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Perturbative Search for Fixed Lines in Large N Gauge Theories

A. Dymarsky, I. R. Klebanov, R. Roiban

TL;DR

This work investigates whether large-$N$ non-supersymmetric orbifolds of $\mathcal{N}=4$ SYM can possess a fixed line of conformal points emanating from the origin in the space of double-trace couplings. By computing one-loop beta functions for twisted double-trace operators across various orbifolds and accounting for $U(1)$ decoupling, the authors show that, in the examples studied, there is no fixed line through the origin at weak coupling; many beta functions remain positive or lack real zeros. The results illuminate how double-trace deformations propagate in orbifold theories, connect to AdS/CFT expectations about tachyon condensation, and motivate exploring special constructions (e.g., product orbifolds or $U(1)$-decoupled theories) where a fixed line might arise. The work therefore constrains perturbative non-supersymmetric large-$N$ conformality and highlights a potential link between gauge theory stability and closed-string tachyon dynamics.

Abstract

The logarithmic running of marginal double-trace operators is a general feature of 4-d field theories containing scalar fields in the adjoint or bifundamental representation. Such operators provide leading contributions in the large N limit; therefore, the leading terms in their beta functions must vanish for a theory to be large N conformal. We calculate the one-loop beta functions in orbifolds of the N=4 SYM theory by a discrete subgroup Gamma of the SU(4) R-symmetry, which are dual to string theory on AdS_5 x S^5/Gamma. We present a general strategy for determining whether there is a fixed line passing through the origin of the coupling constant space. Then we study in detail some classes of non-supersymmetric orbifold theories, and emphasize the importance of decoupling the U(1) factors. Among our examples, which include orbifolds acting freely on the S^5, we do not find any large N non-supersymmetric theories with fixed lines passing through the origin. Connection of these results with closed string tachyon condensation in AdS_5 x S^5/Gamma is discussed.

Perturbative Search for Fixed Lines in Large N Gauge Theories

TL;DR

This work investigates whether large- non-supersymmetric orbifolds of SYM can possess a fixed line of conformal points emanating from the origin in the space of double-trace couplings. By computing one-loop beta functions for twisted double-trace operators across various orbifolds and accounting for decoupling, the authors show that, in the examples studied, there is no fixed line through the origin at weak coupling; many beta functions remain positive or lack real zeros. The results illuminate how double-trace deformations propagate in orbifold theories, connect to AdS/CFT expectations about tachyon condensation, and motivate exploring special constructions (e.g., product orbifolds or -decoupled theories) where a fixed line might arise. The work therefore constrains perturbative non-supersymmetric large- conformality and highlights a potential link between gauge theory stability and closed-string tachyon dynamics.

Abstract

The logarithmic running of marginal double-trace operators is a general feature of 4-d field theories containing scalar fields in the adjoint or bifundamental representation. Such operators provide leading contributions in the large N limit; therefore, the leading terms in their beta functions must vanish for a theory to be large N conformal. We calculate the one-loop beta functions in orbifolds of the N=4 SYM theory by a discrete subgroup Gamma of the SU(4) R-symmetry, which are dual to string theory on AdS_5 x S^5/Gamma. We present a general strategy for determining whether there is a fixed line passing through the origin of the coupling constant space. Then we study in detail some classes of non-supersymmetric orbifold theories, and emphasize the importance of decoupling the U(1) factors. Among our examples, which include orbifolds acting freely on the S^5, we do not find any large N non-supersymmetric theories with fixed lines passing through the origin. Connection of these results with closed string tachyon condensation in AdS_5 x S^5/Gamma is discussed.

Paper Structure

This paper contains 21 sections, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General Considerations
  3. Double-trace correction for general orbifolds
  4. Examples of non-freely acting orbifolds
  5. A Non-Supersymmetric $\mathbb{Z}_2$ Example
  6. A Non-supersymmetric $\mathbb{Z}_3$ orbifold
  7. General non-freely acting $\mathbb{Z}_k$ orbifolds
  8. A class of orbifolds with $SU(3)$ symmetry
  9. A non-supersymmetric $\mathbb{Z}_5$ orbifold
  10. The ${\cal N}=1$ supersymmetric $\mathbb{Z}_3$ orbifold
  11. $\mathbb{Z}_6$
  12. $\mathbb{Z}_k$ with $SU(3)$ symmetry
  13. Discussion
  14. Orbifold traces of Dirac matrices
  15. Renormalization of a single-trace operator
  16. ...and 6 more sections

Figures (2)

  • Figure 1: The quiver diagram for the non-supersymmetric freely acting $\mathbb{Z}_5$ orbifold.
  • Figure 2: The quiver diagram for the non-supersymmetric freely acting $\mathbb{Z}_6$ orbifold.