such noncommutative instantons and its relation to the standard counting invariants of 𝑋.

In the following we will describe the computation of BPS indices of stable D-brane states via instanton counting in certain noncommutative and 𝑞-deformations of gauge theories on branes in various dimensions. We will pay particular attention to three noncompact examples which each arise in the context of Type IIA string theory. (1) D6-D2-D0 bound states in D6-brane gauge theory—These compute Donaldson-Thomas invariants and describe atomic configurations in a melting crystal model [6]. This also provides a solid example of a (topological) gauge theory/string theory duality. The counting of noncommutative instantons in the pertinent topological gauge theory is described in detail in [7, 8]. (2) D4-D2-D0 bound states in D4-brane gauge theory—These count black hole microstates and allow us to probe the OSV conjecture. Their generating functions also appear to be intimately related to the two-dimensional rational conformal field theory. (3) D2-D0 bound states in D2-brane gauge theory—These compute Gromov-Witten invariants of local curves. Instanton counting in the two-dimensional gauge theory on the base of the fibration is intimately related to instanton counting in the four-dimensional gauge theory obtained by wrapping supersymmetric D4-branes around certain noncompact four-cycles 𝐶, and also to the enumeration of flat connections in Chern-Simons theory on the boundary of 𝐶. These interrelationships are explored in detail in [9–13].

These counting problems provide a beautiful hierarchy of relationships between topological string theory/gauge theory in six dimensions, four-dimensional supersymmetric gauge theories, Chern-Simons theory in three dimensions, and a certain 𝑞-deformation of two-dimensional Yang-Mills theory. They are also intimately related to two-dimensional conformal field theory.

2. Topological String Theory

The basic setting in which to describe all gauge theories that we will analyse in this paper within a unified framework is through topological string theory, although many aspects of these models are independent of their connection to topological strings. In this section, we briefly discuss some physical and mathematical aspects of topological string theory, and how they naturally relate to the gauge theories that we are ultimately interested in. Further details about topological string theory can be found in, for example, [14, 15], or in [16] which includes a more general introduction. Introductory and advanced aspects of toric geometry are treated in the classic text [17] and in the reviews [18, 19]. The standard reference for the sheaf theory that we use is the book [20], while a more physicist-geared introduction with applications to string theory can be found in the review [21].

2.1. Topological Strings and Gromov-Witten Theory

Topological string theory may be regarded as a theory whose state space is a “subspace” of that of the full physical Type II string theory. It is designed so that it can resolve the mathematical problem of counting maps 𝑓∶Σ_𝑔→𝑋 (2.1) from a closed oriented Riemann surface Σ_𝑔 of genus 𝑔 into some target space 𝑋. In the physical Type II theory, any harmonic map 𝑓, with respect to chosen metrics on Σ_𝑔 and 𝑋, is allowed. They are described by solutions to second-order partial differential equations, the Euler-Lagrange equations obtained from a variational principle based on a sigma-model. The simplification supplied by topological strings is that one replaces this sigma-model on the worldsheet Σ_𝑔 by a two-dimensional topological field theory, which can be realized as a topological twist of the original 𝒩=2 superconformal field theory. In this reduction, the state space descends to its BRST cohomology defined with respect to the 𝒩=2 supercharges, which naturally carries a Frobenius algebra structure. This defines a consistent quantum theory if and only if the target space 𝑋 is a Calabi-Yau threefold, that is, a complex Kähler manifold of dimension dim_ℂ(𝑋)=3 with trivial canonical holomorphic line bundle 𝐾_𝑋, or equivalently trivial first Chern class 𝑐₁(𝑋)∶=𝑐₁(𝐾_𝑋)=0. We fix a closed nondegenerate Kähler (1,1)-form 𝜔 on 𝑋.

The corresponding topological string amplitudes 𝐹_𝑔 have interpretations in compactifications of Type II string theory on the product of the target space 𝑋 with four-dimensional Minkowski space ℝ^3,1. For instance, at genus zero the amplitude 𝐹₀ is the prepotential for vector multiplets of 𝒩=2 supergravity in four dimensions. The higher genus contributions 𝐹_𝑔, 𝑔≥1 correspond to higher derivative corrections of the schematic form 𝑅²𝑇^2𝑔−2, where 𝑅 is the curvature and 𝑇 is the graviphoton field strength. We will now explain how to compute the amplitudes 𝐹_𝑔. There are two types of topological string theories that we consider in turn.

2.1.1. A-Model

The A-model topological string theory isolates symplectic structure aspects of the Calabi-Yau threefold 𝑋. It is built on holomorphically embedded curves (2.1). The holomorphic string maps 𝑓 in this case are called worldsheet instantons. They are classified topologically by their homology classes 𝛽=𝑓_∗[Σ_𝑔]∈𝐻₂(𝑋,ℤ). (2.2) With respect to a basis 𝑆_𝑖 of two cycles on 𝑋, one can write 𝛽=^𝑏2(𝑋)_𝑖=1𝑛_𝑖[𝑆_𝑖], (2.3) where the Betti number 𝑏₂(𝑋) is the rank of the second homology group 𝐻₂(𝑋,ℤ), and 𝑛_𝑖∈ℤ. Due to the topological nature of the sigma-model, the string theory functional integral localizes equivariantly (with respect to the BRST cohomology) onto contributions from worldsheet instantons.

The sum over all maps can be encoded in a generating function 𝐹^𝑋_t𝑜p(𝑔_𝑠,𝐐) which depends on the string coupling 𝑔_𝑠 and a vector of variables 𝐐=(𝑄₁,…,𝑄_𝑏2(𝑋)) defined as follows. Let 𝑡_𝑖=_𝑆𝑖𝜔 (2.4) be the complex Kähler parameters of 𝑋 with respect to the basis 𝑆_𝑖. They appear in the values of the sigma-model action evaluated on a worldsheet instanton. For an instanton in curve class (2.3), the corresponding Boltzmann weight is 𝐐^𝛽∶=^𝑏2(𝑋)_𝑖=1(𝑄_𝑖)^𝑛𝑖with𝑄_𝑖∶=e^−𝑡𝑖. (2.5) Then the quantum string theory is described by a genus expansion of the free energy 𝐹^𝑋_top(𝑔_𝑠,𝐐)=^∞_𝑔=0𝑔_𝑠^2𝑔−2𝐹_𝑔(𝐐) (2.6) weighted by the Euler characteristic 𝜒(Σ_𝑔)=2−2𝑔 of Σ_𝑔, where the genus 𝑔 contribution to the statistical sum is given by 𝐹_𝑔(𝐐)=_{𝛽∈𝐻2(𝑋,ℤ)}𝑁_𝑔,𝛽(𝑋)𝐐^𝛽, (2.7) and in this formula the classes 𝛽≠0 correspond to worldsheets of genus 𝑔. The numbers 𝑁_𝑔,𝛽(𝑋) are called the Gromov-Witten invariants of 𝑋 and they “count”, in an appropriate sense, the number of worldsheet instantons (holomorphic curves) of genus 𝑔 in the two-homology class 𝛽. They can be defined as follows.

A worldsheet instanton (2.1) is said to be stable if the automorphism group Aut(Σ_𝑔,𝑓) is finite. Let 𝔐_𝑔(𝑋,𝛽) be the (compactified) moduli space of isomorphism classes of stable holomorphic maps (2.1) from connected genus 𝑔 curves to 𝑋 representing 𝛽. This is the instanton moduli space onto which the path integral of topological string theory localizes. It is a proper Deligne-Mumford stack over ℂ which generalizes the moduli space 𝔐_𝑔 of “stable” curves of genus 𝑔. While the dimension of 𝔐_𝑔 is 3𝑔−3, the moduli space 𝔐_𝑔(𝑋,𝛽) is in general reducible and of impure dimension, as all possible stable maps occur. However, there is a perfect obstruction theory [22] which generically has virtual dimension (1−𝑔)(dim_ℂ(𝑋)−3)+_𝛽𝑐₁(𝑋). (2.8) When 𝑋 is a Calabi-Yau threefold, this integer vanishes and there is a virtual fundamental class [22] [𝔐_𝑔(𝑋,𝛽)]^vir∈𝐶𝐻₀(𝔐_𝑔(𝑋,𝛽)) (2.9) in the degree zero Chow group. In this case, we define 𝑁_𝑔,𝛽(𝑋)∶=_{[𝔐𝑔(𝑋,𝛽)]vir}1, (2.10) and so the Gromov-Witten invariants give the “virtual numbers” of worldsheet instantons. One generically has 𝑁_𝑔,𝛽(𝑋)∈ℚ due to the orbifold nature of the moduli stack 𝔐_𝑔(𝑋,𝛽). One can also define invariants by integrating the Euler class of an obstruction bundle over 𝔐_𝑔(𝑋,𝛽). There are precise recipes for computing the Gromov-Witten invariants 𝑁_𝑔,𝛽(𝑋) for toric varieties 𝑋.

2.1.2. B-Model

The B-model topological string theory isolates complex structure aspects of the Calabi-Yau threefold 𝑋. It enumerates the constant string maps which send the entire surface Σ_𝑔 to a fixed point in 𝑋, and hence have trivial curve class 𝛽=0. The Gromov-Witten invariants in this case are completely understood. There is a natural isomorphism 𝔐_𝑔(𝑋,0)≅𝔐_𝑔×𝑋, (2.11) and the degree zero Gromov-Witten invariants 𝑁_𝑔,0(𝑋) involve only the classical cohomology ring 𝐻^•(𝑋,ℚ) and “Hodge integrals” over the moduli spaces of Riemann surfaces 𝔐_𝑔 defined as follows.

There is a canonical stack line bundle 𝔏→𝔐_𝑔 with fibre 𝑇^∗_Σ𝑔 over the moduli point [Σ_𝑔], the cotangent space of Σ_𝑔 at some fixed point. We define the tautological class 𝜓 to be the first Chern class of 𝔏, 𝜓∶=𝑐₁(𝔏)∈𝐻²(𝔐_𝑔,ℚ). The Hodge bundle 𝔈→𝔐_𝑔 is the complex vector bundle of rank 𝑔 whose fibre over a point Σ_𝑔 is the space 𝐻⁰(Σ_𝑔,𝐾_Σ𝑔) of holomorphic sections of the canonical line bundle 𝐾_Σ𝑔→Σ_𝑔. Let 𝜆_𝑗∶=𝑐_𝑗(𝔈)∈𝐻^2𝑗(𝔐_𝑔,ℚ). A Hodge integral over 𝔐_𝑔 is an integral of products of the classes 𝜓 and 𝜆_𝑗.

Explicit expressions for 𝑁_𝑔,0(𝑋) for generic threefolds 𝑋 are then obtained as follows. Let {𝛾_𝑎}_𝑎∈𝐴 be a basis for 𝐻^•(𝑋,ℤ) (modulo torsion), and let 𝐷₂⊂𝐴 index the generators of degree two. Then one has 𝑁_0,0(𝑋)=_{𝑎𝑖∈𝐴}13!_𝑋(𝛾_𝑎1⌣𝛾_𝑎2⌣𝛾_𝑎3𝑁),_1,0(𝑋)=−_𝑎∈𝐷2124_𝑋𝛾_𝑎⌣𝑐₂𝑁(𝑋),_𝑔≥2,0(𝑋)=(−1)^𝑔2_𝑋(𝑐₃(𝑋)−𝑐₁(𝑋)⌣𝑐₂(𝑋))_𝔐𝑔𝜆³_𝑔−1, (2.12) where the Hodge integral can be expressed in terms of Bernoulli numbers as _𝔐𝑔𝜆³_𝑔−1=|𝐵_2𝑔|2𝑔|𝐵_2𝑔−2|12𝑔−2.(2𝑔−2)! (2.13) Note that 𝑐₁(𝑋)=0 above when 𝑋 is Calabi-Yau.

Thus we know how to compute everything in the B-model, and it is completely under control. Our main interest is thus in extending these computations to the A-model. In analogy with the above considerations, one can note that there is a natural forgetful map 𝜋∶𝔐_𝑔(𝑋,𝛽)→𝔐_𝑔,(𝑓,Σ_𝑔)↦Σ_𝑔, (2.14) and then reduce any integral over 𝔐_𝑔(𝑋,𝛽) to 𝔐_𝑔 using the corresponding Gysin push-forward map 𝜋_!. However, this is difficult to do explicitly in most cases. The Gromov-Witten theory of 𝑋 is the study of tautological intersections in the moduli spaces 𝔐_𝑔(𝑋,𝛽). There is a string duality between the A-model and the B-model which is related to homological mirror symmetry.

2.2. Open Topological Strings

An open topological string in 𝑋 is described by a holomorphic embedding 𝑓 : Σ_𝑔,ℎ→𝑋 of a curve Σ_𝑔,ℎ of genus 𝑔 with ℎ holes. A D-brane in 𝑋 is a choice of Dirichlet boundary condition on these string maps, which ensures that the Cauchy problem for the Euler-Lagrange equations on Σ_𝑔,ℎ locally has a unique solution. They correspond to Lagrangian submanifolds 𝐿 of the Calabi-Yau threefold 𝑋, that is, 𝜔|_𝐿=0. If 𝜕Σ_𝑔,ℎ=𝜎₁∪⋯∪𝜎_ℎ, then we consider holomorphic maps such that 𝑓(𝜎_𝑖)⊂𝐿. (2.15) This defines open string instantons, which are labelled by their relative homology classes 𝑓_∗[Σ_𝑔,ℎ]=𝛽∈𝐻₂(𝑋,𝐿). (2.16) If we assume that 𝑏₁(𝐿)=1, so that 𝐻₁(𝐿,ℤ) is generated by a single nontrivial one-cycle 𝛾, then 𝑓_∗[𝜎_𝑖]=𝑤_𝑖𝛾, (2.17) where 𝑤_𝑖∈ℤ, 𝑖=1,…,ℎ, are the winding numbers of the boundary maps 𝑓|_𝜎𝑖.

The free energy of the A-model open topological string theory at genus 𝑔 is given by 𝐹_𝐰,𝑔(𝐐)=_𝛽𝑁_{𝐰,𝑔,𝛽}(𝑋)𝐐^𝛽, (2.18) where 𝐰=(𝑤₁,…,𝑤_ℎ) and the numbers 𝑁_{𝐰,𝑔,𝛽}(𝑋) are called relative Gromov-Witten invariants. To incorporate all topological sectors, in addition to the string coupling 𝑔_𝑠 weighting the Euler characteristics 𝜒(Σ_𝑔,ℎ)=2−2𝑔−ℎ, we introduce an 𝑁×𝑁 Hermitean matrix 𝑉 to weight the different winding numbers. This matrix is associated to the holonomy of a gauge connection (Wilson line) on the D-brane. Then, taking into account that the holes are indistinguishable, the complete genus expansion of the generating function is 𝐹^𝑋_top(𝑔_𝑠,𝐐;𝑉)=^∞∞_{𝑔=0ℎ=1𝐰∈ℤℎ}1𝑔ℎ!^{𝑠2𝑔−2+ℎ}𝐹_𝐰,𝑔(𝐐)^ℎ_𝑖=1Tr(𝑉^𝑤𝑖). (2.19) The traces are computed by formally taking the limit 𝑁→∞ and expanding in irreducible representations 𝑅 of the D-brane gauge group 𝑈(∞).

2.3. Black Hole Microstates and D-Brane Gauge Theory

When 𝑋 is a Calabi-Yau threefold, certain BPS black holes on 𝑋×ℝ^3,1 can be constructed by D-brane engineering. D-branes in 𝑋 correspond to submanifolds of 𝑋 equipped with vector bundles with connection, the Chan-Paton gauge bundles, and they carry charges associated with the Chern characters of these bundles. This data defines a class in the differential K-theory of 𝑋, which provides a topological classification of D-branes in 𝑋.

The microscopic black hole entropy can be computed by counting stable bound states of D0–D2–D4–D6 branes wrapping holomorphic cycles of 𝑋 with the following configurations: (i)D6-brane charge 𝒬₆,(ii)D4-branes wrapping an ample divisor [𝐶]=^𝑏2(𝑋)_𝑖=1𝒬^𝑖₄[𝐶_𝑖]∈𝐻₄(𝑋,ℤ) (2.20) with respect to a basis of four-cycles 𝐶_𝑖, 𝑖=1,…,𝑏₄(𝑋)=𝑏₂(𝑋), of 𝑋,(iii) D2-branes wrapping a two-cycle [𝑆]=^𝑏2(𝑋)_𝑖=1𝒬^𝑖₂[𝑆_𝑖]∈𝐻₂(𝑋,ℤ), (2.21)(iv) D0-brane charge 𝒬₀.

These D-brane charges give the black hole its charge quantum numbers. If we consider large enough numbers of D-branes in this system, then they form bound states which become large black holes with smooth event horizons, that can be counted and therefore account for the microscopic black hole entropy. In this scenario, 𝑝^𝐼=(𝒬₆,𝒬^𝑖₄) are interpreted as magnetic charges and 𝑞_𝐼=(𝒬₀,𝒬^𝑖₂) as electric charges. The thermal partition function defined via a canonical ensemble for the D0 and D2 branes with chemical potentials 𝜇^𝐼=(𝜙⁰,𝜙²_𝑖), and a microcanonical ensemble for the D4 and D6 branes, is given by 𝑍_BH(𝒬₆,𝓠₄,𝝓²,𝜙⁰)=_{𝒬0,𝒬𝑖2}Ω(𝒬₀,𝓠₂,𝓠₄,𝒬₆)e^{−𝒬0𝜙0−𝒬𝑖2𝜙2𝑖}, (2.22) where Ω is the degeneracy of BPS states with the given D-brane charges.

As we mentioned in Section 2.1., the closed topological string amplitudes 𝐹_𝑔 are related to supergravity quantities on Minkowski spacetime ℝ^3,1. The fact that the genus zero free energy 𝐹₀ for topological strings on 𝑋 is a prepotential for BPS black hole charges in 𝒩=2 supergravity determines the entropy 𝑆_BH(𝑝,𝑞) of an extremal black hole as a Legendre transformation of 𝐹₀, provided that one fixes the charge moduli by the attractor mechanism. The genus zero topological string amplitude 𝐹₀ is a homogeneous function of degree two in the 𝒩=2 vector multiplet fields 𝑋^𝐼. The black hole entropy in the supergravity approximation is then 𝑆_BH(𝑝,𝑞)=𝜇^𝐼𝑞_𝐼−Im𝐹₀(𝑋^𝐼=𝑝^𝐼+i𝜇^𝐼), (2.23) where the chemical potentials 𝜇^𝐼 are determined by the charges 𝑝^𝐼 and 𝑞_𝐼 by solving the equation 𝑞_𝐼=𝜕Im𝐹₀𝜕𝜇^𝐼. (2.24)

Further analyses of the entropy of 𝒩=2 BPS black holes on ℝ^3,1 have been extended to higher genus and suggest the relationship 𝑍_BH(𝒬₆,𝓠₄,𝝓²,𝜙⁰)=|𝑍^𝑋_top(𝑔_𝑠,𝐐)|² (2.25) between the black hole partition function (2.22) and the topological string partition function 𝑍^𝑋_top(𝑔_𝑠,𝐐)=exp𝐹^𝑋_top(𝑔_𝑠,𝐐), (2.26) where the moduli on both sides of this equation are related by their fixing at the attractor point 𝑔_𝑠=4𝜋i(i/𝜋)𝜙⁰+𝒬₆,𝑡_𝑖=2𝜙²_𝑖+i𝒬^𝑖₄(i/𝜋)𝜙⁰+𝒬₆. (2.27) The remarkable relationship (2.25) is called the OSV conjecture [1]. It provides a means of using the perturbation expansion of topological strings and Gromov-Witten theory to compute black hole entropy to all orders. Alternatively, although the evidence for this proposal is derived for large black hole charge, the left-hand side of the expression (2.25) makes sense for finite charges and in some cases is explicitly computable in closed form. It can thus be used to define nonperturbative topological string amplitudes, and hence a nonperturbative completion of a string theory.

In the following, we will focus on the computation of the black hole partition function (2.22). The fact that this partition function is computable in a D-brane gauge theory will then give a physical interpretation of the enumerative invariants of 𝑋 in terms of black hole entropy. Suppose that we have a collection of D𝑝-branes wrapping a submanifold 𝑀_𝑝+1⊂𝑋, with dim_ℝ(𝑀_𝑝+1)=𝑝+1 and Chan-Paton gauge field strength 𝐹. D-branes are charged with respect to supergravity differential form fields, the Ramond-Ramond fields, which are also classified topologically by differential K-theory. Recall that such an array couples to all 𝑛-form Ramond-Ramond fields 𝐶_(𝑛) through anomalous Chern-Simons couplings _{𝑀𝑝+1𝑛≥0}𝐶_(𝑛)∧Trexp(2𝜋𝛼^′𝐹), (2.28) where 𝛼^′ is the string length. In particular, these couplings contain all terms _𝑀𝑝+1𝐶_{(𝑝+1−2𝑚)}∧Tr(𝐹^𝑚), (2.29) and so the topological charge ch_𝑚(ℰ) of a Chan-Paton gauge bundle ℰ→𝑀_𝑝+1 on a D𝑝-brane is equivalent to D(𝑝−2𝑚)-brane charge. A prominent example of this, which will be considered in detail later on, is the coupling ∫_𝑀𝑝+1𝐶_(𝑝−3)∧Tr(𝐹∧𝐹). For 𝑝=3, this shows that the counting of D4-D0 brane bound states is equivalent to the enumeration of instantons on the four-dimensional part of the D4-brane in 𝑋. The remaining sections of this paper look at these relationships from the point of view of various BPS configurations of these D-branes. We will study the enumeration problems from the point of view of gauge theories on the D-branes in order of decreasing dimensionality, stressing the analogies between each description.

3. D6-Brane Gauge Theory and Donaldson-Thomas Invariants

In this section we will look at a single D6-brane (𝒬₆=1) and turn off all D4-brane charges (𝒬^𝑖₄=0). We will discuss various physical theories which are modelled by the D6-brane gauge theory in this case, but otherwise have no a priori relation to string theory. These will include a tractable model for quantum gravity and the statistical mechanics of certain atomic crystal configurations. From the perspective of enumerative geometry, these partition functions will compute the Donaldson-Thomas theory of 𝑋.

3.1. Kähler Quantum Gravity

We will construct a model of quantum gravity on any Kähler threefold 𝑋, which will motivate the sorts of counting problems that we consider in this section. The partition function is defined by 𝑍=_{quantized𝜔}e^−𝑆, (3.1) where 1𝑆=𝑔²_𝑠𝑋13!𝜔∧𝜔∧𝜔. (3.2) The sum is over “quantized” Kähler two-forms on 𝑋, in the following sense. We decompose the “macroscopic” form 𝜔 into a fixed “background” Kähler two-form 𝜔₀ on 𝑋 and the curvature 𝐹 of a holomorphic line bundle ℒ over 𝑋 as 𝜔=𝜔₀+𝑔_𝑠𝐹. (3.3) To satisfy the requirement that there are no D4-branes in 𝑋, we impose the fluctuation condition _𝛽𝐹=0 (3.4) for all two-cycles 𝛽∈𝐻₂(𝑋,ℤ).

Substituting (3.3) together with (3.4) into (3.2) gives the action 1𝑆=𝑔²_𝑠13!_𝑋𝜔³₀+12_𝑋𝐹∧𝐹∧𝜔₀+𝑔_𝑠𝑋13!𝐹∧𝐹∧𝐹. (3.5) The statistical sum (3.1) thus becomes (dropping an irrelevant constant term) 𝑍=_{linebundlesℒ}𝑞^{ch3𝑏(ℒ)2(𝑋)}_𝑖=1(𝑄_𝑖)^{∫𝐶𝑖ch2(ℒ)}, (3.6) where 𝑞=−e^−𝑔𝑠, 𝑄_𝑖=e^−𝑡𝑖, and ch_𝑚(ℒ) denotes the 𝑚th Chern character of the given line bundle ℒ→𝑋. Note the formal similarity with the A-model topological string partition function constructed in Section 2.1. However, there is a problem with the way in which we have thus far set up this model. The fluctuation condition (3.4) on 𝐹 implies ch₂(ℒ)=ch₃(ℒ)=0. Hence only trivial line bundles can contribute to the sum (3.6), and the partition function is trivial.

The resolution to this problem is to enlarge the range of summation in (3.6) to include singular gauge fields and ideal sheaves. Namely, we take 𝐹 to correspond to a singular 𝑈(1) gauge field 𝐴 on 𝑋. This can be realized in two (related) possible ways: (1)we can make a singular gauge field 𝐴 nonsingular on the blow-up 𝑋→𝑋 (3.7) of the target space, obtained by blowing up the singular points of 𝐴 on 𝑋 into copies of the complex projective plane ℙ². This means that the quantum gravitational path integral induces a topology change of the target space 𝑋. This is referred to as “quantum foam” in [23, 24], or (2) we can relax the notion of line bundle to ideal sheaf. Ideal sheaves lift to line bundles on 𝑋. However, there are “more” sheaves on 𝑋 than blow-ups 𝑋 of 𝑋.

In this paper we will take the second point of view. Recall that torsion-free sheaves ℰ on 𝑋 can be defined by the property that they sit inside short exact sequences of sheaves of the form 0→ℰ→ℱ→𝒮_𝑍→0, (3.8) where ℱ is a holomorphic vector bundle on 𝑋, and 𝒮_𝑍 is a coherent sheaf supported at the singular points 𝑍⊂𝑋 of a gauge connection 𝐴 of ℱ. Applying the Chern character to (3.8) and using its additivity on exact sequences give ch_𝑚(ℰ)=ch_𝑚(ℱ)−ch_𝑚(𝒮_𝑍) (3.9) for each 𝑚. Thus torsion-free sheaves ℰ fail to be vector bundles at singular points of gauge fields, and including the singular locus can reinstate the nontrivial topological quantum numbers that we desired above.

