Holomorphic Curves and Gauge Theory

Goals: Physicists have many examples of gauge theories that are very different yet, in some limit, yield equivalent quantum field theories. In mathematics, one example is now well-understood: Taubes' proof that a certain limit of the Seiberg-Witten equations concentrate (as solutions of a non-linear PDE) along sets that are completely characterized by Gromov-Witten invariants (solutions of a completely different PDE). The course will introduce both sides of this correspondence, including the needed geometry and analysis, outline Taubes' proof, and examine the concentration phenomenon.

Prerequisites: Familiarity with manifolds (vector fields, differential forms, tangent and tensor bundles). It will be very useful, but not strictly necessary, to have an understanding of Hilbert and Banach spaces (e.g. Chapters 3-5 of Rudin's ``Real and Complex Analysis'') and some knowledge of PDEs.

Homework:  

Lecture Notes:   Primer: Section 1   

Preliminary list of topics:

1. Introduction to gauge theories.
2. Vortices on complex curves.
3. Complex curves and Deligne-Mumford spac.e
4. J-holomorphic maps and Gromov-Witten invariants.
5. Seiberg-Witten equations and their moduli space.
6. Taubes' concentration phenomenon and the proof that SW=Gr.

Helpful reference books:

• Seiberg-Witten and Gromov invariants for symplectic 4-manifolds, by Cliford Taubes, edited by Richard Wentworth.
• J-Holomorphic Curves and Symplectic Geometry by D.McDuff and D. Salomon.