33443115
Wall Crossing as Seen by Matrix Models
The number of BPS bound states of D-branes on a Calabi-Yau manifold depends on two sets of data, the BPS charges and the stability conditions. For D0 and D2-branes bound to a single D6-brane wrapping a Calabi-Yau 3-fold X, both are naturally related to the Kähler moduli space $${{\mathcal M}(X)}$$ . We construct unitary one-matrix models which count such BPS states for a class of toric Calabi-Yau manifolds at infinite ’t Hooft coupling. The matrix model for the BPS counting on X turns out to give the topological string partition function for another Calabi-Yau manifold Y, whose Kähler moduli space $${{\mathcal M}(Y)}$$ contains two copies of $${{\mathcal M}(X)}$$ , one related to the BPS charges and another to the stability conditions. The two sets of data are unified in $${{\mathcal M}(Y)}$$ . The matrix models have a number of other interesting features. They compute spectral curves and mirror maps relevant to the remodeling conjecture. For finite ’t Hooft coupling they give rise to yet more general geometry $${\widetilde{Y}}$$ containing Y.
119669187
A Note on Dimer Models and McKay Quivers
We give one formulation of a procedure of Hanany and Vegh (J High Energy Phys 0710(029):35, 2007) which takes a lattice polygon as an input and produces a set of isoradial dimer models. We study the case of lattice triangles in detail and discuss the relation with coamoebas following Feng et al. (Adv Theor Math Phys 12(3):489–545, 2008).
15310503
Homological mirror symmetry for toric orbifolds of toric del Pezzo surfaces
Abstract We formulate a conjecture which describes the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables in terms of a pair consisting of a consistent dimer model and a perfect matching on it. We prove this conjecture in some cases, and obtain homological mirror symmetry for quotient stacks of toric del Pezzo surfaces by finite subgroups of the torus as a corollary.
10042817
Crystal Melting and Toric Calabi-Yau Manifolds
We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau threefold. The three-dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary non-compact toric Calabi-Yau manifold. We point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the Donaldson-Thomas theory.
118488616
String theory origin of bipartite SCFTs
A bstractWe provide a string theory embedding for $ \mathcal{N} $ = 1 superconformal field theories defined by bipartite graphs inscribed on a disk. We realize these theories by exploiting the close connection with related $ \mathcal{N} $ = 2 generalized (Ak−1, An−1) Argyres-Douglas theories. The $ \mathcal{N} $ = 1 theory is obtained from spacetime filling D5-branes wrapped on an algebraic curve and NS5-branes wrapped on special Lagrangians of $ {{\mathbb{C}}^2} $ which intersect along the BPS flow lines of the corresponding $ \mathcal{N} $ = 2 Argyres-Douglas theory. Dualities of the $ \mathcal{N} $ = 1 field theory follow from geometric deformations of the brane configuration which leave the UV boundary conditions fixed. In particular we show how to recover the classification of IR fixed points from cells of the totally non-negative Grassmannian $ \mathrm{Gr}_{k,n+k}^{\mathrm{tnn}} $. Additionally, we present evidence that in the 3D theory obtained from dimensional reduction on a circle, VEVs of line operators given by D3-branes wrapped over faces of the bipartite graph specify a coordinate system for $ \mathrm{Gr}_{k,n+k}^{\mathrm{tnn}} $.
55014255
The Non-commutative Topological Vertex and Wall Crossing Phenomena
We propose a generalization of the topological vertex, which we call the "non-commutative topological vertex". This gives open BPS invariants for a toric Calabi-Yau manifold without compact 4-cycles, where we have D0/D2/D6-branes wrapping holomorphic 0/2/6-cycles, as well as D2-branes wrapping disks whose boundaries are on D4-branes wrapping non-compact Lagrangian 3-cycles. The vertex is defined combinatorially using the crystal melting model proposed recently, and depends on the value of closed string moduli at infinity. The vertex in one special chamber gives the same answer as that computed by the ordinary topological vertex. We prove an identify expressing the non-commutative topological vertex of a toric Calabi-Yau manifold X as a specialization of the closed BPS partition function of an orbifold of X, thus giving a closed expression for our vertex. We also clarify the action of the Weyl group of an affine A_L Lie algebra on chambers, and comment on the generalization of our results to the case of refined BPS invariants.
125828668
High-Energy Proton Proton Elastic Scattering and Multiple Scattering Effect in Fresnel Approximation