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Large-N reduction in QCD-like theories with massive adjoint fermions

Tatsuo Azeyanagi, Masanori Hanada, Mithat Unsal, Ran Yacoby

TL;DR

The paper demonstrates that large-N QCD-like theories with massive adjoint fermions admit a large-small volume equivalence under unbroken center symmetry, enabling Eguchi-Kawai reduction from small to large volumes. It extends volume independence to finite temperature, linking thermodynamics on ${\mathbb R}^3 \times S^1$ to a reduced unitary matrix model, and confirms these ideas nonperturbatively via Monte Carlo simulations, showing $1/N$-level corrections and effective suppression with twisted boundary conditions. Twisted reductions yield accurate confinement/deconfinement signals on small lattices, and the approach offers a nonperturbative formulation of noncommutative Yang-Mills theory, with broader implications for orientifold equivalences and the conformal window. Overall, the work provides a practical, nonperturbative framework for studying YM dynamics and phase structure using reduced models with controlled finite-$N$ corrections.

Abstract

Large-N QCD with heavy adjoint fermions emulates pure Yang-Mills theory at long distances. We study this theory on a four- and three-torus, and analytically argue the existence of a large-small volume equivalence. For any finite mass, center symmetry unbroken phase exists at sufficiently small volume and this phase can be used to study the large-volume limit through the Eguchi-Kawai equivalence. A finite temperature version of volume independence implies that thermodynamics on R^3 x S^1 can be studied via a unitary matrix quantum mechanics on S^1, by varying the temperature. To confirm this non-perturbatively, we numerically study both zero- and one-dimensional theories by using Monte-Carlo simulation. Order of finite-N corrections turns out to be 1/N. We introduce various twisted versions of the reduced QCD which systematically suppress finite-N corrections. Using a twisted model, we observe the confinement/deconfinement transition on a 1^3 x 2-lattice. The result agrees with large volume simulations of Yang-Mills theory. We also comment that the twisted model can serve as a non-perturbative formulation of the non-commutative Yang-Mills theory.

Large-N reduction in QCD-like theories with massive adjoint fermions

TL;DR

The paper demonstrates that large-N QCD-like theories with massive adjoint fermions admit a large-small volume equivalence under unbroken center symmetry, enabling Eguchi-Kawai reduction from small to large volumes. It extends volume independence to finite temperature, linking thermodynamics on to a reduced unitary matrix model, and confirms these ideas nonperturbatively via Monte Carlo simulations, showing -level corrections and effective suppression with twisted boundary conditions. Twisted reductions yield accurate confinement/deconfinement signals on small lattices, and the approach offers a nonperturbative formulation of noncommutative Yang-Mills theory, with broader implications for orientifold equivalences and the conformal window. Overall, the work provides a practical, nonperturbative framework for studying YM dynamics and phase structure using reduced models with controlled finite- corrections.

Abstract

Large-N QCD with heavy adjoint fermions emulates pure Yang-Mills theory at long distances. We study this theory on a four- and three-torus, and analytically argue the existence of a large-small volume equivalence. For any finite mass, center symmetry unbroken phase exists at sufficiently small volume and this phase can be used to study the large-volume limit through the Eguchi-Kawai equivalence. A finite temperature version of volume independence implies that thermodynamics on R^3 x S^1 can be studied via a unitary matrix quantum mechanics on S^1, by varying the temperature. To confirm this non-perturbatively, we numerically study both zero- and one-dimensional theories by using Monte-Carlo simulation. Order of finite-N corrections turns out to be 1/N. We introduce various twisted versions of the reduced QCD which systematically suppress finite-N corrections. Using a twisted model, we observe the confinement/deconfinement transition on a 1^3 x 2-lattice. The result agrees with large volume simulations of Yang-Mills theory. We also comment that the twisted model can serve as a non-perturbative formulation of the non-commutative Yang-Mills theory.

Paper Structure

This paper contains 26 sections, 59 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Review of original Eguchi-Kawai proposal
  4. Perturbative fluctuations and one-loop potential
  5. Nonperturbative fluctuations and size of eigenvalue bunch
  6. Eguchi-Kawai equivalence for massive QCD(Adj)
  7. QCD(Adj) at zero temperature on $T^4$ at small $L$
  8. $N_f^D=0$ (bosonic)
  9. $N_f^D=1/2$ (single Majorana)
  10. $N_f^D\ge 1$: Uniformization of eigenvalue distribution
  11. QCD(Adj) at finite temperature on asymmetric $T^3 \times S^1$
  12. $N_f^D=0$ (bosonic)
  13. $N_f^D=1/2$ (single Majorana)
  14. $N_f^D\ge 1$
  15. Lattice model and Monte-Carlo simulation
  16. ...and 11 more sections

Figures (5)

  • Figure 1: The scales in the problem. Left panel: $m\gtrsim \lambda_{0d}^{1/4}$ and mass is important. Right panel: $m\lesssim \lambda_{0d}^{1/4}$ and mass is negligible with respect to quantum fluctuations. We can always realize the latter case by taking the size of the four-torus sufficiently small, but still $O(N^0)$. See the text for explanations.
  • Figure 2:
  • Figure 4: Expectation values of the Wilson loop $\langle|W|\rangle$ in QCD(Adj) at $b=0.50$, $N=25$. Clear hysteresis can be seen.
  • Figure 5:
  • Figure 8: