Four-dimensional wall-crossing via three-dimensional field theory
Davide Gaiotto, Gregory W. Moore, Andrew Neitzke
备注：
62 pages; v2: typos fixed, references added, minor improvements to exposition and notation; v3: typos fixed and small additions, including remark on TBA
We give a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories. In the process we give an exact description of the BPS instanton corrections to the hyperkahler metric of the moduli space of the theory on R^3 x S^1. The wall-crossing formula reduces to the statement that this metric is continuous. Our construction of the metric uses a four-dimensional analogue of the two-dimensional tt* equations.
We give an elementary physical derivation of the Kontsevich-Soibelman wall crossing formula, valid for any theory with a 4d N=2 supergravity description. Our argument leads to a slight generalization of the formula, which relates monodromy to the BPS spectrum.
We consider BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S^1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS s…
We study the twisted index of 3d $\mathcal{N}=2$ supersymmetric gauge theories on $S^1 \times Σ$ in the presence of a real FI parameter deformation. This parameter induces a 1d FI parameter for the effective supersymmetric quantum mechanics on $S^1$. Using supersymmetric localisation, the twisted index can be expressed as a contour integral. We show that the contour prescription is modified in the presence of the 1d FI parameter, leading to wall-crossing phenomena for the twisted index. In particular, we derive a general wall-crossing formula for abelian gauge theories. We also examine the ori…
We test a recently proposed wall-crossing formula for the change of the Hilbert space of BPS states in d=4,N=2 theories. We study decays of D4D2D0 systems into pairs of D4D2D0 systems and we show how the wall-crossing formula reproduces results of Goettsche and Yoshioka on wall-crossing behavior of the moduli of slope-stable holomorphic bundles over holomorphic surfaces. Our comparison shows very clearly that the moduli space of the D4D2D0 system on a rigid surface in a Calabi-Yau is not the same as the moduli space of torsion free sheaves, even when worldhseet instantons are neglected. Moreov…
We calculate the leading weak-coupling instanton contribution to the moduli-space metric of N=2 supersymmetric Yang-Mills theory with gauge group SU(2) compactified on R^3 x S^1. The results are in precise agreement with the semiclassical expansion of the exact metric recently conjectured by Gaiotto, Moore and Neitzke based on considerations related to wall-crossing in the corresponding four-dimensional theory.
In this paper we propose definitions and examples of categorical enhancements of the data involved in the $2d$-$4d$ wall-crossing formulas which generalize both Cecotti-Vafa and Kontsevich-Soibelman motivic wall-crossing formulas.
We study the BPS states of a D6-brane wrapping the conifold and bound to collections of D2 and D0 branes. We find that in addition to the complexified Kahler parameter of the rigid sphere it is necessary to introduce an extra real parameter to describe BPS partition functions and marginal stability walls. The supergravity approach to BPS state-counting gives a simple derivation of results of Szendroi concerning Donaldson-Thomas theory on the noncommutative conifold. This example also illustrates some interesting limitations on the supergravity approach to BPS state-counting and wall-crossing.
BPS states in N=4 supersymmetric SU(N) gauge theories in four dimensions can be represented as planar string networks with ends lying on D3-branes. We introduce several protected indices which capture information on the spectrum and various quantum numbers of these states, give their wall crossing formula and describe how using the wall crossing formula we can compute all the indices at all points in the moduli space.
We study the wall-crossing phenomena of D4-D2-D0 bound states with two units of D4-brane charge on the resolved conifold. We identify the walls of marginal stability and evaluate the discrete changes of the BPS indices by using the Kontsevich-Soibelman wall-crossing formula. In particular, we find that the field theories on D4-branes in two large radius limits are properly connected by the wall-crossings involving the flop transition of the conifold. We also find that in one of the large radius limits there are stable bound states of two D4-D2-D0 fragments.
