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Seiberg-Witten differentials on the Hitchin base

Ugo Bruzzo, Peter Dalakov

Abstract

In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss--Manin) derivative of the Seiberg--Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration. Dedicated to Tony Pantev on the occasion of his 60th birthday.

Seiberg-Witten differentials on the Hitchin base

Abstract

In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss--Manin) derivative of the Seiberg--Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration. Dedicated to Tony Pantev on the occasion of his 60th birthday.
Paper Structure (14 sections, 6 theorems, 62 equations)

This paper contains 14 sections, 6 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Embedding of the Cameral Curve
  4. Local Description
  5. Objects, Associated with Roots
  6. Liouville Form
  7. Background: Two Results
  8. A Result of Hurtubise and Markman
  9. A Result of Hertling, Hoevenaars and Posthuma
  10. Proof of Theorem A
  11. Examples
  12. $SL_2(\mathbb{C})$
  13. $SL_3(\mathbb{C})$
  14. $G_2$

Key Result

Proposition 2.1

The normal bundle of $\widetilde{X}_b\subseteq M$ is

Theorems & Definitions (8)

  • Example 1.1
  • Example 1.2
  • Proposition 2.1
  • Proposition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.1: hertling_hoev_posthum, Proposition 8.2