As we will discuss in detail in this section, this construction is realized explicitly by considering a noncommutative gauge theory on the target space 𝑋=ℂ³. We will see that the instanton solutions of gauge theory on a noncommutative deformation ℂ³_𝜃 are described in terms of ideals ℐ in the polynomial ring ℂ[𝑧¹,𝑧²,𝑧³]. For generic 𝑋, the global object that corresponds locally to an ideal is an ideal sheaf, which in each coordinate patch 𝑈_𝛼⊂𝑋 is described as an ideal ℐ_𝑈𝛼 in the ring 𝒪_𝑈𝛼 of holomorphic functions on 𝑈_𝛼. More abstractly, an ideal sheaf is a rank one torsion-free sheaf ℰ with 𝑐₁(ℰ)=0. This is a purely commutative description, since the holomorphic functions on ℂ³ form a commutative subalgebra of ℂ³_𝜃 for the Moyal deformation that we will consider. Thus the desired singular gauge field configurations will be realized explicitly in terms of noncommutative instantons [23, 24].

3.2. Crystal Melting and Random Plane Partitions

As we will see, the counting of ideal sheaves is in fact equivalent to a combinatorial problem, which provides an intriguing connection between the Kähler quantum gravity model of Section 3.1. and a particular statistical mechanics model [6]. Consider a cubic crystal

located on the lattice ℤ³_≥0⊂ℝ³. Suppose that we start heating the crystal at its outermost right corner. As the crystal melts, we remove atoms, depicted symbolically here by boxes, and arrange them into stacks of boxes in the positive octant. Owing to the rules for arranging the boxes according to the order in which they melt, this configuration defines a plane partition or a three-dimensional Young diagram.

Removing each atom from the corner of the crystal contributes a factor 𝑞 = e^−𝜇/𝑇 to the Boltzmann weight, where 𝜇 is the chemical potential and 𝑇 is the temperature.

Let us define more precisely the combinatorial object that we have constructed, which generalizes the usual notion of partition and Young tableau. A plane partition is a semiinfinite rectangular array of nonnegative integers 𝜋𝜋=^1,1𝜋_1,2𝜋_1,3…𝜋_2,1𝜋_2,2𝜋_2,3…𝜋_3,1𝜋_3,2𝜋_3,3…⋮⋮⋮ (3.10) such that 𝜋_𝑖,𝑗≥𝜋_𝑖+1,𝑗, and 𝜋_𝑖,𝑗≥𝜋_𝑖,𝑗+1 for all 𝑖,𝑗≥1. We may regard a partition 𝜋 as a three-dimensional Young diagram, in which we pile 𝜋_𝑖,𝑗 cubes vertically at the (𝑖,𝑗)th position in the plane as depicted above. The volume of a plane partition |𝜋|=_{𝑖,𝑗≥1}𝜋_𝑖,𝑗 (3.11) is the total number of cubes. The diagonal slices of 𝜋 are the partitions (𝜋_{𝑖,𝑖+𝑚})_𝑖≥1, 𝑚≥0, obtained by cutting the three-dimensional Young diagram with planes, and they represent a sequence of ordinary partitions (Young tableaux) 𝜆=(𝜆₁,𝜆₂,…), with 𝜆_𝑖≥𝜆_𝑖+1 for all 𝑖≥1. Here 𝜆_𝑖≥0 is the length of the 𝑖th row of the Young diagram, viewed as a collection of unit squares, and only finitely many 𝜆_𝑖 are nonzero.

The counting problem for random plane partitions can be solved explicitly in closed form. For this, we consider the statistical mechanics in a canonical ensemble in which each plane partition 𝜋 has energy proportional to its volume |𝜋|. The corresponding partition function then gives the generating function for plane partitions𝑍∶=_𝜋𝑞^|𝜋|=^∞_𝑁=0𝑝𝑝(𝑁)𝑞^𝑁=^∞_𝑛=11(1−𝑞^𝑛)^𝑛=∶𝑀(𝑞), (3.12) where 𝑝𝑝(𝑁) is the number of plane partitions 𝜋 with |𝜋|=𝑁 boxes. The function 𝑀(𝑞) is called the MacMahon function.

3.3. Six-Dimensional Cohomological Gauge Theory

We will now describe a 𝑈(1) gauge theory formulation of the above statistical models [7, 8, 24]. If we gauge-fix the residual symmetry of the quantized Kähler gravity action (3.5), we obtain the action1𝑆=2_𝑋(d_𝐴Φ∧⋆d_𝐴Φ+|𝐹^2,0|²+|𝐹^1,1|²1)+2_𝑋(𝐹∧𝐹∧𝜔₀+𝑔_𝑠3𝐹∧𝐹∧𝐹), (3.13) where d_𝐴=d−i[𝐴,−] is the gauge covariant derivative acting on the complex scalar field Φ, ⋆ denotes the Hodge operator with respect to the Kähler metric of 𝑋, and 𝐹=d𝐴 is the curvature two-form which has the Kähler decomposition 𝐹=𝐹^2,0+𝐹^1,1+𝐹^0,2. The field theory defined by this action arises in three (related) instances such as: (1)a topological twist of maximally supersymmetric Yang-Mills theory in six dimensions, (2)the dimensional reduction of supersymmetric Yang-Mills theory in ten dimensions on 𝑋,(3)the low-energy effective field theory on a D6-brane wrapping 𝑋 in Type IIA string theory, with D2 and D0 brane sources.

The gauge theory has a BRST symmetry [25, 26] and its partition function localizes at the BRST fixed points described by the equations 𝐹^2,0=0=𝐹^0,2,𝐹^1,1∧𝜔₀∧𝜔₀d=0,_𝐴Φ=0. (3.14)(3.15)(3.16) These equations also describe three (related) quantities: (i) the Donaldson-Uhlenbeck-Yau (DUY) equations expressing Mumford-Takemoto slope stability of holomorphic vector bundles over 𝑋 with finite characteristic classes, (ii) BPS solutions in the gauge theory which correspond to (generalized) instantons,(iii) bound states of D0–D2 branes in a single D6-brane wrapping 𝑋.

Recall that (3.14) and (3.15) are a special instance of the Hermitean Yang-Mills equations in which a constant 𝜆 is added to the right-hand side of (3.15). These equations arise in compactifications of heterotic string theory. The condition that the compactification preserves at least one unbroken supersymmetry requires 𝜆=0. These are the natural BPS conditions on a Kähler manifold (𝑋,𝜔₀) which generalize the usual self-duality equations in four dimensions.

The localization of the gauge theory partition function 𝑍 onto the corresponding instanton moduli space 𝔐_𝑋 can be written symbolically as [24, 26] 𝑍=_𝔐𝑋𝑒(𝔑_𝑋), (3.17) where 𝑒(𝔑_𝑋) is the Euler class of the obstruction bundle 𝔑_𝑋 whose fibres are spanned by the zero modes of the antighost fields. The zero modes of the fermion fields in the full supersymmetric extension of (3.13) [25, 26] are in correspondence with elements in the cohomology groups of the twisted Dolbeault complex Ω^0,0(𝑋,ad𝒫)−−−−−^𝜕𝐴−→Ω^0,1(𝑋,ad𝒫)−−−−−^𝜕𝐴−→Ω^0,2(𝑋,ad𝒫)−−−−−^𝜕𝐴−→Ω^0,3(𝑋,ad𝒫) (3.18) with ad𝒫 the adjoint gauge bundle over 𝑋. By incorporating the gauge fields, one can rewrite this complex in the form [24] Ω^0,0Ω(𝑋,ad𝒫)→^0,1⊕Ω(𝑋,ad𝒫)_0,3(𝑋,ad𝒫)→Ω^0,2(𝑋,ad𝒫), (3.19) which describes solutions of the DUY equations up to linearized complex gauge transformations. The morphism Ω^0,3(𝑋,ad𝒫)→Ω^0,2(𝑋,ad𝒫) here is responsible for the appearance of the obstruction bundle in (3.17) [24, 26].

In order for the integral (3.17) to be well defined, we need to choose a compactification of 𝔐_𝑋. In light of our earlier discussion, we will take this to be the Gieseker compactification, that is, the moduli space of ideal sheaves on 𝑋. The corresponding variety 𝔐_𝑋 stratifies into components Hilb_𝑛,𝛽(𝑋) given by the Hilbert scheme of points and curves in 𝑋, parameterizing isomorphism classes of ideal sheaves ℰ with ch₁(ℰ)=𝑐₁(ℰ)=0, ch₂(ℰ)=−𝛽, and ch₃(ℰ)=−𝑛. The partition function (3.17) is the generating function for the number of D0-D2 brane bound states in the D6-brane wrapping 𝑋. Mathematically, these are the Donaldson-Thomas invariants of 𝑋. We will define this moduli space integration, and hence these invariants, more precisely in Section 3.9.

3.4. Localization in Toric Geometry

Toric varieties provide a large class of algebraic varieties in which difficult problems in algebraic geometry can be reduced to combinatorics. Much of this paper will be concerned with these geometries as they possess symmetries which facilitate computations, particularly those involving moduli space integrations. Let us start by recalling some basic notions from toric geometry. Below we give the pertinent definitions specifically in the case of varieties of complex dimension three, the case of immediate interest to us, but they extend to arbitrary dimensions in the obvious ways.

A smooth complex threefold 𝑋 is called a toric manifold if it densely contains a (complex algebraic) torus 𝑇³ and the natural action of 𝑇³ on itself (by translations) extends to the whole of 𝑋. Basic examples are the torus 𝑇³ itself, the affine space ℂ³, and the complex projective space ℙ³. If in addition 𝑋 is Calabi-Yau, then 𝑋 is necessarily noncompact.

One of the great virtues of working with toric varieties 𝑋 is that their geometry can be completely described by combinatorial data encoded in a toric diagram. The toric diagram is a graph consisting of the following ingredients: (i)a set of vertices 𝑓 which are the fixed points of the 𝑇³-action on 𝑋, such that 𝑋 can be covered by 𝑇³-invariant open charts homeomorphic to ℂ³,(ii)a set of edges 𝑒 which are 𝑇³-invariant projective lines ℙ¹⊂𝑋 joining particular pairs of fixed points 𝑓₁, 𝑓₂, (iii) a set of “gluing rules” for assembling the ℂ³ patches together to reconstruct the variety 𝑋. In a neighbourhood of each edge 𝑒, 𝑋 looks like the normal bundle over the corresponding ℙ¹. Since this normal bundle is a holomorphic bundle of rank two and every bundle over ℙ¹ is a sum of line bundles (by the Grothendieck-Birkhoff theorem), it is of the form 𝒪_ℙ¹(−𝑚₁)⊕𝒪_ℙ¹(−𝑚₂) (3.20) for some integers 𝑚₁,𝑚₂. The normal bundle in this way determines the local geometry of 𝑋 near the edge 𝑒 via the transition function (𝑤₁,𝑤₂,𝑤₃)↦(𝑤₁⁻¹,𝑤₂𝑤^−𝑚1₁,𝑤₃𝑤^−𝑚2₁) (3.21) between the corresponding affine patches (going from the north pole to the south pole of the associated ℙ¹). In the Calabi-Yau case, the Chern numbers 𝑐₁(𝑋)=0 and 𝑐₁(ℙ¹)=2 imply the condition 𝑚₁+𝑚₂=2.

For an open toric manifold 𝑋, we can exploit the toric symmetries to regularize the infrared singularities on the instanton moduli space 𝔐_𝑋 by “undoing” the 𝑇³-rotations by gauge transformations [5]. In this way we will compute our moduli space integrals by using techniques from equivariant localization, which in the present context will be refered to as toric localization. Recall that the bosonic sector of the topologically twisted theory comprises a gauge connection 𝐴_𝑖 and a complex Higgs field Φ. In particular, the supercharges contain a scalar 𝑄 and a vector 𝑄_𝑖. Generically, only 𝑄 is conserved and can be used to define the topological twist of the gauge theory. If the threefold 𝑋 has symmetries then one can also use 𝑄_𝑖. In the generic formulation of the theory, one only considers the scalar topological charge 𝑄 and restricts attention to gauge-invariant observables. But in the present case one can also use the linear combination 𝑄_Ω=𝑄+𝜖_𝑎Ω^𝑎_𝑖𝑗𝑥^𝑖𝑄^𝑗, (3.22) where 𝜖^𝑎 are the parameters of the isometric action of 𝑇³⊂𝑈(3) on the Kähler space ℂ³, and Ω^𝑎=Ω^𝑎_𝑖𝑗𝑥^𝑗(𝜕/𝜕𝑥_𝑖) are vector fields which generate 𝑆𝑂(6) rotations of ℂ³≅ℝ⁶. In this case we also consider observables which are only gauge-invariant up to a rotation. This means that the new observables are equivariant differential forms and the BRST charge 𝑄_Ω can be interpreted as an equivariant differential d+𝜄_Ω on the space of field configurations, where 𝜄_Ω acts by contraction with the vector field Ω.

This procedure modifies the action and the equations of motion by mixing gauge invariance with rotations. This set of modifications can sometimes be obtained by defining the gauge theory on an appropriate supergravity background called the “Ω-background”. In particular, the fixed point equation (3.16) is modified to d_𝐴Φ=𝜄_Ω𝐹. (3.23) The set of equations (3.14), (3.15), and (3.23) minimizes the action of the cohomological gauge theory in the Ω-background and describes 𝑇³-invariant instantons (or, as we will see, ideal sheaves). In particular, there is a natural lift of the toric action to the instanton moduli space 𝔐_𝑋. We will henceforth study the gauge theory equivariantly and interpret the truncation of the partition function (3.17) as an equivariant integral over 𝔐_𝑋. This will always mean that we work solely in the Coulomb branch of the gauge theory. Due to the equivariant deformation of the BRST charge, these moduli space integrals can be computed via equivariant localization.

3.5. Equivariant Integration over Moduli Spaces

We now explain the localization formulas that will be used to compute partition functions throughout this paper. Let 𝔐 be a smooth algebraic variety. Then we can define the 𝑇-equivariant cohomology 𝐻^•_𝑇(𝔐,ℚ) as the ordinary cohomology 𝐻^•(𝔐_𝑇,ℚ) of the Borel-Moore homotopy quotient 𝔐_𝑇∶=(𝔐×𝐸𝑇)/𝑇, where 𝐸𝑇=(ℂ^∞⧵{0})^𝑁+𝑘 is a contractible space on which 𝑇=𝑈(1)^𝑁×𝑇^𝑘 acts freely. In the present example of interest, 𝑁=1 and 𝑘=3. Given a 𝑇-equivariant vector bundle 𝔈→𝔐, the quotient 𝔈_𝑇=(𝔈×𝐸𝑇)/𝑇 is a vector bundle over 𝔐_𝑇=(𝔐×𝐸𝑇)/𝑇. The 𝑇-equivariant Euler class of 𝔈 is the invertible element defined by 𝑒_𝑇(𝔈)∶=𝑒(𝔈_𝑇)∈𝐻^•_𝑇(𝔐,ℚ), (3.24) where 𝑒 is the ordinary Euler class for vector bundles (the top Chern class).

Let 𝐵𝑇∶=𝐸𝑇/𝑇=(ℙ^∞)^𝑘+𝑁. Then 𝐸𝑇→𝐵𝑇 is a universal principal 𝑇-bundle, and there is a fibration 𝔐_𝑇→𝐵𝑇 with fibre 𝔐. Integration in equivariant cohomology is defined as the pushforward ∮_𝔐 of the collapsing map 𝔐→pt, which coincides with integration over the fibres 𝔐 of the bundle 𝔐_𝑇→𝐵𝑇 in ordinary cohomology. Let 𝑝_𝑖∶𝐵𝑇→ℙ^∞ for 𝑖=1,…,𝑘 and let 𝑞_𝑙 : 𝐵𝑇→ℙ^∞ for 𝑙=1,…,𝑁 be the canonical projections onto the 𝑖th and 𝑙th factors. Introduce equivariant parameters 𝜖_𝑖=(𝑐₁)_𝑇(𝑝^∗_𝑖𝒪_ℙ^∞(1)), with 𝑡_𝑖=e^𝜖𝑖=(ch_𝑇)₁(𝑝^∗_𝑖𝒪_ℙ^∞(1)) and 𝑎_𝑙=(𝑐₁)_𝑇(𝑞^∗_𝑙𝒪_ℙ^∞(1)), with 𝑒_𝑙=e^𝑎𝑙=(ch_𝑇)₁(𝑞^∗_𝑙𝒪_ℙ^∞(1)).

The Atiyah-Bott localization formula in equivariant cohomology states that _𝔐𝛼=_𝔐𝑇𝛼|_𝔐𝑇𝑒_𝑇(𝔑) (3.25) for any equivariant differential form 𝛼∈𝐻^•_𝑇(𝔐,ℚ), where the complex vector bundle 𝔑→𝔐𝑇 is the normal bundle over the (compact) fixed point submanifold in 𝔐. When 𝔐𝑇 consists of finitely many isolated points 𝑓, this formula is simplified to _𝔐𝛼=_𝑓∈𝔐𝑇𝛼(𝑓)𝑒_𝑇(𝑇_𝑓.𝔐) (3.26) Each term in this sum takes values in the polynomial ring 𝐻^•_𝑇(𝑓,ℚ)=𝐻^•(𝐵𝑇,ℚ)≅ℚ[𝜖₁,…,𝜖_𝑘,𝑎₁,…,𝑎_𝑁] (3.27) in the generators of 𝑇=𝑈(1)^𝑁×𝑇^𝑘. When the manifold 𝔐 is noncompact, integration along the fibre is not a well-defined ℚ-linear map. Nevertheless, when 𝔐𝑇 is compact, we can formally define the equivariant integral ∮_𝔐𝛼 by the right-hand side of the formula (3.25).

Going back to our example, when 𝑋=ℂ³, one has ch₂(ℰ)=0 and the partition function 𝑍 is saturated by contributions from isolated, pointlike instantons (D0-branes) by a formal application of the localization formula (3.26). However, these expressions are all rather symbolic, as we are not guaranteed that the algebraic scheme 𝔐_𝑋 is a smooth variety, that is, the instanton moduli space has a well-defined stable tangent bundle with tangent spaces all of the same dimension. However, the variety 𝔐_𝑋 is generically smooth and there is a well-defined virtual tangent bundle. The moduli space integration (3.26) can then be formally defined by virtual localization in equivariant Chow theory. As discussed in [27], the (stratified components of the) instanton moduli space 𝔐_𝑋 carries a canonical perfect obstruction theory in the sense of [22]. In obstruction theory, the virtual tangent space at a point [ℰ]∈𝔐_𝑋 is given by 𝑇^vir_[ℰ]𝔐_𝑋=Ext¹_𝒪𝑋(ℰ,ℰ)⊖Ext²_𝒪𝑋(ℰ,ℰ), (3.28) where Ext¹_𝒪𝑋(ℰ,ℰ) is the Zariski tangent space and Ext²_𝒪𝑋(ℰ,ℰ) the obstruction space of 𝔐_𝑋 at [ℰ]. Its dimension is given by the difference of Euler characteristics 𝜒(𝒪_𝑋⊗𝒪^∨_𝑋)−𝜒(ℰ⊗ℰ^∨). The kernel of the trace map Ext²_𝒪𝑋(ℰ,ℰ)→𝐻²(𝑋,𝒪_𝑋) (3.29) is the obstruction to smoothness at a point [ℰ] of the moduli space.

The bundles 𝔈_𝑖∶=Ext^𝑖_𝒪𝑋(ℰ,ℰ), 𝑖=1,2 for [ℰ]∈𝔐_𝑋 define a canonical 𝑇^𝑘-equivariant perfect obstruction theory 𝔈_•=(𝔈₁→𝔈₂) (see [22, Section 1]) on the instanton moduli space 𝔐=𝔐_𝑋. In this case, one may construct a virtual fundamental class [𝔐]^vir and apply a virtual localization formula. The general theory is developed in [22] and requires a 𝑇^𝑘-equivariant embedding of 𝔐 in a smooth variety 𝔜. The existence of such an embedding in the present case follows from the stratification of 𝔐_𝑋 into Hilbert schemes of points and curves. Then one can deduce the localization formula over 𝔐 from the known ambient localization formula over the smooth variety 𝔜, as above. In this paper we will only need a special case of this general framework, the virtual Bott residue formula.

We can decompose 𝔈_𝑖 into 𝑇^𝑘-eigenbundles. The scheme theoretic fixed point locus 𝔐^𝑇𝑘 is the maximal 𝑇^𝑘-fixed closed subscheme of 𝔐. It carries a canonical perfect obstruction theory, defined by the 𝑇^𝑘-fixed part of the restriction of the complex 𝔈_• to 𝔐^𝑇𝑘, which may be used to define a virtual structure on 𝔐^𝑇𝑘. The sum of the nonzero 𝑇^𝑘-weight spaces of 𝔈_•|_𝔐𝑇^𝑘 defines the virtual normal bundle 𝔑^vir to 𝔐^𝑇𝑘. Define the Euler class of a virtual bundle 𝔄=𝔄₁⊖𝔄₂ using formal multiplicativity, that is, as the ratio of the Euler classes of the two bundles, 𝑒(𝔄)=𝑒(𝔄₁)/𝑒(𝔄₂). Then the virtual Bott localization formula for the Euler class of a bundle 𝔄 of a rank equal to the virtual dimension of 𝔐 reads [22] _[𝔐]vir𝑒(𝔄)=_{[𝔐𝑇𝑘]vir}𝑒_𝑇^𝑘(𝔄|_𝔐𝑇^𝑘)𝑒_𝑇^𝑘(𝔑^vir), (3.30) where the integration is again defined via pushforward maps. The equivariant Euler classes on the right-hand side of this formula are invertible in the localized equivariant Chow ring of the scheme 𝔐 given by 𝐶𝐻^•_𝑇𝑘(𝔐)⊗_{ℚ[𝜖1,…,𝜖𝑘]}ℚ[𝜖₁,…,𝜖_𝑘]_𝔪, where ℚ[𝜖₁,…,𝜖_𝑘]_𝔪 is the localization of the ring ℚ[𝜖₁,…,𝜖_𝑘] at the maximal ideal 𝔪 generated by 𝜖₁,…,𝜖_𝑘.

If 𝔐 is smooth, then 𝔐^𝑇𝑘 is the nonsingular set theoretic fixed point locus, consisting here of finitely many points [ℰ]. However, in general the formula (3.30) must be understood scheme theoretically, here as a sum over 𝑇^𝑘-fixed closed subschemes of 𝔐 supported at the points [ℰ]∈𝔐^𝑇𝑘 (with 𝑘=3). With 𝜌^𝑖_ℰ : 𝑇^𝑘→End_ℂ(Ext^𝑖_𝒪𝑋(ℰ,ℰ)), 𝑖=1,2, denoting the induced torus actions on the tangent and obstruction bundles on 𝔐, one generically has decompositions Ext^𝑖_𝒪𝑋(ℰ,ℰ)=Ext^𝑖₁(ℰ,ℰ)⊕ker(𝜌^𝑖_ℰ(𝑇^𝑘)), (3.31) where Ext^𝑖₁(ℰ,ℰ) is a 𝑇^𝑘-invariant subspace of Ext^𝑖_𝒪𝑋(ℰ,ℰ). As demonstrated in [27, Section 4.5], the kernel module in (3.31) vanishes. Hence each subscheme here is just the reduced point [ℰ] and the 𝑇^𝑘-fixed obstruction theory at [ℰ] is trivial. Under these conditions, the virtual localization formula (3.30) may be written as _[𝔐]vir𝑒(𝔄)=_{[ℰ]∈𝔐𝑇𝑘}𝑒_𝑇^𝑘(𝔄([ℰ]))𝑒_𝑇^𝑘(𝑇^vir_[ℰ].𝔐) (3.32) The right-hand side of this formula again takes values in the polynomial ring ℚ[𝜖₁,…,𝜖_𝑘]. When Ext⁰_𝒪𝑋(ℰ,ℰ)=Ext²_𝒪𝑋(ℰ,ℰ)=0 for all [ℰ]∈𝔐_𝑋, the moduli space 𝔐_𝑋 is a smooth algebraic variety with the trivial perfect obstruction theory and this equation reduces immediately to the standard localization formula in equivariant cohomology given above. In this paper, we will make the natural choice for the bundle 𝔄, the virtual tangent bundle 𝑇^vir𝔐 itself.

3.6. Noncommutative Gauge Theory

To compute the instanton contributions (3.17) to the partition function of the cohomological gauge theory, we have to resolve the small instanton ultraviolet singularities of 𝔐_𝑋. This can be achieved by replacing the space 𝑋=ℂ³≅ℝ⁶ with its noncommutative deformation ℝ⁶_𝜃 defined by letting the coordinate generators 𝑥^𝑖,𝑖=1,…,6, satisfy the commutation relations of the Weyl algebra [𝑥^𝑖,𝑥^𝑗]=i𝜃^𝑖𝑗, (3.33) where (𝜃^𝑖𝑗)=0𝜃₁−𝜃₁00𝜃₂−𝜃₂00𝜃₃−𝜃₃0 (3.34) is a constant 6×6 skew-symmetric matrix which we take in Jordan canonical form without loss of generality (by a suitable linear transformation of ℝ⁶ if necessary). We will assume that 𝜃₁,𝜃₂,𝜃₃>0 for simplicity. The noncommutative polynomial algebra 𝒜=ℂ[𝑥¹,𝑥²,𝑥³]⟨[𝑥^𝑖,𝑥^𝑗]−i𝜃^𝑖𝑗⟩ (3.35) is regarded as the “algebra of functions” on the noncommutative space ℝ⁶_𝜃.

