Academic Editor: M. N. Hounkonnou
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 . To define such invariants, we need moduli spaces that are varieties rather than algebraic stacks. The standard method is to choose a polarization on and restrict attention to semistable sheaves. If is a Kähler manifold, then a natural choice of polarization is provided by a fixed Kähler two-form on . 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 , which can be defined using virtual cycles of the pertinent moduli spaces and are invariant under deformations of . 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 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 [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 .
In the following we will describe the computation of BPS indices of stable D-brane states via instanton counting in certain noncommutative and -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. 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]. 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. 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 , and also to the enumeration of flat connections in Chern-Simons theory on the boundary of . 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 -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
from a closed oriented Riemann surface of genus into some target space . In the physical Type II theory, any harmonic map with respect to chosen metrics on and , 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 by a two-dimensional topological field theory, which can be realized as a topological twist of the original superconformal field theory. In this reduction, the state space descends to its BRST cohomology defined with respect to the supercharges, which naturally carries a Frobenius algebra structure. This defines a consistent quantum theory if and only if the target space is a Calabi-Yau threefold, that is, a complex Kähler manifold of dimension with trivial canonical holomorphic line bundle , or equivalently trivial first Chern class . We fix a closed nondegenerate Kähler -form on .
The corresponding topological string amplitudes have interpretations in compactifications of Type II string theory on the product of the target space with four-dimensional Minkowski space . For instance, at genus zero the amplitude is the prepotential for vector multiplets of supergravity in four dimensions. The higher genus contributions , correspond to higher derivative corrections of the schematic form , where is the curvature and is the graviphoton field strength. We will now explain how to compute the amplitudes . 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 . It is built on holomorphically embedded curves (2.1). The holomorphic string maps in this case are called worldsheet instantons. They are classified topologically by their homology classes
With respect to a basis of two cycles on , one can write
where the Betti number is the rank of the second homology group and . 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 which depends on the string coupling and a vector of variables defined as follows. Let
be the complex Kähler parameters of with respect to the basis 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
Then the quantum string theory is described by a genus expansion of the free energy
weighted by the Euler characteristic of where the genus contribution to the statistical sum is given by
and in this formula the classes correspond to worldsheets of genus . The numbers are called the Gromov-Witten invariants of and they “count”, in an appropriate sense, the number of worldsheet instantons (holomorphic curves) of genus in the two-homology class They can be defined as follows.
A worldsheet instanton (2.1) is said to be stable if the automorphism group is finite. Let be the (compactified) moduli space of isomorphism classes of stable holomorphic maps (2.1) from connected genus curves to representing . This is the instanton moduli space onto which the path integral of topological string theory localizes. It is a proper Deligne-Mumford stack over which generalizes the moduli space of “stable” curves of genus . While the dimension of is , the moduli space 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
When is a Calabi-Yau threefold, this integer vanishes and there is a virtual fundamental class [22]
in the degree zero Chow group. In this case, we define
and so the Gromov-Witten invariants give the “virtual numbers” of worldsheet instantons. One generically has due to the orbifold nature of the moduli stack . One can also define invariants by integrating the Euler class of an obstruction bundle over . There are precise recipes for computing the Gromov-Witten invariants for toric varieties .
2.1.2. B-Model
The B-model topological string theory isolates complex structure aspects of the Calabi-Yau threefold . It enumerates the constant string maps which send the entire surface to a fixed point in , and hence have trivial curve class . The Gromov-Witten invariants in this case are completely understood. There is a natural isomorphism
and the degree zero Gromov-Witten invariants involve only the classical cohomology ring and “Hodge integrals” over the moduli spaces of Riemann surfaces defined as follows.
There is a canonical stack line bundle with fibre over the moduli point , the cotangent space of at some fixed point. We define the tautological class to be the first Chern class of , . The Hodge bundle is the complex vector bundle of rank whose fibre over a point is the space of holomorphic sections of the canonical line bundle . Let . A Hodge integral over is an integral of products of the classes and .
Explicit expressions for for generic threefolds are then obtained as follows. Let be a basis for (modulo torsion), and let index the generators of degree two. Then one has
where the Hodge integral can be expressed in terms of Bernoulli numbers as
Note that above when 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
and then reduce any integral over to using the corresponding Gysin push-forward map . However, this is difficult to do explicitly in most cases. The Gromov-Witten theory of is the study of tautological intersections in the moduli spaces . 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 is described by a holomorphic embedding : of a curve of genus with holes. A D-brane in is a choice of Dirichlet boundary condition on these string maps, which ensures that the Cauchy problem for the Euler-Lagrange equations on locally has a unique solution. They correspond to Lagrangian submanifolds of the Calabi-Yau threefold , that is, . If , then we consider holomorphic maps such that
This defines open string instantons, which are labelled by their relative homology classes
If we assume that , so that is generated by a single nontrivial one-cycle , then
where , are the winding numbers of the boundary maps .
The free energy of the A-model open topological string theory at genus is given by
where and the numbers are called relative Gromov-Witten invariants. To incorporate all topological sectors, in addition to the string coupling weighting the Euler characteristics , we introduce an Hermitean matrix 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
The traces are computed by formally taking the limit and expanding in irreducible representations of the D-brane gauge group .
2.3. Black Hole Microstates and D-Brane Gauge Theory
When is a Calabi-Yau threefold, certain BPS black holes on can be constructed by D-brane engineering. D-branes in correspond to submanifolds of 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 , which provides a topological classification of D-branes in .
The microscopic black hole entropy can be computed by counting stable bound states of D0–D2–D4–D6 branes wrapping holomorphic cycles of with the following configurations: (i)D6-brane charge (ii)D4-branes wrapping an ample divisor
with respect to a basis of four-cycles , , of (iii) D2-branes wrapping a two-cycle
(iv) D0-brane charge .
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, are interpreted as magnetic charges and as electric charges. The thermal partition function defined via a canonical ensemble for the D0 and D2 branes with chemical potentials , and a microcanonical ensemble for the D4 and D6 branes, is given by
where is the degeneracy of BPS states with the given D-brane charges.
