Intrinsic non-commutativity of closed string theory

Laurent Freidel,Robert G Leigh,Djordje Minic

doi:10.1007/jhep09(2017)060

Laurent Freidel, Robert G Leigh + Show 1 more

Open Access

https://doi.org/10.1007/jhep09(2017)060

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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 noncommutative. 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 α′ → 0 limit is more subtle than usually assumed.

Highlights

Classical structure of compact stringWe review various details of the free bosonic string. The majority of this material is standard fare, and we use this section to set-up our conventions, following closely the notation in ref. [1]
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 noncommutative
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

Summary

Classical structure of compact string

We review various details of the free bosonic string. The majority of this material is standard fare, and we use this section to set-up our conventions, following closely the notation in ref. [1]. Our work will show that this traditional point of view can be misleading and that is not a generic message of string theory We emphasize that such an interpretation relies heavily on an assumed structure for the zero modes of the string, namely that they can be thought of as coordinates on T , along with their conjugate variables. It is often assumed that the zero mode x plays a preferred role, being interpreted as a coordinate in the target space, while xis immaterial as it does not appear in the action. Neither x nor xappear in the abelian currents dX and the string action density They do enter the theory in the vertex operator algebra, and so we are led to study them more carefully. It is implicit in this analysis that (xL, xR) (equivalently (x, x)) are independent; it is only in the R → ∞ (R → 0) limit that a projection p → 0 (p → 0) on the spectrum is induced, at which point x (x) decouples

The vertex algebra and mutual locality

The symplectic structure of the zero modes

Causality

T-duality

Conclusion