Higgs bundles and exceptional isogenies

Steven B Bradlow,Laura P Schaposnik

doi:10.1186/s40687-016-0062-0

Abstract

We explore relations between Higgs bundles that result from isogenies between low-dimensional Lie groups, with special attention to the spectral data for the Higgs bundles. We focus on isogenies onto $$\mathrm {SO}(4,\mathbb {C})$$ and $$\mathrm {SO}(6,\mathbb {C})$$ and their split real forms. Using fiber products of spectral curves, we obtain directly the desingularizations of the (necessarily singular) spectral curves associated with orthogonal Higgs bundles. In the case of $$\mathrm {SO}(6,\mathbb {C})$$ , our construction can be interpreted as a new description of Recillas’ trigonal construction.

Highlights

Exceptional isomorphisms between low-rank Lie algebras, and the corresponding isogenies between Lie groups, have been a source of fascination since first catalogued by Cartan a century ago ([6, pp. 352–355], see [12, p. 519])
We explore the implications for Higgs bundles of the isogenies between Lie groups of rank 2 and 3: I2 : SL(2, C) × SL(2, C) → SO(4, C); (1)
Higgs bundles over a compact Riemann surface Σ of genus g ≥ 2 were introduced in [13]

Summary

Introduction

Exceptional isomorphisms between low-rank Lie algebras, and the corresponding isogenies between Lie groups, have been a source of fascination since first catalogued by Cartan a century ago ([6, pp. 352–355], see [12, p. 519]). Where E1 ⊗ E2 has orthogonal structure determined by the symplectic structures on Ei. in the case of the rank 2 isogeny I2, on the generic fibers of the Hitchin fibration for the moduli space MSL(2,C)×SL(2,C), our main result is the following (see Propositions 16, 17, 35): Theorem 1 Let Si be the spectral curve of the SL(2, C)-Higgs bundles (Ei, Φi), and Li ∈ Prym(Si, Σ) the corresponding spectral line bundle.

Objectives

Results

Conclusion