Stringy Fuzziness as the Custodian of Time-Space Noncommutativity

TL;DR

The paper investigates whether field theories with time-space noncommutativity, $[x^\mu,x^\nu]=i\theta^{\mu\nu}$, can be obtained as decoupled limits of open-string theory in electric backgrounds. It identifies two branches of critical singularities in the open- and closed-string moduli space, showing that in standard backgrounds the time-space NC scale cannot be separated from intrinsic string fuzziness, a phenomenon termed noncommutative censorship. A formal decoupling is explored via analytic continuation to imaginary dilaton and a light-cone tumbling, which maps to a large-$N$ master-field with a naked singularity at the noncommutativity scale, raising questions about UV consistency of time-space NC field theories. The results suggest that resolving the singularities may require full stringy effects, reinforcing the view that time-space noncommutativity cannot be naively decoupled from string-scale physics.

Abstract

We study aspects of obtaining field theories with noncommuting time-space coordinates as limits of open-string theories in constant electric-field backgrounds. We find that, within the standard closed-string backgrounds, there is an obstruction to decoupling the time-space noncommutativity scale from that of the string fuzziness scale. We speculate that this censorship may be string-theory's way of protecting the causality and unitarity structure. We study the moduli space of the obstruction in terms of the open- and closed-string backgrounds. Cases of both zero and infinite brane tensions as well as zero string couplings are obtained. A decoupling can be achieved formally by considering complex values of the dilaton and inverting the role of space and time in the light cone. This is reminiscent of a black-hole horizon. We study the corresponding supergravity solution in the large-N limit and find that the geometry has a naked singularity at the physical scale of noncommutativity.