Descripción del proyecto
Nuevos conocimientos e invariantes de nudos y tres variedades en una nueva teoría topológica cuántica de campos
La teoría cuántica de campos es el marco teórico para el estudio de la física de partículas. La teoría topológica cuántica de campos (TTCC) describe espacios cuyas propiedades se conservan bajo deformaciones continuas, como una clase exótica de materiales cuánticos (materiales topológicos) descubierta recientemente, entre los que se incluyen los aislantes topológicos. El equipo del proyecto QuantGeomLangTFT, financiado por el Consejo Europeo de Investigación, utilizará su nueva técnica de TTCC-4D, el programa de Langlands de geometría cuántica y otras técnicas de geometría algebraica derivada y grupos cuánticos para construir cuantificaciones de variedades de caracteres. Estas se utilizarán para entender mejor la teoría de la representación geométrica de grupos cuánticos y álgebras de Hecke doblemente afines, así como para desarrollar nuevos invariantes de nudos y tres variedades.
Objetivo
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.
Ámbito científico
- natural sciencesmathematicsapplied mathematicsmathematical physics
- natural sciencesmathematicspure mathematicstopologyalgebraic topology
- natural sciencesphysical sciencesclassical mechanicsstatistical mechanics
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsalgebraalgebraic geometry
Programa(s)
Régimen de financiación
ERC-STG - Starting GrantInstitución de acogida
EH8 9YL Edinburgh
Reino Unido