SL(2, Z) Duality and 4-Dimensional Noncommutative Theories

TL;DR

This paper explores how four-dimensional noncommutative theories derived from D3 branes transform under $SL(2,Z)$ symmetry, revealing that the axion's rationality influences the duality relations and properties of these theories.

Contribution

It provides a detailed analysis of $SL(2,Z)$ duality effects on noncommutative theories, highlighting the role of the axion's value in determining theory equivalences and properties.

Findings

Rational axion values lead to different duality properties than irrational ones.

NCOS and NCYM are generally not dual unless specific conditions are met.

A new light-like NCYM limit with $SL(2,Z)$ duality to ordinary Yang-Mills is identified.

Abstract

We investigate how the four-dimensional noncommutative open string/Yang-Mills theory behaves under a general non-perturbative quantum $SL(2,Z)$ symmetry transformation. We discuss this by considering D3 branes in a constant background of axion, dilaton, and electric and magnetic fields (including both $ {\bf E} \perp {\bf B}$ and {\bf E}$||${\bf B} cases) in the respective decoupling limit. We find that the value of axion, whether rational or irrational, determines the nature of the resulting theory under $SL(2,Z)$ as well as its properties such as the coupling constant and the number of noncommutative directions. In particular, a strongly coupled theory with an irrational value of axion can never be physically equivalent to a weakly coupled theory while this is usually true for a theory with a rational value of axion. A noncommutative Yang-Mills (NCYM) (resulting from D3 branes with pure magnetic flux) is physically equivalent to a noncommutative open string (NCOS) but if the value of axion is irrational, we also have noncommutative space-space directions in addition to the usual noncommutative space-time directions for NCOS. We also find in general that a NCOS cannot be physically equivalent to a NCYM but to another NCOS if the value of axion is irrational. We find another new decoupling limit for possible light-like NCYM whose $SL(2,Z)$ duality is a light-like ordinary Yang-Mills if the value of the axion is rational. Various related questions are also discussed.