#math.gt

[qhd4]

New paper ^.^

our new thing is up on arxiv, just uploaded, coming in the mail tomorrow

[arsphysica]

Quantum Riemann Surfaces in Chern-Simons Theory. (arXiv:1102.4847 [hep-th])

We construct from first principles the operator 'A-hat' that annihilates the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU(2), SL(2,R), or SL(2,C) on a knot complement M. The operator 'A-hat' is a quantization of the knot complement's classical A-polynomial A(l,m). The construction proceeds by decomposing three-manifolds into ideal tetrahedra, and invoking a new, more global understanding of gluing in TQFT to put them back together. We advocate in particular that, properly interpreted, "gluing = symplectic reduction." We also arrive at a new finite-dimensional state integral model for computing the analytically continued "holomorphic blocks" that compose any physical Chern-Simons partition function.

arXiv:1102.4847 [hep-th]

[arsphysica]

Lectures on Foliation Dynamics: Barcelona 2010. (arXiv:1104.4852 [math.DS])

This survey is based on a series of five lectures, given May 3–7, 2010, at the Centre de Recerca Matematica, Barcelona. The goal of the lectures was to present aspects of the theory of foliation dynamical systems which have particular importance for the classification of foliations of compact manifolds. The lectures emphasized intuitive concepts and informal discussion, while taking the reader into topics of active research in this subject. These notes update and expand on the lectures, and include more recent progress. This article also includes an extensive set of references, as well as highlighting many open questions and problems. A set of "homework problems" is also included, one for each day of lecture.

arXiv:1104.4852 [math.DS]

[arsphysica]

Knot Homology from Refined Chern-Simons Theory. (arXiv:1105.5117 [hep-th])

We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold via the refined topological string and the (2,0) theory on N M5 branes. The refined Chern-Simons theory is defined on any three-manifold with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the S and T matrices of Chern-Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern-Simons theory are similar in many ways; for example, the Verlinde formula holds in both. We obtain new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the knot invariants we compute are the Poincare polynomials of the sl(n) knot homology theory. The latter includes the Khovanov-Rozansky knot homology, as a special case. The conjecture passes a number of nontrivial checks. We show that, for a large number of torus knots colored with the fundamental representation of SU(N), our knot invariants agree with the Poincare polynomials of Khovanov-Rozansky homology. As a byproduct, we show that our theory on S^3 has a large-N dual which is the refined topological string on X=O(-1)+O(-1)->P^1; this supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to sl(n) knot homology. We also provide a matrix model description of some amplitudes of the refined Chern-Simons theory on S^3.

arXiv:1105.5117 [hep-th]

[arsphysica]

Chern-Simons Theory and S-duality. (arXiv:1106.4550 [hep-th])

We study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example, the space of flat SL(2,C) connections on M is identified with the space of supersymmetric vacua in a dual 3d gauge theory. The hidden symmetry "hbar -> - (4 pi^2)/hbar" of SL(2) Chern-Simons theory can be identified as the S-duality transformation of N=4 super-Yang-Mills theory (obtained by compactifying the five-brane theory on a torus); whereas the mapping class group action in Chern-Simons theory on a three-manifold M with boundary C is realized as S-duality in 4d N=2 super-Yang-Mills theory associated with the Riemann surface C. We illustrate these symmetries by considering simple examples of 3-manifolds that include knot complements and punctured torus bundles, on the one hand, and mapping cylinders associated with mapping class group transformations, on the other. A generalization of mapping class group actions further allows us to study the transformations between several distinguished coordinate systems on the phase space of Chern-Simons theory, the SL(2) Hitchin moduli space.

arXiv:1106.4550 [hep-th]

[arsphysica]

Khovanov Homology. (arXiv:1107.1524 [math.GT])

This paper is an introduction to Khovanov homology.

arXiv:1107.1524 [math.GT]

[arsphysica]

Computing Khovanov-Rozansky homology and defect fusion (arXiv:1108.1081 [math.QA])

We compute the categorified sl(N) link invariants as defined by Khovanov and Rozansky, for various links and various values of N. This direct computation is made tractable by a general and constructive method for reducing tensor products of matrix factorisations to finite rank, a process which can also be interpreted as the fusion of defects in B-twisted Landau-Ginzburg models. We implement this method on the computer and show how to use it in practice.

arXiv:1108.1081 [math.QA]

[arsphysica]

Gauge Theories Labelled by Three-Manifolds (arXiv:1108.4389 [hep-th])

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that S^3_b partition functions of two mirror 3d N=2 gauge theories are equal. Three-dimensional N=2 field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional N=2 SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.

arXiv:1108.4389 [hep-th]