QuantGeomLangTFT
Financiado
The Quantum Geometric Langlands Topological Field Theory
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 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.
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Duración del proyecto: 72 meses
Fecha Inicio: 2015-05-29
Fecha Fin: 2021-05-31
Fecha Fin: 2021-05-31
Línea de financiación: concedida
El organismo H2020 notifico la concesión del proyecto el día 2021-05-31
Línea de financiación objetivo
El proyecto se financió a través de la siguiente ayuda:
ERC-StG-2014:
ERC Starting Grant
Cerrada
hace 10 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
Perfil tecnológico
TRL
4-5
315.48K€ |
Participante