Cluster Partition Function and Invariants of 3-manifolds

Abstract

We review some recent developments in Chern-Simons theory on a hyperbolic 3-manifold $M$ with complex gauge group $G$. We focus on the case $G=SL(N,\mathbb{C})$ and with $M$ a knot complement. The main result presented in this note is the cluster partition function, a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral. We also review various applications and open questions regarding the cluster partition function and some of its relation with string theory.

26 pages, 1 figure, contribution to the proceedings of the workshop "Non-abelian Gauged Linear Sigma Model and Geometric Representation Theory" held at BICMR (Jun. 2015). To appear in a special issue of Chinese Annals of Mathematics, Series B; v2: reference added

References in corpus (3)

• N=2 supersymmetric theories on squashed three-sphere
• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant