Hopf Algebra Symmetry and String Theory

TL;DR

This paper explores how Hopf algebra structures can unify string quantization and spacetime symmetries, revealing how certain symmetries are deformed or preserved through Drinfeld twists in string theory.

Contribution

It introduces a unified Hopf algebra framework for string quantization and spacetime symmetries, employing Drinfeld twists to characterize symmetry deformation and invariance.

Findings

Twisted Hopf algebra describes deformed spacetime diffeomorphisms.

Unbroken symmetries are twist invariant Hopf subalgebras.

Broken symmetries are realized as twisted Hopf algebras.

Abstract

We investigate the Hopf algebra structure in string worldsheet theory and give a unified formulation of the quantization of string and the space-time symmetry. We reformulate the path integral quantization of string as a Drinfeld twist at the worldsheet level. The coboundary relation shows that the Drinfeld twist defines a module algebra which is equivalent to operators with normal ordering. Upon applying the twist, the space-time diffeomorphism is deformed into a twisted Hopf algebra, while the Poincar\'e symmetry is unchanged. This suggests a characterization of the symmetry: unbroken symmetries are twist invariant Hopf subalgebras, while broken symmetries are realized as twisted ones. We provide arguments that relate this twisted Hopf algebra to symmetries in path integral quantization.