We derive the wall crossing formula for the decay of a quarter BPS dyon into a pair of half-BPS dyons by analyzing the quantum dynamics of multi-centered black holes in N=4 supersymmetric string theories. Our analysis encompasses the cases where the final decay products are non-primitive dyons. The results are in agreement with the microscopic formula for the dyon spectrum in the special case of heterotic string theory on T^6.
In this paper, we show how to get matrix models corresponding to the refined BPS states partition functions of $\mathbb{C}^3$, resolved conifold and $\mathbb{C}^3/\mathbb{Z}_2$ by inserting the identity operator at a proper position in the fermionic expression of the refined BPS states partition functions.
We describe how categorical BPS data including chain complexes of solitons, CPT pairings, and interior amplitudes jump across a wall of marginal stability in two-dimensional $\mathcal{N}=(2,2)$ models. We show that our jump formulas hold if and only if the $A_{\infty}$-categories of half-BPS branes constructed on either side of the wall are homotopy equivalent. These results can be viewed as categorical enhancements of the Cecotti-Vafa wall-crossing formula.
We study the relation between the instanton counting on ALE spaces and the BPS state counting on a toric Calabi-Yau three-fold. We put a single D4-brane on a divisor isomorphic to A_{N-1}-ALE space in the Calabi-Yau three-fold, and evaluate the discrete changes of BPS partition function of D4-D2-D0 states in the wall-crossing phenomena. In particular, we find that the character of affine SU(N) algebra naturally arises in wall-crossings of D4-D2-D0 states. Our analysis is completely based on the wall-crossing formula for the d=4, N=2 supersymmetric theory obtained by dimensionally reducing the …
We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an N=2 supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactified on a circle, we get a 3d theory with a supersymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkahler space. The 2d-4d wall-crossing formula can be interpreted as a smoothn…
In this paper we describe the Seiberg-Witten invariants, which have been introduced by Witten, for manifolds with $b_+=1$. In this case the invariants depend on a chamber structure, and there exists a universal wall crossing formula. For every Kähler surface with $p_g=0$ and $q$=0, these invariants are non-trivial for all $Spin^c(4)$-structures of non-negative index.
We study K-theoretic GLSM invariants with one-dimensional gauge group and introduce elliptic central charges that depend on an elliptic cohomology class called an elliptic brane and a choice of level structure. These central charges have an integral representation related to an interpolation problem for the elliptic brane and satisfy natural q-difference equations. Integral representaions lead to the wall crossing statement for the central charges in two different phases of the GLSM. We explain how the orbifold structure leads to differences between wall crossing of the usual (K-theoretic) and…
The wall crossing formula of Kontsevich and Soibelman gives an implicit relation between the BPS indices on two sides of the wall of marginal stability by equating two symplectomorphisms constructed from the indices on two sides of the wall. The wall crossing formulae of Manschot, Pioline and the author give two apparently different explicit expressions for the BPS index on one side of the wall in terms of the BPS indices on the other side. We prove the equivalence of all the three formulae.
We explore the physics of two-body decay of BPS states using semiclassical analysis to construct explicit solutions that illustrate the main features of wall crossing, for both ordinary and framed BPS states. In particular we recover the primitive wall-crossing formula from an asymptotic analysis of certain Dirac-type operators on monopole moduli spaces. Along the way we give an asymptotic metric for the moduli space of singular monopoles, analogous to the Gibbons-Manton and Lee-Weinberg-Yi metrics for the moduli space of smooth monopoles, and we find evidence for the existence of stable non-B…
We study the interplay between wall-crossing in four-dimensional gauge theory and instanton contributions to the moduli space metric of the same theory on $\mathbb{R}^{3}\times S^{1}$. We consider $\mathcal{N}=2$ SUSY Yang--Mills with gauge group SU(n) and focus on walls of marginal stability which extend to weak coupling. By comparison with explicit field theory results we verify the Kontsevich--Soibelman formula for the change in the BPS spectrum at these walls and check the smoothness of the metric in the corresponding compactified theory. We also verify in detail the predictions for the on…
We discuss the wall-crossing of the BPS bound states of a non-compact holomorphic D4-brane with D2 and D0-branes on the conifold. We use the Kontsevich-Soibelman wall-crossing formula and analyze the BPS degeneracy in various chambers. In particular we obtain a relation between BPS degeneracies in two limiting attractor chambers related by a flop transition. Our result is consistent with known results and predicts BPS degeneracies in all chambers.
