Modular Invariance and the Odderon

TL;DR

This paper discovers a new modular invariance symmetry in the equations describing the odderon, a three-gluon exchange process in QCD, providing a novel geometric perspective on its Hamiltonian structure.

Contribution

It introduces a modular invariance symmetry for odderon equations, linking the Hamiltonian to the moduli space of elliptic curves with a specific sign constraint.

Findings

Identifies a modular invariance symmetry in odderon equations

Reveals a geometric structure involving elliptic curves

Provides a new framework for understanding odderon dynamics

Abstract

We identify a new symmetry for the equations governing odderon amplitudes, corresponding in the Regge limit of QCD to the exchange of 3 reggeized gluons. The symmetry is a modular invariance with respect to the unique normal subgroup of sl(2,Z) {\,} of index 2. This leads to a natural description of the Hamiltonian and conservation-law operators as acting on the moduli space of elliptic curves with a fixed ``sign'': elliptic curves are identified if they can be transformed into each other by an {\em even} number of Dehn twists.