As we mentioned in Section 2.1., the closed topological string amplitudes are related to supergravity quantities on Minkowski spacetime . The fact that the genus zero free energy for topological strings on is a prepotential for BPS black hole charges in supergravity determines the entropy of an extremal black hole as a Legendre transformation of , provided that one fixes the charge moduli by the attractor mechanism. The genus zero topological string amplitude is a homogeneous function of degree two in the vector multiplet fields . The black hole entropy in the supergravity approximation is then
where the chemical potentials are determined by the charges and by solving the equation
Further analyses of the entropy of BPS black holes on have been extended to higher genus and suggest the relationship
between the black hole partition function (2.22) and the topological string partition function
where the moduli on both sides of this equation are related by their fixing at the attractor point
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 in terms of black hole entropy. Suppose that we have a collection of D-branes wrapping a submanifold , with and Chan-Paton gauge field strength . 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 -form Ramond-Ramond fields through anomalous Chern-Simons couplings
where is the string length. In particular, these couplings contain all terms
and so the topological charge of a Chan-Paton gauge bundle on a D-brane is equivalent to D-brane charge. A prominent example of this, which will be considered in detail later on, is the coupling For , 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 . 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 () and turn off all D4-brane charges (). 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 .
3.1. Kähler Quantum Gravity
We will construct a model of quantum gravity on any Kähler threefold , which will motivate the sorts of counting problems that we consider in this section. The partition function is defined by
where
The sum is over “quantized” Kähler two-forms on , in the following sense. We decompose the “macroscopic” form into a fixed “background” Kähler two-form on and the curvature of a holomorphic line bundle over as
To satisfy the requirement that there are no D4-branes in , we impose the fluctuation condition
for all two-cycles .
Substituting (3.3) together with (3.4) into (3.2) gives the action
The statistical sum (3.1) thus becomes (dropping an irrelevant constant term)
where , , and denotes the th Chern character of the given line bundle . 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 implies . 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 to correspond to a singular gauge field on . This can be realized in two (related) possible ways: (1)we can make a singular gauge field nonsingular on the blow-up
of the target space, obtained by blowing up the singular points of on into copies of the complex projective plane This means that the quantum gravitational path integral induces a topology change of the target space . 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 . However, there are “more” sheaves on than blow-ups of .
In this paper we will take the second point of view. Recall that torsion-free sheaves on can be defined by the property that they sit inside short exact sequences of sheaves of the form
where is a holomorphic vector bundle on , and is a coherent sheaf supported at the singular points of a gauge connection of . Applying the Chern character to (3.8) and using its additivity on exact sequences give
for each . Thus torsion-free sheaves 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 . We will see that the instanton solutions of gauge theory on a noncommutative deformation are described in terms of ideals in the polynomial ring . For generic , the global object that corresponds locally to an ideal is an ideal sheaf, which in each coordinate patch is described as an ideal in the ring of holomorphic functions on . More abstractly, an ideal sheaf is a rank one torsion-free sheaf with . This is a purely commutative description, since the holomorphic functions on form a commutative subalgebra of 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 . 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 = to the Boltzmann weight, where is the chemical potential and 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
such that and for all . We may regard a partition as a three-dimensional Young diagram, in which we pile cubes vertically at the th position in the plane as depicted above. The volume of a plane partition
is the total number of cubes. The diagonal slices of are the partitions , obtained by cutting the three-dimensional Young diagram with planes, and they represent a sequence of ordinary partitions (Young tableaux) , with for all . Here is the length of the th row of the Young diagram, viewed as a collection of unit squares, and only finitely many 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 has energy proportional to its volume . The corresponding partition function then gives the generating function for plane partitions
where is the number of plane partitions with boxes. The function is called the MacMahon function.
3.3. Six-Dimensional Cohomological Gauge Theory
We will now describe a 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 action
where is the gauge covariant derivative acting on the complex scalar field , denotes the Hodge operator with respect to the Kähler metric of , and is the curvature two-form which has the Kähler decomposition . 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 (3)the low-energy effective field theory on a D6-brane wrapping 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
These equations also describe three (related) quantities: (i) the Donaldson-Uhlenbeck-Yau (DUY) equations expressing Mumford-Takemoto slope stability of holomorphic vector bundles over 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 .
Recall that (3.14) and (3.15) are a special instance of the Hermitean Yang-Mills equations in which a constant 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 . These are the natural BPS conditions on a Kähler manifold which generalize the usual self-duality equations in four dimensions.
The localization of the gauge theory partition function onto the corresponding instanton moduli space can be written symbolically as [24, 26]
where is the Euler class of the obstruction bundle 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
with the adjoint gauge bundle over . By incorporating the gauge fields, one can rewrite this complex in the form [24]
which describes solutions of the DUY equations up to linearized complex gauge transformations. The morphism 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 . In light of our earlier discussion, we will take this to be the Gieseker compactification, that is, the moduli space of ideal sheaves on . The corresponding variety stratifies into components given by the Hilbert scheme of points and curves in , parameterizing isomorphism classes of ideal sheaves with , , and . The partition function (3.17) is the generating function for the number of D0-D2 brane bound states in the D6-brane wrapping . Mathematically, these are the Donaldson-Thomas invariants of . 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 is called a toric manifold if it densely contains a (complex algebraic) torus and the natural action of on itself (by translations) extends to the whole of . Basic examples are the torus itself, the affine space , and the complex projective space . If in addition is Calabi-Yau, then is necessarily noncompact.
