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The Quantum Geometric Langlands Topological Field Theory

Total Cost €


EC-Contrib. €






 QuantGeomLangTFT project word cloud

Explore the words cloud of the QuantGeomLangTFT project. It provides you a very rough idea of what is the project "QuantGeomLangTFT" about.

algebra    closed    computing    degenerate    representations    constructing    fundamental    springer    qgl    quantization    hecke    mapping    geometry    once    rasmussen    genus    actions    conjectural    shine    uncovered    manifolds    character    existence    surface    invariants    witten    shende    equipped    moduli    groups    spherical    class    arbitrary    dimensional    quantized    leverage    received    modern    variety    topological    unified    solving    construction    analogs    special    thereby    light    construct    differential    extension    theory    langlands    connections    quantum    dubbed    spaces    parameterizing    torus    representation    unrelated    affine    shedding    knot    previously    recover    module    double    oblomkov    surfaces    varieties    punctured    flat    obtain    realize    interesting    form    first    operators    respectively    algebras    deal    geometric    unity    group    knots    rational    polynomial    verlinde    algebraic    yields    daha    quantizations    elliptic    deep    techniques    root    bundles    cherednik    celebrated    connection   

Project "QuantGeomLangTFT" data sheet

The following table provides information about the project.


Organization address
postcode: EH8 9YL

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country United Kingdom [UK]
 Project website
 Total cost 1˙100˙947 €
 EC max contribution 1˙100˙947 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2014-STG
 Funding Scheme ERC-STG
 Starting year 2015
 Duration (year-month-day) from 2015-06-01   to  2020-05-31


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    THE UNIVERSITY OF EDINBURGH UK (EDINBURGH) coordinator 1˙100˙947.00


 Project objective

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.


year authors and title journal last update
List of publications.
2017 Andrea Appel, Sachin Gautam
An explicit isomorphism between quantum and classical sl(n)
published pages: , ISSN: , DOI:
2018 Samuelson, Peter; Cooper, Benjamin
The Hall Algebras of Surfaces I
published pages: , ISSN: 1474-7480, DOI:
Journal of the Institute of Mathematics of Jussieu to appear 2020-01-24
2016 Martina Balagovic, David Jordan
The Harish-Chandra isomorphism for quantum GL_2
published pages: , ISSN: 1661-6952, DOI:
Journal of Noncommutative Geometry to appear 2020-01-24
2018 Andrea Appel, Valerio Toledano-Laredo
Coxeter categories and quantum groups
published pages: , ISSN: , DOI:
2016 Sabin Cautis, Aaron D. Lauda, Anthony Licata, Peter Samuelson, Joshua Sussan
The Elliptic Hall algebra and the deformed Khovanov Heisenberg category
published pages: , ISSN: 1022-1824, DOI:
Selecta Mathematica to appear 2020-01-24
2017 David Jordan and Monica Vazirani
The rectangular representation of the double affine Hecke algebra via elliptic Schur-Weyl duality
published pages: , ISSN: , DOI:
2018 David Ben-Zvi, Adrien Brochier, David Jordan
Integrating quantum groups over surfaces
published pages: 873-916, ISSN: 1753-8416, DOI: 10.1112/topo.12072
Journal of Topology 11/4 2020-01-24
2017 David Jordan and Noah White
The center of the reflection equation algebra via quantum minors
published pages: , ISSN: , DOI:
2018 Yuri Berest, Peter Samuelson
Affine cubic surfaces and character varieties of knots
published pages: 644-690, ISSN: 0021-8693, DOI: 10.1016/j.jalgebra.2017.11.015
Journal of Algebra 500 2020-01-24
2017 Andrea Appel, Valerio Toledano-Laredo
Uniqueness of quasi-Coxeter structures on Kac-Moody algebras
published pages: , ISSN: , DOI:
2018 David Ben-Zvi, Adrien Brochier, David Jordan
Quantum character varieties and braided module categories
published pages: , ISSN: 1022-1824, DOI: 10.1007/s00029-018-0426-y
Selecta Mathematica 2020-01-24

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