Non-compact Chern-Simons Theory, Positive Representations, and Cluster Varieties

Over the past 30-years, deep connections between Chern–Simons theory, supersymmetric (SUSY) gauge theory, and representation theory of quantum groups, have caused an avalanche of research in mathematics and physics.

In this proposal I use quantum cluster varieties to develop positive representation theory of quantum groups and a non-compact analogue of Chern–Simons theory.

I also obtain new invariants of links and 3-manifolds, and establish new connections between SUSY gauge theories and quantum character varieties.

This proposal builds on my prior work, where I prove fundamental cases of the Fock–Goncharov modular functor conjecture in higher Teichmüller theory, and Gaiotto’s conjecture on the existence of cluster structure on K-theoretic Coulomb branches of 3d N = 4 SUSY gauge theories. The proposal is split into the following four projects:1.

Prove the modular functor conjecture and extend it to a non-compact analogue of Chern–Simons theory. Obtain new powerful invariants of links and 3-manifolds.2.

Develop positive representation theory: construct continuous braided monoidal category from positive representations, prove non-compact Peter–Weyl theorem, obtain explicit formulas for finite-dimensional 6j-symbols, prove that the category of positive representations of quantum groups in type A is equivalent to a fusion category in Toda conformal field theory.3.

Describe cluster structure on K-theoretic Coulomb branches of 3d N = 4 SUSY gauge theories, conjectured by Gaiotto. Obtain cluster structure on spherical double affine Hecke algebra, and Slodowy intersections. Provide an algorithm, identifying certain theories of class S with quiver gauge theories.4.

Relate cluster quantization of character varieties with the topological quantum field theory constructed by Ben-Zvi, Brochier, and Jordan. Use it to obtain a canonical quantization of the A-polynomial.