Spherical geometry and integrable systems

We prove that the cosine law for spherical triangles and spherical tetrahedra defines integrable systems, both in the sense of multidimensional consistency and in the sense of dynamical systems.Comment: 15 pages, 5 figure

SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls

We propose an equivalence of the partition functions of two different 3d gauge theories. On one side of the correspondence we consider the partition function of 3d SL(2,R) Chern-Simons theory on a 3-manifold, obtained as a punctured Riemann surface times an interval. On the other side we have a partition function of a 3d N=2 superconformal field theory on S^3, which is realized as a duality domain wall in a 4d gauge theory on S^4. We sketch the proof of this conjecture using connections with quantum Liouville theory and quantum Teichmuller theory, and study in detail the example of the once-punctured torus. Motivated by these results we advocate a direct Chern-Simons interpretation of the ingredients of (a generalization of) the Alday-Gaiotto-Tachikawa relation. We also comment on M5-brane realizations as well as on possible generalizations of our proposals.Comment: 53+1 pages, 14 figures; v2: typos corrected, references adde

S^3/Z_n partition function and dualities

We investigate S^3/Z_n partition function of N = 2 supersymmetric gauge theories. A gauge theory on the orbifold has degenerate vacua specified by the holonomy. The partition function is obtained by summing up the contributions of saddle points with different holonomies. An appropriate choice of the phase of each contribution is essential to obtain the partition function. We determine the relative phases in the holonomy sum in a few examples by using duality to non-gauge theories. In the case of odd n the phase factors can be absorbed by modifying a single function appearing in the partition function.Comment: 30 pages, 2 figures, added reference

The matrix model version of AGT conjecture and CIV-DV prepotential

Recently exact formulas were provided for partition function of conformal (multi-Penner) beta-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted as Dotsenko-Fateev correlator of screenings and analytically continued in the number of screening insertions, represents generic Virasoro conformal blocks. Actually these formulas describe the lowest terms of the q_a-expansion, where q_a parameterize the shape of the Penner potential, and are exact in the filling numbers N_a. At the same time, the older theory of CIV-DV prepotential, straightforwardly extended to arbitrary beta and to non-polynomial potentials, provides an alternative expansion: in powers of N_a and exact in q_a. We check that the two expansions coincide in the overlapping region, i.e. for the lowest terms of expansions in both q_a and N_a. This coincidence is somewhat non-trivial, since the two methods use different integration contours: integrals in one case are of the B-function (Euler-Selberg) type, while in the other case they are Gaussian integrals.Comment: 27 pages, 1 figur

Holomorphic Blocks in Three Dimensions

We decompose sphere partition functions and indices of three-dimensional N=2 gauge theories into a sum of products involving a universal set of "holomorphic blocks". The blocks count BPS states and are in one-to-one correspondence with the theory's massive vacua. We also propose a new, effective technique for calculating the holomorphic blocks, inspired by a reduction to supersymmetric quantum mechanics. The blocks turn out to possess a wealth of surprising properties, such as a Stokes phenomenon that integrates nicely with actions of three-dimensional mirror symmetry. The blocks also have interesting dual interpretations. For theories arising from the compactification of the six-dimensional (2,0) theory on a three-manifold M, the blocks belong to a basis of wavefunctions in analytically continued Chern-Simons theory on M. For theories engineered on branes in Calabi-Yau geometries, the blocks offer a non-perturbative perspective on open topological string partition functions.Comment: 124 pages, 21 figures. v3: Typos correcte

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