Descrizione del progetto
Una nuova teoria quantistica dei campi topologica: nuove intuizioni e invarianti di nodi e 3-varietà
La teoria quantistica dei campi è il quadro teorico per lo studio della fisica delle particelle. Le teorie quantistiche dei campi topologiche (TQFT, topological quantum field theory) descrivono spazi le cui proprietà si conservano sotto deformazioni continue, come una classe esotica recentemente scoperta di materiali quantistici (materiali topologici), tra cui gli isolanti topologici. Il progetto QuantGeomLangTFT, finanziato dal Consiglio europeo della ricerca, utilizzerà la sua nuova 4D-TQFT, la Quantum Geometric Langlands (QGL) e altre tecniche di geometria algebrica derivata e gruppi quantistici per costruire quantizzazioni di varietà di caratteri. Questi saranno utilizzati per approfondire la teoria delle rappresentazioni geometriche dei gruppi quantistici e delle algebre di Hecke doppie affini, nonché per produrre nuovi invarianti di nodi e di 3-varietà.
Obiettivo
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.
Campo scientifico
- natural sciencesmathematicsapplied mathematicsmathematical physics
- natural sciencesmathematicspure mathematicstopologyalgebraic topology
- natural sciencesphysical sciencesclassical mechanicsstatistical mechanics
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsalgebraalgebraic geometry
Programma(i)
Argomento(i)
Meccanismo di finanziamento
ERC-STG - Starting GrantIstituzione ospitante
EH8 9YL Edinburgh
Regno Unito