We can represent the algebra 𝒜 on the standard Fock module ℋ=ℂ[𝛼^†₁,𝛼^†₂,𝛼^†₃]|0,0,0⟩=^∞⊕_{𝑖,𝑗,𝑘=0}ℂ|𝑖,𝑗,𝑘⟩, (3.36) where the orthonormal basis states |𝑖,𝑗,𝑘⟩ are connected by the action of creation and annihilation operators 𝛼^†_𝑎 and 𝛼_𝑎,𝑎=1,2,3. They obey 𝛼_𝑎|0,0,0⟩=0 and [𝛼^†_𝑎,𝛼_𝑏]=𝛿_𝑎𝑏,[𝛼_𝑎,𝛼_𝑏]=0=[𝛼^†_𝑎,𝛼^†_𝑏]. (3.37) In the Weyl operator realization with the complex combinations of operators 𝑧^𝑎=𝑥^2𝑎−1−i𝑥^2𝑎=2𝜃_𝑎𝛼_𝑎,𝑧^𝑎=𝑥^2𝑎−1+i𝑥^2𝑎=2𝜃_𝑎𝛼^†_𝑎 (3.38) for 𝑎=1,2,3, derivatives of fields are replaced by the inner automorphisms 𝜕_𝑧𝑎1𝑓→2𝜃_𝑎𝛿_𝑎𝑏[𝑧^𝑏,𝑓], (3.39) while spacetime averages are replaced by traces over ℋ according to _ℝ6d⁶𝑥𝑓(𝑥)→(2𝜋)³𝜃₁𝜃₂𝜃₃Tr_ℋ(𝑓). (3.40)

In the noncommutative gauge theory, we introduce the covariant coordinates 𝑋^𝑖=𝑥^𝑖+i𝜃^𝑖𝑗𝐴_𝑗 (3.41) and their complex combinations 𝑍^𝑎=12𝜃_𝑎(𝑋^2𝑎−1+i𝑋^2𝑎) (3.42) for 𝑎=1,2,3. Then the (1,1) and (2,0) components of the curvature two-form can be, respectively, expressed as 𝐹_𝑎𝑏=[𝑍_𝑎,𝑍_𝑏1]+2𝜃_𝑎𝛿_𝑎𝑏,𝐹_𝑎𝑏=[𝑍_𝑎,𝑍_𝑏], (3.43) while the covariant derivatives of the Higgs field become (𝜕_𝐴)_𝑎Φ=[𝑍_𝑎,Φ]. (3.44) The instanton equations (3.14), (3.15), and (3.23) then become algebraic equations [𝑍^𝑎,𝑍^𝑏]=0,[𝑍^𝑎,𝑍^†_𝑎]=3,[𝑍_𝑎,Φ]=𝜖_𝑎𝑍_𝑎. (3.45) These equations describe BPS bound states of the D0–D6 system in a 𝐵-field background, which is necessary for reinstating supersymmetry [28, 29]. In addition, 𝑇³-invariance of the (unique) holomorphic three-form on 𝑋 imposes the Calabi-Yau condition 𝜖₁+𝜖₂+𝜖₃=0. (3.46)

3.7. Instanton Moduli Space

A major technical advantage of introducing the noncommutative deformation is that the instanton moduli space can be constructed explicitly, by solving the noncommutative instanton equations (3.45). First we construct the vacuum solution of the noncommutative gauge theory, with 𝐹=0. It is obtained by setting 𝐴=0 and is given explicitly by harmonic oscillator algebra 𝑍^𝑎=𝛼_𝑎,Φ=³_𝑎=1𝜖_𝑎𝛼^†_𝑎𝛼_𝑎. (3.47)

Other solutions are found via the solution generating technique described in, for example, references [30, 31]. For the general solution, fix an integer 𝑛≥1 and let 𝑈_𝑛 be a partial isometry on the Hilbert space ℋ which projects out all states |𝑖,𝑗,𝑘⟩ with 𝑖+𝑗+𝑘<𝑛. Such an operator satisfies the equations 𝑈^†_𝑛𝑈_𝑛=1−Π_𝑛,𝑈_𝑛𝑈^†_𝑛=1, (3.48) where Π_𝑛=_{𝑖+𝑗+𝑘<𝑛}|𝑖,𝑗,𝑘⟩⟨𝑖,𝑗,𝑘| (3.49) is a Hermitean projection operator onto a finite-dimensional subspace of ℋ. Then we make the ansatz 𝑍^𝑎=𝑈_𝑛𝛼_𝑎𝑓(𝑁)𝑈^†_𝑛,Φ=𝑈_𝑛(³_𝑎=1𝜖_𝑎𝛼^†_𝑎𝛼_𝑎)𝑈^†_𝑛, (3.50) where 𝑓 is a real function of the number operator 𝑁=³_𝑎=1𝛼^†_𝑎𝛼_𝑎. (3.51)

Using standard harmonic oscillator algebra, we can write the DUY equations (3.45) as 𝑈_𝑛(𝑁𝑓²(𝑁−1)−(𝑁+3)𝑓²(𝑁)+3)𝑈^†_𝑛=0. (3.52) This recursion relation has a unique solution with the initial conditions 𝑓(𝑖) =0,𝑖 = 0,1,…,𝑛−1, and the finite energy condition 𝑓(𝑟)→1 as 𝑟→∞. It is given by [32] 𝑓(𝑁)=1−𝑛(𝑛+1)(𝑛+2)(𝑁+1)(𝑁+2)(𝑁+3)(1−Π_𝑛). (3.53) The topological charge of the corresponding noncommutative instanton is ℓ(𝑛)=ch₃i(ℰ)=−6𝜃₁𝜃₂𝜃₃Tr_ℋ1(𝐹∧𝐹∧𝐹)=6𝑛(𝑛+1)(𝑛+2). (3.54) Thus the instanton number is the number of states in ℋ with 𝑁<𝑛, that is, the number of vectors removed by 𝑈_𝑛, or equivalently the rank of the projector Π_𝑛.

The partial isometry 𝑈_𝑛 identifies the full Fock space ℋ=ℂ[𝛼^†₁,𝛼^†₂,𝛼^†₃]|0,0,0⟩ with the subspace ℋ_ℐ=⊕_𝑓∈ℐ𝑓(𝛼^†₁,𝛼^†₂,𝛼^†₃)|0,0,0⟩, (3.55) where ℐ=ℂ⟨𝑤^𝑖₁𝑤^𝑗₂𝑤^𝑘₃∣𝑖+𝑗+𝑘≥𝑛⟩ (3.56) is a monomial ideal of codimension ℓ=ℓ(𝑛) in the polynomial ring ℂ[𝑤₁,𝑤₂,𝑤₃]. The instanton moduli space can thus be identified as the Hilbert scheme 𝔐_𝑋=Hilb_ℓ,0(𝑋)=𝑋^[ℓ] (3.57) of ℓ points in 𝑋=ℂ³. The Hilbert-Chow morphism 𝑋^[ℓ]→Sym^ℓ𝑋(𝑋)=^ℓ𝑆_ℓ (3.58) identifies the Hilbert scheme of points as a crepant resolution of the (coincident point) singularities of the ℓth symmetric product orbifold of 𝑋. The ideal ℐ defines a plane partition 𝜋 with |𝜋|=ℓ boxes given by 𝜋={(𝑖,𝑗,𝑘)∣𝑖,𝑗,𝑘≥1,𝑤₁^𝑖−1𝑤₂^𝑗−1𝑤₃^𝑘−1∉ℐ}. (3.59) Heuristically, this configuration represents instantons which sit on top of each other at the origin of ℂ³, and along its coordinate axes where they asymptote to four-dimensional noncommutative instantons at infinity described by ordinary Young tableaux 𝜆.

3.8. Donaldson-Thomas Theory

We can finally compute the instanton contributions to the partition function of the cohomological gauge theory on any toric Calabi-Yau threefold 𝑋 [7, 8, 24]. Let us start with the case 𝑋=ℂ³. Using (3.54), the contribution of an instanton corresponding to a plane partition 𝜋 contributes a factor exp(−i𝑔_𝑠48𝜋³Tr_ℋℐ(𝐹³))=e^{−𝑔𝑠|𝜋|} (3.60) to the Boltzmann weight appearing in the functional integral. There is also a measure factor which comes from integrating out the bosonic and fermionic fields in the supersymmetric gauge theory. This yields a ratio of fluctuation determinants 𝑍_𝜋=det(adΦ)∏_𝑖<𝑗det(adΦ+𝜖_𝑖+𝜖_𝑗)det(adΦ+𝜖₁+𝜖₂+𝜖₃)∏³_𝑖=1det(adΦ+𝜖_𝑖)=exp(−^∞₀d𝑡𝑡ch_ℐ(𝑡)ch_ℐ(−𝑡)(1−e^𝑡𝜖1)(1−e^𝑡𝜖2)(1−e^𝑡𝜖3)) (3.61) with the normalized character ch_ℐ(𝑡)=³_𝑖=1(1−e^𝑡𝜖𝑖)Tr_ℋℐ(e^𝑡Φ)=1−³_𝑖=1(1−e^𝑡𝜖𝑖)_{(𝑖,𝑗,𝑘)∈𝜋}e^{𝑡(𝜖1(𝑖−1)+𝜖2(𝑗−1)+𝜖3(𝑘−1))}, (3.62) where we have used the solution for Φ in (3.50). Using the Calabi-Yau condition 𝜖₁+𝜖₂+𝜖₃=0, it is easy to see that these determinants cancel up to a sign.

After some computation, one can explicitly determine this sign to get 𝑍_𝜋=𝑍_𝜋=∅⋅(−1)^|𝜋|. (3.63) The contribution 𝑍_∅ from the empty partition is the one-loop perturbative contribution to the functional integral, and hence will be dropped. Then the instanton sum for the partition function is given by 𝑍^ℂ3_DT(𝑞)=_𝜋(−e^−𝑔𝑠)^|𝜋|=_𝜋𝑞^|𝜋| (3.64) which is just the MacMahon function 𝑀(𝑞) with 𝑞=−e^−𝑔𝑠. This is the known formula for the Donaldson-Thomas partition function on ℂ³.

For later use, let us note a convenient resummation formula for this partition function [6]. Using interlacing relations, the sum over plane partitions 𝜋 can be converted into a triple sum over the Young tableaux obtained from the main diagonal slice 𝜆=(𝜋_𝑖,𝑖)_𝑖≥1, together with a sum over pairs of semistandard tableaux of shape 𝜆 obtained by putting 𝑚+1 of them in boxes of the skew diagram associated to the 𝑚th diagonal slice for each 𝑚≥0. The partial sum over each semistandard tableaux coincides with the combinatorial definition of the Schur functions at a particular value, which can be expressed through the hook formula 𝑠_𝜆(𝑞^𝜌)=𝑞^{𝑛(𝜆)+|𝜆|/2}_{(𝑖,𝑗)∈𝜆}11−𝑞^{ℎ(𝑖,𝑗)}, (3.65) where 𝑛(𝜆)=∑_𝑖(𝑖−1)𝜆_𝑖, and ℎ(𝑖,𝑗) is the hook length of the box located at position (𝑖,𝑗) in the Young tableau 𝜆⊂ℤ²_≥0. Then the partition function can be rewritten as a sum over ordinary partitions 𝑍^ℂ3_DT(𝑞)=_𝜆𝑠_𝜆(𝑞^𝜌)². (3.66)

This construction can be generalized to arbitrary toric Calabi-Yau threefolds 𝑋 by using the gluing rules of toric geometry. The two simplest such varieties are described by the toric diagrams

with a single vertex representing 𝑋=ℂ³, whose partition function was computed above and is the basic building block for the generating functions on more complicated geometries, and a single line joining two vertices representing the resolved conifold 𝑋=𝒪_ℙ¹(−1)⊕𝒪_ℙ¹(−1), where the ℙ¹ contribute to the 𝐹∧𝐹∧𝜔₀ term of the gauge theory action (3.13). The 𝑇³-invariant noncommutative 𝑈(1) instantons on 𝑋 correspond to ideal sheaves ℰ and are described by the following combinatorial data: (1) three-dimensional partitions 𝜋_𝑓 at each vertex 𝑓 of the toric diagram, corresponding to monomial ideals ℐ_𝑓⊂ℂ[𝑤₁,𝑤₂,𝑤₃], (2) two-dimensional partitions 𝜆_𝑒 at each edge 𝑒 of the toric diagram, representing the four-dimensional instanton asymptotics of 𝜋_𝑓.

This description requires generalizing the calculation on 𝑋=ℂ³ above to compute the perpendicular partition function 𝑃_{𝜆,𝜇,𝜈}(𝑞) [6], which is defined to be the generating function for three-dimensional partitions with fixed asymptotics 𝜆, 𝜇, and 𝜈 in the three coordinate directions. Such partitions correspond to instantons on ℂ³_𝜃 with nontrivial boundary conditions at infinity along each of the coordinate axes. It can be expressed in terms of skew Schur functions, with 𝑍^ℂ3_DT(𝑞)=𝑃_∅,∅,∅(𝑞).

For the example of the resolved conifold, using the gluing rules one easily computes𝑍^conifold_DT(𝑞,𝑄)=_𝜋𝑓𝑞^{|𝜋𝑓|+∑(𝑖,𝑗)∈𝜆(𝑖+𝑗+1)}(−1)^|𝜆|𝑄^|𝜆|=^∞_𝑛=1(1−(−1)^𝑛𝑞^𝑛𝑄)^𝑛(1−(−1)^𝑛𝑞^𝑛)^2𝑛=𝑀(−𝑞)^2∞_𝑛=1(1−(−𝑞)^𝑛𝑄)^𝑛. (3.67) More generally, with these rules one finds that the instanton partition function is the generating function 𝑍^𝑋_DT(𝑞,𝐐)=_{𝑛∈ℤ𝛽∈𝐻2(𝑋,ℤ)}𝐼_𝑛,𝛽(𝑋)𝑞^𝑛𝐐^𝛽 (3.68) for the Donaldson-Thomas invariants 𝐼_𝑛,𝛽(𝑋)∈ℤ, which are defined as follows. The moduli variety Hilb_𝑛,𝛽(𝑋) of ideal sheaves on 𝑋 is a projective scheme with a perfect obstruction theory. For general threefolds 𝑋, it has virtual dimension [33, Lemma 1] _𝛽𝑐₁(𝑋) (3.69) which coincides with that of 𝔐_𝑔(𝑋,𝛽) from Section 2.1. In the Calabi-Yau case, the virtual dimension is zero, and the corresponding virtual cycle is [Hilb_𝑛,𝛽(𝑋)]^vir∈𝐶𝐻₀(Hilb_𝑛,𝛽(𝑋)). (3.70) Then the Donaldson-Thomas invariants 𝐼_𝑛,𝛽(𝑋)∶=_{[Hilb𝑛,𝛽(𝑋)]vir}1 (3.71) count the virtual numbers of ideal sheaves on 𝑋 with the given Chern character. The right-hand side is defined via equivariant integration, as explained in Section 3.5. The torus action on 𝑋 lifts to the moduli scheme Hilb_𝑛,𝛽(𝑋). The 𝑇-fixed locus Hilb_𝑛,𝛽(𝑋)𝑇 has a 𝑇-equivariant virtual theory with cycle [Hilb_𝑛,𝛽(𝑋)𝑇]^vir∈𝐶𝐻₀(Hilb_𝑛,𝛽(𝑋)𝑇) and virtual normal bundle 𝔑^vir_𝑋 in the equivariant K-theory 𝐾⁰_𝑇(Hilb_𝑛,𝛽(𝑋)𝑇). This construction gives precise meaning to the moduli space integral (3.17) via application of the virtual localization formula in equivariant Chow theory, described in Section 3.5.

3.9. Wall-Crossing Formulas

We will now make contact with Section 2. For the present class of threefolds 𝑋, there is a gauge theory/string theory duality [27]. This follows from the fact that the perpendicular partition function 𝑃_{𝜆,𝜇,𝜈}(𝑞) is related to the Calabi-Yau crystal formulation of the topological vertex 𝐶_{𝜆,𝜇,𝜈}(𝑞)=𝑀(𝑞)⁻¹𝑞^{(1/2)(‖𝜆‖2+‖𝜇‖2+‖𝜈‖2)}𝑃_{𝜆,𝜇,𝜈}(𝑞) (3.72) with ‖𝜆‖²=∑_𝑖𝜆²_𝑖, which are the building blocks for the computation of the generating function for Gromov-Witten invariants using rules analogous to those described in Section 3.8 [14, 15]. Using these relations one can show that the six-dimensional cohomological gauge theory is S-dual to the A-model topological string theory. The respective partition functions are related by 𝑍^𝑋_top(𝑔_𝑠,𝐐)=𝑀(𝑞)^{−𝜒(𝑋)}𝑍^𝑋_DT(𝑞=−e^−𝑔𝑠,𝐐), (3.73) where the Euler characteristic 𝜒(𝑋) of 𝑋 is the number of vertices in its toric diagram. For the conifold example, the gluing rules for the topological vertex yield [14, 15] 𝑍^conifold_top(𝑔_𝑠,𝑄)=_𝜆𝐶_∅,∅,𝜆(𝑞)𝐶_∅,∅,𝜆(𝑞)𝑄^|𝜆|=exp(^∞_𝑛=1𝑄^𝑛𝑛1(𝑞^𝑛/2−𝑞^−𝑛/2)²), (3.74) which should be compared with (3.67). This Gromov-Witten/Donaldson-Thomas correspondence is known to hold for arbitrary toric threefolds [34]. The relationship (3.73) can be thought of as a wall-crossing formula, as we now explain.

The relationship (3.73) is in apparent contradiction with the OSV conjecture (2.25) if we wish to interpret the right-hand side as the generating function 𝑍_BH(1,0,𝜙²,𝜙⁰) of a suitable index for black hole microstates. However, the conjectural relations (2.25) and (3.73) hold in different regimes of validity. The number of BPS particles in four dimensions formed by wrapping supersymmetric bound states of D-branes around holomorphic cycles of 𝑋 depends on the choice of a stability condition, and the BPS countings for different stability conditions are related by wall-crossing formulas. For example, stability of black holes requires that their chemical potentials 𝜇^𝐼 lie in the ranges 𝒬₀𝜙⁰>0 and 𝒬^𝑖₂𝜙²_𝑖>0.

On the other hand, the validity of (3.73) is related to the existence of BPS invariants 𝐵_𝑔,𝛽(𝑋)∈ℤ such that the topological string amplitudes have an expansion given by [35, 36] 𝑍^𝑋_top(𝑔_𝑠,𝐐)=^∞_{𝑔=0𝛽∈𝐻2(𝑋,ℤ)}𝐐^𝛽_{2𝛾∈𝐻(𝑋,ℤ)⧵{0}𝛽=𝑘𝛾}𝐵_𝑔,𝛽1(𝑋)𝑘(2sinh(𝑘𝑔_𝑠2))^2𝑔−2, (3.75) of which the conifold partition function (3.67) is an explicit case. These are partition functions of D6-D2-D0 brane bound states only for certain Kähler moduli. Analyses of Calabi-Yau compactifications of Type II string theory show that the Hilbert spaces of BPS states jump discontinuously across real codimension one walls in the moduli space of vacua, known as walls of marginal stability. The noncommutative instantons do not account for walls of marginal stability extending to infinity. One should instead apply some sort of stability condition (such as Π-stability) to elements of the bounded derived category of coherent sheaves D^𝑏(coh(𝑋)) of the given charge, which gives a topological classification of A-model D-branes on 𝑋. These issues are discussed in more detail in [37–39].

From a mathematical perspective, we can study this phenomenon by looking at framed moduli spaces, which consist of instantons that are trivial “at infinity”. More precisely, we can consider a toric compactification of 𝑋 obtained by adding a compactification divisor 𝐷_∞, and consider sheaves ℱ with a fixed trivialization on 𝐷_∞. The Kähler polarization defined by 𝜔₀ allows us to define the moduli space 𝔐^𝜔0_𝑋=𝔐_𝑋 of stable sheaves. Then the symbolic definition of the gauge theory partition function (3.17) as a particular Euler characteristic can be made precise in the more local definition of Donaldson-Thomas invariants given by [40].

As a scheme with a perfect obstruction theory, the instanton moduli space 𝔐_𝑋 can be viewed locally as the scheme theoretic critical locus of a holomorphic function, the superpotential 𝑊, on a compact manifold 𝒳 with the action of a gauge group 𝒢. 𝔐_𝑋 has virtual dimension zero, and at nonsingular points, the obstruction sheaf 𝔑_𝑋 on 𝔐_𝑋 coincides with the cotangent bundle. Hence if 𝔐_𝑋 were everywhere nonsingular, then the partition function (3.17) would just compute the signed Euler characteristic (−1)^{dimℂ(𝔐𝑋)}𝜒(𝔐_𝑋). At singular points, however, the invariants differ from these characteristics.

There is a constructible function 𝜈∶𝔐_𝑋→ℤ which can be used to define the weighted Euler characteristic 𝜒(𝔐_𝑋,𝜈)∶=_𝑛∈ℤ𝑛⋅𝜒(𝜈⁻¹(𝑛)). (3.76) For sheaves of fixed Chern character, this coincides with the curve-counting invariants 𝐼_𝑛,𝛽(𝑋). At nonsingular points, 𝜈=(−1)^{dimℂ(𝔐𝑋)}, while at singular points it is the more complicated function given by 𝜈(ℰ)=(−1)^{dimℂ(𝒳/𝒢)}(1−MF_𝑊(ℰ)), (3.77) where MF_𝑊(ℰ) is the Milnor fibre of the superpotential 𝑊 at the point corresponding to ℰ. The weighted Euler characteristic is a deformation invariant of 𝑋.

In this approach, one can use topological Euler characteristics to define 𝐼_𝑛,𝛽(𝑋) as invariants associated to moduli varieties of framed sheaves. Fixing 𝛽∈𝐻₂(𝑋,ℤ) and 𝑛∈ℤ, the variety Hilb_𝑛,𝛽(𝑋) equivalently parametrizes isomorphism classes of the following objects: (a) surjections (framings) 𝒪_𝑋→ℱ→0 (3.78) with ch(ℱ)=(1,0,𝛽,𝑛),(b) stable sheaves ℰ with ch(ℰ)=(1,0,−𝛽,−𝑛) and trivial determinant, (c) subschemes 𝑆⊂𝑋 of dimension ≤1 with curve class [𝑆]=𝛽 and holomorphic Euler characteristic 𝜒(𝒪_𝑆)=𝑛.

The equivalences between these three descriptions are described explicitly for 𝑋=ℂ³ in [7, 8].

As we vary the polarization 𝜔₀, the moduli spaces 𝔐^𝜔0_𝑋(𝑋) change and so do the associated counting invariants, leading to a wall-and-chamber structure. The wall-crossing behaviour of the enumerative invariants is studied in [41, 42]. The analog of varying 𝜔₀ for framed sheaves is to consider quotients of the structure sheaf 𝒪_𝑋 in different abelian subcategories of the bounded derived category D^𝑏(coh(𝑋)) of coherent sheaves on 𝑋. The analog of wall-crossing gives the Pandharipande-Thomas theory of stable pairs [43, 44] and the BPS invariants above. For this, the quotients of 𝒪_𝑋 are the stable pairs (ℰ,𝛼), where ℰ is a coherent 𝒪_𝑋-module of pure dimension one with ch₂(ℰ)=−𝛽 and 𝜒(ℰ)=−𝑛, and 𝛼∶𝒪_𝑋→ℰ is a nonzero sheaf map such that coker(𝛼) is of pure dimension zero, together with le Poitier's 𝛿-stability condition for coherent systems. In this case the change of Donaldson-Thomas invariants is described by the Kontsevich-Soibelman wall-crossing formula [42].

To cast these constructions into the language of noncommutative instantons, a proper definition of noncommutative toric manifolds is desired, beyond the heuristic approach presented above whereby only open ℂ³ patches are deformed. Isospectral type deformations of toric geometry, and instantons therein, are investigated in [45]. It may also aid in the classification of 𝑈(𝑁) noncommutative instantons on ℂ³ for rank 𝑁>1, along the lines of what was done in Section 3.7. (See [7, 8] for some explicit examples.) This appears to be related to the problem of defining a nonabelian version of Donaldson-Thomas theory which counts higher-rank torsion-free sheaves, for which no general, appropriate notion of stability is yet known.

4. D4-Brane Gauge Theory and Euler Characteristics

In this section we will take 𝒬₆=0 (no D6-branes) and consider 𝑁 D4-branes wrapping a four-cycle 𝐶⊂𝑋. In this case the worldvolume gauge theory on the D4-branes is the 𝒩=4 Vafa-Witten topologically twisted 𝑈(𝑁) Yang-Mills theory on 𝐶, where the topological twist is generically required in order to realize covariantly constant spinors on a curved geometry. When the gauge theory is formulated on an arbitrary toric singularity 𝐶 in four dimensions, we may regard 𝐶 as a four-cycle inside the Calabi-Yau threefold 𝑋=𝐾_𝐶, and we will obtain an explicit description of the instanton moduli spaces and their Euler characteristics. The precise forms of the partition functions will be amenable to checks of the OSV conjecture (2.25), and hence a description of wall-crossing phenomena.

4.1. 𝒩=4 Supersymmetric Yang-Mills Theory on Kähler Surfaces

Vafa and Witten [46] introduced a topologically twisted version of 𝒩=4 supersymmetric Yang-Mills theory in four dimensions. The twisting procedure modifies the quantum numbers of the fields in the physical theory in such a way that a particular linear combination of the supercharges becomes a scalar. This scalar supercharge is used to define the cohomological field theory and its observables on an arbitrary four-manifold 𝐶. In the following we will only consider the case where 𝐶 is a connected smooth Kähler manifold with Kähler two-form 𝑘₀. When certain conditions are met, the partition function of the twisted gauge theory computes the Euler characteristic of the instanton moduli space.

