35 research outputs found
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