[Paper Review] The Verlinde bundles and the semihomogeneous Wirtinger duality

Dragos Oprea|ArXiv.org|Dec 29, 2008

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper determines the splitting type of Verlinde bundles on the Jacobian of a curve of genus g ≥ 1 by expressing them as direct sums of simple semihomogeneous vector bundles. Using the Fourier-Mukai transform and trace computations of torsion points on generalized theta functions, it establishes a level-rank duality and provides an explicit formula for multiplicities in terms of Verlinde numbers and a genus-g generalization of Jordan’s totient function, confirming compatibility with strange duality and extending known results to higher genus.

ABSTRACT

We determine the splitting type of the Verlinde vector bundles in higher genus in terms of simple semihomogeneous factors. In agreement with strange duality, the simple factors are interchanged by the Fourier-Mukai transform, and their spaces of sections are naturally dual.

Motivation & Objective

• Understand the structure of Verlinde bundles on the Jacobian of a curve of genus g ≥ 1.
• Decompose Verlinde bundles into indecomposable semihomogeneous factors to clarify their splitting type.
• Establish a precise formula for the multiplicities of these factors in terms of number-theoretic and geometric invariants.
• Confirm compatibility with strange duality via Fourier-Mukai transform and level-rank symmetry.
• Generalize genus-1 results to higher genus using a new genus-g Jordan totient symbol.

Proposed method

• Define Verlinde bundles as pushforwards of pluri-theta line bundles via the determinant map on the moduli space of semistable bundles.
• Introduce simple symmetric semihomogeneous bundles $W_{r,k,\xi}$ on the Jacobian, characterized by rank and determinant conditions.
• Use the trace of h-torsion line bundles on the space of generalized theta functions to decompose the Verlinde bundle fibers.
• Apply Riemann-Roch and Chern class computations on moduli spaces to evaluate the trace of torsion actions.
• Introduce a genus-g generalization of Jordan’s totient function to express multiplicities in the decomposition.
• Verify consistency with known cases: coprime rank-level duality, genus 1 results, and level-rank symmetry.

Experimental results

Research questions

• RQ1 What is the explicit splitting type of Verlinde bundles $E_{r,k}$ on the Jacobian of a curve of genus $g \geq 1$?
• RQ2 How do the multiplicities of semihomogeneous factors in the Verlinde bundle decomposition depend on the rank $r$, level $k$, and torsion data?
• RQ3 Does the Fourier-Mukai transform preserve the decomposition into semihomogeneous factors, and how does it relate to strange duality?
• RQ4 Can the trace of a torsion point on the space of generalized theta functions be expressed in terms of Verlinde numbers in higher genus?
• RQ5 What is the role of the genus-g generalization of Jordan’s totient in the multiplicity formula?

Key findings

• The Verlinde bundle $E_{hr,hk}$ splits as a direct sum of semihomogeneous bundles $W_{r,k,\xi}$, indexed by $h$-torsion line bundles $\xi$, with multiplicities given by a formula involving the genus-g Jordan totient symbol.
• The multiplicity $m_\xi(r,k)$ of $W_{r,k,\xi}$ in $E_{hr,hk}$ is explicitly computed as a sum over divisors $\delta$ of $h/\omega$, involving Verlinde numbers and the genus-g totient symbol.
• Trace computations of $h$-torsion line bundles on $H^0(SUX(hr), L^{hk})$ yield a formula matching Verlinde numbers in smaller rank and higher genus, confirming consistency with Beauville and Oprea’s genus-1 results.
• The decomposition is compatible with strange duality: the dual of $W_{r,k,\xi}$ is isomorphic to $W_{k,r,\xi}$, and the multiplicities satisfy $m_\xi(r,k) = m_\xi(k,r)$, confirming level-rank symmetry.
• The formula reduces correctly to known cases: when $\gcd(r,k)=1$, it recovers the coprime splitting $E_{r,k} = \bigoplus W_{r,k}$, and for $g=1$, it matches the genus-1 computation of Oprea.
• An explicit Verlinde formula for $PGL_r$-bundles is derived by averaging the trace formula over $A[d]$, showing agreement with the general formula in [AMW] via a coperiodicity condition on subsets of $\{1,\dots,r+k\}$.