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The Quantum Geometric Langlands Topological Field Theory

Projektbeschreibung

Eine neuartige topologische Quantenfeldtheorie: neue Erkenntnisse und Invarianten von Knoten und 3-Mannigfaltigkeiten

Die Quantenfeldtheorie bildet den theoretischen Rahmen für die Untersuchung der Teilchenphysik. Topologische Quantenfeldtheorien (TQFT) beschreiben Räume, deren Eigenschaften bei kontinuierlichen Verformungen erhalten bleiben, wie z. B. eine kürzlich entdeckte exotische Klasse von Quantenmaterialien (topologische Materialien) einschließlich topologischer Isolatoren. Im Rahmen des vom Europäischen Forschungsrat finanzierten Projekts QuantGeomLangTFT werden dessen neuartige 4D-TQFT, die Quantum Geometric Langlands (QGL) und andere Verfahren der abgeleiteten algebraischen Geometrie und Quantengruppen eingesetzt, um Quantifizierungen von Charaktervarietäten zu konstruieren. Diese werden genutzt, um Einblicke in die geometrische Darstellungstheorie von Quantengruppen und doppelten affinen Hecke-Algebren zu gewinnen und um neue Invarianten von Knoten und 3-Mannigfaltigkeiten zu erzeugen.

Ziel

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.

Finanzierungsplan

ERC-STG - Starting Grant

Gastgebende Einrichtung

THE UNIVERSITY OF EDINBURGH
Netto-EU-Beitrag
€ 1 100 947,50
Adresse
OLD COLLEGE, SOUTH BRIDGE
EH8 9YL Edinburgh
Vereinigtes Königreich

Auf der Karte ansehen

Region
Scotland Eastern Scotland Edinburgh
Aktivitätstyp
Higher or Secondary Education Establishments
Links
Gesamtkosten
€ 1 100 947,50

Begünstigte (1)