One of the great virtues of working with toric varieties 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 which are the fixed points of the -action on , such that can be covered by -invariant open charts homeomorphic to (ii)a set of edges which are -invariant projective lines joining particular pairs of fixed points , , (iii) a set of “gluing rules” for assembling the patches together to reconstruct the variety . In a neighbourhood of each edge , looks like the normal bundle over the corresponding . Since this normal bundle is a holomorphic bundle of rank two and every bundle over is a sum of line bundles (by the Grothendieck-Birkhoff theorem), it is of the form
for some integers . The normal bundle in this way determines the local geometry of near the edge via the transition function
between the corresponding affine patches (going from the north pole to the south pole of the associated ). In the Calabi-Yau case, the Chern numbers and imply the condition .
For an open toric manifold , we can exploit the toric symmetries to regularize the infrared singularities on the instanton moduli space by “undoing” the -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 and a complex Higgs field . In particular, the supercharges contain a scalar and a vector . Generically, only is conserved and can be used to define the topological twist of the gauge theory. If the threefold has symmetries then one can also use . In the generic formulation of the theory, one only considers the scalar topological charge and restricts attention to gauge-invariant observables. But in the present case one can also use the linear combination
where are the parameters of the isometric action of on the Kähler space and are vector fields which generate rotations of . 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 can be interpreted as an equivariant differential on the space of field configurations, where acts by contraction with the vector field .
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 “-background”. In particular, the fixed point equation (3.16) is modified to
The set of equations (3.14), (3.15), and (3.23) minimizes the action of the cohomological gauge theory in the -background and describes -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 . We will henceforth study the gauge theory equivariantly and interpret the truncation of the partition function (3.17) as an equivariant integral over . 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 be a smooth algebraic variety. Then we can define the -equivariant cohomology as the ordinary cohomology of the Borel-Moore homotopy quotient , where is a contractible space on which acts freely. In the present example of interest, and . Given a -equivariant vector bundle , the quotient is a vector bundle over . The -equivariant Euler class of is the invertible element defined by
where is the ordinary Euler class for vector bundles (the top Chern class).
Let Then is a universal principal -bundle, and there is a fibration with fibre . Integration in equivariant cohomology is defined as the pushforward of the collapsing map , which coincides with integration over the fibres of the bundle in ordinary cohomology. Let for and let : for be the canonical projections onto the th and th factors. Introduce equivariant parameters , with and , with .
The Atiyah-Bott localization formula in equivariant cohomology states that
for any equivariant differential form , where the complex vector bundle is the normal bundle over the (compact) fixed point submanifold in . When consists of finitely many isolated points this formula is simplified to
Each term in this sum takes values in the polynomial ring
in the generators of . When the manifold is noncompact, integration along the fibre is not a well-defined -linear map. Nevertheless, when is compact, we can formally define the equivariant integral by the right-hand side of the formula (3.25).
Going back to our example, when , one has and the partition function 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 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 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 carries a canonical perfect obstruction theory in the sense of [22]. In obstruction theory, the virtual tangent space at a point is given by
where is the Zariski tangent space and the obstruction space of at . Its dimension is given by the difference of Euler characteristics . The kernel of the trace map
is the obstruction to smoothness at a point of the moduli space.
The bundles , for define a canonical -equivariant perfect obstruction theory (see [22, Section ]) on the instanton moduli space . In this case, one may construct a virtual fundamental class and apply a virtual localization formula. The general theory is developed in [22] and requires a -equivariant embedding of in a smooth variety . The existence of such an embedding in the present case follows from the stratification of into Hilbert schemes of points and curves. Then one can deduce the localization formula over from the known ambient localization formula over the smooth variety , as above. In this paper we will only need a special case of this general framework, the virtual Bott residue formula.
We can decompose into -eigenbundles. The scheme theoretic fixed point locus is the maximal -fixed closed subscheme of . It carries a canonical perfect obstruction theory, defined by the -fixed part of the restriction of the complex to , which may be used to define a virtual structure on . The sum of the nonzero -weight spaces of defines the virtual normal bundle to . Define the Euler class of a virtual bundle using formal multiplicativity, that is, as the ratio of the Euler classes of the two bundles, . Then the virtual Bott localization formula for the Euler class of a bundle of a rank equal to the virtual dimension of reads [22]
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 given by , where is the localization of the ring at the maximal ideal generated by .
If is smooth, then is the nonsingular set theoretic fixed point locus, consisting here of finitely many points . However, in general the formula (3.30) must be understood scheme theoretically, here as a sum over -fixed closed subschemes of supported at the points (with ). With : , denoting the induced torus actions on the tangent and obstruction bundles on , one generically has decompositions
where is a -invariant subspace of . As demonstrated in [27, Section ], the kernel module in (3.31) vanishes. Hence each subscheme here is just the reduced point and the -fixed obstruction theory at is trivial. Under these conditions, the virtual localization formula (3.30) may be written as
The right-hand side of this formula again takes values in the polynomial ring . When for all , the moduli space 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 , the virtual tangent bundle 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 . This can be achieved by replacing the space with its noncommutative deformation defined by letting the coordinate generators satisfy the commutation relations of the Weyl algebra
where
is a constant skew-symmetric matrix which we take in Jordan canonical form without loss of generality (by a suitable linear transformation of if necessary). We will assume that for simplicity. The noncommutative polynomial algebra
is regarded as the “algebra of functions” on the noncommutative space .
We can represent the algebra on the standard Fock module
where the orthonormal basis states are connected by the action of creation and annihilation operators and . They obey and
In the Weyl operator realization with the complex combinations of operators
for , derivatives of fields are replaced by the inner automorphisms
while spacetime averages are replaced by traces over according to
In the noncommutative gauge theory, we introduce the covariant coordinates
and their complex combinations
for . Then the and components of the curvature two-form can be, respectively, expressed as
while the covariant derivatives of the Higgs field become
The instanton equations (3.14), (3.15), and (3.23) then become algebraic equations
These equations describe BPS bound states of the D0–D6 system in a -field background, which is necessary for reinstating supersymmetry [28, 29]. In addition, -invariance of the (unique) holomorphic three-form on imposes the Calabi-Yau condition
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 . It is obtained by setting and is given explicitly by harmonic oscillator algebra
Other solutions are found via the solution generating technique described in, for example, references [30, 31]. For the general solution, fix an integer and let be a partial isometry on the Hilbert space which projects out all states with . Such an operator satisfies the equations
where
is a Hermitean projection operator onto a finite-dimensional subspace of . Then we make the ansatz
where is a real function of the number operator
Using standard harmonic oscillator algebra, we can write the DUY equations (3.45) as
This recursion relation has a unique solution with the initial conditions = = , and the finite energy condition as . It is given by [32]
The topological charge of the corresponding noncommutative instanton is
Thus the instanton number is the number of states in with , that is, the number of vectors removed by , or equivalently the rank of the projector .
