ementas

Gauge theory and moduli spaces: an introduction

(MAT2425 - Tópicos de Física Matemática I)

Público-alvo: alunos de pós.

Program

• Basics in gauge theory: bundles, connection and curvature.
• Anti-self-duality equations and instantons.
• Moment maps and symplectic reduction.
• Hyper-Kähler geometry and reduction.
• The ADHM equations.
• The moduli space of instantons and Donaldson’s theorems.
• The moduli space of Higgs bundles.
• Non-abelian Hodge correspondence.
• Hitchin integrable system.

Recommended bibliography:

• Geometry of Yang–Mills Fields, M. Atiyah, Accademia Nazionale Dei Lincei Scuola Normale Superiore (1979).
• Construction of instantons, M. Atiyah, V. Drinfeld, N. Hitchin, Y. Manin, Phys. Lett. 65A (1978).
• Instantons and Four-Manifolds, D. Freed, K. Uhlenbeck, Springer–Verlag (1984).
• Integrable systems: Twistors, loop groups, and Riemann surfaces, N. Hitchin, G. Segal, R. Ward, Vol. 4. OUP Oxford (2013).
• Stable bundles and integrable systems, N. Hitchin, Duke Math Journal, 54(1):91-114 (1987).
• The self-duality equations on a Riemann surface, N. Hitchin, Proc. London Math. Soc, 55(3):59-126 (1987).
• An Introduction to Gauge Theory and its Applications, M. Jardim, Publicações Matemáticas, 25 CBM (2017)
• The geometry of four-manifolds, S. Donaldson, P. Kronheimer, OUP Oxford (1990).
• Differential Geometry of Complex Vector Bundles, S. Kobayashi, Princeton University Press (1987).
• Topology, Geometry and Gauge Fields: Interactions, G. Naber, Springer–Verlag.
• The ADHM construction of Yang-Mills instantons, S. Donaldson, 2205.08639