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The Self-Duality Equations on a Riemann Surface and Four-Dimensional Chern-Simons Theory

Roland Bittleston, Lionel Mason, Seyed Faroogh Moosavian

Abstract

We construct a Lagrangian formulation of Hitchin's self-duality equations on a Riemann surface $Σ$ using potentials for the connection and Higgs field. This two-dimensional action is then obtained from a four-dimensional Chern-Simons theory on $Σ\times \mathbb{CP}^1$ with an appropriate choice of meromorphic 1-form on $\mathbb{CP}^1$ and boundary conditions at its poles. We show that the symplectic structure induced by the four-dimensional theory coincides with the canonical symplectic form on the Hitchin moduli space in the complex structure corresponding to the moduli space of Higgs bundles. We further provide a direct construction of Hitchin Hamiltonians in terms of the four-dimensional gauge field. Exploiting the freedom in the choice of the meromorphic one-form, we construct a family of four-dimensional Chern-Simons theories depending on a $\mathbb{CP}^1$-valued parameter. Upon reduction to two dimensions, these descend to a corresponding family of two-dimensional actions on $Σ$ whose field equations are again Hitchin's equations. Furthermore, we obtain a family of symplectic structures from our family of four-dimensional theories and show that they agree with the hyperkähler family of symplectic forms on the Hitchin moduli space, thereby identifying the $\mathbb{CP}^1$-valued parameter with the twistor parameter of the Hitchin moduli space. Our results place Hitchin's equations and their integrable structure within the framework of four-dimensional Chern-Simons theory and make the role of the twistor parameter manifest.

The Self-Duality Equations on a Riemann Surface and Four-Dimensional Chern-Simons Theory

Abstract

We construct a Lagrangian formulation of Hitchin's self-duality equations on a Riemann surface using potentials for the connection and Higgs field. This two-dimensional action is then obtained from a four-dimensional Chern-Simons theory on with an appropriate choice of meromorphic 1-form on and boundary conditions at its poles. We show that the symplectic structure induced by the four-dimensional theory coincides with the canonical symplectic form on the Hitchin moduli space in the complex structure corresponding to the moduli space of Higgs bundles. We further provide a direct construction of Hitchin Hamiltonians in terms of the four-dimensional gauge field. Exploiting the freedom in the choice of the meromorphic one-form, we construct a family of four-dimensional Chern-Simons theories depending on a -valued parameter. Upon reduction to two dimensions, these descend to a corresponding family of two-dimensional actions on whose field equations are again Hitchin's equations. Furthermore, we obtain a family of symplectic structures from our family of four-dimensional theories and show that they agree with the hyperkähler family of symplectic forms on the Hitchin moduli space, thereby identifying the -valued parameter with the twistor parameter of the Hitchin moduli space. Our results place Hitchin's equations and their integrable structure within the framework of four-dimensional Chern-Simons theory and make the role of the twistor parameter manifest.
Paper Structure (28 sections, 2 theorems, 115 equations)

This paper contains 28 sections, 2 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. General Themes
  3. Contributions of the Paper
  4. 2d Action for the Hitchin's Equations (§\ref{["sec:2d action for the Hitchin's equations"]}).
  5. 2d Action for the Hitchin's Equations from 4d CS Theory (§\ref{['sec:4dCS-Hitchin']}--\ref{['sec:symplectic structure, and Hitchin Hamiltonians']}).
  6. $\mathbb{CP}^1$-Family of 2d Actions from 4d CS Theory (§\ref{["sec:CP^1-family of 2d actions associated with the Hitchin's equations"]}).
  7. From 4d Chern--Simons Theory to Hitchin's Equations
  8. 2d Action for the Hitchin's Equations
  9. 4d CS Theory Setup
  10. Derivation of the 2d Action from 4d CS Theory
  11. Reality Conditions.
  12. Potentials.
  13. Reduction of the Action.
  14. $S_0$:
  15. $S_1$:
  16. ...and 13 more sections

Key Result

Proposition 1.1

Take the four-dimensional Chern--Simons theory eq:action 4d CS, introduction with Since $\omega$ has poles at $z=0,\infty$ we impose the boundary conditions Then if the field equations are imposed, there exists a gauge in which the new gauge field $\mathcal{A}$ can be written as and the action eq:action 4d CS, introduction is gauge equivalent to eq:action for Hitchin's equations, introduction f

Theorems & Definitions (4)

  • Proposition 1.1
  • Proposition 1.2
  • Proof
  • Proof