Instantons and fermion condensate in adjoint QCD_2

TL;DR

This paper investigates instantons and fermion condensates in adjoint QCD_2, revealing temperature-dependent behaviors and a paradox for N>2, with implications for understanding nonperturbative effects in gauge theories.

Contribution

It provides a detailed analysis of instanton solutions and fermion condensates in adjoint QCD_2, highlighting differences between N=2 and N>2 cases and discussing paradoxes in the theoretical understanding.

Findings

Instantons exist due to nontrivial topology in adjoint QCD_2.

Fermion condensate persists at high and low temperatures for N=2.

A paradox arises for N>2 regarding condensate presence and zero modes.

Abstract

We show that $QCD_2$ with adjoint fermions involves instantons due to nontrivial $\pi_1[SU(N)/Z_N]~=~Z_N$. At high temperatures, quasiclassical approximation works and the action and the form of effective (with account of quantum corrections) instanton solution can be evaluated. Instanton presents a localized configuration with the size $\propto g^{-1}$. At $N=2$, it involves exactly 2 zero fermion modes and gives rise to fermion condensate $<\bar{\lambda}^a \lambda^a>_T$ which falls off $\propto \exp\{-\pi^{3/2} T/g\}$ at high $T$ but remains finite. At low temperatures, both instanton and bosonization arguments also exhibit the appearance of fermion condensate $<\bar{\lambda}^a \lambda^a>_{T=0} ~\sim ~g$. For $N>2$, the situation is paradoxical. There are $2(N-1)$ fermion zero modes in the instanton background which implies the absence of the condensate in the massless limit. From the other hand, bosonization arguments suggest the appearance of the condensate for any $N$. Possible ways to resolve this paradox (which occurs also in some 4-dim gauge theories) are discussed.