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

Robert Hanson

TL;DR

The paper extends Arinkin–Gaitsgory geometric Eisenstein series from the de Rham side to Dolbeault, Hodge, and twistor moduli within non-abelian Hodge theory, and uses this to decompose a hyperholomorphic BBB-brane theory into cuspidal and Eisenstein components. It introduces Dolbeault/Hodge Eisenstein functors Eis$^{\mathrm{Dol}}_P$ and Eis$^{\mathrm{Hod}}_P$, and a twistor Eis$^{\mathrm{Tw}}_P$ that preserves nilpotent singular supports across all NAH geometries, including a formal Harder–Narasimhan stratification to manage non-compactness. A central achievement is the demonstration that BBB-branes on Twistor$_G$ admit a cuspidal–Eisenstein decomposition compatible with a twistor Langlands program, and that the Dirac–Higgs BBB-brane decomposes consistently under parabolic induction. The work also develops a twistor–Wilson formalism, providing a robust framework for intertwining Eisenstein and Wilson operators, and posits a twistor geometric Langlands conjecture tying spectral BBB-branes to automorphic BAA-branes, in alignment with Kapustin–Witten duality. Overall, the results supply a comprehensive, structurally rich plan to integrate Eisenstein theory, NAH, and hyperkähler-structured branes across de Rham, Dolbeault, Hodge, and twistor perspectives, with explicit, compatible functors and decompositions that advance both mathematical Langlands program and its physical interpretations.

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 hyperkähler 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.

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

TL;DR

The paper extends Arinkin–Gaitsgory geometric Eisenstein series from the de Rham side to Dolbeault, Hodge, and twistor moduli within non-abelian Hodge theory, and uses this to decompose a hyperholomorphic BBB-brane theory into cuspidal and Eisenstein components. It introduces Dolbeault/Hodge Eisenstein functors Eis and Eis, and a twistor Eis that preserves nilpotent singular supports across all NAH geometries, including a formal Harder–Narasimhan stratification to manage non-compactness. A central achievement is the demonstration that BBB-branes on Twistor admit a cuspidal–Eisenstein decomposition compatible with a twistor Langlands program, and that the Dirac–Higgs BBB-brane decomposes consistently under parabolic induction. The work also develops a twistor–Wilson formalism, providing a robust framework for intertwining Eisenstein and Wilson operators, and posits a twistor geometric Langlands conjecture tying spectral BBB-branes to automorphic BAA-branes, in alignment with Kapustin–Witten duality. Overall, the results supply a comprehensive, structurally rich plan to integrate Eisenstein theory, NAH, and hyperkähler-structured branes across de Rham, Dolbeault, Hodge, and twistor perspectives, with explicit, compatible functors and decompositions that advance both mathematical Langlands program and its physical interpretations.

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 hyperkähler 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.
Paper Structure (101 sections, 61 theorems, 306 equations)

This paper contains 101 sections, 61 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. de Rham Cuspidal-Eisenstein decompositions
  4. Non-abelian Hodge theory
  5. Classical limit
  6. Twistors and BBB-branes
  7. Twistor geometric Langlands
  8. Digression on finiteness
  9. Dirac--Higgs complex
  10. Riemann--Hilbert and restricted variation
  11. Analytic sheaves
  12. Wilson operators
  13. Structure
  14. Conventions
  15. Acknowledgements
  16. ...and 86 more sections

Key Result

Theorem 1.1

arinkingaitsgory. $\mathop{\mathrm{IndCoh}}\nolimits_{\mathcal{N}}(\mathop{\mathrm{LocSys}}\nolimits_{G})$ is generatedA dg category $\mathcal{D}$ is generated by a collection of subcategories $\{\mathcal{D}_i\}_{i \in I}$ if the smallest full dg subcategory containing every $\mathcal{D}_i$ is equiv for all parabolic subgroups $P \subset G$.

Theorems & Definitions (111)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 2.1
  • Lemma 2.2
  • ...and 101 more