Recently, using supergravity analysis, a hyperbolic reflection group was found to underlie the structure of wall-crossing, or the discontinuous moduli dependence of the supersymmetric index due to the presence of walls of marginal stability, of the BPS dyons in the N=4, d=4 compactification. In this paper we work in the regime where four-dimensional gravity decouples and we show how the presence of such a group structure can be easily understood as a consequence of the supersymmetry of a system of (p,q) five-brane network, or equivalently the holomorphicity of the Riemann surface wrapped by th…
We study refined and motivic wall-crossing formulas in N=2 supersymmetric gauge theories with SU(2) gauge group and N_f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that "refined = motivic."
Wall-crossing phenomena are ubiquitous in many problems of algebraic geometry and theoretical physics. Various ways to encode the relevant information and the need to track the changes under the variation of parameters lead to rather complicated transformation rules and non-trivial combinatorial problems. In this paper we propose a framework, reminiscent of collections and plethysms in the theory of operads, that conceptualizes those transformation rules. As an application we obtain new streamlined proofs of some existing wall-crossing formulas as well as some new formulas related to attractor…
We formulate Witten index problems for theories with two supercharges in a Majorana doublet, as in $d=3$ $\mathcal N=1$ theories and dimensional reduction thereof. Regardless of spacetime dimensions, the wall-crossing occurs generically, in the parameter space of the real superpotential $W$. With scalar multiplets only, the path integral reduces to a Gaussian one in terms of $dW$, with a winding number interpretation, and allows an in-depth study of the wall-crossing. After discussing the connection to well-known mathematical approaches such as the Morse theory, we move on to Abelian gauge the…
We outline a comprehensive and first-principle solution to the wall-crossing problem in D=4 N=2 Seiberg-Witten theories. We start with a brief review of the multi-centered nature of the typical BPS states and recall how the wall-crossing problem thus becomes really a bound state formation/dissociation problem. Low energy dynamics for arbitrary collections of dyons is derived, from Seiberg-Witten theory, with the proximity to the so-called marginal stability wall playing the role of the small expansion parameter. We find that, surprisingly, the $\mathbb{R}^{3n}$ low energy dynamics of n+1 BPS d…
We study the wall-crossing phenomena of BPS D4-D2-D0 states on the conifold and orbifold C^2/Z_2, from the viewpoint of the quiver quantum mechanics on the D-branes. The Kahler moduli dependence of the BPS index is translated into the FI parameter dependence of the Witten index. The wall-crossing phenomena are related to the Seiberg dualities of the quiver quantum mechanics. All the differences from the D6-D2-D0 case arise from the additional superpotential and "anti-quark" induced by the D4-brane. When the D-branes are on the conifold, the flop transition changes the duality cascade. When the…
We formulate supersymmetric low energy dynamics for BPS dyons in strongly-coupled N=2 Seiberg-Witten theories, and derive wall-crossing formulae thereof. For BPS states made up of a heavy core state and n probe (halo) dyons around it, we derive a reliable supersymmetric moduli dynamics with 3n bosonic coordinates and 4n fermionic superpartners. Attractive interactions are captured via a set of supersymmetric potential terms, whose detail depends only on the charges and the special Kaehler data of the underlying N=2 theories. The small parameters that control the approximation are not electric …
We compute the Witten index of one-dimensional gauged linear sigma models with at least ${\mathcal N}=2$ supersymmetry. In the phase where the gauge group is broken to a finite group, the index is expressed as a certain residue integral. It is subject to a change as the Fayet-Iliopoulos parameter is varied through the phase boundaries. The wall crossing formula is expressed as an integral at infinity of the Coulomb branch. The result is applied to many examples, including quiver quantum mechanics that is relevant for BPS states in $d=4$ ${\mathcal N}=2$ theories.