Let 𝑔_𝑖𝑗 be the Kähler metric of (𝐶,𝑘₀). Then the twisted gauge theory corresponds to the moduli problem associated with the equations𝜎_𝑖𝑗∶=𝐹⁺_𝑖𝑗+14[𝐵⁺_𝑖𝑘,𝐵⁺_𝑗𝑙]𝑔^𝑘𝑙+12[Φ,𝐵⁺_𝑖𝑗𝜅]=0,_𝑖∶=d^𝑗_𝐴𝐵⁺_𝑖𝑗+(d_𝐴)_𝑖Φ=0, (4.1) where 𝐹⁺=(1/2)(𝐹−⋆𝐹) is the self-dual part of the curvature two-form with respect to the Kähler metric. The field space 𝔚 is spanned by a connection 𝐴_𝑖 on a principal 𝐺-bundle 𝒫→𝐶, a scalar field Φ, and a self-dual two-form 𝐵⁺_𝑖𝑗, so that𝔚=𝔄_𝒫×Ω⁰(𝐶,ad𝒫)×Ω^2,+(𝐶,ad𝒫), (4.2) where 𝔄_𝒫 denotes the space of connections on 𝒫 and ad𝒫 is the adjoint bundle of 𝒫. Their superpartners 𝜓_𝑖, 𝜁, and 𝜓⁺_𝑖𝑗 live in the tangent space to 𝔚. Associated with the equations of motion are two multiplets (𝜒⁺_𝑖𝑗,𝐻⁺_𝑖𝑗) and (𝜒_𝑖,𝐻_𝑖) which are sections of the bundle 𝔉=Ω^2,+(𝐶,ad𝒫)⊕Ω¹(𝐶,ad𝒫). (4.3) Schematically, the action of the topological gauge theory is of the form 𝑆={𝑄,Ψ}+_𝐶Tr(𝐹∧𝐹)+_𝐶Tr(𝐹∧𝑘₀), (4.4) where 𝑄 is the scalar supercharge singled out by the twisting procedure. The gauge fermion Ψ is a suitable functional of the fields which contains the term _𝐶√𝑔Tr(𝜒⁺_𝑖𝑗(𝐻^+𝑖𝑗+𝜎^𝑖𝑗)+𝜒_𝑖(𝐻^𝑖+𝜅^𝑖)), (4.5) that makes the gauge theory localize onto the solutions of (4.1).

Geometrically, the partition function can be interpreted as a Mathai-Quillen representative of the Thom class of the bundle 𝔙=𝔚×_𝒢𝔉, where 𝒢=Aut(𝒫) is the group of gauge transformations. Its pullback via the sections in (4.1) gives the Euler class of 𝔙. Under favourable circumstances, appropriate vanishing theorems hold [46] which ensure that each solution of the system (4.1) has Φ=𝐵⁺=0 and corresponds to an instanton, that is, a solution to the self-duality equations 𝐹⁺=0. In this case the gauge theory localizes onto the instanton moduli space 𝔐_𝐶 and the Boltzmann weight gives a representative of the Euler class of the tangent bundle 𝑇𝔐_𝐶. Therefore, the partition function computes moduli space integrals of the form _𝔐𝐶𝑒(𝑇𝔐_𝐶)=𝜒(𝔐_𝐶), (4.6) which gives the Euler characteristic of the instanton moduli space. Since the instanton moduli space is not generally a smooth variety, most of the quantities introduced above can only be defined formally. We will discuss how to define these integrations more precisely later on. In particular, we will allow for nontrivial vacuum expectation values for the Higgs field Φ, in order to define the partition function in the Ω-background as before. We will assume that the vanishing theorems can be extended to this case as well, by replacing the instanton moduli space with its compactification obtained by adding torsion-free sheaves on 𝐶 as before.

The Euler characteristic of instanton moduli space can be computed through the index of the deformation complex associated with 𝒩=4 topological Yang-Mills theory via (4.1). It has the form [47]Ω⁰(𝐶,ad𝒫)^DΩ−−−−−→¹⊕Ω(𝐶,ad𝒫)₀⊕Ω(𝐶,ad𝒫)_2,+(𝐶,ad𝒫)^sΩ−−−−−→^2,+⊕Ω(𝐶,ad𝒫)₁(𝐶,ad𝒫) (4.7) where the first morphism is an infinitesimal gauge transformation dD(𝜙)=^𝐴𝜙[Φ,𝜙][𝐵⁺,𝜙], (4.8) while the second morphism corresponds to the linearization of the sections (𝜎_𝑖𝑗,𝜅_𝑖) given by s(𝜓,𝜁,𝜓⁺)=𝑝⁺d_𝐴𝜓−[𝜓⁺,𝐵⁺]+[𝜓⁺,Φ]+[𝐵⁺,𝜁]+d_𝐴𝜁+[𝜓,Φ]+𝑝⁺d^∗_𝐴𝜓⁺+[𝜓,𝐵⁺], (4.9) with 𝑝⁺ giving the projection of a two-form onto its self-dual part. Under the assumption that all solutions of the original system of (4.1) have Φ=𝐵⁺=0, the complex (4.7) splits into the Atiyah-Hitchin-Singer instanton deformation complex Ω⁰(𝐶,ad𝒫)−−−−−^d𝐴−→Ω¹(𝐶,ad𝒫)−−−−−−−^{𝑝+∘d𝐴}−−→Ω^2,+(𝐶,ad𝒫) (4.10) plus Ω⁰(𝐶,ad𝒫)⊕Ω^2,+(𝐶,ad𝒫)−−−−−−−−−→^{(d𝐴,𝑝+∘d∗𝐴)}Ω¹(𝐶,ad𝒫) (4.11) which is again the instanton deformation complex. One can compute the index of the original complex (4.7) (assuming the Vafa-Witten vanishing theorems) by computing the index of the two complexes above. However, these are equal and contribute with opposite signs. This means that the Euler characteristic of instanton moduli space receives contributions only from isolated points and simply counts the number of such points. On a toric surface 𝐶, this is anticipated from the toric localization formula (3.26) and will be made explicit below.

In the applications to black hole microstate counting, we will consider gauge group 𝐺=𝑈(𝑁). The chemical potential ∫_𝐶𝐶₍₂₎∧Tr(𝐹) for the D2-branes requires taking the Ramond-Ramond field 𝐶₍₂₎ proportional to the two-form 𝑘^𝑖 on 𝐶 which are dual to the basis two-cycle 𝑆_𝑖, in order to get the correct charges. In this case the D0-brane charges 𝒬₀=18𝜋²_𝐶Tr(𝐹∧𝐹) (4.12) correspond to the instanton numbers of the gauge bundle 𝒫, while the D2-brane charges 𝒬^𝑖₂=12𝜋_𝑆𝑖Tr(𝐹) (4.13) correspond to nontrivial magnetic fluxes 𝑐₁(𝒫)≠0 through 𝑆_𝑖. To compute the macroscopic black hole entropy from the counting of the corresponding BPS states in the gauge theory, we introduce observables associated to these sources and compute their gauge theory expectation values using the localization arguments above to get 𝑍_BH(𝑁,𝜙²,𝜙⁰𝜙)=⟨exp(−⁰8𝜋²_𝐶𝜙Tr(𝐹∧𝐹)−^2𝑖2𝜋_C𝑘^𝑖∧Tr(𝐹))⟩_SYM=_{𝒬0,𝒬𝑖2}Ω(𝒬₀,𝓠₂;𝑁)e^{−𝒬0𝜙0−𝒬𝑖2𝜙2𝑖}, (4.14) where Ω(𝒬₀,𝓠₂;𝑁) is the Witten index which computes the Euler characteristic of the moduli space 𝔐_{𝑁,𝓠2,𝒬0}(𝐶) of 𝑈(𝑁) instantons on 𝐶 with Chern invariants 𝑐₁(𝒫)=𝓠₂∈𝐻²(𝑋,ℤ) and −ch₂(𝒫)=𝒬₀𝜈∈𝐻⁴(𝑋,ℤ). Here 𝜈 is the generator of 𝐻⁴(𝑋,ℤ)≅ℤ which is Poincaré dual to a point in 𝑋.

4.2. Toric Localization and the Instanton Moduli Space

Instantons on 𝐶=ℂ² can be described as follows. Consider the quiver

with the single relation 𝑟 specified by the linear combination of paths 𝑟=[𝑏₁,𝑏₂]+𝑖𝑗. (4.15) This is called the ADHM quiver Q_ADHM. The ADHM construction establishes a one-to-one correspondence between stable framed representations of the quiver Q_ADHM in the category Vect_ℂ of finite-dimensional complex vector spaces and framed torsion-free sheaves on the projective plane ℙ². In the rank one case, these are equivalent to ideal sheaves on ℂ², and the correspondence gives an isomorphism with the Hilbert schemes of points on ℂ².

Let 𝑉 and 𝑊 be inner product spaces of complex dimensions 𝑘=𝒬₀ and 𝑁, respectively. The instanton moduli space 𝔐_𝑁,𝑘(ℂ²) can be realized as a hyperKähler quotient by the natural action of 𝑈(𝑘) on the variety consisting of linear operators 𝐵₁,𝐵₂∈Hom_ℂ(𝑉,𝑉),𝐼∈Hom_ℂ(𝑊,𝑉),𝐽∈Hom_ℂ(𝑉,𝑊), (4.16) constrained by the ADHM equations 𝜇_𝑐=[𝐵₁,𝐵₂𝜇]+𝐼𝐽=0,_𝑟=[𝐵₁,𝐵^†₁]+[𝐵₂,𝐵^†₂]+𝐼𝐼^†−𝐽^†𝐽=0. (4.17) On ℂ² one can obtain a better compactification of this moduli space by deforming the gauge theory to a noncommutative field theory, as before [2, 3, 48]. This is equivalent to a modification of the hyperKähler quotient that defines the instanton moduli space, obtained by changing the images of the moment maps of (4.17) to 𝜇_𝑐=0,𝜇_𝑟=𝜁i𝑑_𝑉, (4.18) where 𝜁=𝜃₁+𝜃₂. This quotient gives a compactification of the instanton moduli space 𝔐_𝑁,𝑘(ℂ²) obtained by blowing up its singularities.

The classification of toric fixed points is given in [49], by identifying the instanton moduli space 𝔐_1,𝑘(ℂ²) with the Hilbert scheme of points (ℂ²)^[𝑘]. The fixed points are point-like instantons which are in one-to-one correspondence with Young tableaux 𝜆 having |𝜆|=𝑘 boxes. In the more general case of a 𝑈(𝑁) gauge theory in the Coulomb branch, one takes 𝑁 copies of the 𝑈(1) theory and the fixed points are classified in terms of 𝑁-tuples of Young diagrams 𝜆=(𝜆₁,…,𝜆_𝑁), called 𝑁-coloured Young diagrams. One can show that the fixed points are isolated.

One can describe the local structure of the instanton moduli space by using the ADHM construction at the fixed points of the 𝑇=𝑈(1)^𝑁×𝑇² action. Following [49], one introduces a two-dimensional 𝑇²-module 𝑄 to keep track of the toric action. The operators (𝐵₁,𝐵₂,𝐼,𝐽) corresponding to a fixed point configuration are elements of the 𝑇-modules (𝐵₁,𝐵₂)∈Hom_ℂ(𝑉,𝑉)⊗𝑄,𝐼∈Hom_ℂ(𝑊,𝑉),𝐽∈Hom_ℂ(𝑉,𝑊)⊗²𝑄. (4.19) Then the local structure of the instanton moduli space is described by the complexHom_ℂ(𝑉,𝑉)−−−−^𝜎−→Hom_ℂ(𝑉,𝑉)⊗𝑄⊕Hom_ℂ⊕(𝑊,𝑉)Hom_ℂ(𝑉,𝑊)⊗⋀²𝑄^𝜏−−−−−→Hom_ℂ(𝑉,𝑉)⊗⋀²𝑄 (4.20) which is just a finite-dimensional version of the Atiyah-Hitchin-Singer instanton deformation complex (4.10). The map 𝜎 corresponds to infinitesimal (complex) gauge transformations while 𝜏 is the linearization of the ADHM constraint 𝜇_𝑐=0. In general, the complex (4.20) has three nonvanishing cohomology groups. In our case we can safely assume that 𝐻⁰ and 𝐻² vanish. The only nonvanishing cohomology 𝐻¹ describes field configurations that obey the linearized ADHM constraint 𝜇_𝑐=0 but are not gauge variations. It is thus a local model for the tangent space to the instanton moduli space at each 𝑇-fixed point. Later on we will compute weights of the toric action on the tangent space modelled on (4.20).

The partition function of the 𝑈(1) topologically twisted gauge theory on 𝑋=ℂ² is easily computed. The only nontrivial topological charge is the instanton number 𝑘=−∫_𝑋𝐹∧𝐹 and therefore the partition function has the form 𝑍^ℂ2_𝑈(1)(𝑞)=^∞_𝑘=0𝑞^𝑘𝜒(𝔐_1,𝑘(ℂ²)). (4.21) The expansion parameter can be identified in terms of gauge theory variables 𝑞∶=e^2𝜋i𝜏 with 𝜏=4𝜋i𝑔²_YM+𝜗2𝜋 (4.22) the complexified gauge coupling, which is related to topological string variables 𝑔_𝑠=𝑔²_YM/2 at the attractor point. At a toric fixed point, 𝑘 is identified as the number of boxes in a partition 𝜆. The Euler classes exactly cancel in the localization formula (3.26), and one is left with the sum over fixed points [50] 𝜒(𝔐_1,𝑘(ℂ²))=_{𝜆∶|𝜆|=𝑘}1. (4.23) By Euler's formula, one has 𝜂(𝑞)⁻¹∶=^∞_𝑛=111−𝑞^𝑛=^∞_𝑁=0𝑝(𝑁)𝑞^𝑁, (4.24) where 𝑝(𝑁) is the number of partitions 𝜆=(𝜆₁,𝜆₂,…) (ordinary, two-dimensional Young tableaux) of degree |𝜆|=∑_𝑖𝜆_𝑖=𝑁. The function 𝜂(𝑞) is related to the Dedekind function. It follows that the 𝑈(1) partition function 𝑍^ℂ2_𝑈(1)(𝑞)=𝜂(𝑞)⁻¹ (4.25) is the generating function for two-dimensional Young diagrams.

This construction can be easily generalized to the nonabelian case. The fixed points are now 𝑁-coloured Young tableaux 𝜆=(𝜆₁,…,𝜆_𝑁) corresponding to a partition of the instanton number 𝑘=(𝑘₁,…,𝑘_𝑁). The instanton action is again equal to 𝑘, where the additional factor of 𝑁 arising from the sum over colours 𝑙 = 1,…,𝑁 cancels with the normalization of the 𝐹∧𝐹 term which carries a factor 1/𝑁 (the inverse of the dual Coxeter number of the gauge group 𝐺=𝑈(𝑁)). The Euler characteristic of instanton moduli space is now 𝜒(𝔐_𝑁,𝑘(ℂ²))=_{𝜆∶|𝜆|=𝑘}1 (4.26) with |𝜆|∶=∑_𝑙|𝜆_𝑙|. The 𝑈(𝑁) partition function reduces to 𝑁 copies of the 𝑈(1) partition function 𝑍^ℂ2_𝑈(𝑁)(𝑞)=(𝑍^ℂ2_𝑈(1)(𝑞))^𝑁. (4.27) This factorization follows from the fact that after toric localization, the gauge symmetry 𝑈(𝑁)→𝑈(1)^𝑁 is broken to the maximal torus. The Coulomb phase corresponds to well-separated D4-branes, but the topological nature of the gauge theory ensures that the partition function is independent of the Higgs moduli representing the lengths of open strings stretching between D-branes. In the rest of this section, we will extend these constructions to generic toric surfaces 𝑋.

4.3. Hirzebruch-Jung Spaces

Our main example will be the most general toric singularity in four dimensions, which defines a class of toric Calabi-Yau (hence open) surfaces known as Hirzebruch-Jung spaces 𝐶=𝐶(𝑝,𝑛). They are determined by two relatively prime positive integers 𝑝 and 𝑛 with 𝑝>𝑛. Consider the quotient singularity ℂ²/Γ_(𝑝,𝑛), with the generator of the cyclic group Γ_(𝑝,𝑛)≅ℤ_𝑝 acting on (𝑧,𝑤)∈ℂ² as (𝑧,𝑤)↦(e^{2𝜋i𝑛/𝑝}𝑧,e^2𝜋i/𝑝𝑤). (4.28) This orbifold has an 𝐴_𝑝,𝑛 singularity at the origin of ℂ². Then 𝐶(𝑝,𝑛) is defined to be the minimal resolution of the 𝐴_𝑝,𝑛 singularity by a chain of ℓ exceptional divisors 𝑆_𝑖≅ℙ¹ whose intersection numbers are summarized in the intersection matrix 𝐂=−𝑒₁10…01−𝑒₂1…001−𝑒₃…0⋮⋮⋮⋱⋮000…−𝑒_ℓ, (4.29) which is called a generalized Cartan matrix. The divisors thus only intersect transversally with their nearest neighbours in the chain. The self-intersection numbers 𝑒_𝑖≥2 of the spheres ℙ¹ of the blow-up are determined from the continued fraction expansion 𝑝𝑛=𝑒₁−1𝑒₂−1𝑒₃−1⋱𝑒_ℓ−1−1𝑒_ℓ. (4.30) Let us consider two particular well-known instances of these spaces.

4.3.1. Local ℙ¹

Setting 𝑛=1, the space 𝐶(𝑝,1) can be identified with the total space of the holomorphic line bundle 𝒪_ℙ¹(−𝑝) over ℙ¹ of degree −𝑝, with ℓ=1 and 𝑒₁=𝑝. In this case, 𝑆=ℙ¹ is the zero section divisor. In the context of topological string theory, such four cycles appear in the “local” Calabi-Yau threefolds 𝑋 which are regarded as neighbourhoods of a holomorphically embedded rational curve in a compact Calabi-Yau threefold, that is, as the normal bundle 𝒩→ℙ¹. Since 𝒩 is a holomorphic vector bundle of rank two over ℙ¹ and the Calabi-Yau condition implies 𝑐₁(𝒩)=−𝜒(ℙ¹)=−2, it follows that 𝑋 is the total space of a bundle of the form 𝒪_ℙ¹(−𝑝)⊕𝒪_ℙ¹(𝑝−2)→ℙ¹.

4.3.2. 𝐴_𝑝−1 ALE Space

The complex surface 𝐶=𝐶(𝑝,𝑝−1) is an example of an asymptotically locally Euclidean (ALE) space. This means that 𝐶 carries a scalar flat Kähler metric 𝑔 such that (𝐶,𝑔) is complete, and there exists a compact set 𝐾 such that 𝐶⧵𝐾≅(ℝ⁴⧵𝐵_𝑅)/ℤ_𝑝. Here ℤ_𝑝⊂𝑂(4) acts freely on ℝ⁴⧵𝐵_𝑅 and the metric 𝑔 approximates the flat Euclidean metric on ℝ⁴. Such a coordinate system is called a coordinate system at infinity. We regard ℤ_𝑝⊂𝑈(2) acting on (𝑧,𝑤)∈ℂ²≅ℝ⁴ as described above (with 𝑛=𝑝−1), and the complex structure 𝐼 on 𝐶 approximates that on ℂ²=ℝ⁴. As the resolution of the Klein singularity ℂ²/ℤ_𝑝, 𝐶(𝑝,𝑝−1) contains a chain of ℓ=𝑝−1 projective lines ℙ¹, each with self-intersection number 𝑒_𝑖=2. In this case, the intersection matrix 𝐂 coincides with the Cartan matrix of the 𝐴_𝑝−1 Dynkin diagram.

4.4. Instantons on ALE Spaces

We begin by describing in some detail the instanton moduli space in the case of the 𝐴_𝑝−1 ALE spaces, for which a rigorous construction is known. 𝑈(𝑁) instantons on ALE spaces are given by the ADHM construction. Since the topological gauge theory is invariant under blow-ups of the surface (using blow-up formulas), one can do the instanton computation on the orbifold ℂ²/Γ where Γ=Γ_{(𝑝,𝑝−1)}≅ℤ_𝑝. This result is at the heart of the McKay correspondence which provides a one-to-one correspondence between irreducible representations of the orbifold group Γ and tautological bundles over the exceptional divisors of the minimal resolution 𝐶=𝐶(𝑝,𝑝−1).

Since 𝐶 is noncompact, the instanton moduli space 𝔐_𝐶 must be defined with respect to connections which have appropriate asymptotic decay at infinity. We will describe this in more generality later on in terms of framed moduli spaces of torsion-free sheaves. These connections correspond to instantons of finite energy and are asymptotic to flat connections with 𝐹=0. In particular, there are solutions which have fractional first Chern class and are related to instantons that asymptote to flat connections with nontrivial holonomy at the boundary of 𝐶, which is topologically the Lens space 𝐿(𝑝,𝑝−1)=𝑆³/Γ. The flat connections are classified by homomorphisms 𝜌∶𝜋₁(𝐶)→𝑈(𝑁), where 𝜋₁(𝐶)=Γ≅ℤ_𝑝. The asymptotic connection at infinity is thus labelled by irreducible representations (𝑘₀,𝑘₁,…,𝑘_𝑝−1) of the orbifold group ℤ_𝑝, with ∑_𝑖𝑘_𝑖=𝑁, and are given explicitly by 𝜌_𝑘(e^2𝜋i/𝑝)=e^{2𝜋i𝑘/𝑝}, (4.31) where 𝑘=0,1,…,𝑝−1.

Starting from the ADHM construction on ℂ² outlined in Section 4.2., one constructs its Γ-invariant decomposition. Consider the universal scheme 𝒵⊂𝐶×ℂ² given by the correspondence diagram (4.32)

The tautological bundle on 𝐶 is defined by ℛ∶=𝑞_1∗𝒪_𝒵. (4.33) Under the action of Γ on 𝒵,ℛ transforms in the regular representation and can thus be decomposed into irreducible representations ℛ=⊕^𝑝−1_𝑘=0ℛ_𝑘⊗𝜌_𝑘,ℛ_𝑘=Hom_Γ(ℛ,𝜌_𝑘). (4.34) By the McKay correspondence, the bundles ℛ₀=𝒪_𝐶,ℛ₁,…,ℛ_𝑝−1 form the canonical integral basis of the K-theory group 𝐾⁰(𝐶) constructed in [51].

In this case, we take 𝑄≅ℂ² to be a module on which the regular representation of Γ acts. We also take Γ⊂𝑆𝑈(2) so that the determinant representation is trivial as a Γ-module, that is, ∧²𝑄⊗_Γℛ≅ℛ. The two vector spaces 𝑉 and 𝑊 which feature in the ADHM construction have a natural grading under the action of the orbifold group Γ given by 𝑉=⊕^𝑝−1_𝑘=0𝑉_𝑘⊗𝜌_𝑘,𝑊=⊕^𝑝−1_𝑘=0𝑊_𝑘⊗𝜌_𝑘. (4.35) The modification of (4.20) is given by [52]Hom_Γ(ℛ^∗,𝑉)−−−−^𝜎−→Hom_Γ(ℛ^∗,𝑄⊗_Γ⊕𝑉)Hom_Γ(ℛ^∗,𝑊)^𝜏−−−−−→Hom_Γ(ℛ^∗,𝑉), (4.36) and the condition that the sequence (4.36) is a complex is equivalent to the (generalized) ADHM equations. After imposing a certain stability condition, this construction realizes the instanton moduli space as a quiver variety 𝔐(𝑉,𝑊).

This construction identifies two distinct types of instanton contributions to the ALE partition function, which we consider in turn. As before, after toric localization the gauge symmetry breaks as 𝑈(𝑁)→𝑈(1)^𝑁 and the 𝑈(𝑁) partition function factorizes as 𝑍^ALE_𝑈(𝑁)(𝑞,𝐐)=(𝑍^ALE_𝑈(1)(𝑞,𝐐))^𝑁. (4.37) It therefore suffices to focus on the 𝑈(1) case in the following.

4.4.1. Regular Instantons

Regular instantons on 𝐴_𝑝−1 live in the regular representation 𝑘₀=𝑘₁=⋯=𝑘_𝑝−1=𝑘 of the orbifold group Γ=ℤ_𝑝. They correspond to D0-branes moving freely on 𝐶 with 𝑝 orbifold images away from the orbifold point. For gauge group 𝑈(1), the moduli space is given by specifying 𝐾 = 𝑘𝑝 points on 𝐶 up to permutations. Hence the moduli space 𝔐^𝑈(1)_r𝑒𝑔(𝐶) of regular 𝑈(1) instantons on 𝐶 is isomorphic to the Hilbert scheme 𝐶^[𝐾]. The generating function for the Euler numbers of the instanton moduli spaces can then be computed explicitly by applying Göttsche's formula to get 𝑍^𝑈(1)_reg(𝑞)=^∞_𝐾=0𝑞^𝐾𝜒(𝐶^[𝐾])=𝜂(𝑞)^−𝑝. (4.38) The 𝑈(𝑁) partition function is the 𝑁th power of this quantity. Heuristically, we may think of this formula as originating by covering 𝐶 with 𝑝=𝜒(𝐶) open charts to get 𝑝 copies of 𝑈(𝑁) instantons on ℂ², each contributing 𝜂(𝑞)^−𝑁. This can be demonstrated rigorously on any toric surface 𝐶 by a localization computation [53, 54].

