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Phase Transitions of Large N Orbifold Gauge Theories

Yasuaki Hikida

TL;DR

This work analyzes the phase structure of N=4 U(N) super Yang–Mills theory on R × S^3/Z_k at large N in the zero coupling limit, revealing first-order deconfinement-type transitions among holonomy vacua despite the free theory. It develops a matrix-model formulation of the partition function, computes critical temperatures and Casimir energies for different holonomies and fermion spin structures, and shows that at high temperature the free energy becomes universal across vacua. The authors then map these gauge-theory results to a dual gravity description in AdS_5/Z_k, discussing Hawking–Page transitions, the T-dual NS5-brane pictures, and localized tachyon condensation leading to Eguchi–Hanson solitons, with the soliton mass relating closely to Casimir energies. The study highlights how vacuum structure in orbifold gauge theories mirrors geometric transitions and tachyon dynamics in the gravity dual, and points to future work on nonzero ’t Hooft coupling and the continuation to strong coupling. Overall, it extends the AdS/CFT correspondence to orbifolds with multi-vacuum structure and provides quantitative links between gauge-theory vacua and gravitational geometries.

Abstract

We study the phase structures of N=4 U(N) super Yang-Mills theories on R x S^3/Z_k with large N. The theory has many vacua labelled by the holonomy matrix along the non-trivial cycle on S^3/Z_k, and for the fermions the periodic and the anti-periodic boundary conditions can be assigned along the cycle. We compute the partition functions of the orbifold theories and observe that phase transitions occur even in the zero 't Hooft coupling limit. With the periodic boundary condition, the vacua of the gauge theory are dual to various arrangements of k NS5-branes. With the anti-periodic boundary condition, transitions between the vacua are dual to localized tachyon condensations. In particular, the mass of a deformed geometry is compared with the Casimir energy for the dual vacuum. We also obtain an index for the supersymmetric orbifold theory.

Phase Transitions of Large N Orbifold Gauge Theories

TL;DR

This work analyzes the phase structure of N=4 U(N) super Yang–Mills theory on R × S^3/Z_k at large N in the zero coupling limit, revealing first-order deconfinement-type transitions among holonomy vacua despite the free theory. It develops a matrix-model formulation of the partition function, computes critical temperatures and Casimir energies for different holonomies and fermion spin structures, and shows that at high temperature the free energy becomes universal across vacua. The authors then map these gauge-theory results to a dual gravity description in AdS_5/Z_k, discussing Hawking–Page transitions, the T-dual NS5-brane pictures, and localized tachyon condensation leading to Eguchi–Hanson solitons, with the soliton mass relating closely to Casimir energies. The study highlights how vacuum structure in orbifold gauge theories mirrors geometric transitions and tachyon dynamics in the gravity dual, and points to future work on nonzero ’t Hooft coupling and the continuation to strong coupling. Overall, it extends the AdS/CFT correspondence to orbifolds with multi-vacuum structure and provides quantitative links between gauge-theory vacua and gravitational geometries.

Abstract

We study the phase structures of N=4 U(N) super Yang-Mills theories on R x S^3/Z_k with large N. The theory has many vacua labelled by the holonomy matrix along the non-trivial cycle on S^3/Z_k, and for the fermions the periodic and the anti-periodic boundary conditions can be assigned along the cycle. We compute the partition functions of the orbifold theories and observe that phase transitions occur even in the zero 't Hooft coupling limit. With the periodic boundary condition, the vacua of the gauge theory are dual to various arrangements of k NS5-branes. With the anti-periodic boundary condition, transitions between the vacua are dual to localized tachyon condensations. In particular, the mass of a deformed geometry is compared with the Casimir energy for the dual vacuum. We also obtain an index for the supersymmetric orbifold theory.

Paper Structure

This paper contains 16 sections, 92 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Orbifold gauge theories
  3. Partition function
  4. Path integral formulation
  5. Phase transitions of the gauge theories
  6. Critical temperatures
  7. Casimir energies
  8. Just above the critical temperatures
  9. High temperature behaviors
  10. Dual gravity description
  11. Hawking-Page transition
  12. Phase transitions of the orbifolds
  13. Localized tachyon condensation
  14. Conclusion and discussions
  15. The partition function of single scalar particle
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Free energies $F(T)/N^2$ as functions of $T$ in the cases with the periodic and the anti-periodic boundary conditions and with $k=4,6$. The solid lines are for the trivial holonomy case and the dotted lines are for the ${\mathbb Z}_k$ symmetric case.