Algebraic Geometry Seminar

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Higgs Bundles, Spectral Data, and Isomorphisms Among Low Dimensional Lie Groups

Speaker: Steve Bradlow, University of Illinois at Urbana-Champaign

Location: Warren Weaver Hall 201

Date: Tuesday, March 24, 2015, 3:30 p.m.

Synopsis:

We will explore some interesting relations among Higgs bundles, from the point of view of spectral data, that result from special isomorphisms among low dimensional Lie algebras and Lie groups.

Higgs bundles provide an algebro-geometric description of surface group representations into complex reductive Lie groups, and also into their real forms, say G. The defining data sets for such G-Higgs bundles include a Riemann surface Σ, a holomorphic principal bundle E→Σ, and a Higgs field Φ which is a holomorphic section of an associated vector bundle. Alternatively, the defining data can be encoded in a ramified cover S→Σ (the spectral curve) and a line bundle in the Jacobian of the spectral curve. If two groups, say G₁ and G₂, are related by a group homomorphism, one can expect the corresponding Higgs bundles and their spectral data sets to inherit induced relationships. We will explore this phenomenon in the case of isogenies resulting from accidental isomorphisms among low dimensional Lie algebras.