4.4.2. Fractional Instantons

To each irreducible representation (𝑘₀,𝑘₁,…,𝑘_𝑝−1) of the orbifold group ℤ_𝑝, there corresponds a fractional instanton which is stuck at the orbifold points. It has no moduli (or orbifold images) and can be regarded as a state in which open strings ending on the same D0-brane are projected out by the action of the orbifold group. They carry magnetic fluxes through the ℙ¹'s of the minimal resolution, and correspond to self-dual 𝑈(1) gauge connections with curvatures 𝐹=−2𝜋i𝑢_𝑖𝑐₁(ℛ_𝑖), (4.39) where 𝑢_𝑖∈ℤ and ℛ_𝑖=𝒪_ℙ¹(𝑒_𝑖) are the tautological line bundles. The Chern classes 𝑐₁(ℛ_𝑖), 𝑖=1,…,𝑝−1, form a basis of 𝐻²(𝐶,ℤ). Fractional instantons can thus be thought of as Dirac monopoles on the two-spheres of the orbifold resolution.

The corresponding intersection numbers are given by _𝐶𝑐₁(ℛ_𝑖)∧𝑐₁(ℛ_𝑗)=−(𝐂⁻¹)_𝑖𝑗,_𝑆𝑖𝑐₁(ℛ_𝑗)=𝛿_𝑖𝑗. (4.40) Since 𝐶 is noncompact, the intersection matrix 𝐂 is not necessarily unimodular, and the corresponding instanton charges can be fractional. The contribution of fractional instantons to the supersymmetric Yang-Mills action with observables is thus given by 𝑆_frac=−i𝜏4𝜋_𝐶𝐹∧𝐹−i𝜙²_𝑗2𝜋_𝐶𝐹∧𝑐₁(ℛ_𝑗)=−𝜋i𝜏(𝐂⁻¹)^𝑖𝑗𝑢_𝑖𝑢_𝑗+𝑧_𝑖𝑢_𝑖 (4.41) with 𝑧_𝑖=(𝐂⁻¹)_𝑖𝑗𝜙²_𝑗. Setting 𝐮∶=(𝑢₁,…,𝑢_𝑝−1) and identifying 𝑄_𝑖=e^−𝑧𝑖 using the attractor mechanism, we find that the contribution of 𝑈(1) fractional instantons to the full partition function is given by a theta-function 𝑍^𝑈(1)_frac(𝑞,𝐐)=_{𝐮∈ℤ𝑝−1}𝑞^{(1/2)𝐮⋅𝐂−1𝐮}𝐐^𝐮 (4.42) on a Riemann surface of genus 𝑔=𝑝−1 and period matrix 𝜏𝐂⁻¹.

4.5. Instantons on Local ℙ¹

Let us now discuss what is known beyond the ALE case, in the instance that 𝐶 is the total space of the holomorphic line bundle 𝒪_ℙ¹(−𝑝) [55]. In this case an ADHM construction is not available. Nevertheless, much of the construction in the ALE case carries through, and by introducing weighted Sobolev norms, one shows that 𝔐_𝐶 is a smooth Kähler manifold with torsion-free homology groups which vanish in odd degrees. Let us consider some explicit examples.

The zero section of 𝒪_ℙ¹(−𝑝), considered as a divisor 𝑆=ℙ¹ of 𝐶, produces a line bundle ℒ→𝐶 such that 𝑐₁(ℒ) is the generator of 𝐻²(𝐶,ℤ)=ℤ. It has a unique self-dual connection asymptotic to the trivial connection. Let ℂ=𝐶×ℂ be the trivial line bundle over 𝐶. Set ℰ=ℂ⊕ℒ, and let 𝔐(ℰ) be the moduli space of self-dual connections on ℰ which are asymptotic to the trivial connection at infinity. Then dim_ℝ(𝔐(ℰ))=2𝑝. Since 𝐻₁(𝐶,ℝ)=0, using Morse theory one can show that the only nonvanishing homology groups of the instanton moduli space are 𝐻₀(𝔐(ℰ),ℝ)=𝐻₂(𝔐(ℰ),ℝ)=ℝ.

Alternatively, set ℰ=ℒ⊕ℒ^∨, and let 𝔐_(𝑘)(ℰ), 𝑘=0,1,…,𝑝−1, be the moduli space of self-dual connections asymptotic to 𝜌_𝑘⊕𝜌^∗_𝑘. Then for any 𝑝>2,dim_ℝ(𝔐_(𝑘)(ℰ))=2, while for 𝑝=2 (whereby 𝒪_ℙ¹(−2) coincides with the 𝐴₁ ALE space), one has dim_ℝ(𝔐_(𝑘)(ℰ))=4. In particular, for 𝑝=2 there is a diffeomorphism 𝔐_(𝑘)(ℰ)≅𝑇^∗ℙ¹ with 𝐻₀(𝔐_(𝑘)(ℰ),ℝ)=𝐻₂(𝔐_(𝑘)(ℰ),ℝ)=ℝ, while for 𝑝=4 one has 𝔐_(𝑘)(ℰ)≅𝐵². Instantons on local ℙ¹ will be studied in more generality later on in terms of moduli spaces of framed torsion-free sheaves.

4.6. Wall-Crossing Formulas

In [56] it was suggested that the structure of the instanton partition function on ALE spaces can be extrapolated to give a general result valid for all Hirzebruch-Jung surfaces 𝐶=𝐶(𝑝,𝑛). Thus one postulates the form of the 𝑈(𝑁) partition function𝑍^𝐶_𝑈(𝑁)(𝑞,𝐐)=(𝑍^𝑈(1)_reg(𝑞)𝑍^𝑈(1)_frac(𝑞,𝐐))^𝑁=1𝜂(𝜏)^{𝑁𝜒(𝐶)}_{𝐮∈ℤ2𝑁𝑏(𝐶)}𝑞^{(1/2)𝐮⋅𝐂−1𝐮}𝐐^𝐮, (4.43) where 𝐮∶=^𝑁_𝑙=1𝐮_𝑙. (4.44) In the ALE case, one has 𝜒(𝐴_𝑝−1)=𝑝 and 𝑏₂(𝐴_𝑝−1)=𝑝−1. The evidence for this formula is supported by calculations in the reduction to 𝑞-deformed two-dimensional Yang-Mills theory [13, 57], which captures the contributions from fractional instantons on the exceptional divisors. It was even conjectured to hold in the case when 𝐶 is a compact toric surface, at least in the region of moduli space where the instanton charges are large. Though the regular and fractional instantons are always readily constructed exactly as in the ALE case, the issue is whether or not this formula takes into account all of the instanton contributions, and in which regions of moduli space the factorization into 𝑈(1) partition functions holds. Later on we will give more precise meanings to these “asymptotic charge regions” in terms of moduli spaces of torsion-free sheaves.

Curiously, the formula (4.43) coincides with the chiral torus partition function of a conformal field theory in two dimensions with central charge 𝑐=𝑁𝜒(𝐶), that is, of 𝑁𝜒(𝐶) free bosons, with 𝑁𝑏₂(𝐶) of them being compact. The compact degrees of freedom live in a torus determined by the lattice 𝐻²(𝐶,ℤ) with the bilinear form 𝐂. The appearence of this two-dimensional field theory can be understood in M-theory, wherein the D4-D2-D0 brane quantum mechanics lifts to a (4,0) two-dimensional superconformal field theory on an M5-brane worldvolume [58]. This is reminscent of the recent conjectural relations between four-dimensional superconformal gauge theories and two-dimensional Liouville conformal field theories [59, 60]. When 𝐶 is an 𝐴_𝑝−1 ALE space and the gauge group is 𝑆𝑈(𝑁), there is yet another conformal field theoretic interpretation [46, 55, 61, 62]. In this case, the level 𝑁 affine 𝔰𝔲(𝑝) Lie algebra acts on the cohomology ring of the instanton moduli space, and the partition function (with appropriate local curvature and signature corrections inserted) coincides with the character of the affine Kac-Moody algebra given by 𝑍^ALE_{𝑆𝑈(𝑁)}(𝑞,𝐐)=^∞_{𝑛=0𝐮∈ℤ𝑝−1}Ω(𝑛,𝐮)𝑞^{𝑛−𝑐/24}𝐐^𝐮=Tr_ℋ(𝑞^{𝐿0−𝑐/24}𝐐^𝐉0), (4.45) where 𝐮=𝑐₁(𝒫), 𝑛=𝑐₂(𝒫)=𝑘+(1/2)𝐮⋅𝐂𝐮, and Ω(𝑛,𝐮) are the degeneracies of BPS states with the specified quantum numbers. Here ℋ is the Hilbert space on which the chiral algebra acts, which can be represented in terms of free fermion or boson conformal field theories with extended symmetry generators 𝐉₀ [61]. The 𝑈(1) partition function is also expressed as a 𝔲(1)₁ character in [61, 62].

Let us now compare the instanton partition function (4.43) with the black hole partition function. Microscopic black hole entropy formulas for BPS bound states of D0-D2-D4 branes in Type IIA supergravity are readily available on compact Calabi-Yau threefolds 𝑋 [58], in the large volume limit and when contributions from worldsheet instantons are negligible. In this limit we can expand the cycle [𝐶] as in (2.20). Let 𝛼_𝑖, 𝑖=1,…,𝑏₂(𝑋), be an integral basis of two cocycles for 𝐻²(𝑋,ℤ) dual to the four-cycle[𝐶_𝑖] with the intersection numbers 𝐷_𝑖𝑗𝑘=16_𝑋𝛼_𝑖∧𝛼_𝑗∧𝛼_𝑘,𝑐_2,𝑖=_𝑋𝛼_𝑖∧𝑐₂(𝑋), (4.46) and let 𝐷^𝑖𝑗 be the matrix inverse of 𝐷_𝑖𝑗=𝐷_𝑖𝑗𝑘𝒬^𝑘₄. The genus zero topological string amplitude 𝐹₀ can be expressed as 𝐹₀=𝐷_𝑖𝑗𝑘𝑋^𝑖𝑋^𝑗𝑋^𝑘𝑋⁰. (4.47) With 𝒬₀=(1/8𝜋²)∫_𝐶𝐹∧𝐹 and 𝒬_𝑖=𝒬^𝑖₂=(𝐂⁻¹)^𝑖𝑗𝑢_𝑗, the black hole entropy is given by [58] 𝑆_BH(𝒬₀,𝓠,𝓠₄)=2𝜋(𝐷_𝑖𝑗𝑘𝒬^𝑖₄𝒬^𝑗₄𝒬^𝑘₄+16𝑐_2,𝑖𝒬^𝑖₄)(𝒬₀+1𝐷12^𝑖𝑗𝒬_𝑖𝒬_𝑗). (4.48) We would now like to interpret the gauge theory partition function (4.43) as the corresponding black hole partition function 𝑍_BH(𝓠₄,𝝓²,𝜙⁰). For this, we expand it as 𝑍^𝐶_𝑈(𝑁)(𝑞,𝐐)=_{𝒬0,𝒬𝑖}Ω(𝒬₀,𝓠,𝓠₄)e^{−𝒬0𝜙0−𝓠⋅𝜙2}. (4.49) Then Cardy's formula gives the black hole entropy as 𝑆_BH(𝒬₀,𝓠,𝓠₄)=logΩ(𝒬₀,𝓠,𝓠₄). (4.50) This expression agrees [56] with the macroscopic supergravity result (4.48) for the Bekenstein-Hawking-Wald entropy in the large 𝒬₀ limit. Wall-crossing issues in a similar context are discussed in more detail in [63–67].

In parallel to what we did in Section 3.9., in order to explore the pertinent wall-crossing formulas in this instance, we will now consider moduli spaces which parametrize isomorphism classes of the following objects: (a) surjections (framings) 𝒪_𝐶→ℱ→0 of torsion-free sheaves with ch(ℱ)=(𝑁,𝑑,−𝑘),(b) stable torsion-free sheaves ℰ on 𝐶 with ch(ℰ)=(𝑁,𝑑,𝑘),(c) closed subschemes 𝑆⊂𝐶 of dimension ≤1 with dual curve class [𝑆]^∨=𝑑 and holomorphic Euler characteristic 𝜒(𝒪_𝑆)=𝑘.

In contrast to the six-dimensional situation, in four dimensions the connections between these three classes of objects are somewhat more subtle. We will examine each of them in turn and how they compare with the gauge theory results we have thus far obtained.

4.7. Moduli Spaces of Framed Instantons

We begin with Point (a) at the end of Section 4.6. Let 𝐶 be a smooth, quasiprojective, open, toric surface. We will assume that 𝐶 admits a projective compactification 𝐶, that is, 𝐶 is a smooth, compact, projective, toric surface with a smooth divisor ℓ_∞⊂𝐶 (called the “line at infinity”) which is a 𝑇²-invariant ℙ¹ in 𝐶, and such that 𝐶=𝐶⧵ℓ_∞. We will also require that ℓ_∞⋅ℓ_∞>0, in addition to ℓ_∞≅ℙ¹. The difference between the counting of framed instantons on the compact toric surface 𝐶 (with boundary condition at “infinity”) and of unframed instantons on the open toric surface 𝐶 is a universal perturbative contribution, which will be dropped here. Since 𝐶 is compact, these calculations will only capture the contributions from instantons with integer charges on 𝐶. We will mention later on how to incorporate the contributions from fractional instantons on 𝐶.

Let us fix two numbers 𝑁∈ℕ and 𝑘∈ℚ, and an integer cohomology class 𝑑∈𝐻²(𝐶,ℤ). Let 𝔐_{𝑁,𝑑,𝑘}(𝐶) be the framed moduli space consisting of isomorphism classes [ℰ] of torsion-free sheaves ℰ on 𝐶 such that (1)ℰ has the following topological Chern invariants: 𝑁=ch₀(ℰ)=rank(ℰ),𝑑=ch₁(ℰ)=𝑐₁(ℰ),𝑘=−_𝐶ch₂(ℰ); (4.51)(2)ℰ is locally-free in a neighbourhood of ℓ_∞, and there is an isomorphism ℰ|_ℓ∞≅𝒪_ℓ^⊕𝑁_∞ called the “framing at infinity”.

These topological conditions imply that the moduli space 𝔐_{𝑁,𝑑,𝑘}(𝐶) is nonempty only when 𝑑|_ℓ∞=𝑁𝑐₁(𝒪_ℓ∞)=0, (4.52) so that 𝑑 defines a class 𝑑∈𝐻²_cpt(𝐶,ℤ) in the compactly supported cohomology of 𝐶. Furthermore, from ch₂(ℰ)=−𝑐₂(ℰ)+(1/2)𝑐₁(ℰ)∧𝑐₁(ℰ), the last relation in (4.51) can be written as 𝑘=_𝐶(𝑐₂1(ℰ)−2𝑑∧𝑑). (4.53) Framed sheaves yield stable pairs [68], analogous to those described in Section 3.9., after suitable choices of polarization on 𝐶 and stability parameter.

We write ℰ(−ℓ_∞)∶=ℰ⊗𝒪_𝐶(−ℓ_∞). At a given point [ℰ]∈𝔐_{𝑁,𝑑,𝑘}(𝐶), the space of reducible connections is Ext⁰_𝒪𝐶(ℰ,ℰ(−ℓ_∞))=Hom_𝒪𝐶(ℰ,ℰ(−ℓ_∞)), the Zariski tangent space is Ext¹_𝒪𝐶(ℰ,ℰ(−ℓ_∞)), and the obstruction space is Ext²_𝒪𝐶(ℰ,ℰ(−ℓ_∞)). The cohomology of the instanton deformation complex is greatly simplified by the fact that in this case [69] Ext⁰_𝒪𝐶(ℰ,ℰ(−ℓ_∞))=Ext²_𝒪𝐶(ℰ,ℰ(−ℓ_∞))=0. (4.54) Using (4.54) and the Riemann-Roch theorem, one shows [69] that the moduli space 𝔐_{𝑁,𝑑,𝑘}(𝐶) is a smooth quasiprojective variety of (complex) dimension 2𝑁𝑘+𝑑², where 𝑑²∶=∫_𝐶𝑑∧𝑑, whose tangent space at a point [ℰ] is isomorphic to the vector space Ext¹_𝒪𝐶(ℰ,ℰ(−ℓ_∞)).

We will now describe a natural torus invariant subspace of the instanton moduli space. The 𝑇²-invariance of ℓ_∞ implies that the pullback of the 𝑇²-action on 𝐶 defines an action on 𝔐_{𝑁,𝑑,𝑘}(𝐶). There is also an action of the diagonal maximal torus 𝑇^𝑁 of 𝐺𝐿(𝑁,ℂ) on the framing. Altogether we get an action of the complex algebraic torus 𝑇=𝑇²×𝑇^𝑁 on 𝔐_{𝑁,𝑑,𝑘}(𝐶) [69]. We are interested in the fixed point set 𝔐_{𝑁,𝑑,𝑘}(𝐶)𝑇 of this torus action.

An isomorphism class [ℰ]∈𝔐_{𝑁,𝑑,𝑘}(𝐶) is fixed by the 𝑇^𝑁-action if and only if it decomposes as ℰ=ℰ₁⊕⋯⊕ℰ_𝑁,ℰ_𝑙∈𝔐_1,𝑑,𝑘(𝐶) (4.55) such that ℰ_𝑙|_ℓ∞ is mapped to the 𝑙th factor 𝒪_ℓ∞ of 𝒪_ℓ^⊕𝑁_∞ under the framing isomorphism. Since the double dual ℰ_𝑙^∨∨ is a line bundle which is trivial on ℓ_∞, it is equal to 𝒪_𝐶(𝐷_𝑙) for some divisor 𝐷_𝑙⊂𝐶 disjoint from ℓ_∞. Via the natural injection ℰ_𝑙⊂ℰ_𝑙^∨∨=𝒪_𝐶(𝐷_𝑙), the sheaf ℰ_𝑙 is thus equal to ℐ_𝑙(𝐷_𝑙)=ℐ_𝑙⊗𝒪_𝐶(𝐷_𝑙) for some ideal sheaf ℐ_𝑙 of a zero-dimensional subscheme 𝑍_𝑙⊂𝐶, also disjoint from ℓ_∞. If ℰ is also fixed by the 𝑇²-action, then so are 𝐷_𝑙, ℐ_𝑙, and 𝑍_𝑙.

The supports of the 𝑇²-invariant subschemes 𝑍_𝑙 and 𝐷_𝑙 can be represented explicitly in terms of the toric geometry of 𝐶. Recall that this is described in terms of an underlying toric graph Δ(𝐶) whose vertices are in bijective correspondence with the 𝑇²-fixed points in 𝐶, and two vertices are joined by an edge 𝑒 if and only if the corresponding fixed points are connected by a 𝑇²-invariant ℙ¹. Let 𝑉(𝐶) and 𝐸(𝐶), respectively, denote the set of vertices and edges of Δ(𝐶). Then since Δ(𝐶) is a chain, one has |𝐸(𝐶)|=|𝑉(𝐶)|−1. The number of vertices 𝑛=|𝑉(𝐶)| is also the number of two-cones in the toric fan of 𝐶 and it is related to the Euler characteristic 𝜒(𝐶) of the surface as 𝑛=𝜒(𝐶)=𝜒(𝐶)−2. (4.56) We denote by 𝑝_𝑓 the 𝑇²-invariant point in 𝐶 corresponding to the vertex 𝑓∈𝑉(𝐶) and by ℓ_𝑒 the 𝑇²-invariant line ℙ¹ corresponding to the edge 𝑒∈𝐸(𝐶).

Since the zero-cycles 𝑍_𝑙 are not supported on ℓ_∞, they must be contained in the fixed point set 𝑉(𝐶) in 𝐶. Thus each 𝑍_𝑙 is a union of subschemes 𝑍^𝑓_𝑙,𝑓∈𝑉(𝐶) supported at the 𝑇²-fixed points 𝑝_𝑓∈𝐶. If we choose a local coordinate system (𝑥,𝑦)∈ℂ² in a patch 𝑈_𝑓 around 𝑝_𝑓, then the 𝑇²-invariant ideal of 𝑍^𝑓_𝑙 in the coordinate ring ℂ[𝑥,𝑦] of 𝑈_𝑓≅ℂ² is generated by the 𝑇²-eigenfunctions with nontrivial characters, which are monomials 𝑥^𝑖𝑦^𝑗, and hence 𝑍^𝑓_𝑙 corresponds to a Young diagram 𝜆^𝑓_𝑙 with |𝜆^𝑓_𝑙| boxes (with monomial 𝑥^𝑖𝑦^𝑗 placed at (𝑖+1,𝑗+1)). The ideal is spanned by monomials outside the Young diagram. Likewise, since the 𝑇²-invariant two-cycles 𝐷_𝑙 are disjoint from the line at infinity, they are supported along the edges ℓ_𝑒≅ℙ¹,𝑒∈𝐸(𝐶).

It follows that the fixed point set 𝔐_{𝑁,𝑑,𝑘}(𝐶)𝑇 is parameterized by finitely many 𝑁-tuples (𝐷,𝝀)=((𝐷₁,𝝀₁),…,(𝐷_𝑁,𝝀_𝑁)), (4.57) where 𝐷_𝑙∈𝐻₂(𝐶,ℤ)≅⊕_{𝑒∈𝐸(𝐶)}ℤ[ℓ_𝑒] (4.58) are 𝑇²-invariant divisors in 𝐶 and 𝝀_𝑙=(𝜆^𝑓_𝑙)_{𝑓∈𝑉(𝐶)} (4.59) is a vector of Young tableaux with |𝝀_𝑙|∶=∑_{𝑓∈𝑉(𝐶)}|𝜆^𝑓_𝑙| boxes. The Young tableaux parametrize the contributions from regular pointlike D0-brane instantons (freely moving inside 𝐶). The divisors parametrize the contributions from D2-brane instantons (wrapping 𝐷_𝑙 with appropriate units of magnetic flux). We will see that these two types of contributions factorize completely. We can write the topological invariants of a generic element (4.57) in 𝔐_{𝑁,𝑑,𝑘}(𝐶)𝑇 in terms of this combinatorial data. Since 𝑐₁(ℐ_𝑙)=0, the constraint 𝑑=𝑐₁(ℰ) can be written as 𝑑=^𝑁_𝑙=1𝑐₁(𝒪_𝐶(𝐷_𝑙)), (4.60) whereas (4.53) becomes 𝑘=^𝑁_𝑙=1|𝝀_𝑙|+_𝐶(_𝑙<𝑙′𝑐₁(𝒪_𝐶(𝐷_𝑙))∧𝑐₁(𝒪_𝐶(𝐷_𝑙′1))−2𝑑∧𝑑). (4.61) The Young diagram box sum in (4.61) gives the length of the singularity set of the sheaf (4.55).

To make contact with our previous formulas, let us rewrite these constraints in a more explicit parametrization. Define the invertible, symmetric, integer-valued intersection matrix 𝐂=(𝐂_𝑒,𝑒′)_{𝑒,𝑒′∈𝐸(𝐶)} between lines of the toric graph Δ(𝐶) as 𝐂_𝑒,𝑒′∶=ℓ_𝑒⋅ℓ_𝑒′. (4.62) Each divisor ℓ_𝑒 is canonically associated to a 𝑇²-equivariant holomorphic line bundle ℒ_𝑒∶=𝒪_𝐶(ℓ_𝑒) whose first Chern class 𝑐₁(ℒ_𝑒)∈𝐻²_cpt(𝐶,ℤ) is Poincaré dual to ℓ_e with _ℓ𝑒𝑐₁(ℒ_𝑒′)=𝐂_𝑒,𝑒′,_𝐶𝑐₁(ℒ_𝑒)∧𝑐₁(ℒ_𝑒′)=𝐂_𝑒,𝑒′. (4.63) Note that these intersection numbers differ from (4.40), that is, the basis of line bundles ℒ_𝑒 does not coincide with the tautological line bundles ℛ_𝑖. Using (4.58) we can write 𝑐₁(𝒪_𝐶(𝐷_𝑙))=∑_{𝑒∈𝐸(𝐶)}𝑢^𝑒_𝑙𝑐₁(𝐿_𝑒) with 𝑢^𝑒_𝑙∈ℤ. Then the constraint (4.60) can be expressed as 𝑑=^𝑁_{𝑙=1𝑒∈𝐸(𝐶)}𝑢^𝑒_𝑙𝑐₁(ℒ_𝑒). (4.64) Combined with (4.61) and using (4.63), this gives 𝑘=^𝑁_𝑙=1|𝝀_𝑙|+_𝑙<𝑙′𝑢^𝑒_𝑙𝐂_𝑒,𝑒′𝑢^𝑒′_𝑙′−12^𝑁_{𝑙,𝑙′=1}𝑢^𝑒_𝑙𝐂_𝑒,𝑒′𝑢^𝑒′_𝑙′=^𝑁_𝑙=1|𝝀_𝑙1|−2^𝑁_𝑙=1𝑢^𝑒_𝑙𝐂_𝑒,𝑒′𝑢^𝑒′_𝑙. (4.65)

The Vafa-Witten-Nekrasov partition function on the toric surface C is now defined as the 𝑇-equivariant index 𝑍^𝐶_𝑈(𝑁)(𝑞,𝐐)^fr=_𝑘∈ℚ𝑞^𝑘_{𝑑∈𝐻2cpt(𝐶,ℤ)}𝐐^𝑑𝜒(𝔐_{𝑁,𝑑,𝑘}(𝐶)), (4.66) where as before 𝑞=e^2𝜋i𝜏 with 𝜏 the complexified gauge coupling constant and 𝐐^𝑑∶=_{𝑒∈𝐸(𝐶)}(𝑄_𝑒)^{(𝐂−1)′𝑒,𝑒∫ℓ𝑒′𝑑} (4.67) for a collection of formal variables 𝑄_𝑒,𝑒∈𝐸(𝐶) and a given cohomology class 𝑑∈𝐻²_cpt(𝐶,ℤ). From the localization formula (3.25), we see that the 𝑇-equivariant Euler class cancels in (3.26) at the fixed points. This cancellation in the fluctuation determinants is a consequence of (4.54) which implies that the obstruction bundle is trivial, and there is an isomorphism between the tangent and normal bundles over the fixed point set 𝔐_{𝑁,𝑑,𝑘}(𝐶)𝑇 in the instanton moduli space. Because of this cancellation, we see that the contribution from each critical point of the gauge theory is independent of the equivariant parameters 𝜖_𝑖 and 𝑎_𝑙. In the sum over critical points, we can replace the sum over 𝑘∈ℚ using (4.65) by sums over Young diagrams 𝝀 and over the integers 𝐮=(𝐮₁,…,𝐮_𝑁) and 𝐮_𝑙=(𝑢^𝑒_𝑙)_{𝑒∈𝐸(𝐶)}∈ℤ^𝑛−1 representing magnetic fluxes through ℓ_𝑒 on D2-branes wrapping the divisors 𝐷_𝑙. The sum over 𝑑∈𝐻²_cpt(𝐶,ℤ) may then be saturated using (4.64).

