We review free fermion, melting crystal, and matrix model representations of wall-crossing phenomena on local, toric Calabi-Yau manifolds. We consider both unrefined and refined BPS counting of closed BPS states involving D2- and D0-branes bound to a D6-brane, as well as open BPS states involving open D2-branes ending on an additional D4-brane. Appropriate limit of these constructions provides, among the others, matrix model representation of refined and unrefined topological string amplitudes.
This paper is devoted to some aspects of counting of BPS states in a system of D
The BPS states we are interested in, and which we will refer to as
The BPS generating functions which we consider are intimately related to topological string amplitudes on corresponding Calabi-Yau spaces. This relation is most transparent in the physical derivation discussed in Section
In more detail, we will consider generating functions of D2 and D0-branes bound to a single D6-brane of the following form:
If there is an additional D4-brane which spans a Lagrangian submanifold inside the Calabi-Yau space, in addition to the above
One more important aspect of BPS counting is referred to as
As an exemplary and, hopefully, inspiring application of the entire formalism presented in this paper, in the final Section
The literature on the topics presented in this paper is extensive and still growing, and we unavoidably mention just a fraction of important developments. The relation between Donaldson-Thomas invariants for the noncommutative chamber of the conifold was first found by Szendrői [
In parallel to the above-mentioned mathematical activity, wall-crossing phenomena for local Calabi-Yau manifolds were analyzed from physical viewpoint. The analysis of nontrivial BPS counting for the conifold was described by Jafferis and Moore in [
Let us also mention some other, related works devoted to crystals and free fermions. The fermionic construction of MacMahon function for
The plan of this paper is as follows. In Section
In this section, we introduce generating functions of BPS states of D-branes in toric Calabi-Yau manifolds. Our task in the rest of this paper is to provide interpretation of these generating functions in terms of free fermions, melting crystals, and matrix models. These generating functions can be derived using wall-crossing formulas, as was done first in the unrefined [
We start this section by reviewing the M-theory derivation of (unrefined) closed and open BPS generating functions. Then, to get acquainted with a crystal interpretation of these generating functions, we discuss their crystal interpretation in simple cases of
We start by considering a system of D2 and D0-branes bound to a single D6-brane in type IIA string theory. It can be reinterpreted in M-theory as follows [
Under the above conditions, the counting of D6-D2-D0 bound states is reinterpreted in terms of a gas of particles arising from M2-branes wrapped on cycles
To be more precise, an identification as a topological string partition function or its square arises if
In what follows we denote BPS generating functions in chambers with positive
The above structure can be generalized by including in the initial D6-D2-D0 configuration additional D4-branes wrapping Lagrangian cycles in the internal Calabi-Yau manifold and extending in two space-time dimensions [
Lifting this system to M-theory, we obtain a background of the form
From the M-theory perspective, we are interested in counting the net degeneracies of M2-branes ending on this M5-brane
The BPS generating functions we are after are given by a trace over the Fock space built by the oscillators of the second quantized field
Similarly as in the closed string case, the above degeneracies can be related to open topological string amplitudes, rewritten in [
Similarly as in the closed string case, there are also a few particularly interesting chambers to consider. For example, in the extreme chamber corresponding to Im
Closed BPS generating functions (
On the other hand, the MacMahon function is a generating function of plane partitions, that is, three-dimensional generalization of Young diagrams. These plane partitions represent the simplest three-dimensional crystal model, namely, they can be identified with stacks of unit cubes filling the positive octant of
Plane partitions represent melting crystal configurations of
The conifold provides another simple, yet nontrivial example of toric Calabi-Yau manifold. It consists of two
Infinite pyramids with one and four balls in the top row, with generating functions given, respectively, by
Finite pyramids with
To write down explicitly BPS generating functions for the conifold in various chambers, we can take advantage of their relation to the topological string amplitude (
Using the relation (
In the second set of chambers, we have
Above, we presented just the simplest examples of crystal models. Using fermionic formulation presented below, one can find other crystal models for arbitrary toric geometry without compact four cycles. Let us also mention that those models can be equivalently expressed in terms dimers. In particular, the operation of enlarging the crystal, as in the conifold pyramids, corresponds to so-called
In this section, we introduce some mathematical background on which the main results presented in this paper rely. In Section
Some introductory material on toric Calabi-Yau manifolds, from the perspective relevant for mirror symmetry and topological string thoery, can be found, for example, in [
Toric graphs for
A closed loop in a toric diagram represents a compact four cycles in the geometry. As follows from the reasoning in Section
Let us denote independent
As explained in Section
Formalism of free fermions in two dimensions is well known [
The states in the free fermion Fock space are created by the (anticommuting) modes of the fermion field
Relation between Young diagrams and states in the Fermi sea.
We introduce vertex operators
The operator
We also introduce various colors
In matrix model theory, or theory of random matrices, one is interested in properties of various ensembles of matrices. Excellent reviews of random matrix theory can be found for example in [
In the context of BPS counting and topological strings, unitary ensembles of matrices of infinite size arise. In this case, the matrix model simplifies to the integral over eigenvalues
It has been observed in several contexts that topological strings on toric manifolds can be related to matrix models, whose spectral curves take form of the so-called mirror curves. Mirror curves arise for manifolds which are mirror to toric Calabi-Yau manifolds [
Toric diagrams for
One of the first relations between topological strings for toric manifolds and matrix models was encountered in [
One of our aims is to provide matrix model interpretation of BPS counting. It is natural to expect such an interpretation in view of an intimate relation between BPS counting and topological string theory discussed in Section
Having introduced all the ingredients above, we are now ready to present fermionic formulation of BPS counting. To start with, in Section
As explained in Section
Slicing of a plane partition (a) into a sequence of interlacing two-dimensional partitions (b). A sequence of
Toric Calabi-Yau manifolds represented by a triangulation of a strip. There are
In what follows, we present a formalism which allows to generalize this computation to a large class of chambers, for arbitrary toric geometry without compact four cycles.
