Abstract
We review and elaborate on certain aspects of the connections between instanton counting in maximally supersymmetric gauge theories and the computation of enumerative invariants of smooth varieties. We study in detail three instances of gauge theories in six, four, and two dimensions which naturally arise in the context of topological string theory on certain noncompact threefolds. We describe how the instanton counting in these gauge theories is related to the computation of the entropy of supersymmetric black holes and how these results are related to wall-crossing properties of enumerative invariants such as Donaldson-Thomas and Gromov-Witten invariants. Some features of moduli spaces of torsion-free sheaves and the computation of their Euler characteristics are also elucidated.
1. Introduction
Topological theories in physics usually relate BPS quantities to geometrical invariants of the underlying manifolds on which the physical theory is defined. For the purposes of the present article, we will focus on two particular and well-known instances of this. The first is instanton counting in supersymmetric gauge theories in four dimensions, which gives the Seiberg-Witten and Donaldson-Witten invariants of four-manifolds. The second is topological string theory, which is related to the enumerative geometry of Calabi-Yau threefolds and computes, for example, Gromov-Witten invariants, Donaldson-Thomas invariants, Gopakumar-Vafa BPS invariants, and key aspects of Kontsevich's homological mirror symmetry conjecture.
From a physical perspective, these topological models are not simply of academic interest, but they also serve as exactly solvable systems which capture the physical content of certain sectors of more elaborate systems with local propagating degrees of freedom. Such is the case for the models we will consider in this paper, which are obtained as topological twists of a given physical theory. The topologically twisted theories describe the BPS sectors of physical models, and compute nonperturbative effects therein. For example, for certain supersymmetric charged black holes, the microscopic Bekenstein-Hawking-Wald entropy is computed by the Witten index of the relevant supersymmetric gauge theory. This is equivalent to the counting of stable BPS bound states of D-branes in the pertinent geometry, and is related to invariants of threefolds via the OSV conjecture [1].
From a mathematical perspective, we are interested in counting invariants associated to moduli spaces of coherent sheaves on a smooth complex projective variety . 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 tha