Putting everything together, and using (4.64) and (4.63) to write _ℓ𝑒𝑑=^𝑁_𝑙=1𝐂_𝑒,𝑒′𝑢^𝑒′_𝑙, (4.68) the partition function (4.66) thereby becomes 𝑍^𝐶_𝑈(𝑁)(𝑞,𝐐)^fr=_{𝝀𝐮∈ℤ𝑁(𝑛−1)}𝑞^{|𝐮𝝀|−(1/2)𝐮⋅𝐂}𝐐^𝐮, (4.69) where we have used (4.44). The sums over Young tableaux decouple, and for each vertex 𝑓∈𝑉(𝐶) and each 𝑙=1,…,𝑁, they produce a factor of the Euler function 𝜂(𝑞)⁻¹. Then we can bring (4.69) into the final form 𝑍^𝐶_𝑈(𝑁)(𝑞,𝐐)^fr=1𝜂(𝑞)^{𝑁𝜒(𝐶)}_{𝐮∈ℤ2𝑁𝑏(𝐶)}𝑞^{𝐮−(1/2)𝐮⋅𝐂}𝐐^𝐮, (4.70) where we recall that 𝜒(𝐶)=|𝑉(𝐶)|=𝑛 and 𝑏₂(𝐶)=|𝐸(𝐶)|=𝑛−1.

This formula differs from (4.43) in that only integral values of the first Chern class are permitted in (4.70). In the case of an 𝐴_𝑝−1 singularity, fractional first Chern classes can be incorporated by constructing instead the moduli space of torsion-free sheaves on the orbifold compactification 𝐶^orb=𝐶∪ℓ_∞̃ of the hyper-Kähler ALE space 𝐶, where ℓ_∞̃=ℙ¹/Γ [52, 55, 70, 71]. In a neighbourhood of infinity, we can approximate 𝐶 by ℙ²/Γ with the singularity at the origin resolved. More precisely, we obtain the divisor ℓ_∞̃ by gluing together the trivial bundle 𝒪_𝐶 on 𝐶 with the line bundle 𝒪_ℙ²_/Γ(1) on ℙ²/Γ. The latter bundle has a Γ-equivariant structure such that the map 𝒪_𝐶→𝒪_ℙ²_/Γ(1) is Γ-equivariant. Let us examine some examples which are covered by the analysis above.

4.7.1. Affine Plane

Let 𝐶=ℂ². Then 𝐶=ℙ² with ℓ_∞=[0,𝑧₁,𝑧₂]≅ℙ¹ and intersection number ℓ_∞⋅ℓ_∞=1>0. Since 𝐻²(ℙ²,ℤ)≅ℤ in this case, the constraint (4.52) implies 𝑑=0. The instanton moduli space 𝔐_𝑁,𝑘(ℂ²) in this case is a smooth variety of complex dimension 2𝑁𝑘. Furthermore, 𝑛=𝜒(ℙ²)−2=3−2=1. The partition function (4.66) computed with fixed 𝑑=0 and 𝑛=1 is thus given by 𝑍^ℂ2_𝑈(𝑁)(𝑞)^fr=𝜂(𝑞)^−𝑁, (4.71) which coincides with the instanton partition function on ℂ² described in Section 4.2.

4.7.2. Local ℙ¹

Let 𝐶=𝐶_𝑝, 𝑝>0, be the total space of the holomorphic line bundle 𝒪_ℙ¹(−𝑝) of degree −𝑝 over ℙ¹. Then 𝐶=𝔽_𝑝∶=ℙ(𝒪_ℙ¹(−𝑝)⊕𝒪_ℙ¹) is the 𝑝th Hirzebruch surface. Let ℓ₀=ℙ(0⊕𝒪_ℙ¹)≅ℙ¹ and ℓ_∞=ℙ(𝒪_ℙ¹(−𝑝)⊕0)≅ℙ¹. Then ℓ₀⋅ℓ₀=−𝑝<0, ℓ_∞⋅ℓ_∞=𝑝>0, and ℓ₀⋅ℓ_∞=0 so that the line ℓ₀ does not pass through the “line at infinity”. From the constraint (4.52) and the second Betti number 𝑏₂(𝐶_𝑝)=1, it follows that 𝑑=𝑚𝑐₁(𝒪_𝐶𝑝(ℓ₀)) for some 𝑚∈ℤ. In this case, the instanton moduli space 𝔐_{𝑁,𝑚,𝑘}(𝐶_𝑝) is a smooth variety of complex dimension 2𝑁𝑘−𝑝𝑚². The positivity of this dimension is a constraint on the integral classes which ensures that the moduli space is non-empty. One has 𝜒(𝔽_𝑝)=1+2+1=4, and hence 𝑛=𝜒(𝔽_𝑝)−2=2. The partition function (4.70) in this case becomes 𝑍^𝐶𝑝_𝑈(𝑁)(𝑞,𝑄)^fr=1𝜂(𝑞)^2𝑁_𝑢∈ℤN𝑞^{𝑢(𝑝/2)𝑢⋅}𝑄^𝑢 (4.72) with 𝑢=𝑢₁+⋯+𝑢_𝑁.

Like the ALE spaces, one can work instead with a stack compactification 𝐶^orb of the total space of the bundle 𝒪_ℙ¹(−𝑝) obtained by adding a divisor ℓ_∞̃≅ℓ_∞/Γ. The resulting variety is a toric Deligne-Mumford stack whose coarse space is the Hirzebruch surface 𝔽_𝑝 [72]. With this compactification, one produces framed sheaves with fractional first Chern classes 𝑑=(𝑚/𝑝)𝑐₁(𝒪_𝐶𝑝(ℓ₀)), 𝑚∈ℤ, and in this case the partition function takes the form [72] 𝑍^𝐶𝑝_𝑈(𝑁)(𝑞,𝑄)^orb𝜃=(³(𝑣/𝑝∣𝜏/𝑝)𝜂(𝑞)²)^𝑁 (4.73) anticipated by [13, 56], where 𝑞=e^2𝜋i𝜏, 𝑄=e^2𝜋i𝑣, and 𝜃₃(𝑣∣𝜏)=∑_𝑛∈ℤ𝑞^(1/2)𝑛2𝑄^𝑛 is a Jacobi elliptic function.

On these spaces, one also has a version of the Hitchin-Kobayashi correspondence which makes contact with the realization of instantons as self-dual connections. Namely, 𝑆𝑈(𝑁)-instantons on 𝐶_𝑝 are in one-to-one correspondence with holomorphic bundles ℰ of rank 𝑁 on 𝐶_𝑝 with 𝑐₁(ℰ)=0 together with a framing at infinity [73]. When 𝑝=2, these spaces coincide with the 𝐴₁ ALE space.

4.7.3. Compact Surfaces

We will now examine situations under which the formula (4.70) holds in the case of compact surfaces, providing rigorous justification for some of the conjectural formulas of [56]. Let 𝐶 be a compact, smooth, projective toric surface. Let 𝑐_∞ be a generic point in 𝐶 which is disjoint from the torus invariant lines ℓ_𝑒, 𝑒∈𝐸(𝐶) of 𝐶. Let 𝔐_{𝑁,𝑑,𝑘}(𝐶,𝑐_∞) be the moduli space of isomorphism classes [ℰ] of torsion-free coherent sheaves ℰ on 𝐶 with topological Chern invariants as in (4.51), together with a “framing” at the point 𝑐_∞. When ℰ is locally free, this framing means a choice of basis for the fibre space ℰ_𝑐∞≅ℂ^𝑁. When ℰ is not locally free, the framing is defined with respect to a locally free resolution ℰ^•→ℰ→0.

Let 𝜎∶(𝐶,ℓ_∞̂)→(𝐶,𝑐_∞) be the blow-up of 𝐶 at 𝑐_∞. Then 𝐶 is also a compact, smooth, projective toric surface. We will suppose that the exceptional divisor ℓ_∞̂=𝜎⁻¹(𝑐_∞)≅ℙ¹ has positive self-intersection ℓ_∞̂⋅ℓ_∞̂>0. Let 𝔐^fr_{𝑁,𝑑,𝑘}(𝐶) be the moduli space of isomorphism classes [ℰ] of torsion-free sheaves ℰ on 𝐶 which are framed on the line ℓ_∞̂ as above. The blow-up map 𝜎 determines mutually inverse sheaf morphisms [ℰ]↦[ℰ]=[𝜎_∗ℰ] and [ℰ]↦[ℰ]=[𝜎^∗ℰ], and hence an explicit isomorphism 𝔐_{𝑁,𝑑,𝑘}(𝐶,𝑐_∞)≅𝔐^fr_{𝑁,𝑑,𝑘}(𝐶). (4.74) It follows from above that the moduli space 𝔐_{𝑁,𝑑,𝑘}(𝐶,𝑐_∞) is thus a smooth variety of complex dimension 2𝑁𝑘+𝑑², where 𝑑²∶=∫_𝐶𝑑∧𝑑. Using the isomorphism (4.74), the instanton partition function on 𝐶 can be computed from the blow-up formula 𝑍^𝐶_𝑈(𝑁)(𝑞,𝐐)^fr𝐶=𝑍^⧵_∞𝑈(ℓ_𝑁)̂(𝑞,𝐐)^fr, (4.75) where the right-hand side is given by the formula (4.70).

As an explicit example, consider the complex projective plane 𝐶=ℙ². Let 𝑧_∞ be a generic point on ℙ² disjoint from the line ℓ_∞=[0,𝑧₁,𝑧₂]. Then the instanton moduli space 𝔐_{𝑁,𝑑,𝑘}(ℙ²,𝑧_∞) is a smooth variety of complex dimension 2𝑁𝑘+𝑑², where 𝑑∈ℤ. Let 𝜎∶ℙ²→ℙ² be the blowup of ℙ² at 𝑧_∞ with exceptional divisor ℓ_∞̂=𝜎⁻¹(𝑧_∞)≅ℙ¹. Then there are 𝑛=𝜒(ℙ²⧵ℓ_∞̂)=𝜒(ℙ²)=3 maximal two cones and 𝑏₂(ℙ²⧵ℓ_∞̂)=𝑏₂(ℙ²)=1 edge in the toric graph Δ(ℙ²⧵ℓ_∞̂)=Δ(ℙ²). Since ℓ_∞⋅ℓ_∞=1 in this case, the partition function (4.75) gives the 𝑈(𝑁) formula 𝑍^ℙ2_𝑈(𝑁)(𝑞,𝑄)^fr=1𝜂(𝑞)^3𝑁_{𝑢∈ℤ𝑁}𝑞^{𝑢−(1/2)𝑢⋅}𝑄^𝑢 (4.76) with 𝑢=𝑢₁+⋯+𝑢_𝑁. For 𝑁=1 this agrees with the 𝑈(1) gauge theory partition function on ℙ² derived in [54, Section 5.1].

4.8. Stability

Next we address Point (b) at the end of Section 4.6. If one wishes to relax the requirement of framing, then one must carefully analyse stability issues in order to obtain well-defined instanton moduli spaces. Let 𝐶 be a smooth, quasiprojective toric surface. Again we fix invariants 𝑁∈ℕ, 𝑘∈ℚ, and 𝑑∈𝐻²_cpt(𝐶,ℤ). Let 𝔐_{𝑁,𝑑,𝑘}(𝐶) be the unframed moduli space consisting of isomorphism classes [ℰ] of semistable torsion-free coherent sheaves ℰ on 𝐶 such that ℰ has topological Chern invariants 𝑁=ch₀(ℰ)=rank(ℰ),𝑑=ch₁(ℰ)=𝑐₁(ℰ),𝑘=−_𝐶ch₂(ℰ). (4.77)

The notion of “semistability” considers sheaves with fixed Hilbert polynomial. In this case, a general result of Maruyama [74, 75] constructs the algebraic scheme 𝔐_{𝑁,𝑑,𝑘}(𝐶) and shows that it is projective. In particular, it admits a 𝑇-equivariant embedding into a smooth variety. In the case that 𝐶 is a polarized surface, that is, it admits a smooth distribution in its tangent bundle 𝑇𝐶 which is integrable and Lagrangian (in an appropriate sense), we can formulate the notion of semistability in terms of the more familiar slope semistability following [76]. Let ℒ be a fixed ample line bundle on 𝐶. For a coherent torsion-free sheaf ℰ on 𝐶, define the polynomial 𝜌_ℰ(𝑛)=𝜒(ℰ⊗ℒ^⊗𝑛)rank(ℰ) (4.78) for 𝑛∈ℕ₀. Then ℰ is said to be ℒ-semistable if 𝜌_ℱ(𝑛)≤𝜌_ℰ(𝑛) (4.79) for all 𝑛≫0 whenever ℱ is a coherent subsheaf of ℰ. In this case, Gieseker [76] constructs 𝔐_{𝑁,𝑑,𝑘}(𝐶) as a projective 𝑄𝑢𝑜𝑡 scheme. The slope of ℰ is the rational number 𝜇(ℰ)=deg(ℰ),rank(ℰ) (4.80) where the degree of ℰ is defined using the polarization as deg(ℰ)=_𝐶𝑐₁(ℰ)∧𝑐₁(ℒ). (4.81) An application of the Riemann-Roch theorem shows [76] 𝜌_ℰ𝑛(𝑛)=2_𝐶(𝑐₁(ℒ)∧(𝑛𝑐₁(ℒ)+𝑐₁(𝐶))+2ch₂(ℰ)+𝑐₁(ℰ)∧𝑐₁(𝐶)rank(ℰ))+𝜇(ℰ)+𝜒(𝒪_𝐶). (4.82) It follows that 𝜌_ℰ(𝑛)≥𝜌_ℱ(𝑛) for 𝑛≫0 if and only if 𝜇(ℰ)≥𝜇(ℱ), and hence ℒ-semistability is equivalent to the usual quasi-BPS instanton equations in this case. In physics applications, one is interested in instances where 𝐶 is a Kähler surface. In this case one can use the Kähler polarization and take 𝑐₁(ℒ) to be the Kähler two-form 𝑘₀ in (4.32).

The following result computes the expected dimension of the instanton moduli space in this case.

Lemma 4.1. The virtual dimension of 𝔐_{𝑁,𝑑,𝑘}(𝐶) equals 2𝑁𝑘+𝑑²−(𝑁²−1)𝜒(𝒪_𝐶), where 𝑑²∶=∫_𝐶𝑑∧𝑑 and 𝜒(𝒪_𝐶) is the holomorphic Euler characteristic of 𝐶.

Proof. As discussed in Section 3.5., the dimension of the virtual tangent space 𝑇^vir_[ℰ]𝔐_{𝑁,𝑑,𝑘}(𝐶)=Ext¹_𝒪𝐶(ℰ,ℰ)⊖Ext²_𝒪𝐶(ℰ,ℰ) to the instanton moduli space at a point [ℰ]∈𝔐_{𝑁,𝑑,𝑘}(𝐶) in obstruction theory is the difference of Euler characteristics 𝜒(𝒪_𝐶⊗𝒪^∨_𝐶)−𝜒(ℰ⊗ℰ^∨). The latter quantity may be computed for ℰ locally free by using the Hirzebruch-Riemann-Roch theorem to write 𝜒(ℰ⊗ℰ^∨)=_𝐶ch(ℰ⊗ℰ^∨)∧Td(𝐶). (4.83) Let 𝜈 be the generator of 𝐻⁴(𝐶,ℤ)≅ℤ which is Poincaré dual to a point in 𝐶. Then using (4.77) one computes the Chern character ch(ℰ⊗ℰ^∨)=ch(ℰ)∧ch(ℰ^∨)=(𝑁+𝑑−𝑘𝜈)∧(𝑁−𝑑−𝑘𝜈)=𝑁²−𝑑∧𝑑−2𝑁𝑘𝜈. (4.84) The Todd characteristic class of the tangent bundle of 𝐶 is given in terms of Chern classes of 𝑇𝐶 as 1Td(𝐶)=1+2𝑐₁1(𝐶)+12(𝑐₁(𝐶)∧𝑐₁(𝐶)+𝑐₂(𝐶)). (4.85) We may thus write (4.83) as 𝜒(ℰ⊗ℰ^∨)=_𝐶(𝑁²12(𝑐₁(𝐶)∧𝑐₁(𝐶)+𝑐₂(𝐶))−𝑑∧𝑑−2𝑁𝑘𝜈)=−2𝑁𝑘−𝑑²+𝑁²_𝐶Td(𝐶). (4.86) From the Hirzebruch-Riemann-Roch formula and ch(𝒪_𝐶)=1, one has 𝜒(𝒪_𝐶)=_𝐶Td(𝐶)=𝜒(𝒪_𝐶⊗𝒪^∨_𝐶) (4.87) and the result follows when ℰ is a bundle. When ℰ is not locally free, we use a locally free resolution ℰ^•→ℰ→0 along with additivity of the Chern character.

This result shows that the expected dimension of the instanton moduli space for 𝑁=1 coincides with the dimension of the framed moduli space 𝔐_1,𝑑,𝑘(𝐶) of Section 4.7. Indeed, torsion-free sheaves of rank one are always stable in the sense explained above. Since any torsion-free sheaf decomposes as the product ℰ=ℐ⊗ℒ of an ideal sheaf of a zero-dimensional subscheme and a line bundle, the moduli space factorizes into a product 𝔐_1,𝑑,𝑘(𝐶)=𝔐_{1,0,𝑘+𝑑}²_/2(𝐶)×𝔐_{1,𝑑,−𝑑}²_/2(𝐶), (4.88) where 𝔐_1,0,𝐾(𝐶)≅𝐶^[𝐾] is the Hilbert scheme of 𝐾 points on 𝐶 which is a smooth variety of dimension 2𝐾, and the (zero-dimensional) Picard group 𝔐_{1,𝑑,−𝑑}²_/2(𝐶)≅Pic^𝑑(𝐶) parameterizes fractional instantons. The factorization of the 𝑈(1) gauge theory partition function into contributions from regular and fractional instantons then follows from the multiplicativity of the Euler class under tensor product of (tangent) bundles. For compact toric surfaces 𝐶, it is given by the formula (4.70) with 𝑁=1. In the case of Hirzebruch-Jung surfaces, the contributions of fractional charges in this case are shown in [54] to arise from the noncompact prime divisors of 𝐶 via linear equivalences. It amounts to identifying the generators 𝑐₁(ℛ_𝑖) of the Kähler cone in 𝐶𝐻¹(𝐶)⊗ℚ with the duals to the exceptional divisors 𝐷_𝑖 which generate the Mori cone in 𝐶𝐻^cpt₁(𝐶)⊗ℚ. For the ALE spaces, the moduli space 𝔐_{1,0,𝑘+𝑑}²_/2(𝐶) coincides with the quiver variety of the ADHM construction (with suitable stability conditions) [77].

The situation for higher rank 𝑁>1 is much more complicated. In this case, slope-stability does not seem to properly account for walls of marginal stability extending to infinity which describe wall-crossing behaviour of the partition functions counting D4-D2-D0 brane bound states on Calabi-Yau manifolds 𝑋 with ℎ^1,1(𝑋)>1 [64]. As discussed in Section 3.9., the physical theory is described by moduli spaces of stable objects in the derived category D^𝑏(coh(𝑋)), as the observed D-brane decays are impossible in the abelian category coh(𝑋) of coherent sheaves on 𝑋.

4.9. Perpendicular Partition Functions and Universal Sheaves

Finally, we come to the last Point (c) at the end of Section 4.6. For 𝑈(1) gauge theory, this relationship is analysed in detail in [54]. In this case a four-dimensional analog of the topological vertex formalism can be developed. On the gauge theory side, the vertex contributions should be computed by a version of the perpendicular partition function of Section 3.8., that is, the generating function 𝑃^𝑚1,𝑚2_𝑈(𝑁)(𝑞) for instantons on 𝐶=ℂ² with fixed asymptotics such that 𝑍^ℂ2_𝑈(𝑁)(𝑞)=𝑃^0,0_𝑈(𝑁)(𝑞), while the edge contributions should be read off from the character of the corresponding universal sheaf. In the remainder of this section, we will describe in detail the instanton moduli space with boundary conditions specified by integers 𝑚₁, and 𝑚₂, and show that the fixed point loci of the induced 𝑇-action are enumerated by two-dimensional Young diagrams with asymptotics 𝑚₁,and𝑚₂. This gives the four-dimensional version of the gauge theory gluing rules of Section 3.8. and also a first principle derivation of the empirical vertex rules of [54, Section 4.3].

4.9.1. Instanton Moduli Spaces on 𝔽₀

We will first describe how to generate instantons with nontrivial first Chern class using our previous formalism. For this, rather than working with the projective plane 𝐶=ℙ², it is more convenient to work with the “two-point compactification” of 𝐶=ℂ² given by a product of two projective lines 𝔽₀=ℙ¹_𝑧×ℙ¹_𝑤, where the labels will be used to keep track of each of the two factors. This variety can be identified as the zeroth Hirzebruch surface 𝔽₀=ℙ(𝒪_ℙ¹⊕𝒪_ℙ¹), that is, the projective compactification of the trivial line bundle 𝒪_ℙ¹=ℂ×ℙ¹, which is again a toric surface. For (𝑡_𝑧,𝑡_𝑤)∈𝑇², the toric action is described by the automorphism 𝐹_{𝑡𝑧,𝑡𝑤} of 𝔽₀ defined by 𝐹_{𝑡𝑧,𝑡𝑤}([𝑧₀,𝑧₁];[𝑤₀,𝑤₁])=([𝑧₀,𝑡_𝑧𝑧₁];[𝑤₀,𝑡_𝑤𝑤₁]). (4.89) Let 𝑝_𝑧 : 𝔽₀→ℙ¹_𝑧 and 𝑝_𝑤 : 𝔽₀→ℙ¹_𝑤 be the canonical projections, and 𝑖_𝑧∶ℙ¹_𝑧≅ℙ¹_𝑧×pt↪𝔽₀ and 𝑖_𝑤 : ℙ¹_𝑤≅pt×ℙ¹_𝑤↪𝔽₀ the natural inclusions. There are now two independent distinguished lines at infinity fixed by this 𝑇² action, given by the pushforwards ℓ_𝑧=(𝑖_𝑧)_∗ℙ¹_𝑧 and ℓ_𝑤=(𝑖_𝑤)_∗ℙ¹_𝑤, for which the respective factors of the action of 𝑇²=ℂ^∗×ℂ^∗ reduce to the standard ℂ^∗-action on ℓ_𝑧,ℓ_𝑤≅ℙ¹. These two divisors have intersection products ℓ_𝑧⋅ℓ_𝑧=ℓ_𝑤⋅ℓ_𝑤=0 and ℓ_𝑧⋅ℓ_𝑤=1, and the canonical divisor is 𝐾_𝔽0=−2ℓ_𝑧−2ℓ_𝑤. They generate the Picard group of 𝔽₀, and hence induce a bigrading on line bundles over 𝔽₀. For 𝑚_𝑧,𝑚_𝑤∈ℤ, we write 𝒪_𝔽0(𝑚_𝑧,𝑚_𝑤)∶=𝒪_𝔽0(𝑚_𝑧ℓ_𝑧+𝑚_𝑤ℓ_𝑤)=𝑝^∗_𝑧𝒪_ℙ¹_𝑧(𝑚_𝑧)⊗𝑝^∗_𝑤𝒪_ℙ¹_𝑤(𝑚_𝑤). (4.90) By Künneth's theorem, the nontrivial cohomology groups of 𝔽₀ are given by 𝐻⁰(𝔽₀,ℤ)=ℤ,𝐻²(𝔽₀,ℤ)=ℤ[𝜉_𝑧]⊕ℤ[𝜉_𝑤],𝐻⁴(𝔽₀,ℤ)=ℤ[𝜉_𝑧∧𝜉_𝑤], (4.91) where 𝜉_𝑧=𝑐₁(𝒪_𝔽0(1,0))=𝑝^∗_𝑧𝑐₁(𝒪_ℙ¹_𝑧(1)) and 𝜉_𝑤=𝑐₁(𝒪_𝔽0(0,1))=𝑝^∗_𝑤𝑐₁(𝒪_ℙ¹_𝑤(1)).

Let 𝔐_{𝑁;(𝑚𝑧,𝑚𝑤);𝑘}(𝔽₀) be the moduli space of isomorphism classes [ℰ] of torsion-free sheaves ℰ on 𝔽₀ with topological Chern invariants as in (4.51), where 𝑑=𝑚_𝑧𝜉_𝑧+𝑚_𝑤𝜉_𝑤, and nontrivial framings along the two independent “directions at infinity” are prescribed by two fixed isomorphisms ℰ|_ℓ𝑧≅𝑊⊗𝒪_ℙ¹_𝑧(𝑚_𝑧) and ℰ|_ℓ𝑤≅𝑊⊗𝒪_ℙ¹_𝑤(𝑚_𝑤), where 𝑊 is a fixed 𝑁-dimensional complex vector space. We will say that such a sheaf is “trivialized at infinity” if it is equipped with two isomorphisms ℰ|_ℓ𝑧≅𝑊⊗𝒪_ℙ¹_𝑧 and ℰ|_ℓ𝑤≅𝑊⊗𝒪_ℙ¹_𝑤. The moduli space of trivialized sheaves is denoted 𝔐_𝑁,𝑘(𝔽₀)∶=𝔐_{𝑁;(0,0);𝑘}(𝔽₀). These moduli spaces carry an obvious induced action of the torus 𝑇=𝑇²×𝑇^𝑁, and the following formal arguments show that the instanton counting in these moduli spaces is the same as before.

