Innovating Works

NCST

Financiado
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 propos... 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. ver más
30/06/2026
UEDIN
1M€
Perfil tecnológico estimado
Duración del proyecto: 68 meses Fecha Inicio: 2020-10-16
Fecha Fin: 2026-06-30

Línea de financiación: concedida

El organismo H2020 notifico la concesión del proyecto el día 2020-10-16
Línea de financiación objetivo El proyecto se financió a través de la siguiente ayuda:
ERC-2020-STG: ERC STARTING GRANTS
Cerrada hace 5 años
Presupuesto El presupuesto total del proyecto asciende a 1M€
Líder del proyecto
THE UNIVERSITY OF EDINBURGH No se ha especificado una descripción o un objeto social para esta compañía.
Perfil tecnológico TRL 4-5