We explore 1d vortex dynamics of 3d supersymmetric Yang-Mills theories, as inferred from factorization of exact partition functions. Under Seiberg-like dualities, the 3d partition function must remain invariant, yet it is not a priori clear what should happen to the vortex dynamics. We observe that the 1d quivers for the vortices remain the same, and the net effect of the 3d duality map manifests as 1d Wall-Crossing phenomenon; Although the vortex number can shift along such duality maps, the ranks of the 1d quiver theory are unaffected, leading to a notion of fundamental vortices as basic bui…
We show that the BPS spectrum of pure SU(3) four-dimensional super Yang-Mills with N=2 supersymmetry exhibits a surprising phenomenon: there are regions of the Coulomb branch where the growth of the BPS degeneracies with the charge is exponential. We show this using spectral networks and independently using wall-crossing formulae and quiver methods. The computations using spectral networks provide a very nontrivial example of how these networks determine the four-dimensional BPS spectrum. We comment on some physical implications of the wild spectrum: for example, exponentially many field-theor…
We study the quantum effects on the Coulomb branch of N=2 SU(2) supersymmetric Yang-Mills with fundamental matters compactified on R^3 x S^1, and extract the explicit perturbative and leading non-perturbative corrections to the moduli space metric predicted from the recent work of Gaiotto, Moore and Neitzke on wall-crossing [1]. We verify the predicted metric by computing the leading weak coupling instanton contribution to the four fermion correlation using standard field theory techniques, and demonstrate perfect agreement. We also demonstrate how previously known three dimensional quantities…
This paper summarizes recent developments in the theory of Bogomol'nyi-Prasad-Sommerfield (BPS) state counting and the wall crossing phenomena, emphasizing in particular the role of the statistical mechanical model of crystal melting. This paper is divided into two parts, which are closely related to each other. In the first part, we discuss the statistical mechanical model of crystal melting counting BPS states. Each of the BPS state contributing to the BPS index is in one-to-one correspondence with a configuration of a molten crystal, and the statistical partition function of the melting cry…
When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on $R^3 \times S^1$ are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these two spaces are related under a general duality between, on one hand, quaternion-Kahler manifolds with a quaternionic isometry and, on the other hand, hyperkahler manifolds with a rotational isometry, further equipped with a hyperholomorphic circle bundle with a connection. We sh…
Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show the compatibility of this equation with anomaly equations previously observed in the context of N=4 topological Yang-Mills theory on P^2 and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The non-…
We prove formula between Nekrasov partition functions defined from stable and co-stable ADHM data for the plane following method by Nakajima-Yoshioka based on the theory of wall-crossing formula developed by Mochizuki. This formula is similar to conjectures by Itoh-Maruyoshi-Okuda for $A_{1}$ singularity.
By embedding N=2 gauge theories in string theory and utilizing string dualities we map the counting of BPS states with arbitrary electric and magnetic charges to computations of an A-model topological string on an associated geometry constructed from the data of the SW curve. We show how the conjecture of Kontsevich and Soibelman regarding wall crossing, as well as a more refined version which captures the spin content of BPS states, is a natural consequence. Chern-Simons theory realized on A-branes and a twistorial construction play key roles.
