ITGP midterm activity
Dates: From March to June 2012
Place: Centre de Recerca Matemàtica
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Ogranising Committee
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"
Activities Organized
Weekly seminar
International school on Geometry and Physics, Geometry and Quantization of Moduli Spaces
March 26 to 30, 2012
Master class and Workshop on Topological Quantum Field Theories
April 19 to 27, 2012
Master class and Workshop on Geometry of Surface Group Representations
May 9 to 17, 2012
Conference on Geometry and Quantization of Moduli Spaces (2012 VBAC Conference)
June 18 to 22, 2012
Programme Description
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.
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