Geometric Eisenstein series in non-abelian Hodge theory and hyperholomorphic branes from supersymmetry

TL;DR

This paper extends the geometric Eisenstein series framework to non-abelian Hodge theory, enabling a decomposition of hyperholomorphic sheaves related to branes and dualities in supersymmetric gauge theories.

Contribution

It generalizes Eisenstein series constructions to coherent sheaves on various moduli spaces, linking non-abelian Hodge theory with geometric Langlands and supersymmetry.

Findings

Decomposition of hyperholomorphic sheaves into cuspidal and Eisenstein parts.

Extension of Eisenstein series to Dolbeault, Hodge, and twistor moduli.

Connection between brane dualities and geometric Langlands conjecture.

Abstract

Using geometric Eisenstein series, foundational work of Arinkin and Gaitsgory constructs cuspidal-Eisenstein decompositions for ind-coherent nilpotent sheaves on the de Rham moduli of local systems. This article extends these constructions to coherent (not ind-coherent) nilpotent sheaves on the Dolbeault, Hodge and twistor moduli from non-abelian Hodge theory. We thus account for Higgs bundles, Hodge filtrations and hyperk\"ahler rotations of local systems. In particular, our constructions are shown to decompose a hyperholomorphic sheaf theory of so-called BBB-branes into cuspidal and Eisenstein components. Our work is motivated, on the one hand, by the `classical limit' or `Dolbeault geometric Langlands conjecture' of Donagi and Pantev, and on the other, by attempts to interpret Kapustin and Witten's physical duality between BBB-branes and BAA-branes in 4D supersymmetric Yang--Mills theories as a mathematical statement within the geometric Langlands program.