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The Prym-Hitchin Connection and Anti-Invariant Level-Rank Duality

Thomas Baier, Michele Bolognesi, Johan Martens, Christian Pauly

TL;DR

The paper extends Hitchin’s non-abelian theta framework to higher-rank Prym varieties arising from unramified double covers, constructing a Prym-Hitchin flat projective connection on bundles of non-abelian theta functions. It develops the geometry of higher-rank anti-invariant Prym moduli, proves key codimension and cohomology results to support the heat-operator construction, and derives a Prym–Hitchin symbol that yields a flat connection. A Prym analogue of Laszlo’s theorem is established, linking the Prym–Hitchin connection with a twisted WZW connection, and the authors formulate and verify an anti-invariant level-rank duality, proving it at level one and showing flatness across all levels via conformal-embedding techniques. The results provide a robust framework for understanding dualities and flat structures in Prym settings, with potential implications for the monodromy and geometric representation theory in non-abelian theta contexts.

Abstract

We construct a "Hitchin-type" connection on bundles of non-abelian theta functions on higher-rank Prym varieties, for unramified double covers of curves. We formulate a version of level-rank duality in this Prym setting (building on work of Zelaci), show it holds for level one, and establish that the duality respects the flat connections at all levels.

The Prym-Hitchin Connection and Anti-Invariant Level-Rank Duality

TL;DR

The paper extends Hitchin’s non-abelian theta framework to higher-rank Prym varieties arising from unramified double covers, constructing a Prym-Hitchin flat projective connection on bundles of non-abelian theta functions. It develops the geometry of higher-rank anti-invariant Prym moduli, proves key codimension and cohomology results to support the heat-operator construction, and derives a Prym–Hitchin symbol that yields a flat connection. A Prym analogue of Laszlo’s theorem is established, linking the Prym–Hitchin connection with a twisted WZW connection, and the authors formulate and verify an anti-invariant level-rank duality, proving it at level one and showing flatness across all levels via conformal-embedding techniques. The results provide a robust framework for understanding dualities and flat structures in Prym settings, with potential implications for the monodromy and geometric representation theory in non-abelian theta contexts.

Abstract

We construct a "Hitchin-type" connection on bundles of non-abelian theta functions on higher-rank Prym varieties, for unramified double covers of curves. We formulate a version of level-rank duality in this Prym setting (building on work of Zelaci), show it holds for level one, and establish that the duality respects the flat connections at all levels.

Paper Structure

This paper contains 26 sections, 33 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Connections of Hitchin type
  4. (Projective) connections and Atiyah algebroids
  5. Flat morphisms between connections
  6. Projective connections from symbol maps
  7. Hitchin's connection
  8. Geometry of higher-rank Prym varieties
  9. The Prym--Hitchin system
  10. The Prym--Hitchin system for a fixed double cover
  11. Codimension estimates in the moduli space of anti-invariant Higgs bundles
  12. Results on some cohomology spaces of N
  13. The Prym--Hitchin system for a family of double covers
  14. The Prym--Hitchin connection
  15. A Laszlo theorem for anti-invariant bundles
  16. ...and 11 more sections

Key Result

Theorem 2.3.1

With $L$ and $\pi:\mathcal{M}\rightarrow S$ as before, if the following conditions hold for a given candidate symbol map $\rho: T_S \rightarrow \pi_\ast \mathop{\mathrm{Sym}}\nolimits^2 T_{\mathcal{M}/S}$: then there exists a unique projective heat operator $D$ on $L$ with symbol $\rho$.

Theorems & Definitions (65)

  • Theorem 2.3.1: Van Geemen -- De Jong,vangeemen.dejong:1998, BBMP:2020
  • Theorem 2.3.2: hitchin:1990 BBMP:2020
  • Theorem 2.4.1: hitchin:1990,BBMP:2020
  • Remark 3.1.1
  • Theorem 3.2.1: Zelaci
  • Remark 3.2.2
  • Proposition 3.2.3
  • proof
  • Proposition 3.2.4
  • proof
  • ...and 55 more