We wish to reformulate BPS counting in the fermionic language in a way in which we associate to each toric manifold a fermionic state, such that the BPS generating function can be expressed as an overlap of two such states, generalizing
In what follows we use the notation introduced in Section
Now we can associate several operators to a given toric manifold. Firstly, we define
In addition, we define the above-mentioned
We often use a simplified notation when the argument of the above operators is
Above we associated operators
The states
In the previous, section we associated to toric manifolds the states
On the other hand, the value of the field
Therefore, schematically, the generating functions in chambers with
Consider a chamber characterized by positive
In the second case, we consider the positive value of
Now we consider negative
In the last case, we consider positive
In the previous section, we found a free fermion representation of D6-D2-D0 generating functions. The fermionic correlators which reproduce BPS generating functions automatically provide melting crystal interpretation of these functions [
The crystal interpretation is a consequence of the fact that all operators used in the construction of states
To get more insight about a geometric structure of a crystal, it is convenient to introduce the following graphical representation. We associate various arrows to the vertex operators, as shown in Figure
Assignment of arrows.
Let us reconsider
Toric diagram for
The crystal structure can be read off from a sequence of arrows associated to
Now we consider the resolution of
Toric diagram for the resolution of
If we turn on an arbitrary
The crystals corresponding to
We already presented pyramid crystals for the conifold in Section
(a): toric diagram for the conifold and arrow representation of
The quantum states (
Therefore, the fermionic correlators take form
Conifold crystal in the chamber with positive
In this section, we explain how matrix model formalism can be applied to analyze BPS counting functions. In the first part, Section
Let us explain how to relate fermionic representation of BPS amplitudes, introduced in Section
There is a large freedom in choosing the value of
Brane associated to the external leg of a toric diagram (of a conifold in this particular case). Closed string parameter is denoted by
In this section, we illustrate the relation between BPS counting and matrix models in case of the noncommutative chamber for for the conifold, we obtain a representation of the pyramid partition generating function ( for
In this section, we illustrate how matrix model techniques can be used in the context of models which arise for BPS counting. We focus on the conifold matrix model in arbitrary closed BPS chamber
Now we wish to analyze the matrix model
Using the expansion of the quantum dilogarithm
Now we wish to solve the model (
The above curve is given by a symmetric function of
The spectral curve for the conifold matrix model (
In the BPS counting problem we are interested in, as follows from the form of the identity operator (
Firstly, we see that not only the spectral and mirror curves agree, but moreover the matrix integral (
Secondly, it is natural to conjecture that the total partition function of the matrix model, for finite
Finally, we also note that in the limit
In this section, we finally consider arbitrary closed and open BPS chamber, so that matrix models take most general form. Analyzing the case of
We recall that the open topological string amplitude for a brane in
Now we present how the result (
Factorization of
Here we illustrate a relation between matrix models and open BPS generating functions for the conifold, related to a brane associated to the external leg of a toric diagram, as in Figure
This result arises from matrix model viewpoint as follows. We take advantage of the results of Section
Factorization of the conifold pyramid which leads to open BPS generating functions. The size of the pyramid
As another example, we consider open BPS counting functions for resolved
On the other hand, using results of Section
In the last section, we turn our attention to so-called refined BPS amplitudes, and explain how to incorporate the effect of such refinement in the fermionic formalism and matrix models, following [
Our aim in this section is to construct refined crystal and matrix models, which would encode refined BPS generating functions, and in particular (in the commutative chamber) refined topological string amplitudes. We note that an additional motivation to find such models arises from the AGT conjecture [
Let us also note that in this section we consider the same set of walls as in the unrefined case. More general walls, along which only refined BPS states decay, may also exist [
In this section, we use the following refined notation. The string coupling
Let us present now refined BPS generating functions for some Calabi-Yau spaces. For For the resolved conifold, refined generating functions were computed in [ For a resolution of Generating functions for an arbitrary toric geometry, in for the noncommutative chamber, are given (as in the unrefined case) by the modulus square of the (instanton part of the) refined topological string amplitude
In the nonrefined case to a geometry consisting of
Refined plane partitions which count D6-D0 bound states in
Now we wish to follow the idea of Section
In Section
Here we construct fermionic states
Our claim now is that the refined BPS generating function can be written as
To prove (
We can now extend the fermionic representation to nontrivial chambers, for simplicity restricting our considerations to the case of a conifold and a resolution of
Refined pyramid crystal for the conifold, in the chamber corresponding to
Refined pyramid crystal for the resolution of
The assignment of colors is determined as follows. All stones on one side of the crystal are encoded in
Now the crystal with
Finally, all stones on the right side of the crystal have again the same light or dark color, so that the corresponding state is
We can now commute away all weight operators in the above expressions, using commutation relations from Section
In the refined case, one can associate matrix models to refined generating functions in the same way as described in Section
For arbitrary geometry, in the noncommutative chamber, refined matrix model integrand can be expressed in terms of the refined theta function
We obtain a refined matrix model for
Finally we find matrix models for the refined conifold. Starting with the representation (
The author thanks Robbert Dijkgraaf, Hirosi Ooguri, Cumrun Vafa, and Masahito Yamazaki for discussions and collaboration on related projects. This research was supported by the DOE Grant DE-FG03-92ER40701FG-02 and the European Commission under the Marie-Curie International Outgoing Fellowship Programme. The contents of this publication reflect only the views of the author and not the views of the funding agencies.