Leon Takhtajan

Synopsis

Character varieties appear in many areas of mathematics. Character varieties of surface groups inlclude Teichmüller spaces, spaces of projective structures on surfaces, moduli spaces of flat connections, hoilomorphic vector bundles and Higgs bundles. In context of Hamiltonian action, character varieties are examples of symplectic reduction. In mathematical physics, character varieties appear in the context of Yang-Mills and Chern-Simons theories and are related to many interesting constructions.

Lectures

TuTh 11:30am-12:50pm, Physics P 127.

Instructor

Leon Takhtajan

Office hours: TuTh 2:00pm-3:30pm and by appointment.

Course description

The course will cover the following topics (if time permits):

• Character varieties as GIT quotients Hom(Γ,G)//G, where Γ is a surface group and G is a reductive Lie group (real or complex).
• Special examples: G=PSL(2,R) (Teichmüller theory), G=PSL(2,C) (moduli of projective structures) and G=SU(n) (moduli of flat connections/holomorphic vector bundles).
• Complex and symplectic structure of character varieties. Goldman symplectic form and Ralph Fox free differential calculus.
• Complex analytic structure of Teichmüller spaces according to Ahlfors-Bers (using quasiconformal mappings and Beltrami equation) and Weil (using Kodaira-Spencer deformation of complex structures theory).
• Weil-Petersson metric on Teichmüller spaces. Proof that it is a Käler metric and the study of its curvature properties.
• Goldman symplectic form as symplectic form of Weil-Petersson metric.
• Goldman symplectic form in case G=PSL(2,C) and canonical symplectic form on the cotangent bundle to the Teichmüller space.
• Moduli spaces of holomorphic vector bundles over a Riemann surface and Narasimhan-Seshadri theorem.
• Character varieties for orbifold surfaces in case G=PSL(2,C).

Textbook

There will be no assigned textbooks. Each week a reference will be given on the Brightspace to research papers and monographs related to the material covered.

Course materials

0. Pleminary lecture notes updated, 4/24/2024

L.T. course lecture notes draft

1. Surveys of general properties of representation and character varieties

A. Maret, Lectures on character varieties

A. Sikora, Character varieties

A. Sikora, Character varieties of abelian groups

2. Classic papers

M.S. Narasimhan and C.S. Seshadri, Holomorphic vector bundles on a compact Riemann surface

W.M. Goldman, The symplectic nature of fundamental groups of surfaces

3. Stability and vector bundles

D. Mumford, Projective invariants of projective structures and applications

G. Faltings, Lectures on vector bundles on curves (see Theorem 5)

4. Narasimhan-Seshadri theorem

M.S. Narasimhan and C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface

S.K. Donaldson, A new proof of a theorem of Narasimhan and Seshadri

5. Deformation theory

K. Kodaira, Complex manifolds and deformations of complex structures

S. Nakano, Parametrization of a family of bundles

C. Schnell, Deformations of complex structures

6. Cauchy-Riemann operators, moduli space of stable vector bundles and special unitary connections

M.F. Atiyah & R. Bott, Yang-Mills equations over Riemann surfaces

S. Kobayshi, Differential geometry of complex vector bundles

L.T. & P. Zograf, On the geometry of moduli spaces of vector bundles over a Riemann surface

7. PSL(2,R) character variety and Teichmüller spaces

W.M. Goldman, Discontinuous groups and the Euler class. PhD thesis

W.M. Goldman, Topological components of spaces of representations

L.V. Ahlfors, Lectures on Quasiconformal Mappings: Second Edition

8. Goldman symplectic form: details

R.H. Fox, Free differential calculus. I: Derivation in a free group ring

K. Guruprasad & C.S. Rajan, Group cohomology and the symplectic structure on the moduli space of representations

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