Theta Functions for $\SL(n)$ versus $\GL(n)$

TL;DR

This paper establishes a formula relating the dimensions of sections of theta bundles over moduli spaces of vector bundles on a curve, revealing a deep connection between different moduli spaces and proposing a duality conjecture.

Contribution

It provides a simple formula linking the spaces of sections of theta bundles over two types of moduli spaces of bundles on a curve, and introduces a conjectural duality between these spaces.

Findings

Derived a formula relating dimensions of sections of theta bundles

Identified a proportional relationship involving gcd of rank and degree

Proposed a duality conjecture for these spaces of sections

Abstract

Over a smooth complex projective curve $C$ of genus $g$ let $\M (n,d)$ be the moduli space of semistable bundles of rank $n$ and degree $d$ on $C$, and $\SM (n,L)$, the moduli space of those bundles whose determinant is isomorphic to a fixed line bundle $L$ over $C$. Let $\theta_F$ and $\theta$ be theta bundles over these two moduli spaces. We prove a simple formula relating their spaces of sections: if $h=\gcd (n,d)$ is the greatest common divisor of $n$ and $d$, and $L\in \Pic ^d(C)$, then $$\dim H^0(\SM (n,L), \theta^k) \cdot k^g=\dim H^0(\M(n,d),\theta_F^k)\cdot h^g.$$ We also formulate a conjectural duality between these two types of spaces of sections.