The partial isometry identifies the full Fock space with the subspace
where
is a monomial ideal of codimension in the polynomial ring . The instanton moduli space can thus be identified as the Hilbert scheme
of points in . The Hilbert-Chow morphism
identifies the Hilbert scheme of points as a crepant resolution of the (coincident point) singularities of the th symmetric product orbifold of . The ideal defines a plane partition with boxes given by
Heuristically, this configuration represents instantons which sit on top of each other at the origin of , and along its coordinate axes where they asymptote to four-dimensional noncommutative instantons at infinity described by ordinary Young tableaux .
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 [7, 8, 24]. Let us start with the case . Using (3.54), the contribution of an instanton corresponding to a plane partition contributes a factor
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
with the normalized character
where we have used the solution for in (3.50). Using the Calabi-Yau condition , it is easy to see that these determinants cancel up to a sign.
After some computation, one can explicitly determine this sign to get
The contribution 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
which is just the MacMahon function with . This is the known formula for the Donaldson-Thomas partition function on .
For later use, let us note a convenient resummation formula for this partition function [6]. Using interlacing relations, the sum over plane partitions can be converted into a triple sum over the Young tableaux obtained from the main diagonal slice , together with a sum over pairs of semistandard tableaux of shape obtained by putting of them in boxes of the skew diagram associated to the th diagonal slice for each . 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
where , and is the hook length of the box located at position in the Young tableau . Then the partition function can be rewritten as a sum over ordinary partitions
This construction can be generalized to arbitrary toric Calabi-Yau threefolds by using the gluing rules of toric geometry. The two simplest such varieties are described by the toric diagrams
with a single vertex representing 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 , where the contribute to the term of the gauge theory action (3.13). The -invariant noncommutative instantons on correspond to ideal sheaves and are described by the following combinatorial data: (1) three-dimensional partitions at each vertex of the toric diagram, corresponding to monomial ideals , (2) two-dimensional partitions at each edge of the toric diagram, representing the four-dimensional instanton asymptotics of .
This description requires generalizing the calculation on above to compute the perpendicular partition function [6], which is defined to be the generating function for three-dimensional partitions with fixed asymptotics , , and in the three coordinate directions. Such partitions correspond to instantons on with nontrivial boundary conditions at infinity along each of the coordinate axes. It can be expressed in terms of skew Schur functions, with .
For the example of the resolved conifold, using the gluing rules one easily computes
More generally, with these rules one finds that the instanton partition function is the generating function
for the Donaldson-Thomas invariants , which are defined as follows. The moduli variety of ideal sheaves on is a projective scheme with a perfect obstruction theory. For general threefolds , it has virtual dimension [33, Lemma ]
which coincides with that of from Section 2.1. In the Calabi-Yau case, the virtual dimension is zero, and the corresponding virtual cycle is
Then the Donaldson-Thomas invariants
count the virtual numbers of ideal sheaves on with the given Chern character. The right-hand side is defined via equivariant integration, as explained in Section 3.5. The torus action on lifts to the moduli scheme . The -fixed locus has a -equivariant virtual theory with cycle and virtual normal bundle in the equivariant K-theory . 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 , there is a gauge theory/string theory duality [27]. This follows from the fact that the perpendicular partition function is related to the Calabi-Yau crystal formulation of the topological vertex
with 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
where the Euler characteristic of is the number of vertices in its toric diagram. For the conifold example, the gluing rules for the topological vertex yield [14, 15]
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 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 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 lie in the ranges and .
On the other hand, the validity of (3.73) is related to the existence of BPS invariants such that the topological string amplitudes have an expansion given by [35, 36]
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 -stability) to elements of the bounded derived category of coherent sheaves of the given charge, which gives a topological classification of A-model D-branes on . 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 obtained by adding a compactification divisor , and consider sheaves with a fixed trivialization on . The Kähler polarization defined by allows us to define the moduli space 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 can be viewed locally as the scheme theoretic critical locus of a holomorphic function, the superpotential , on a compact manifold with the action of a gauge group . has virtual dimension zero, and at nonsingular points, the obstruction sheaf on coincides with the cotangent bundle. Hence if were everywhere nonsingular, then the partition function (3.17) would just compute the signed Euler characteristic . At singular points, however, the invariants differ from these characteristics.
There is a constructible function which can be used to define the weighted Euler characteristic
For sheaves of fixed Chern character, this coincides with the curve-counting invariants . At nonsingular points, , while at singular points it is the more complicated function given by
where is the Milnor fibre of the superpotential at the point corresponding to . The weighted Euler characteristic is a deformation invariant of .
In this approach, one can use topological Euler characteristics to define as invariants associated to moduli varieties of framed sheaves. Fixing and , the variety equivalently parametrizes isomorphism classes of the following objects: (a) surjections (framings)
with (b) stable sheaves with and trivial determinant, (c) subschemes of dimension with curve class and holomorphic Euler characteristic .
The equivalences between these three descriptions are described explicitly for in [7, 8].