We initiate the study of wall crossing phenomena in orientifolds of local toric Calabi-Yau 3-folds from a topological string perspective. For this purpose, we define a notion of real Donaldson-Thomas partition function at the large volume, orbifold and non-commutative point in Kaehler moduli space. As a byproduct, we refine the constant map contribution to the partition function of the real topological string on a local toric background. We conjecture the general relation between the real large volume and real non-commutative/orbifold Donaldson-Thomas partition function of orientifolds of loca…
We derive supersymmetric quantum mechanics of n BPS objects with 3n position degrees of freedom and 4n fermionic partners with SO(4) R-symmetry. The potential terms, essential and sufficient for the index problem for non-threshold BPS states, are universal, and 2(n-1) dimensional classical moduli spaces M_n emerge from zero locus of the potential energy. We emphasize that there is no natural reduction of the quantum mechanics to M_n, contrary to the conventional wisdom. Nevertheless, via an index-preserving deformation that breaks supersymmetry partially, we derive a Dirac index on M_n as the …
The appearance of a generalized (or Borcherds-) Kac-Moody algebra in the spectrum of BPS dyons in N=4, d=4 string theory is elucidated. From the low-energy supergravity analysis, we identify its root lattice as the lattice of the T-duality invariants of the dyonic charges, the symmetry group of the root system as the extended S-duality group PGL(2,Z) of the theory, and the walls of Weyl chambers as the walls of marginal stability for the relevant two-centered solutions. This leads to an interpretation for the Weyl group as the group of wall-crossing, or the group of discrete attractor flows. F…
We consider wall-crossing phenomena associated to the counting of D2-branes attached to D4-branes wrapping lagrangian cycles in Calabi-Yau manifolds, both from M-theory and matrix model perspective. Firstly, from M-theory viewpoint, we review that open BPS generating functions in various chambers are given by a restriction of the modulus square of the open topological string partition functions. Secondly, we show that these BPS generating functions can be identified with integrands of matrix models, which naturally arise in the free fermion formulation of corresponding crystal models. A parame…
Starting from N=2 supersymmetric theories in 2 dimensions, we formulate a novel time-dependent supersymmetric quantum theory where the R-charge is twisted along the time. The invariance of the supersymmetric index under variations of the action for these theories leads to two predictions: In the IR limit it predicts how the degeneracy of BPS states change as we cross the walls of marginal stability. On the other hand, its equivalence with the UV limit relates this to the spectrum of the U(1) R-charges of the Ramond ground states at the conformal point. This leads to a conceptually simple deriv…
Families of Bridgeland stability conditions induce families of stability data, wall-crossing structures and scattering diagrams on the motivic Hall algebra. These structures can be transferred to the quantum torus if the stability conditions of the family have global dimension at most 2. We show that geometric stability conditions on surfaces with nef anticanonical bundle have global dimension 2 and we study the resulting family of stability data. We formulate a conjecture relating this family to the family of stability data associated with a quiver with potential and we verify this conjecture…
We introduce moduli spaces of stable perverse coherent systems on small crepant resolutions of Calabi-Yau 3-folds and consider their Donaldson-Thomas type counting invariants. The stability depends on the choice of a component (= a chamber) in the complement of finitely many lines (= walls) in the plane. We determine all walls and compute generating functions of invariants for all choices of chambers when the Calabi-Yau is the resolved conifold. For suitable choices of chambers, our invariants are specialized to Donaldson-Thomas, Pandharipande-Thomas and Szendroi invariants.
We study spectrum of D=2 N=(2,2) QED with N+1 massive charged chiral multiplets, with care given to precise supermultiplet countings. In the infrared the theory flows to CP^N model with twisted masses, where we construct generic flavored kink solitons for the large mass regime, and study their quantum degeneracies. These kinks are qualitatively different and far more numerous than those of small mass regime, with features reminiscent of multi-pronged (p,q) string web, complete with the wall-crossing behavior. It has been also conjectured that spectrum of this theory is equivalent to the hyperm…
We determine the generating functions of 1/4 BPS dyons in a class of 4d $\mathcal{N}=4$ string vacua arising as CHL orbifolds of $K3 \times T^2$, a classification of which has been recently completed. We show that all such generating functions obey some simple physical consistency conditions that are very often sufficient to fix them uniquely. The main constraint we impose is the absence of unphysical walls of marginal stability: discontinuities of 1/4 BPS degeneracies can only occur when 1/4 BPS dyons decay into pairs of 1/2 BPS states. Formally, these generating functions in spacetime can be…
In this paper, we use an M-theory model to conjecture the refined reminiscence of the OSV formula connecting the refined topological string partition function with the refined BPS states partition function for the toric Calabi-Yau threefolds without any compact four cycles. Further, we show how to use the vertex operators in 2d free fermions to reproduce the refined BPS states partition function for the $\mathbb{C}^3$ case and the wall-crossing formulas of the refined BPS states partition function for the resolved conifold and $\mathcal{O}(-2)\oplus\mathcal{O}\rightarrow \mathbb{P}^1$ cases.
