Quantum Wall Crossing in N=2 Gauge Theories

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."Comment: 24 pages, 4 figure

\Omega-deformation of B-twisted gauge theories and the 3d-3d correspondence

We study \Omega-deformation of B-twisted gauge theories in two dimensions. As an application, we construct an \Omega-deformed, topologically twisted five-dimensional maximally supersymmetric Yang-Mills theory on the product of a Riemann surface $\Sigma$ and a three-manifold $M$, and show that when $\Sigma$ is a disk, this theory is equivalent to analytically continued Chern-Simons theory on $M$. Based on these results, we establish a correspondence between three-dimensional $\mathcal{N} = 2$ superconformal theories and analytically continued Chern-Simons theory. Furthermore, we argue that there is a mirror symmetry between {\Omega}-deformed two-dimensional theories.Comment: 26 pages. v2: the discussion on the boundary condition for vector multiplet improved, and other minor changes mad

Multiple D4-D2-D0 on the Conifold and Wall-crossing with the Flop

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.Comment: 24 pages, 4 figures; v2: typos corrected, minor changes, a reference adde

K-Decompositions and 3d Gauge Theories

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2-Lagrangian, meaning that the K_2-avatar of the symplectic form restricts to zero. We construct an open part of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)-connections on surfaces. Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2-isotropic if the boundary satisfies some topological constraints (Theorem 4.2). In some cases this implies that L_K(M) is K_2-Lagrangian. For general M, we extend a classic result of Neumann-Zagier on symplectic properties of PGL(2) gluing equations to reduce the K_2-Lagrangian property to a combinatorial claim. Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M. This extends known constructions for K=2. Just as for K=2, the theories T_K[M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f=1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T_K[M] grow cubically in K.Comment: 121 pages + 2 appendices, 80 figures; Version 2: reorganized mathematical perspective, swapped Sections 3 and

Wall-crossing, open BPS counting and matrix models

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 parameter specifying a choice of an open BPS chamber has a natural, geometric interpretation in the crystal model. These results extend previously known relations between open topological string amplitudes and matrix models to include chamber dependence.Comment: 25 pages, 8 figures, published versio

Generalized Toda Theory from Six Dimensions and the Conifold

Recently, a physical derivation of the Alday-Gaiotto-Tachikawa correspondence has been put forward. A crucial role is played by the complex Chern-Simons theory arising in the 3d-3d correspondence, whose boundary modes lead to Toda theory on a Riemann surface. We explore several features of this derivation and subsequently argue that it can be extended to a generalization of the AGT correspondence. The latter involves codimension two defects in six dimensions that wrap the Riemann surface. We use a purely geometrical description of these defects and find that the generalized AGT setup can be modeled in a pole region using generalized conifolds. Furthermore, we argue that the ordinary conifold clarifies several features of the derivation of the original AGT correspondence.Comment: 27+2 pages, 3 figure