As we vary the polarization , the moduli spaces 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 for framed sheaves is to consider quotients of the structure sheaf in different abelian subcategories of the bounded derived category of coherent sheaves on . 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 are the stable pairs , where is a coherent -module of pure dimension one with and , and is a nonzero sheaf map such that is of pure dimension zero, together with le Poitier's -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 patches are deformed. Isospectral type deformations of toric geometry, and instantons therein, are investigated in [45]. It may also aid in the classification of noncommutative instantons on for rank , 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 (no D6-branes) and consider D4-branes wrapping a four-cycle . In this case the worldvolume gauge theory on the D4-branes is the Vafa-Witten topologically twisted Yang-Mills theory on , 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 in four dimensions, we may regard as a four-cycle inside the Calabi-Yau threefold , 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. Supersymmetric Yang-Mills Theory on Kähler Surfaces
Vafa and Witten [46] introduced a topologically twisted version of 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 . In the following we will only consider the case where is a connected smooth Kähler manifold with Kähler two-form . When certain conditions are met, the partition function of the twisted gauge theory computes the Euler characteristic of the instanton moduli space.
Let be the Kähler metric of . Then the twisted gauge theory corresponds to the moduli problem associated with the equations
where is the self-dual part of the curvature two-form with respect to the Kähler metric. The field space is spanned by a connection on a principal -bundle , a scalar field , and a self-dual two-form , so that
where denotes the space of connections on and is the adjoint bundle of . Their superpartners , , and live in the tangent space to . Associated with the equations of motion are two multiplets and which are sections of the bundle
Schematically, the action of the topological gauge theory is of the form
where is the scalar supercharge singled out by the twisting procedure. The gauge fermion is a suitable functional of the fields which contains the term
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 , where is the group of gauge transformations. Its pullback via the sections in (4.1) gives the Euler class of . Under favourable circumstances, appropriate vanishing theorems hold [46] which ensure that each solution of the system (4.1) has and corresponds to an instanton, that is, a solution to the self-duality equations . In this case the gauge theory localizes onto the instanton moduli space and the Boltzmann weight gives a representative of the Euler class of the tangent bundle . Therefore, the partition function computes moduli space integrals of the form
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 , in order to define the partition function in the -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 as before.
The Euler characteristic of instanton moduli space can be computed through the index of the deformation complex associated with topological Yang-Mills theory via (4.1). It has the form [47]
where the first morphism is an infinitesimal gauge transformation
while the second morphism corresponds to the linearization of the sections given by
with 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 , the complex (4.7) splits into the Atiyah-Hitchin-Singer instanton deformation complex
plus
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 , 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 . The chemical potential for the D2-branes requires taking the Ramond-Ramond field proportional to the two-form on which are dual to the basis two-cycle in order to get the correct charges. In this case the D0-brane charges
correspond to the instanton numbers of the gauge bundle , while the D2-brane charges
correspond to nontrivial magnetic fluxes through . 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
where is the Witten index which computes the Euler characteristic of the moduli space of instantons on with Chern invariants and . Here is the generator of which is Poincaré dual to a point in .
4.2. Toric Localization and the Instanton Moduli Space
Instantons on can be described as follows. Consider the quiver
with the single relation specified by the linear combination of paths
This is called the ADHM quiver . The ADHM construction establishes a one-to-one correspondence between stable framed representations of the quiver in the category of finite-dimensional complex vector spaces and framed torsion-free sheaves on the projective plane . In the rank one case, these are equivalent to ideal sheaves on , and the correspondence gives an isomorphism with the Hilbert schemes of points on .
Let and be inner product spaces of complex dimensions and , respectively. The instanton moduli space can be realized as a hyperKähler quotient by the natural action of on the variety consisting of linear operators
constrained by the ADHM equations
On 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
where . This quotient gives a compactification of the instanton moduli space obtained by blowing up its singularities.
The classification of toric fixed points is given in [49], by identifying the instanton moduli space with the Hilbert scheme of points . The fixed points are point-like instantons which are in one-to-one correspondence with Young tableaux having boxes. In the more general case of a gauge theory in the Coulomb branch, one takes copies of the theory and the fixed points are classified in terms of -tuples of Young diagrams , called -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 action. Following [49], one introduces a two-dimensional -module to keep track of the toric action. The operators corresponding to a fixed point configuration are elements of the -modules
Then the local structure of the instanton moduli space is described by the complex
which is just a finite-dimensional version of the Atiyah-Hitchin-Singer instanton deformation complex (4.10). The map corresponds to infinitesimal (complex) gauge transformations while is the linearization of the ADHM constraint . In general, the complex (4.20) has three nonvanishing cohomology groups. In our case we can safely assume that and vanish. The only nonvanishing cohomology describes field configurations that obey the linearized ADHM constraint but are not gauge variations. It is thus a local model for the tangent space to the instanton moduli space at each -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 topologically twisted gauge theory on is easily computed. The only nontrivial topological charge is the instanton number and therefore the partition function has the form
The expansion parameter can be identified in terms of gauge theory variables with
the complexified gauge coupling, which is related to topological string variables at the attractor point. At a toric fixed point, is identified as the number of boxes in a partition . The Euler classes exactly cancel in the localization formula (3.26), and one is left with the sum over fixed points [50]
By Euler's formula, one has
where is the number of partitions (ordinary, two-dimensional Young tableaux) of degree . The function is related to the Dedekind function. It follows that the partition function
is the generating function for two-dimensional Young diagrams.
This construction can be easily generalized to the nonabelian case. The fixed points are now -coloured Young tableaux corresponding to a partition of the instanton number . The instanton action is again equal to , where the additional factor of arising from the sum over colours = cancels with the normalization of the term which carries a factor (the inverse of the dual Coxeter number of the gauge group ). The Euler characteristic of instanton moduli space is now
with The partition function reduces to copies of the partition function
This factorization follows from the fact that after toric localization, the gauge symmetry 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 .
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 . They are determined by two relatively prime positive integers and with . Consider the quotient singularity , with the generator of the cyclic group acting on as
This orbifold has an singularity at the origin of . Then is defined to be the minimal resolution of the singularity by a chain of exceptional divisors whose intersection numbers are summarized in the intersection matrix
which is called a generalized Cartan matrix. The divisors thus only intersect transversally with their nearest neighbours in the chain. The self-intersection numbers of the spheres of the blow-up are determined from the continued fraction expansion
Let us consider two particular well-known instances of these spaces.
