Geometry and Quantization of Moduli Spaces

 Scientific description: This Research Program will be centered on the study of the geometry of algebraic moduli spaces, mostly (but not exclusively) associated to compact Riemann surfaces. The scope is intended to be wide, ranging from questions on the topology (cohomology, stable homotopy, etc.) of moduli spaces to problems on their quantization (Verlinde algebra, Hitchin connection, etc.), including also problems on dynamics (e.g. action of the mapping class group of the surface), quantum cohomology (Atiyah–Floer conjecture, Geometric Langlands program) and other areas. Most of the moduli spaces to be studied in the Research Program will be either instances or related to moduli spaces of flat connections and Higgs bundles on Riemann surfaces.

Background and objectives: Geometric structures on moduli spaces of Higgs bundles. Higgs bundles over Riemann surfaces were introduced by Hitchin in 1987 in the study of the self-duality Yang-Mills equations. The moduli space of Higgs bundles over Riemann surfaces has a tremendously rich geometric structure and is a remarkable object from the point of view of the theory of completely integrable systems (of which it is an example), hyperkähler geometry, topology (notably through its identification with moduli spaces of local systems), differential geometry (it provides generalizations of Teichmüller spaces for a number of different geometric structures on Riemann surfaces), number theory (it plays a crucial role in the recent proof of Langland’s fundamental lemma for automorphic forms by Chˆu) and mathematical physics (notably through the work of Kapustin and Witten).

This is a sample of problems on Higgs bundles moduli spaces which will be studied in the Research Program:

• Using Higgs bundles, Hitchin has introduced higher analogues of Teichmüller spaces inside the representation varieties of higher rank groups which are split real forms of a simple Lie groups. One would like to find geometric structures on surfaces which are parametrized by the Teichmüller component.

• To study the topology of the Higgs bundles moduli spaces (for example, to count the number of their connected components; or, more generally, to compute its Betti numbers).

• To study the dependence of the geometry of surface Higgs moduli space (which, as a topological space, can be identified with the moduli space of representations of the fundamental group of a surface Σ) on the Riemann surface structure on Σ. This includes in particular understanding the action of the mapping class group action on the moduli space of local systems.

• To study Fock and Goncharov’s extensions of the Thurston spaces of measured foliations/laminations to their higher Teichmüller spaces using tropical geometry (an approach which was implicit in Morgan and Shalen’s work on classical Teichmüller space).

Quantization of moduli spaces. The geometric quantization of moduli spaces of flat G-connections on a 2-dimensional manifold with G a compact Lie group is well understood.

By results of Narasimhan–Seshadri and Ramanathan, Teichmüller space is a natural parameter space of Kähler structures on the moduli spaces, and geometric quantization defines a finite rank vector bundle over Teichmüller space. This bundle carries a projectively flat connection by results by Axelrod, Della Pietra, Hitchin and Witten. To describe the quantum operators one can rely on general results on geometric quantization of compact Kaehler manifolds, using Berezin–Toeplitz operators. Some problems on quantization of moduli spaces which will be studied in the Research Program:

• The relation between the Berezin-Topelitz operators and Hitchin’s projectively flat connection has been understood by the work of Andersen, leading to a proof of Turaev’s asymptotic faithfulness conjecture, a proof that the mapping class groups do not have Kazhdan’s property T, as well as to mapping class group invariant deformation quantization of the SU(N)-moduli spaces. A very interesting problem is to extend this program to singular moduli spaces.

• Different techniques have been used to define quantizations of moduli spaces of local systems with non-compact structure group (such as quantum group techniques, deformation quantization using Vassiliev invariants, or skein theory). An interesting problem is to understand the relation between all these different quantizations. Understanding relations to quantizations of the Higgs bundle moduli space is also of fundamental importance.

• Witten proposed that TQFT’s could be constructed by applying geometric quantization to the moduli spaces of flat SU(N)-connections, but a full construction is still missing. A very interesting problem is to provide a complete gauge theory construction of the Reshetikhin-Turaev TQFT. A satisfactory solution to this problem should include a truly 3-dimensional geometric construction of the boundary states of the theory.

Other questions. These include: Verlinde spaces and twisted K-theory, dynamics and Teichmüller theory, Riemannian geometry of moduli spaces of vortices.

Perspectives of the Programme:

The central aims of the programme are to bring together experts in various aspects on the geometry and quantization of moduli spaces and related areas, to advance these topics, and to introduce research students and post-docs to the wealth of ideas and problems in them. As stated above, the interdependence of the topics we have identified is crucial to the development of the theory, and a major goal is to develop these ideas further. The programme will include an advanced course, two workshops, a final conference, as well as a regular seminar.

Luis Álvarez-Cónsul, ICMAT-CSIC Madrid

Steven Bradlow, University of Illinois at Urbana-Champaign

William Goldman, University of Maryland

Peter Gothen, Universidade do Porto

Ignasi Mundet i Riera, Universitat de Barcelona, “Chairman”

Chiu-Chu Liu Columbia University
Luis Alvarez-Cónsul ICMAT/CSIC Madrid
Thomas Baier Universidade do Porto
Philip Boalch CNRS
Steven Bradlow University of Illinois at Urbana
Champaign Leticia Brambila CIMAT A.C.
Ugo Bruzzo International School for
Advanced Studies (SISSA)
Marc Burger ERCOM
Marc Burger ERCOM
François Costantino Aarhus University
Georgios Daskalopoulos Brown University
Joergen Ellegaard Aarhus University
David Fernández ICMAT/CSIC Madrid 
Oscar García-Prada CSIC,  William Goldman University of Maryland 
Eduardo Gonzalez University of Massachusetts 
Eduardo Gonzalez University of Massachusetts 
Olivier Guichard Université Paris-Sud 
Jochen Heinloth University of Amsterdam 
Benjamin Himpel Aarhus University 
Victoria Hoskins Oxford University 
Rinat Kashaev Université de Genève
Alastair D. King University of Bath 
Herbert Lange Emmy-Noether-Zentrum
Alina Marian Northeastern University 
Cristina Martinez Universitat Autonòma de
Barcelona  Gregor Masbaum
Institut de Mathématiques de Jussieu
Gregor Masbaum Institut de Mathématiques de
Jussieu 
Brendan McLellan Aarhus University 
Ignasi Mundet Universitat de Barcelona 
Madumbai S. Narasimhan Indian Institute of Science 
Peter Newstead University of Liverpool 
Hiraku Nozawa Universidade de Santiago de Compostela
André Oliveira Centre of Mathematics of UTAD 16/04/2012 19/05/2012
Andreas Ott Max Planck Institute for Mathematics 
Bertrand Patureau-Mirand Université Paris
Christian Pauly Université de Nice Sophia
Antipolis Thang Tu Quoc Georgia Institute of Technology 
Sundararaman Ramanan Chennai Mathematical Institute 
Claudia Reynoso Universidad de Guanajuato 
José I. Royo Universidad del País Vasco 
Darío Sánchez Universidad de Salamanca 
Florent Schaffhauser Universidad de Los Andes 
Alexander Schmitt Freie Universität Berlin Carlos Simpson Université de Nice Sophia
Antipolis Constantin Teleman
University of California at Berkeley 
Domingo Toledo University of Utah 
Richard A. Wentworth University of Maryland 
Peter Zograf Steklov Mathematical Institute

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