Two-Dimensional Instantons with Bosonization and Physics of Adjoint $QC{D}_2$

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 $QC{D}_2$ 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 $QC{D}_2$ 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 $N\ge 4$.

36 pages, LaTeX. References added

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