4.3.1. Local
Setting , the space can be identified with the total space of the holomorphic line bundle over of degree , with and . In this case, is the zero section divisor. In the context of topological string theory, such four cycles appear in the “local” Calabi-Yau threefolds which are regarded as neighbourhoods of a holomorphically embedded rational curve in a compact Calabi-Yau threefold, that is, as the normal bundle . Since is a holomorphic vector bundle of rank two over and the Calabi-Yau condition implies , it follows that is the total space of a bundle of the form .
4.3.2. ALE Space
The complex surface is an example of an asymptotically locally Euclidean (ALE) space. This means that carries a scalar flat Kähler metric such that is complete, and there exists a compact set such that . Here acts freely on and the metric approximates the flat Euclidean metric on . Such a coordinate system is called a coordinate system at infinity. We regard acting on as described above (with ), and the complex structure on approximates that on . As the resolution of the Klein singularity , contains a chain of projective lines , each with self-intersection number . In this case, the intersection matrix coincides with the Cartan matrix of the Dynkin diagram.
4.4. Instantons on ALE Spaces
We begin by describing in some detail the instanton moduli space in the case of the ALE spaces, for which a rigorous construction is known. 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 where . This result is at the heart of the McKay correspondence which provides a one-to-one correspondence between irreducible representations of the orbifold group and tautological bundles over the exceptional divisors of the minimal resolution .
Since is noncompact, the instanton moduli space 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 . 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 , which is topologically the Lens space . The flat connections are classified by homomorphisms , where . The asymptotic connection at infinity is thus labelled by irreducible representations of the orbifold group , with and are given explicitly by
where .
Starting from the ADHM construction on outlined in Section 4.2., one constructs its -invariant decomposition. Consider the universal scheme given by the correspondence diagram 
(4.32)
The tautological bundle on is defined by
Under the action of on transforms in the regular representation and can thus be decomposed into irreducible representations
By the McKay correspondence, the bundles form the canonical integral basis of the K-theory group constructed in [51].
In this case, we take to be a module on which the regular representation of acts. We also take so that the determinant representation is trivial as a -module, that is, . The two vector spaces and which feature in the ADHM construction have a natural grading under the action of the orbifold group given by
The modification of (4.20) is given by [52]
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 .
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 and the partition function factorizes as
It therefore suffices to focus on the case in the following.
4.4.1. Regular Instantons
Regular instantons on live in the regular representation of the orbifold group . They correspond to D0-branes moving freely on with orbifold images away from the orbifold point. For gauge group , the moduli space is given by specifying = points on up to permutations. Hence the moduli space of regular instantons on is isomorphic to the Hilbert scheme . The generating function for the Euler numbers of the instanton moduli spaces can then be computed explicitly by applying Göttsche's formula to get
The partition function is the th power of this quantity. Heuristically, we may think of this formula as originating by covering with open charts to get copies of instantons on , each contributing . This can be demonstrated rigorously on any toric surface by a localization computation [53, 54].
4.4.2. Fractional Instantons
To each irreducible representation of the orbifold group 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 's of the minimal resolution, and correspond to self-dual gauge connections with curvatures
where and are the tautological line bundles. The Chern classes , form a basis of . 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
Since is noncompact, the intersection matrix 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
with . Setting and identifying using the attractor mechanism, we find that the contribution of fractional instantons to the full partition function is given by a theta-function
on a Riemann surface of genus and period matrix .
4.5. Instantons on Local
Let us now discuss what is known beyond the ALE case, in the instance that is the total space of the holomorphic line bundle [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 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 , considered as a divisor of , produces a line bundle such that is the generator of . It has a unique self-dual connection asymptotic to the trivial connection. Let be the trivial line bundle over . Set , and let be the moduli space of self-dual connections on which are asymptotic to the trivial connection at infinity. Then . Since , using Morse theory one can show that the only nonvanishing homology groups of the instanton moduli space are .
Alternatively, set , and let , be the moduli space of self-dual connections asymptotic to . Then for any , while for (whereby coincides with the ALE space), one has . In particular, for there is a diffeomorphism with , while for one has . Instantons on local 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 . Thus one postulates the form of the partition function
where
In the ALE case, one has and . The evidence for this formula is supported by calculations in the reduction to -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 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 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 , that is, of free bosons, with of them being compact. The compact degrees of freedom live in a torus determined by the lattice with the bilinear form . 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 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 is an ALE space and the gauge group is , there is yet another conformal field theoretic interpretation [46, 55, 61, 62]. In this case, the level affine 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
where , , and are the degeneracies of BPS states with the specified quantum numbers. Here 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 [61]. The partition function is also expressed as a 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 [58], in the large volume limit and when contributions from worldsheet instantons are negligible. In this limit we can expand the cycle as in (2.20). Let , be an integral basis of two cocycles for dual to the four-cycle with the intersection numbers
and let be the matrix inverse of . The genus zero topological string amplitude can be expressed as
With and , the black hole entropy is given by [58]
We would now like to interpret the gauge theory partition function (4.43) as the corresponding black hole partition function . For this, we expand it as
Then Cardy's formula gives the black hole entropy as
This expression agrees [56] with the macroscopic supergravity result (4.48) for the Bekenstein-Hawking-Wald entropy in the large 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) of torsion-free sheaves with (b) stable torsion-free sheaves on with ,(c) closed subschemes of dimension with dual curve class and holomorphic Euler characteristic .
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 be a smooth, quasiprojective, open, toric surface. We will assume that admits a projective compactification , that is, is a smooth, compact, projective, toric surface with a smooth divisor (called the “line at infinity”) which is a -invariant in , and such that . We will also require that , in addition to . The difference between the counting of framed instantons on the compact toric surface (with boundary condition at “infinity”) and of unframed instantons on the open toric surface is a universal perturbative contribution, which will be dropped here. Since is compact, these calculations will only capture the contributions from instantons with integer charges on . We will mention later on how to incorporate the contributions from fractional instantons on .
