Geometry of the moduli space of Higgs bundles

Abstract

This thesis contains work which appeared in several papers. Additionally to the results in the papers it contains a detailed introduction and some further proofs and remarks. The dissertation gives a description of the topology and symplectic and algebraic geometry of Hitchin's hyperkaehler moduli space M of rank 2 Higgs bundles with fixed determinant of odd degree over a fixed Riemann surface. After the long introduction it describes a compactification of M in great detail, using symplectic cutting (math.AG/9804083). Examining the downward Morse flow of a natural circle action on M it shows the vanishing of intersection numbers (math.AG/9805071). Examining the upward Morse flow it explains a set of generators of the cohomology ring and a conjectured explicit description of the cohomology ring (which was proven in math.AG/0003094). Then finally it introduces the resolution tower for M, and shows that its direct limit is homotopically equivalent with the classifying space of the gauge group. In turn it yields another proof of the generation theorem (as in math.AG/0003093) and also yields a purely algebraic geometric proof of the Mumford conjecture about the cohomology ring of the moduli space of rank 2 stable bundles on curves. It finishes by proving homotopy stabilizations in the resolution tower analogously to the Atiyah-Jones conjecture.

arXiv commentauthor's 1998 PhD thesis, 116 pages

References in corpus (2)

• Dyon - Monopole Bound States, Self-Dual Harmonic Forms on the Multi-Monopole Moduli Space, and SL(2,Z) Invariance in String Theory
• The cohomology rings of abelian symplectic quotients

Cited by in corpus (18)

• Homotopy groups of moduli spaces of representations
• Stratifications on the Moduli Space of Higgs Bundles
• Singular Geometry and Higgs Bundles in String Theory
• The moduli space of flat SU(2)-bundles over a nonorientable surface
• Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles and complex variations of Hodge structure
• Hyperkahler analogues of Kahler quotients
• Quantization via Mirror Symmetry
• Mirror symmetry with branes by equivariant Verlinde formulae
• Mirror symmetry for Nahm branes
• Non-Abelian Hodge Theory and Related Topics
• Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces
• Asymptotic Geometry of the Moduli Space of Parabolic $SL(2,\mathbb{C})$-Higgs Bundles
• Simpson Filtration and Oper Stratum Conjecture
• Co-Higgs bundles on P^1
• Branes and DAHA Representations
• Hodge structure on the cohomology of the moduli space of Higgs bundles
• Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles
• Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current