Two-Dimensional Instantons with Bosonization and Physics of Adjoint $QCD_2$

TL;DR

This paper analyzes instanton effects in the bosonized form of adjoint QCD2, revealing the conditions for equivalence with fermionic models, and discusses the spontaneous symmetry breaking and phase transitions in the system.

Contribution

It establishes the precise conditions under which bosonized and fermionic models of adjoint QCD2 are equivalent, including gauge group embeddings and instanton contributions.

Findings

Bosonized model matches fermionic model with specific gauge-invariant WZNW action.

Small fermion mass expansion of partition function involves only even powers.

For odd N, discrete chiral symmetry is spontaneously broken, leading to a fermion condensate.

Abstract

We evaluate partition functions $Z_I$ in topologically nontrivial (instanton) gauge sectors in the bosonized version of the Schwinger model and in a gauged WZNW model corresponding to $QCD_2$ with adjoint fermions. We show that the bosonized model is equivalent to the fermion model only if a particular form of the WZNW action with gauge-invariant integrand is chosen. For the exact correspondence, it is necessary to integrate over the ways the gauge group $SU(N)/Z_N$ is embedded into the full $O(N^2 - 1)$ group for the bosonized matter field. For even $N$, one should also take into account the contributions of both disconnected components in $O(N^2 - 1)$. In that case, $Z_I \propto m^{n_0}$ for small fermion masses where $2n_0$ coincides with the number of fermion zero modes in a particular instanton background. The Taylor expansion of $Z_I/m^{n_0}$ in mass involves only even powers of $m$ as it should. The physics of adjoint $QCD_2$ is discussed. We argue that, for odd $N$, the discrete chiral symmetry $Z_2 \otimes Z_2$ present in the action is broken spontaneously down to $Z_2$ and the fermion condensate $<\bar{\lambda} \lambda>_0$ is formed. The system undergoes a first order phase transition at $T_c = 0$ so that the condensate is zero at an arbitrary small temperature. It is not yet quite clear what happens for even $N \geq 4$.