Proposition 4.2. There is a natural 𝑇-equivariant birational equivalence between the moduli spaces 𝔐_𝑁,𝑘(𝔽₀)≅𝔐_𝑁,𝑘(ℂ²). (4.92)

Proof. Represent 𝔽₀ as a divisor in ℙ²×ℙ¹ via the embedding ℙ¹_𝑧↪ℙ², [𝑧₀,𝑧₁]↦[𝑧₀,𝑧₁,𝑧₁]. Let ℙ² be the surface obtained as the blowup 𝑝 of a pair of points on the line at infinity ℓ_∞⊂ℙ² (see Section 4.7.). Note that ℓ_∞ is disjoint from the image of the line ℓ_𝑧. Then there is a correspondence diagram (4.93)where 𝑞 is the blowup of the intersection point ℓ_𝑧⋅ℓ_𝑤 on 𝔽₀=ℙ¹_𝑧×ℙ¹_𝑤. The corresponding Fourier-Mukai functors 𝑞_∗𝑝^∗ and 𝑝_∗𝑞^∗ determine equivalences on the categories of coherent sheaves over ℙ² and 𝔽₀, and hence induce mutually inverse rational maps between the corresponding sets of isomorphism classes of trivially framed torsion-free sheaves.

Proposition 4.3. For each fixed (𝑚_𝑧,𝑚_𝑤)∈ℤ², there is a natural 𝑇-equivariant bijection between the moduli spaces 𝔐_𝑁,𝑘(𝔽₀)≅𝔐_{𝑁;(𝑁𝑚𝑧,𝑁𝑚𝑤);𝑘(𝑚𝑧,𝑚𝑤)}(𝔽₀), (4.94) where 𝑘(𝑚_𝑧,𝑚_𝑤)=𝑘−𝑁𝑚_𝑧𝑚_𝑤. (4.95)

Proof. Given [ℰ]∈𝔐_𝑁,𝑘(𝔽₀), define ℰ(𝑚_𝑧,𝑚_𝑤)=ℰ⊗𝒪_𝔽0(𝑚_𝑧,𝑚_𝑤). (4.96) By (4.51) (with 𝑑=0), one has ch(ℰ)=𝑁−𝑘𝜉_𝑧∧𝜉_𝑤, (4.97) and using multiplicativity of the Chern character we compute ch(ℰ(𝑚_𝑧,𝑚_𝑤))=(𝑁−𝑘𝜉_𝑧∧𝜉_𝑤)∧(1+𝑚_𝑧𝜉_𝑧)∧(1+𝑚_𝑤𝜉_𝑤)=𝑁+𝑁(𝑚_𝑧𝜉_𝑧+𝑚_𝑤𝜉_𝑤)+(𝑁𝑚_𝑧𝑚_𝑤−𝑘)𝜉_𝑧∧𝜉_𝑤. (4.98) This therefore gives a map [ℰ]↦[ℰ(𝑚_𝑧,𝑚_𝑤)] on 𝔐_𝑁,𝑘(𝔽₀)→𝔐_{𝑁;(𝑁𝑚𝑧,𝑁𝑚𝑤);𝑘(𝑚𝑧,𝑚𝑤)}(𝔽₀). An identical calculation shows that the map ℰ(𝑚_𝑧,𝑚_𝑤)↦ℰ(𝑚_𝑧,𝑚_𝑤)⊗𝒪_𝔽0(−𝑚_𝑧,−𝑚_𝑤) is its inverse. These maps clearly induce bijections between (𝑚_𝑧,𝑚_𝑤)-framings and trivializations at infinity.

Proposition 4.2 shows that an instanton gauge bundle ℰ with trivial asymptotics on 𝔽₀ can again be regarded as an element [ℰ]∈𝔐_𝑁,𝑘(ℂ²). For fixed 𝑚_𝑧,𝑚_𝑤∈ℤ, Proposition 4.3 shows that the counting of torsion-free sheaves in the moduli spaces 𝔐_𝑁,𝑘(𝔽₀) and 𝔐_{𝑁;(𝑁𝑚𝑧,𝑁𝑚𝑤);𝑘(𝑚𝑧,𝑚𝑤)}(𝔽₀) coincides. In other words, the number of instantons with fixed nontrivial asymptotics matches the Young tableau count, as the number of sheaves (4.96) is independent of the integers 𝑚_𝑧,𝑚_𝑤. Proposition 4.3 also reproduces the formula [54, Section 4.2.2] for the renormalized volume of Young tableaux (reproduced above for 𝑁=1) as the induced shift in instanton charge in (4.95) of the sheaves (4.96).

The bijection of Proposition 4.2 is not necessarily a diffeomorphism. In fact, we have not yet shown that 𝔐_𝑁,𝑘(𝔽₀) is a smooth variety. Naively, one may try to argue this by using the formalism of [69], as our sheaves are trivialized on the divisor ℓ_∞=ℓ_𝑧+ℓ_𝑤 which has positive self-intersection ℓ_∞⋅ℓ_∞=2, and 𝔽₀⧵ℓ_∞≅ℂ². However, this divisor is not irreducible; in particular, it is not a ℙ¹. Note that formally applying the arguments which led to (4.72) in the limit 𝑝=0 would produce the partition function 𝑍^𝔽0_𝑈(𝑁)(𝑞)=(𝑍^ℂ2_𝑈(𝑁)(𝑞))², (4.99) as the Hirzebruch surface has 𝑛=𝜒(𝔽₀)−2=2. Although the generating function (4.99) is “doubled”, the instanton counting problems on 𝔽₀ and on ℂ² are nevertheless identical. A rigorous derivation of the formula (4.99) using the techniques of Section 4.7. follows once we establish that the equivalence of Proposition 4.2 is an isomorphism of underlying smooth varieties. Recall from Section 4.7. that the complex dimension of the instanton moduli space 𝔐_𝑁,𝑘(ℂ²) is 2𝑁𝑘.

Proposition 4.4. The moduli space 𝔐_𝑁,𝑘(𝔽₀) is a smooth quasiprojective variety of complex dimension 2𝑁𝑘.

Proof. As usual, the trivialization condition at infinity guarantees Gieseker semistability and hence quasi-projectivity, as discussed in Section 4.8. Using the divisor ℓ_∞=ℓ_𝑧+ℓ_𝑤 of 𝔽₀, one constructs the framed moduli space 𝔐_𝑁,𝑘(𝔽₀) as in [68] with tangent spaces Ext¹_𝒪𝔽0(ℰ,ℰ(−ℓ_∞)) and obstruction spaces given by Ext²_𝒪𝔽0(ℰ,ℰ(−ℓ_∞)). By reference [78, Section 5], one has Ext⁰_𝒪𝔽0(ℰ,ℰ(−1,−1))=Ext²_𝒪𝔽0(ℰ,ℰ(−1,−1))=0 (4.100) and hence 𝔐_𝑁,𝑘(𝔽₀) is smooth. The dimension of the tangent spaces is thus equal to minus the Euler characteristic −𝜒(ℰ,ℰ(−1,−1)), which for ℰ locally free may be calculated by using the Hirzebruch-Riemann-Roch theorem to write 𝜒(ℰ,ℰ(−1,−1))=_𝔽0ch(ℰ^∨⊗ℰ(−1,−1))∧Td(𝔽₀). (4.101) Using (4.97) and (4.98), one computes the Chern character ch(ℰ^∨⊗ℰ(−1,−1))=ch(ℰ^∨)∧ch(ℰ(−1,−1))=(𝑁−𝑘𝜉_𝑧∧𝜉_𝑤)∧(𝑁−𝑁(𝜉_𝑧+𝜉_𝑤)+(𝑁−𝑘)𝜉_𝑧∧𝜉_𝑤)=𝑁²−𝑁²(𝜉_𝑧+𝜉_𝑤)+(𝑁²−2𝑁𝑘)𝜉_𝑧∧𝜉_𝑤. (4.102) The Todd characteristic class and holomorphic Euler characteristic are given by (4.85) and (4.87) with 𝐶=𝔽₀, and we may thus write (4.101) as 𝜒(ℰ,ℰ(−1,−1))=_𝔽0(𝑁²12(𝑐₁(𝔽₀)∧𝑐₁(𝔽₀)+𝑐₂(𝔽₀𝑁))−²2(𝜉_𝑧+𝜉_𝑤)∧𝑐₁(𝔽₀)+(𝑁²−2𝑁𝑘)𝜉_𝑧∧𝜉_𝑤)=𝑁²𝜒(𝒪_𝔽0𝑁)−²2_𝔽0𝑐₁(𝔽₀)∧(𝜉_𝑧+𝜉_𝑤)+𝑁²−2𝑁𝑘. (4.103) One has 𝜒(𝒪_𝔽0)=1 since 𝑏₂(𝔽₀)=2=ℎ^1,1(𝔽₀). Since 𝜉_𝑧,𝜉_𝑤∈𝐻²(𝔽₀,ℤ) are the Poincaré duals of [ℓ_𝑧],[ℓ_𝑤]∈𝐻₂(𝔽₀,ℤ), with ℓ_𝑧,ℓ_𝑤≅ℙ¹ and ℓ_𝑧⋅ℓ_𝑧=ℓ_𝑤⋅ℓ_𝑤=0, the remaining integral in (4.103) may be computed as _𝔽0𝑐₁(𝔽₀)∧(𝜉_𝑧+𝜉_𝑤)=2_ℙ1𝑐₁(ℙ¹)=4, (4.104) where in the last step we used the fact that the holomorphic tangent bundle of ℙ¹ can be identified with 𝒪_ℙ¹(2). Putting everything together, we find 𝜒(ℰ,ℰ(−1,−1))=−2𝑁𝑘 (4.105) and the result follows when ℰ is a bundle. If ℰ is not locally free, then we simply consider a locally free resolution ℰ^•→ℰ→0 and use additivity of the Chern characteristic class.

In a similar vein, the equivalence established in Proposition 4.3 as it stands is only a bijective correspondence. The moduli space for nontrivial asymptotics is an example of the moduli spaces of “decorated sheaves” described in [20, Section 4.B], with the (𝑚_𝑧,𝑚_𝑤)-framed sheaves on 𝔽₀ providing examples of semistable framed modules. The moduli space 𝔐_{𝑁;(𝑁𝑚𝑧,𝑁𝑚𝑤);𝑘(𝑚𝑧,𝑚𝑤)}(𝔽₀) is thus a projective scheme with a universal family, that is, it is fine, and so possesses a universal sheaf. We will now argue that the moduli spaces 𝔐_{𝑁;(𝑁𝑚𝑧,𝑁𝑚𝑤);𝑘(𝑚𝑧,𝑚𝑤)}(𝔽₀) are smooth and diffeomorphic to one another for all 𝑚_𝑧,𝑚_𝑤∈ℤ. Then the computations of Section 4.7. can be repeated by integrating over 𝔐_{𝑁;(𝑁𝑚𝑧,𝑁𝑚𝑤);𝑘(𝑚𝑧,𝑚𝑤)}(𝔽₀) using (4.95) and Göttsche's formula to get the desired perpendicular partition function 𝑃^{𝑚𝑧,𝑚𝑤}_𝑈(𝑁)(𝑞)=𝑞^{−𝑁𝑚𝑧𝑚𝑤}𝜂(𝑞)^−2𝑁. (4.106) For 𝑁=1, this yields a rigorous derivation of the vertex rules of [54].

Let 𝔐(𝔽₀) be the moduli space of isomorphism classes of pairs ℰ=(ℰ,𝛼), where ℰ is an (𝑚_𝑧,𝑚_𝑤)-framed torsion-free sheaf on 𝔽₀ and 𝛼∶𝒪_𝔽0(𝑚_𝑧,𝑚_𝑤)^⊕𝑁→ℰ is the surjective morphism induced by the framing. Consider the family of pairs ℱ=(ℱ,𝛽) consisting of a coherent sheaf ℱ on 𝔽₀ together with a homomorphism 𝛽∶𝒪_𝔽0(𝑚_𝑧,𝑚_𝑤)^⊕𝑟→ℱ for some 𝑟∈ℕ. These pairs define objects of an abelian category coh_𝑓(𝔽₀) with the obvious morphisms between pairs induced by morphisms of coherent sheaves. Any (𝑚_𝑧,𝑚_𝑤)-framed torsion-free sheaf ℰ on 𝔽₀ clearly defines an object of coh_𝑓(𝔽₀), with 𝑟=𝑁 and ch(ℱ)=ch(ℰ).

Given the abelian category coh_𝑓(𝔽₀), one can derive functors, and hence compute sheaf cohomology in coh_𝑓(𝔽₀). We denote the corresponding Ext-functors by Ext^𝑖_𝑓. Then Ext²_𝑓(ℰ,ℰ) is the obstruction space for 𝔐(𝔽₀) at ℰ, while Ext¹_𝑓(ℰ,ℰ) is the tangent space to 𝔐(𝔽₀) at ℰ. Generally, for any two objects ℱ,𝒢 of coh_𝑓(𝔽₀), the group Ext¹_𝑓(ℱ,𝒢) can be computed in a purely algebraic way in terms of equivalence classes of extensions (4.107)

This definition of Ext¹_𝑓(ℱ,𝒢) is equivalent to the usual definition in terms of a twisted Dolbeault complex, and in particular it gives the deformation theory of the moduli space 𝔐(𝔽₀). Furthermore, there is a spectral sequence connecting it to the ordinary Ext-groups Ext^𝑖_𝒪𝔽0(ℱ,𝒢).

The line bundle 𝒪_𝔽0(𝑚_𝑧,𝑚_𝑤) is a projective 𝒪_𝔽0-module and is therefore flat, whence the tensor product map defined by (4.96) preserves exact sequences such as (4.107). It follows that the tangent spaces Ext¹_𝑓(ℰ,ℰ) are the same for all 𝑚_𝑧,𝑚_𝑤∈ℤ. Thus by Proposition 4.4, the moduli spaces 𝔐(𝔽₀) are all smooth and diffeomorphic to one another. This is consistent with the fact that stability (either slope stability of torsion-free sheaves [76] or stability of framed modules [20, Section 4.B]) is preserved by twisting with the line bundles 𝒪_𝔽0(𝑚_𝑧,𝑚_𝑤).

4.9.2. Universal Sheaves

We now construct the universal sheaf for the standard framing on 𝔽₀, and compute its 𝑇-equivariant character. Since the moduli space 𝔐_𝑁,𝑘(𝔽₀) is fine [68], there exists a universal sheaf 𝔈→𝔽₀×𝔐_𝑁,𝑘(𝔽₀), that is, a torsion-free sheaf 𝔈 such that 𝔈|_𝔽0×[ℰ]≅ℰ for every point [ℰ]∈𝔐_𝑁,𝑘(𝔽₀), unique up to tensor product with a (unique) line bundle. There are two natural vector bundles over 𝔐_𝑁,𝑘(𝔽₀) associated to a universal sheaf 𝔈. Firstly, there is the framing bundle 𝑊=𝐻⁰(𝔽₀,𝔈|_ℓ∞) of rank 𝑁 given by the fibre at infinity. Secondly, there is the bundle 𝑉 of “Dirac zero modes”, whose fibre at a given locally free sheaf [ℰ]∈𝔐_𝑁,𝑘(𝔽₀) restricted to ℝ⁴≅ℂ²≅𝔽₀⧵ℓ_∞ is the space of 𝐿²-solutions to the Dirac equation on ℝ⁴ in the background of the fundamental representation of the instanton gauge field corresponding to [ℰ]|_ℂ². It is constructed explicitly as follows. Let 𝜋₁ and 𝜋₂ be the canonical projections of 𝔽₀×𝔐_𝑁,𝑘(𝔽₀) onto the first and second factors, respectively. Then the Dirac bundle is 𝑉=𝑅¹𝜋_2∗(𝔈⊗𝜋^∗₁𝒪_𝔽0(−1,−1)), (4.108) where 𝑅¹𝜋_2∗ is the first right derived functor of the pushforward functor 𝜋_2∗. Its fibre over a point [ℰ]∈𝔐_𝑁,𝑘(𝔽₀) is the vector space 𝐻¹(𝔽₀,ℰ(−1,−1)).

Proposition 4.5. The Dirac bundle 𝑉 is a vector bundle of rank 𝑘 over 𝔐_𝑁,𝑘(𝔽₀).

Proof. By reference [78, Section 5], one has 𝐻⁰(𝔽₀,ℰ(−1,−1))=𝐻²(𝔽₀,ℰ(−1,−1))=0 (4.109) and hence 𝑉 is a vector bundle of rank equal to −𝜒(ℰ(−1,−1)). For ℰ locally free, the Hirzebruch-Riemann-Roch theorem, together with (4.98), (4.85) with 𝐶=𝔽₀, (4.104), and 𝜒(𝒪_𝔽0)=1, givesrank(𝑉)=−_𝔽0ch(ℰ(−1,−1))∧Td(𝔽₀)=−_𝔽0(𝑁−𝑁(𝜉_𝑧+𝜉_𝑤)+(𝑁−𝑘)𝜉_𝑧∧𝜉_𝑤)1∧(1+2𝑐₁(𝔽₀1)+12(𝑐₁(𝔽₀)∧𝑐₁(𝔽₀)+𝑐₂(𝔽₀)))=−𝑁𝜒(𝒪_𝔽0𝑁)+2_𝔽0𝑐₁(𝔽₀)∧(𝜉_𝑧+𝜉_𝑤)+𝑘−𝑁=𝑘. (4.110) Again, the statement for generic torsion-free sheaves [ℰ]∈𝔐_𝑁,𝑘(𝔽₀) follows by considering a locally free resolution ℰ^•→ℰ→0.

By definition, (4.89) and the construction of Section 4.7., the 𝑇-equivariant characters of the framing and Dirac bundles regarded as 𝑇-modules, at a fixed point in 𝔐_𝑁,𝑘(𝔽₀)𝑇 labelled by an 𝑁-coloured Young tableau 𝜆=(𝜆₁,…,𝜆_𝑁), are given by ch_𝑇(𝑊)(𝜆)=^𝑁_𝑙=1𝑒_𝑙,ch_𝑇(𝑉)(𝜆)=^𝑁_𝑙=1𝑒_{𝑙(𝑖,𝑗)∈𝜆𝑙}𝑡_𝑧^𝑖−1𝑡_𝑤^𝑗−1, (4.111) where 𝑒_𝑙=e^𝑎𝑙. Let us now describe how the ADHM construction is modified in this case. This has been worked out in [78, Section 5]. The linear algebraic data is formally the same as for the analysis of framed instantons on ℙ² [49], defined by linear operators 𝐵_𝑧∈Hom(𝑉,𝑉),𝐵_𝑤∈Hom(𝑉,𝑉),𝐼∈Hom(𝑊,𝑉),𝐽∈Hom(𝑉,𝑊). (4.112) Then any framed torsion-free sheaf [ℰ]∈𝔐_𝑁,𝑘(𝔽₀) can be represented as the middle cohomology group of the complex 0→𝑉⊗𝒪_𝔽0(−1,−1)−−−^𝜎−→𝑉⊗𝒪_𝔽0⊕(−1,0)𝑉⊗𝒪_𝔽0⊕(0,−1)𝑊⊗𝒪_𝔽0^𝜏−−−→𝑉⊗𝒪_𝔽0→0 (4.113) which is exact at the first and last terms, where 𝑧𝜎=⁰𝐵_𝑧−𝑧₁𝑤₀𝐵_𝑤−𝑤₁𝑧₀𝑤₀𝐽,𝜏=−(𝑤₀𝐵_𝑤−𝑤₁)𝑧₀𝐵_𝑧−𝑧₁𝐼. (4.114)

The virtual 𝑇-equivariant bundle defined by the cohomology of the complex (4.113) gives a representative of the isomorphism class of the universal sheaf 𝔈 in the 𝑇-equivariant K-theory group 𝐾⁰_𝑇(𝔽₀×𝔐_𝑁,𝑘(𝔽₀)) as 𝔈=(𝒪_𝔽0⊠𝑊)⊕(𝑆⁻⊝𝑆⁺)⊠𝑉, (4.115) where 𝑆⁺=𝒪_𝔽0(−1,−1)⊕𝒪_𝔽0,𝑆⁻=𝒪_𝔽0(−1,0)⊕𝒪_𝔽0(0,−1) (4.116) are 𝑇-equivariant bundles over 𝔽₀ which, after tensoring with a line bundle of degree one, restrict to the usual positive/negative chirality spinor bundles over ℝ⁴≅ℂ²≅𝔽₀⧵ℓ_∞. In the topologically twisted gauge theory, fermion fields become differential forms and these bundles are identified with the bundles of even/odd holomorphic forms over 𝔽₀ [79]. By (4.89), the holomorphic line bundles (4.90) have 𝑇-equivariant characters ch_𝑇(𝒪_𝔽0(𝑚_𝑧,𝑚_𝑤))=𝑡^−𝑚𝑧_𝑧𝑡^−𝑚𝑤_𝑤, (4.117) and consequently, ch_𝑇(𝑆⁺)=1+𝑡_𝑧𝑡_𝑤,ch_𝑇(𝑆⁻)=𝑡_𝑧+𝑡_𝑤. (4.118) Using (4.111) and (4.118), we may thus compute the character of the universal sheaf (4.115) at a fixed point in the instanton moduli space as ch_𝑇(𝔈)(𝜆)=ch_𝑇(𝑊)(𝜆)+(ch_𝑇(𝑆⁻)−ch_𝑇(𝑆⁺))ch_𝑇=(𝑉)(𝜆)_𝑁𝑙=1𝑒_𝑙(1−(1−𝑡_𝑧)(1−𝑡_𝑤)_{(𝑖,𝑗)∈𝜆𝑙}𝑡_𝑧^𝑖−1𝑡_𝑤^𝑗−1), (4.119) which coincides with the standard expression for instantons on ℝ⁴ [80].

For completeness, let us record here the 𝑇-equivariant Chern character of the tangent bundle 𝑇𝔐_𝑁,𝑘(𝔽₀) over the instanton moduli space at the torus fixed points. As the only nonvanishing cohomology group of (4.20) is a model of the tangent space, it can be computed as the equivariant index of this complex. For this, let us recall some combinatorial definitions. Let 𝜆 be a Young diagram. Define the arm and leg lengths of a box (𝑖,𝑗)∈𝜆, respectively, by 𝐴_𝜆(𝑖,𝑗)=𝜚_𝑖−𝑗,𝐿_𝜆(𝑖,𝑗)=𝜚^𝑡_𝑗−𝑖, (4.120) where 𝜚_𝑖 is the length of the 𝑖th column of 𝜆 and 𝜚^𝑡_𝑗 is the length of the 𝑗th row of 𝜆. At a fixed point 𝜆, the character can then be expressed after some algebra in terms of the characters of the representation as ch_𝑇(𝑇𝔐_𝑁,𝑘(𝔽₀))(𝜆)=^𝑁_𝑙,𝑚=1𝑒_𝑙𝑒_𝑚⁻¹(_{(𝑖,𝑗)∈𝜆𝑙}𝑡^{−𝐿𝜆𝑙}_𝑧^(𝑖,𝑗)𝑡^𝐴𝜆𝑚_𝑤^{(𝑖,𝑗)+1}+_{(𝑖,𝑗)∈𝜆𝑚}𝑡^𝐿𝜆𝑙_𝑧^{(𝑖,𝑗)+1}𝑡^{−𝐴𝜆𝑚}_𝑤^(𝑖,𝑗)). (4.121) Again this coincides with the standard result [50, 80, 81]. As in [82], after toric localization one has ch_𝑇(𝑇𝔐_𝑁,𝑘(𝔽₀))=−_{𝔽0×𝔐𝑁,𝑘(𝔽0)}ch_𝑇(𝔈)∧ch_𝑇(𝔈^∨)∧Td_𝑇(𝔽₀) (4.122) at the 𝑇-fixed points 𝜆, up to a universal perturbative contribution (the character of 𝑊⊗𝑊^∗). This expression formally generalizes to generic toric surfaces [83] and is used to construct vertex gluing rules below. From the top Chern class, one can also straightforwardly extract the equivariant Euler classes 𝑒_𝑇(𝑇𝔐_𝑁,𝑘(𝔽₀))(𝜆)=^𝑁_𝑙,𝑚=1𝑛^𝜆_𝑙,𝑚(𝜖_𝑧,𝜖_𝑤,𝑎), (4.123) where 𝑛^𝜆_𝑙,𝑚(𝜖_𝑧,𝜖_𝑤,𝑎)=_{(𝑖,𝑗)∈𝜆𝑙}(𝑎_𝑚−𝑎_𝑙−𝐿_𝜆𝑚(𝑖,𝑗)𝜖_𝑧+(𝐴_𝜆𝑙(𝑖,𝑗)+1)𝜖_𝑤)×_{(𝑖′,𝑗′)∈𝜆𝑚}(𝑎_𝑚−𝑎_𝑙+(𝐿_{𝜆𝑙(𝑖′,𝑗′)}+1)𝜖_𝑧−𝐴_𝜆𝑚(𝑖^′,𝑗^′)𝜖_𝑤). (4.124)

4.9.3. Framed Modules on 𝔽₀ from Dirac Modules on ℙ¹

For the case of nontrivial (𝑚_𝑧,𝑚_𝑤)-framings, let 𝜄(𝑚_𝑧,𝑚_𝑤)∶𝔐_𝑁,𝑘(𝔽₀)→𝔐_{𝑁;(𝑁𝑚𝑧,𝑁𝑚𝑤);𝑘(𝑚𝑧,𝑚𝑤)}(𝔽₀) (4.125) be the isomorphism induced by the map [ℰ]↦[ℰ(𝑚_𝑧,𝑚_𝑤)]=[ℰ⊗𝒪_𝔽0(𝑚_𝑧,𝑚_𝑤)], and let 𝔈 be a universal sheaf on 𝔽₀×𝔐_𝑁,𝑘(𝔽₀). Then the torsion-free sheaf 𝔈(𝑚_𝑧,𝑚_𝑤)=(id_𝔽0×𝜄(𝑚_𝑧,𝑚_𝑤)⁻¹)^∗(𝔈) (4.126) is a universal sheaf on 𝔽₀×𝔐_{𝑁;(𝑁𝑚𝑧,𝑁𝑚𝑤);𝑘(𝑚𝑧,𝑚𝑤)}(𝔽₀). The equivariant character of 𝔈(𝑚_𝑧,𝑚_𝑤) at the fixed points of the toric action modifies the decompositions (4.111) to allow for “propagators” which appear along the edges ℙ¹ of the toric diagram Δ(𝐶) of a generic toric surface 𝐶. These modifications of the 𝑇-equivariant character of the universal sheaf corresponding to the instanton sheaves (4.96) are determined via shifts by the ℂ^∗-equivariant character of modules of solutions to the Dirac equation on ℙ¹. We will now describe these modules explicitly. As we have seen, the bundle 𝒪_ℙ¹(𝑚) of degree 𝑚∈ℤ over the Riemann sphere ℙ¹ is the crucial ingredient in generating instantons with nontrivial framings. After choosing a hermitean metric on the fibres of 𝒪_ℙ¹(𝑚), it is the holomorphic line bundle underlying the standard Dirac monopole line bundle of topological charge 𝑚. The form of the Dirac operator 𝐷_𝑚 in the background of the corresponding monopole gauge potential is well known and can be conveniently described in the following way [32].

