Description du projet
Une nouvelle théorie quantique des champs topologiques: nouvelles perspectives et invariants des nœuds et des 3-variétés
La théorie quantique des champs est le cadre théorique de l’étude de la physique des particules. Les théories quantiques des champs topologiques (TQFT) décrivent des espaces dont les propriétés sont préservées par des déformations continues, comme une classe exotique de matériaux quantiques (matériaux topologiques) récemment découverte, comprenant les isolants topologiques. Le projet QuantGeomLangTFT, financé par le Conseil européen de la recherche, utilisera sa nouvelle approche 4D-TQFT, le Quantum Geometric Langlands (QGL) et d’autres techniques de géométrie algébrique dérivée et de groupes quantifiés pour élaborer des quantifications de variétés de caractères. Celles-ci seront utilisées pour mieux comprendre la théorie des représentations géométriques des groupes quantiques et des algèbres de Hecke doublement affines, et pour produire de nouveaux invariants des nœuds et des 3-variétés.
Objectif
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.
Champ scientifique
- natural sciencesmathematicsapplied mathematicsmathematical physics
- natural sciencesmathematicspure mathematicstopologyalgebraic topology
- natural sciencesphysical sciencesclassical mechanicsstatistical mechanics
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsalgebraalgebraic geometry
Programme(s)
Thème(s)
Régime de financement
ERC-STG - Starting GrantInstitution d’accueil
EH8 9YL Edinburgh
Royaume-Uni