Let us fix two numbers and , and an integer cohomology class . Let be the framed moduli space consisting of isomorphism classes of torsion-free sheaves on such that (1) has the following topological Chern invariants:
(2) is locally-free in a neighbourhood of and there is an isomorphism called the “framing at infinity”.
These topological conditions imply that the moduli space is nonempty only when
so that defines a class in the compactly supported cohomology of . Furthermore, from the last relation in (4.51) can be written as
Framed sheaves yield stable pairs [68], analogous to those described in Section 3.9., after suitable choices of polarization on and stability parameter.
We write . At a given point , the space of reducible connections is , the Zariski tangent space is , and the obstruction space is . The cohomology of the instanton deformation complex is greatly simplified by the fact that in this case [69]
Using (4.54) and the Riemann-Roch theorem, one shows [69] that the moduli space is a smooth quasiprojective variety of (complex) dimension , where , whose tangent space at a point is isomorphic to the vector space .
We will now describe a natural torus invariant subspace of the instanton moduli space. The -invariance of implies that the pullback of the -action on defines an action on . There is also an action of the diagonal maximal torus of on the framing. Altogether we get an action of the complex algebraic torus on [69]. We are interested in the fixed point set of this torus action.
An isomorphism class is fixed by the -action if and only if it decomposes as
such that is mapped to the th factor of under the framing isomorphism. Since the double dual is a line bundle which is trivial on , it is equal to for some divisor disjoint from . Via the natural injection , the sheaf is thus equal to for some ideal sheaf of a zero-dimensional subscheme also disjoint from . If is also fixed by the -action, then so are , and .
The supports of the -invariant subschemes and can be represented explicitly in terms of the toric geometry of . Recall that this is described in terms of an underlying toric graph whose vertices are in bijective correspondence with the -fixed points in , and two vertices are joined by an edge if and only if the corresponding fixed points are connected by a -invariant . Let and respectively, denote the set of vertices and edges of . Then since is a chain, one has . The number of vertices is also the number of two-cones in the toric fan of and it is related to the Euler characteristic of the surface as
We denote by the -invariant point in corresponding to the vertex and by the -invariant line corresponding to the edge .
Since the zero-cycles are not supported on , they must be contained in the fixed point set in . Thus each is a union of subschemes supported at the -fixed points . If we choose a local coordinate system in a patch around , then the -invariant ideal of in the coordinate ring of is generated by the -eigenfunctions with nontrivial characters, which are monomials and hence corresponds to a Young diagram with boxes (with monomial placed at ). The ideal is spanned by monomials outside the Young diagram. Likewise, since the -invariant two-cycles are disjoint from the line at infinity, they are supported along the edges .
It follows that the fixed point set is parameterized by finitely many -tuples
where
are -invariant divisors in and
is a vector of Young tableaux with boxes. The Young tableaux parametrize the contributions from regular pointlike D0-brane instantons (freely moving inside ). The divisors parametrize the contributions from D2-brane instantons (wrapping 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 in terms of this combinatorial data. Since , the constraint can be written as
whereas (4.53) becomes
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 between lines of the toric graph as
Each divisor is canonically associated to a -equivariant holomorphic line bundle whose first Chern class is Poincaré dual to with
Note that these intersection numbers differ from (4.40), that is, the basis of line bundles does not coincide with the tautological line bundles . Using (4.58) we can write with . Then the constraint (4.60) can be expressed as
Combined with (4.61) and using (4.63), this gives
The Vafa-Witten-Nekrasov partition function on the toric surface is now defined as the -equivariant index
where as before with the complexified gauge coupling constant and
for a collection of formal variables and a given cohomology class . From the localization formula (3.25), we see that the -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 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 and . In the sum over critical points, we can replace the sum over using (4.65) by sums over Young diagrams and over the integers and representing magnetic fluxes through on D2-branes wrapping the divisors . The sum over may then be saturated using (4.64).
Putting everything together, and using (4.64) and (4.63) to write
the partition function (4.66) thereby becomes
where we have used (4.44). The sums over Young tableaux decouple, and for each vertex and each they produce a factor of the Euler function . Then we can bring (4.69) into the final form
where we recall that and .
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 singularity, fractional first Chern classes can be incorporated by constructing instead the moduli space of torsion-free sheaves on the orbifold compactification of the hyper-Kähler ALE space , where [52, 55, 70, 71]. In a neighbourhood of infinity, we can approximate by with the singularity at the origin resolved. More precisely, we obtain the divisor by gluing together the trivial bundle on with the line bundle on . The latter bundle has a -equivariant structure such that the map is -equivariant. Let us examine some examples which are covered by the analysis above.
4.7.1. Affine Plane
Let Then with and intersection number . Since in this case, the constraint (4.52) implies . The instanton moduli space in this case is a smooth variety of complex dimension . Furthermore, . The partition function (4.66) computed with fixed and is thus given by
which coincides with the instanton partition function on described in Section 4.2.
4.7.2. Local
Let , be the total space of the holomorphic line bundle of degree over . Then is the th Hirzebruch surface. Let and . Then , , and so that the line does not pass through the “line at infinity”. From the constraint (4.52) and the second Betti number , it follows that for some . In this case, the instanton moduli space is a smooth variety of complex dimension . The positivity of this dimension is a constraint on the integral classes which ensures that the moduli space is non-empty. One has , and hence . The partition function (4.70) in this case becomes
with .
Like the ALE spaces, one can work instead with a stack compactification of the total space of the bundle obtained by adding a divisor . The resulting variety is a toric Deligne-Mumford stack whose coarse space is the Hirzebruch surface [72]. With this compactification, one produces framed sheaves with fractional first Chern classes , , and in this case the partition function takes the form [72]
anticipated by [13, 56], where , and 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, -instantons on are in one-to-one correspondence with holomorphic bundles of rank on with together with a framing at infinity [73]. When these spaces coincide with the 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 be a compact, smooth, projective toric surface. Let be a generic point in which is disjoint from the torus invariant lines , of . Let be the moduli space of isomorphism classes of torsion-free coherent sheaves on with topological Chern invariants as in (4.51), together with a “framing” at the point . When is locally free, this framing means a choice of basis for the fibre space . When is not locally free, the framing is defined with respect to a locally free resolution .