Consider the degeneracies of BPS bound states of one D6 brane wrapping Calabi-Yau X with D0 branes and D2 branes. When we include D4-branes wrapping Lagrangian cycle L in addition, D2-branes can end on them. These give rise to new bound states in the d=2, N=(2,2) theory of the D4 branes. We call these "open" BPS states, in contrast to closed BPS states that arise from D-branes without boundaries. Lifting this to M-theory, we show that the generating function is captured by free Fock space spanned by M2-brane particles ending on M5 branes wrapping L. This implies that the open BPS bound states …
We describe wall-crossing for local, toric Calabi-Yau manifolds without compact four-cycles, in terms of free fermions, vertex operators, and crystal melting. Firstly, to each such manifold we associate two states in the free fermion Hilbert space. The overlap of these states reproduces the BPS partition function corresponding to the non-commutative Donaldson-Thomas invariants, given by the modulus square of the topological string partition function. Secondly, we introduce the wall-crossing operators which represent crossing the walls of marginal stability associated to changes of the B-field …
BPS spectrum with finite number of states are found for higher rank four dimensional N=2 theory engineered from six dimensional A_{N-1} (2,0) theory on a Riemann surface with various kinds of defects. The wall crossing formula is interpreted as the quantum dilogarithm identity. Various methods including quiver representation theory, maximal green mutation, and cluster algebra are used extensively. The spectral generator and its refined version for the higher rank theory are written down using the explicit spectrum information. The finite chamber has an interesting N^3 behavior in the large N l…
We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincare polynomials of the moduli spaces for the curve classes $β$ having arithmetic genus at most 2. We formulate a conjecture that these Poincare polynomials are divisible by the Poincare polynomials of $((-K_S).β-1)$-dimensional projective space. This conjecture motivates upcoming work on log BPS numbers.
Noncommutative Donaldson-Thomas invariants for abelian orbifold singularities can be studied via the enumeration of instanton solutions in a six-dimensional noncommutative N=2 gauge theory; this construction is based on the generalized McKay correspondence and identifies the instanton counting with the counting of framed representations of a quiver which is naturally associated to the geometry of the singularity. We extend these constructions to compute BPS partition functions for higher-rank refined and motivic noncommutative Donaldson-Thomas invariants in the Coulomb branch in terms of gauge…
We study the BPS spectrum of four-dimensional $\mathcal{N}=2$ supersymmetric Yang-Mills theory with gauge group $SU(2)$ and a massive adjoint hypermultiplet, which has an extremely intricate structure with infinite spectrum in all chambers of its Coulomb moduli space, and is not well understood. We build on previous results by employing the BPS quiver description of the spectrum, and explore the qualitative features in detail using numerical techniques. We find novel and unexpected behaviour in the form of wall-crossings involving interactions between BPS particles with negative electric-magne…
We introduce a higher rank analog of the Joyce-Song theory of stable pairs. Given a nonsingular projective Calabi-Yau threefold $X$, we define the higher rank Joyce-Song pairs given by ${O}^{r}_{X}(-n)\rightarrow F$ where $F$ is a pure coherent sheaf with one dimensional support, $r>1$ and $n\gg 0$ is a fixed integer. We equip the higher rank pairs with a Joyce-Song stability condition and compute their associated invariants using the wallcrossing techniques in the category of weakly semistable objects.
