Skip to main content

The Quantum Geometric Langlands Topological Field Theory

Objective

We will use modern techniques in derived algebraic geometry, topological field theory and quantum groups to construct quantizations of character varieties, moduli spaces parameterizing G-bundles with flat connection on a surface. We will leverage our construction to shine new light on the geometric representation theory of quantum groups and double affine Hecke algebras (DAHA's), and to produce new invariants of knots and 3-manifolds.

Our previous research has uncovered strong evidence for the existence of a novel construction of quantum differential operators -- and their extension to higher genus surfaces -- in terms of a four-dimensional topological field theory, which we have dubbed the Quantum Geometric Langlands (QGL) theory. By construction, the QGL theory of a surface yields a quantization of its character variety; quantum differential operators form just the first interesting example. We thus propose the following long-term projects:

1. Build higher genus analogs of DAHA's, equipped with mapping class group actions -- thereby solving a long open problem -- by computing QGL theory of arbitrary surfaces; recover quantum differential operators and the (non-degenerate, spherical) DAHA of G, respectively, from the once-punctured and closed two-torus.
2. Obtain a unified construction of both the quantized A-polynomial and the Oblomkov-Rasmussen-Shende invariants, two celebrated -- and previously unrelated -- conjectural knot invariants which have received a great deal of attention.
3. By studying special features of our construction when the quantization parameter is a root of unity, realize the Verlinde algebra as a module over the DAHA, shedding new light on fundamental results of Cherednik and Witten.
4. Develop genus one, and higher, quantum Springer theory -- a geometric approach to constructing representations of quantum algebras -- with deep connections to rational and elliptic Springer theory, and geometric Langlands program.

Field of science

  • /natural sciences/physical sciences/classical mechanics/statistical mechanics
  • /natural sciences/mathematics/applied mathematics/mathematical physics
  • /natural sciences/mathematics/pure mathematics/topology/algebraic topology
  • /natural sciences/mathematics/pure mathematics/geometry
  • /engineering and technology/electrical engineering, electronic engineering, information engineering/electronic engineering/computer hardware/quantum computer
  • /natural sciences/mathematics/pure mathematics/algebra/algebraic geometry

Call for proposal

ERC-2014-STG
See other projects for this call

Funding Scheme

ERC-STG - Starting Grant

Host institution

THE UNIVERSITY OF EDINBURGH
Address
Old College, South Bridge
EH8 9YL Edinburgh
United Kingdom
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 100 947,50

Beneficiaries (1)

THE UNIVERSITY OF EDINBURGH
United Kingdom
EU contribution
€ 1 100 947,50
Address
Old College, South Bridge
EH8 9YL Edinburgh
Activity type
Higher or Secondary Education Establishments