# Intrinsic non-commutativity of closed string theory

## TL;DR

This paper demonstrates that closed string theory inherently exhibits non-commutativity due to cocycle operators, revealing a subtle effect rooted in the string's monodromies and zero-mode algebra, impacting the classical limit.

## Contribution

It identifies and explains the intrinsic non-commutativity in closed string theory arising from cocycle operators and monodromies, a previously overlooked aspect.

## Key findings

- Non-commutativity is linked to cocycle operators in vertex operators.
- This non-commutativity stems from the Polyakov action with windings.
- The effect persists even in trivial backgrounds and affects the classical limit.

## Abstract

We show that the proper interpretation of the cocycle operators appearing in the physical vertex operators of compactified strings is that the closed string target is non-commutative. We track down the appearance of this non-commutativity to the Polyakov action of the flat closed string in the presence of translational monodromies (i.e., windings). In view of the unexpected nature of this result, we present detailed calculations from a variety of points of view, including a careful understanding of the consequences of mutual locality in the vertex operator algebra, as well as a detailed analysis of the symplectic structure of the Polyakov string. We also underscore why this non-commutativity was not emphasized previously in the existing literature. This non-commutativity can be thought of as a central extension of the zero-mode operator algebra, an effect set by the string length scale -- it is present even in trivial backgrounds. Clearly, this result indicates that the $\alpha'\to 0$ limit is more subtle than usually assumed.