Given homogeneous coordinates [𝑧₀,𝑧₁] on ℙ¹, let 𝑦=𝑧₁/𝑧₀ denote stereographic coordinates on the northern hemisphere. Then the twisted Dirac operator in the monopole background of magnetic charge 𝑚∈ℤ is given by 𝐷_𝑚=0𝐷⁻_𝑚𝐷⁺_𝑚0, (4.127) where 𝐷⁺_𝑚=12[(1+𝑦𝜕𝑦)𝜕𝑦−12(𝑚+1)𝑦],𝐷⁻_𝑚1=−2[(1+𝑦𝜕𝑦)+1𝜕𝑦2(𝑚−1)𝑦]. (4.128) These operators act on sections of the chiral/antichiral spinor line bundles associated to the twisted holomorphic spinor bundles 𝒪_ℙ¹(±1)⊗𝒪_ℙ¹(𝑚). We are interested in the subspaces of zero modes 𝑆^±_𝑚=ker𝐷^±_𝑚. (4.129)

The action of 𝑡∈ℂ^∗ is implemented by the automorphism 𝐹_𝑡 of ℙ¹ defined by 𝐹_𝑡(𝑦,𝑦)=(𝑡𝑦,𝑡⁻¹𝑦). (4.130) The irreducible representations 𝑇^𝑖≅ℂ of ℂ^∗ are labelled by their weights 𝑖∈ℤ and are defined by 𝑧↦𝑡⋅𝑧=𝑡^𝑖𝑧 for 𝑡∈ℂ^∗ and 𝑧∈ℂ. The corresponding ℂ^∗-eigenspace decomposition of the modules (4.129) can be described for all 𝑚∈ℤ as follows.

Theorem 4.6. The isotopical decompositions of the spinor modules 𝑆^±_𝑚 over ℙ¹, as ℂ^∗-modules, are given by 𝑆⁺_𝑚=⊕^|𝑚|_𝑖=1𝑇^𝑖−1,𝑆⁻_𝑚𝑆={0}for𝑚<0,_−𝑚=⊕^𝑚_𝑖=1𝑇^{−(𝑖−1)},𝑆⁺_𝑚={0}for𝑚>0. (4.131)

Proof. The solutions of the Dirac equation are given by 𝐿²-solutions of the differential equations 𝐷^±_𝑚𝜓^±_𝑚=0 (4.132) for the spinors 𝜓^±_𝑚∈ker𝐷^±_𝑚. The line bundle 𝒪_ℙ¹(𝑚) has holomorphic transition function 𝑦^𝑚 transforming sections from the northern hemisphere to the southern hemisphere of ℙ¹, which after unitary reduction to a hermitean line bundle becomes (𝑦/𝑦)^𝑚/2. By using these transition functions, it is easy to see that the only solutions of the equations (4.132) which are regular on both the northern and southern hemispheres are of the form 𝜓⁺_𝑚=1(1+𝑦𝑦)^{(|𝑚|−1)/2|𝑚|}_𝑖=1𝜉_𝑖𝑦^𝑖−1,𝜓⁻_𝑚𝜓=0for𝑚<0,_−𝑚=1(1+𝑦𝑦)^{𝑚(𝑚−1)/2}_𝑖=1𝜉̃_𝑖𝑦^𝑖−1,𝜓⁺_𝑚=0for𝑚>0, (4.133) with constant coefficients 𝜉_𝑖,𝜉̃_𝑖∈ℂ. The result now follows from (4.130).

We can use Theorem 4.6 to compute the ℂ^∗-equivariant characters of the spinor modules (4.129). One finds ch_ℂ^∗(𝑆⁻_𝑚)=^𝑚_𝑖=1𝑡^{−(𝑖−1)}=1−𝑡^−𝑚1−𝑡⁻¹ (4.134) for 𝑚>0, while ch_ℂ^∗(𝑆⁺_𝑚)=^|𝑚|_𝑖=1𝑡^𝑖−1=1−𝑡^|𝑚|1−𝑡 (4.135) for 𝑚<0. In the nonequivariant limit 𝑡→1, the characters (4.134) and (4.135) reproduce the known index of the Dirac operator, index(𝐷_𝑚)=−𝑚, in the monopole background. These characters shift the character (4.119). For example, with 𝑚_𝑧,𝑚_𝑤>0 and 𝑁=1, one has ch_𝑇(𝔈(𝑚_𝑧,𝑚_𝑤))(𝜆)=ch_𝑇(𝔈)(𝜆)+(1−𝑡_𝑧)ch_ℂ^∗_𝑧(𝑆⁻_𝑚𝑧)+(1−𝑡_𝑤)ch_ℂ^∗_𝑤(𝑆⁻_𝑚𝑤). (4.136) This completes the derivation of the gluing rules of [54, Section 4.3].

5. D2-Brane Gauge Theory and Gromov-Witten Invariants

In this final section, we study the reduction of the four-dimensional gauge theories of Section 4 on local curves and examine in detail the example of local ℙ¹ which has been extensively described from the point of view of Vafa-Witten theory. We begin with a somewhat heuristic description of how these two-dimensional supersymmetric gauge theories are induced. Then we proceed to a more formal topological field theory formalism which systematically computes topological string amplitudes and Gromov-Witten invariants. Finally, we briefly address wall-crossing issues once again from a physical standpoint.

5.1. 𝑞-Deformed Two-Dimensional Yang-Mills Theory

One of the simplest classes of examples are the noncompact local Calabi-Yau threefolds which are fibred over curves. If Σ_𝑔→𝑋 is a holomorphically embedded curve of genus 𝑔 in a Calabi-Yau threefold, then the holomorphic tangent bundle restricts to Σ_𝑔 as 𝑇Σ_𝑔⊕𝒩_Σ𝑔, where the normal bundle 𝒩_Σ𝑔 is a holomorphic bundle of rank two over Σ_𝑔. By the Calabi-Yau condition 𝑐₁(𝑋)=0, one has 𝑐₁(𝒩_Σ𝑔)=−𝜒(Σ_𝑔)=2𝑔−2. Thus in a neighbourhood of Σ_𝑔, the manifold 𝑋 looks like the total space of a holomorphic bundle 𝒩→Σ_𝑔 of rank two with 𝑐₁(𝒩)=2𝑔−2. Hence we consider the total space of the bundle 𝑋=𝑋_𝑝=𝒪_Σ𝑔(𝑝+2𝑔−2)⊕𝒪_Σ𝑔(−𝑝) (5.1) over Σ_𝑔. Since 𝒪_Σ𝑔(𝑝) is a holomorphic line bundle over Σ_𝑔 of degree 𝑝, one has 𝑐₁(𝑋_𝑝)=(𝑝+2𝑔−2)+(−𝑝)=2𝑔−2 (5.2) as required. For genus 𝑔=0, this is just the example of local ℙ¹ which was introduced in Section 4.3.

In our applications to black hole microstate counting, we count bound states of D4-D2-D0 branes in 𝑋_𝑝 with 𝑁 D4-branes wrapping the four-cycle 𝐶_p which is the total space of the line bundle 𝒪_Σ𝑔(−𝑝)→Σ_𝑔, together with D2-branes wrapping the base Riemann surface Σ_𝑔 (embedded in 𝐶 and 𝑋_𝑝 as the zero section). One can then localize the path integral of the 𝒩=4 topological gauge theory on 𝐶_𝑝 to the ℂ^∗-invariant modes along the fibre of 𝒪_Σ𝑔(−𝑝). The result is an effective 𝑈(𝑁) gauge theory on Σ_𝑔 whose action is given by [84] 1𝑆=𝑔_𝑠Σ𝑔𝜗Tr(Φ𝐹)+𝑔_𝑠Σ𝑔𝑝Tr(Φ𝑘)−2𝑔_𝑠Σ𝑔Tr(Φ²𝑘), (5.3) where 𝜗=𝑔_𝑠𝜙²/2𝜋 and the last term is a mass deformation which originates in four dimensions due to the nontriviality of the bundle 𝒪_Σ𝑔(−𝑝) [85]. The scalar field Φ is given by the holonomy of the four-dimensional gauge connection at infinity Φ(𝑧)=_{𝒪Σ𝑔(−𝑝)𝑧}𝐹(𝑧)=_{𝑆1𝑧,|𝑢|=∞}𝐴, (5.4) where 𝑧∈Σ_𝑔 and 𝑢∈𝒪_Σ𝑔(−𝑝).

The action (5.3) is just the BF-theory representation of two-dimensional Yang-Mills theory, with the usual 𝐹²-term coming from performing the gaussian integral over Φ in the functional integral. If Φ were arbitrary, then we could use diagonalization techniques to reduce the partition function to the usual heat kernel expansion of two-dimensional Yang-Mills theory. However, the identification of Φ as a holonomy means that it should be treated as a periodic field, and the change from a Lie algebra-valued variable to a group-like variable affects the Jacobian of the path integral measure that arises from diagonalization. This modifies the usual Migdal expansion of two-dimensional gauge theory to give the partition function 𝑍^𝐶𝑝_𝑈(𝑁)(𝑞,𝑄)=_𝑅dim_𝑞(𝑅)^2−2𝑔𝑞^{(𝑝/2)𝐶2(𝑅)}𝑄^𝐶1(𝑅), (5.5) where 𝑞=e^−𝑔𝑠 as before and 𝑄=e^i𝜗. The sum runs over unitary irreducible representations 𝑅=(𝑅₁,…,𝑅_𝑁) of the gauge group 𝑈(𝑁) with weights 𝑅_𝑖, first and second Casimir invariants 𝐶₁(𝑅)=∑_𝑖𝑅_𝑖 and 𝐶₂(𝑅)=∑_𝑖𝑅²_𝑖, and quantum dimension dim_𝑞(𝑅)=_{1≤𝑖<𝑗≤𝑁}[𝑅_𝑖−𝑅_𝑗+𝑗−𝑖]_𝑞[𝑗−𝑖]_𝑞, (5.6) where the 𝑞-number is defined by [𝑛]_𝑞𝑞∶=^𝑛/2−𝑞^−𝑛/2𝑞^1/2−𝑞^−1/2 (5.7) with [𝑛]_𝑞=𝑛+𝑂(𝑞−1) for 𝑞→1. This reduced two-dimensional gauge theory is a 𝑞-deformation of ordinary Yang-Mills theory.

5.2. Topological Field Theory

We will now give a more precise definition of this gauge theory using techniques of two-dimensional topological field theory. This formalism can then be used to compute topological string amplitudes. The idea is that the cutting and pasting of base Riemann surfaces is equivalent to the cutting and pasting of the corresponding local Calabi-Yau threefolds, by either adding or cancelling off D-branes corresponding to the boundaries in the topological vertex formalism [84, 86]. The operations of gluing manifolds satisfy all axioms of a two-dimensional topological field theory. By computing the open string topological A-model amplitudes on a few Calabi-Yau manifolds, we then get all others by gluing. In the following we focus on the genus zero case Σ₀=ℙ¹ for simplicity and illustrative purposes.

When the base of the fibration is the sphere, we will only need the basic Calabi-Yau cap amplitude corresponding to a disk 𝐷, represented symbolically as

where the levels 𝑘₁=𝑒(ℒ₁) and 𝑘₂=𝑒(ℒ₂) label the degrees in 𝐻²(𝐷,𝜕𝐷) of a pair of line bundles ℒ₁⊕ℒ₂→𝐷, and the unitary matrix 𝑈 labels the holonomy of a gauge connection on 𝜕𝐷≅𝑆¹. Under concatenation the levels add. The basic cap amplitude is defined by

where 𝑞=e^−𝑔𝑠 as before, the sum over 𝑅 runs through the space of symmetric functions (Young tableaux), that is, irreducible representations of 𝑆𝑈(∞), and ch_𝑅(𝑈) is the character of the holonomy 𝑈 in the representation 𝑅. The quantity 𝑑_𝑞(𝑅)∶=_{(𝑖,𝑗)∈𝑅}1[ℎ(𝑖,𝑗)]_𝑞 (5.8) is the quantum dimension of the symmetric group representation corresponding to the Young diagram of 𝑅, and 𝑘_𝑅=2_{(𝑖,𝑗)∈𝑅}(𝑖−𝑗). (5.9) The quantity 𝑑_𝑞(𝑅)𝑞^{−𝑘𝑅/4}=∶𝐶_𝑅,∅,∅(𝑞)=𝑊_𝑅,∅(𝑞) (5.10) is the topological vertex amplitude for the A-model topological string theory with a single stack of D-branes on the noncompact Calabi-Yau threefold 𝑋=ℂ³. Similarly, define the cap amplitude

For the gluing rules, we sew two open Riemann surfaces Σ_𝐿 and Σ_𝑅 together along their common boundary to get the Riemann surface Σ_𝐿∪𝑅. For this, the orientations of the corresponding boundary circles must be opposite. Reversing the orientation acts on the boundary holonomy as 𝑈→𝑈⁻¹, and hence the gluing rules read 𝑍(Σ_𝐿∪𝑅)=_{𝑆𝑈(∞)}d𝑈𝑍(Σ_𝐿)(𝑈)𝑍(Σ_𝑅)(𝑈⁻¹), (5.11) where d𝑈 is the bi-invariant Haar measure and the integrals can be evaluated explicitly by using the orthogonality relations for characters _{𝑆𝑈(∞)}d𝑈ch_𝑅1(𝑈)ch_𝑅2(𝑈⁻¹)=𝛿_𝑅1,𝑅2. (5.12) We will also need the Calabi-Yau trinion (or “pants”) amplitudes

Let us point out two particularly noteworthy aspects of this construction thus far. Firstly, the construction defines a functor of tensor categories 𝑍∶𝑆_ℒ1,ℒ2→Rep, (5.13) where Rep is the representation category of 𝑆𝑈(∞) and 𝑆_ℒ1,ℒ2 is the geometric tensor category defined as follows. The objects of 𝑆_ℒ1,ℒ2 are compact oriented one-manifold 𝑌, that is, disjoint unions of oriented circles. A morphism 𝑌₁→𝑌₂ between two objects of 𝑆_ℒ1,ℒ2 is a triple (𝑊,𝑌₁,𝑌₂), where 𝑊 is an oriented cobordism between 𝑌₁ and 𝑌₂, that is, a smooth oriented two-manifold with boundary 𝜕𝑊=𝑌₁∐(−𝑌₂), and the complex line bundles ℒ₁,ℒ₂ over 𝑊 are trivialized on 𝜕𝑊. This is analogous to the Baum-Douglas description of D-branes in K-homology [87]. There is a natural notion of equivalence provided by boundary preserving, oriented diffeomorphisms 𝑓 : 𝑊→𝑊^′ with ℒ_𝑖≅𝑓^∗ℒ^′_𝑖, 𝑖=1,2. Composition of morphisms is given by concatenation of cobordism and gluing of bundles along the concatenation using the trivializations. For a connected cobordism 𝑊, we can label the isomorphism classes of a pair of line bundles (ℒ₁,ℒ₂) by the levels (𝑘₁,𝑘₂), where 𝑘_𝑖=𝑒(ℒ_𝑖)∈𝐻²(𝑊,𝜕𝑊). Under concatenation, these levels add.

Secondly, the ℂ^∗-action on ℙ¹ lifts to an action of the torus 𝑇=(ℂ^∗)² on 𝑋_𝑝, via the natural scaling action on the fibres. The Gromov-Witten invariants in this case are defined via the virtual localization formula as a residue integral over the 𝑇-fixed point locus, along the lines explained in Section 3.5. A stable map to 𝑋_𝑝 which is 𝑇-invariant factors through the zero section. It follows that there is a natural isomorphism of moduli spaces 𝔐_𝑔(𝑋_𝑝,𝛽)^𝑇=𝔐_𝑔(ℙ¹,𝛽), with 𝛽=𝑑∈ℤ, and consequently [𝔐_𝑔(𝑋_𝑝,𝑑)^𝑇]^vir=[𝔐_𝑔(ℙ¹,𝑑)]^vir. (5.14) This equality of virtual fundamental cycles implies that the invariants constructed by integrating over each moduli space coincide. It means that topological string theory on 𝑋_𝑝 is equivalent to a field theory on ℙ¹, which is just the reduction we argued in Section 5.1. In this case the Gromov-Witten invariants of 𝑋_𝑝 correspond to degree 𝑑 Hurwitz numbers of ℙ¹.

Returning to our computations, we get the annulus (or tube) amplitude by contracting the cap and trinion amplitudes to get

Finally, we compute the amplitude of the fibration 𝑋_𝑝→ℙ¹ by using (5.14) and the gluing rules. In order to get the appropriate bundle degrees (𝑝−2,−𝑝), we glue 𝑝 tubes between two caps to get (5.15)

For 𝑝=1, the threefold 𝑋₁≅𝐾_ℙ^1/2₁⊕𝐾_ℙ^1/2₁ is the resolved conifold, and this formula is a 𝑞-deformation of the classical Hurwitz formula counting unramified covers of the Riemann sphere ℙ¹.

In fact, the quantity 𝑑_𝑞(𝑅) is analogously the 𝑁→∞ limit of the quantum dimension dim_𝑞(𝑅) of the 𝑈(𝑁) representation with the same Young tableau. This follows by Schur-Weyl reciprocity which writes 𝑆𝑈(𝑁) representations in terms of representations of the symmetric group 𝑆_𝑁. Let 𝑅 be a representation corresponding to a Young diagram 𝜆 with row lengths 𝜆_𝑖, 𝑖=1,…,𝑑(𝜆), where 𝜆₁≥𝜆₂≥⋯, and 𝑑(𝜆) is the number of rows in 𝜆. Set 𝜇=𝑞^𝑁, and define another 𝑞-number by [𝑛]_𝜇=𝜇^1/2𝑞^𝑛/2−𝜇^−1/2𝑞^−𝑛/2𝑞^1/2−𝑞^−1/2. (5.16) Then the quantum dimension of 𝑅 can be written as dim_𝑞(𝑅)=_{1≤𝑖<𝑗≤𝑑(𝜆)}[𝜆_𝑖−𝜆_𝑗+𝑗−𝑖]_𝑞[𝑗−𝑖]_𝑞^𝑑(𝜆)_𝑖=1∏^{𝜆𝑖−𝑖}_{𝑣=1−𝑖}[𝑣]_𝜇∏^𝜆𝑖_𝑣=1[𝑣−𝑖+𝑑(𝜆)]_𝑞. (5.17) This is a Laurent polynomial in 𝜇^±1/2 whose coefficients are rational functions of 𝑞^±1/2. The leading power of 𝜇 is |𝜆|/2, and the coefficient of this power is the rational function of 𝑞^±1/2 given by 𝑞^{−𝑘𝑅/4}_{1≤𝑖<𝑗≤𝑑(𝜆)}[𝜆_𝑖−𝜆_𝑗+𝑗−𝑖]_𝑞[𝑗−𝑖]_𝑞^{𝑑(𝜆)𝜆}_𝑖=1^𝑖_𝑣=11[𝑣−𝑖+𝑑(𝜆)]_𝑞=𝑑_𝑞(𝑅)𝑞^{−𝑘𝑅/4}. (5.18) As before, this is just the topological vertex amplitude 𝐶_𝑅,∅,∅(𝑞). The limit 𝑞→1 gives the ordinary dimension of the representation 𝑅.

Generalizing the generating function (5.15) thereby gives 𝑍_𝑞YM(ℙ¹)=_𝑅dim_𝑞(𝑅)²𝑞^{(𝑝/2)𝐶2(𝑅)}, (5.19) where now the sum runs over irreducible representations 𝑅 of 𝑈(𝑁) and 𝐶₂(𝑅)=𝑘_𝑅+𝑁(𝑅) is the second Casimir invariant of 𝑅. By the attractor mechanism, the Kähler modulus of ℙ¹ is given by 𝑡=(𝑁/2)(𝑝−2)𝑔_𝑠. In the limit 𝑞→1, this two-dimensional field theory coincides with ordinary Yang-Mills theory on the sphere ℙ¹. Note that this is a weak-coupling limit 𝑔_𝑠→0, with 𝑝𝑔_𝑠=𝑔²_YM𝐴 and 𝐴 the area of ℙ¹. Using similar constructions as above, it is possible to formulate this gauge theory directly as a two-dimensional topological field theory, without any reference to the extrinsic bundle structure of the local threefold 𝑋_𝑝. Explicit computations of the corresponding Gromov-Witten invariants of 𝑋_𝑝 can be found in [9–11].

5.3. Wall-Crossing Formulas

Finally, let us briefly comment on the relationship to black hole entropy counting. Reinstating the 𝜗-angle as in (5.5), the attractor mechanism fixes the Kähler modulus of Σ₀=ℙ¹ as 𝑋𝑡=2𝜋i¹𝑋⁰=(𝑝−2)𝑁𝑔_𝑠2−i𝜗. (5.20) The large 𝑁 limit of the partition function (5.5) should possess this modulus. The modulus squared structure anticipated by the OSV relation (2.25) is given in this limit by [84] 𝑍_𝑞YM(ℙ¹)=_{𝑟∈ℤ𝑅1,𝑅2}𝑍^𝑞YM+_𝑅1,𝑅2(𝑡+𝑝𝑔_𝑠𝑟)𝑍^𝑞YM−_𝑅1,𝑅2(𝑡+𝑝𝑔_𝑠𝑟), (5.21) where the sum over 𝑟 runs through 𝑈(1) charges in the local decomposition 𝑈(𝑁)∼𝑈(1)×𝑆𝑈(𝑁) of the gauge group, the second sum runs through pairs of irreducible 𝑆𝑈(𝑁) representations 𝑅₁,𝑅₂, and 𝑍^𝑞YM+_𝑅1,𝑅2(𝑡) is the perturbative topological string amplitude on 𝑋_𝑝 with two stacks of D-branes in the fibre. The conjugate amplitude is 𝑍^𝑞YM−_𝑅1,𝑅2(𝑡)=(−1)^{𝐶1(𝑅1)+𝐶1(𝑅2)}𝑍^𝑞YM+_{𝑅𝑡1,𝑅𝑡2}(𝑡). Thus in this case the OSV relation (2.25) is modified to the symbolic form 𝑍_BH=_{𝑟∈ℤ𝑏}|𝑍^(𝑏,𝑟)_top|², (5.22) where the first sum runs over Ramond-Ramond fluxes 𝑟 through the base Riemann surface Σ_𝑔 while the second sum is over fibre D-branes which carry additional moduli 𝑡̂ measuring their distances from the base Σ_𝑔 along the noncompact fibre directions.

Further analysis of the relationships with 𝒩=4 Yang-Mills theory on 𝐶_𝑝 is studied in [9, 10, 12, 13, 57], where it is observed that the instanton expansions of the two-dimensional and four-dimensional gauge theory partition functions differ by perturbative contributions. These extra factors arise from the noncompactness of the surface 𝐶_𝑝. When 𝐶_𝑝 is the resolution of an 𝐴_𝑝,𝑛 singularity, its boundary is the three-dimensional Lens space 𝐿(𝑝,𝑛)=𝑆³/Γ_(𝑝,𝑛). The perturbative factors can then be identified as the partition function of Chern-Simons gauge theory on the boundary and arise as a consequence of the fact that the two-dimensional gauge theory implicitly integrates over all boundary conditions. Once these boundary contributions are stripped, the enumeration of instantons in the two-dimensional and four-dimensional gauge theories coincide [13].

Acknowledgments

It is a pleasure to thank M. Cirafici, E. Gasparim, A.-K. Kashani-Poor, A. Maciocia, and A. Sinkovics for enjoyable discussions and collaborations on some of the results presented here. This work was supported in part by Grant ST/G000514/1 “String Theory Scotland” from the UK Science and Technology Facilities Council.