16 - Higgs Bundles in the Vector Representation

Summary

Dedicated to Peter Newstead on the occasion of his 65th birthday.

Introduction

A Higgs bundle, as introduced by the author 20 years ago [H1], consists of a holomorphic bundle E over a compact Riemann surface ∑ together with a section Φ ∈ H⁰ (∑, End E ⊗ K) (called the Higgs field) satisfying a certain stability condition. This can be generalized to a holomorphic principal G-bundle P over ∑ together with a section Φ of the bundle Ad(P) ⊗ K. The moduli space of such pairs has a very rich geometry and featured recently in the physical derivation by A.Kapustin and E.Witten of the central ideas of the geometric Langlands programme [KW]. A further paper by Kapustin [K] initiates a parallel discussion where the Higgs field Φ takes values in R(P) ⊗ K for a representation more general than the adjoint. In particular, he considers the case where G = SU(n) and R consists of k copies of the n-dimensional vector representation. This means that we have a rank n vector bundle E with trivial determinant and Φ = (ϕ₁, …, ϕ_k) ∈ H⁰ (∑, E ⊗ K) ⊗ C^k.

The study of the algebraic geometry of these moduli spaces already exists in the literature under the name of stable k-pairs [BDGW], [BDW]. These papers, however, generally differ from the current problem in two ways. One is the type of moduli space considered.