Let be the blow-up of at . Then is also a compact, smooth, projective toric surface. We will suppose that the exceptional divisor has positive self-intersection . Let be the moduli space of isomorphism classes of torsion-free sheaves on which are framed on the line as above. The blow-up map determines mutually inverse sheaf morphisms and , and hence an explicit isomorphism
It follows from above that the moduli space is thus a smooth variety of complex dimension , where . Using the isomorphism (4.74), the instanton partition function on can be computed from the blow-up formula
where the right-hand side is given by the formula (4.70).
As an explicit example, consider the complex projective plane . Let be a generic point on disjoint from the line . Then the instanton moduli space is a smooth variety of complex dimension , where . Let be the blowup of at with exceptional divisor . Then there are maximal two cones and edge in the toric graph . Since in this case, the partition function (4.75) gives the formula
with . For this agrees with the gauge theory partition function on derived in [54, Section ].
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 be a smooth, quasiprojective toric surface. Again we fix invariants , and . Let be the unframed moduli space consisting of isomorphism classes of semistable torsion-free coherent sheaves on such that has topological Chern invariants
The notion of “semistability” considers sheaves with fixed Hilbert polynomial. In this case, a general result of Maruyama [74, 75] constructs the algebraic scheme and shows that it is projective. In particular, it admits a -equivariant embedding into a smooth variety. In the case that is a polarized surface, that is, it admits a smooth distribution in its tangent bundle 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 be a fixed ample line bundle on . For a coherent torsion-free sheaf on , define the polynomial
for . Then is said to be -semistable if
for all whenever is a coherent subsheaf of . In this case, Gieseker [76] constructs as a projective scheme. The slope of is the rational number
where the degree of is defined using the polarization as
An application of the Riemann-Roch theorem shows [76]
It follows that for if and only if and hence -semistability is equivalent to the usual quasi-BPS instanton equations in this case. In physics applications, one is interested in instances where is a Kähler surface. In this case one can use the Kähler polarization and take to be the Kähler two-form 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 equals , where and is the holomorphic Euler characteristic of .
Proof. As discussed in Section 3.5., the dimension of the virtual tangent space to the instanton moduli space at a point in obstruction theory is the difference of Euler characteristics . The latter quantity may be computed for locally free by using the Hirzebruch-Riemann-Roch theorem to write
Let be the generator of which is Poincaré dual to a point in . Then using (4.77) one computes the Chern character
The Todd characteristic class of the tangent bundle of is given in terms of Chern classes of as
We may thus write (4.83) as
From the Hirzebruch-Riemann-Roch formula and , one has
and the result follows when is a bundle. When is not locally free, we use a locally free resolution along with additivity of the Chern character.
This result shows that the expected dimension of the instanton moduli space for coincides with the dimension of the framed moduli space 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 of an ideal sheaf of a zero-dimensional subscheme and a line bundle, the moduli space factorizes into a product
where is the Hilbert scheme of points on which is a smooth variety of dimension , and the (zero-dimensional) Picard group parameterizes fractional instantons. The factorization of the 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 , it is given by the formula (4.70) with . 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 via linear equivalences. It amounts to identifying the generators of the Kähler cone in with the duals to the exceptional divisors which generate the Mori cone in . For the ALE spaces, the moduli space coincides with the quiver variety of the ADHM construction (with suitable stability conditions) [77].
The situation for higher rank 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 with [64]. As discussed in Section 3.9., the physical theory is described by moduli spaces of stable objects in the derived category , as the observed D-brane decays are impossible in the abelian category of coherent sheaves on .
4.9. Perpendicular Partition Functions and Universal Sheaves
Finally, we come to the last Point (c) at the end of Section 4.6. For 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 for instantons on with fixed asymptotics such that , 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 and , and show that the fixed point loci of the induced -action are enumerated by two-dimensional Young diagrams with asymptotics . 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 ].
4.9.1. Instanton Moduli Spaces on
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 , it is more convenient to work with the “two-point compactification” of given by a product of two projective lines , 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 , that is, the projective compactification of the trivial line bundle , which is again a toric surface. For , the toric action is described by the automorphism of defined by
Let : and : be the canonical projections, and and : the natural inclusions. There are now two independent distinguished lines at infinity fixed by this action, given by the pushforwards and , for which the respective factors of the action of reduce to the standard -action on . These two divisors have intersection products and , and the canonical divisor is . They generate the Picard group of , and hence induce a bigrading on line bundles over . For , we write
By Künneth's theorem, the nontrivial cohomology groups of are given by
where and .
Let be the moduli space of isomorphism classes of torsion-free sheaves on with topological Chern invariants as in (4.51), where and nontrivial framings along the two independent “directions at infinity” are prescribed by two fixed isomorphisms and , where is a fixed -dimensional complex vector space. We will say that such a sheaf is “trivialized at infinity” if it is equipped with two isomorphisms and . The moduli space of trivialized sheaves is denoted . These moduli spaces carry an obvious induced action of the torus , 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 -equivariant birational equivalence between the moduli spaces
Proof. Represent as a divisor in via the embedding , . Let be the surface obtained as the blowup of a pair of points on the line at infinity (see Section 4.7.). Note that is disjoint from the image of the line . Then there is a correspondence diagram
(4.93)where is the blowup of the intersection point on . The corresponding Fourier-Mukai functors and determine equivalences on the categories of coherent sheaves over and , 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 , there is a natural -equivariant bijection between the moduli spaces
where
Proof. Given , define
By (4.51) (with ), one has
and using multiplicativity of the Chern character we compute