Behavior of the two-dimensional model with holomorphic superpotential choice s = −1

Determine what occurs in the two-dimensional worldvolume Wess–Zumino model when choosing s = −1 in the superpotential defined by ∂W/∂φ1 = g(φ1^2 − φ2^2) and ∂W/∂φ2 = −2g φ1 φ2; in particular, ascertain whether the auxiliary noise fields continue to satisfy Wick’s theorem without anomalies or whether anomalies arise in this holomorphic case.

Background

The authors construct two-dimensional noise fields F1 and F2 whose quadratic combination reproduces kinetic and potential terms up to total derivatives and crossterms. For the choice s = 1, the crossterms become total derivatives and Monte Carlo simulations show no anomalies in the noise-field correlators.

For s = −1, the superpotential is holomorphic but SO(2) invariance is potentially broken by crossterms, and the authors explicitly note that it remains unresolved what happens in this case.