Probing the holomorphic anomaly of the $D=2, \mathcal{N}=2$, Wess-Zumino model on the lattice

TL;DR

This paper investigates the holomorphic anomaly in a two-dimensional ,  supersymmetric Wess-Zumino model on the lattice, demonstrating that fluctuations prevent anomaly formation due to the preservation of continuous symmetries.

Contribution

It introduces a generalized Langevin framework for scalar field theories, linking supersymmetry, fluctuations, and anomalies, and shows that in two dimensions, continuous symmetries protect against holomorphic anomalies.

Findings

Fluctuations prevent holomorphic anomalies in 2D supersymmetric models.

Supersymmetry can be probed via correlation function identities.

Continuous symmetries cannot be broken in two dimensions, safeguarding against anomalies.

Abstract

We study a generalization of the Langevin equation, that describes fluctuations, of commuting degrees of freedom, for scalar field theories with worldvolumes of arbitrary dimension, following Parisi and Sourlas and correspondingly generalizes the Nicolai map. Supersymmetry appears inevitably, as defining the consistent closure of system+fluctuations and it can be probed by the identities satisfied by the correlation functions of the noise fields, sampled by the action of the commuting fields. This can be done effectively, through numerical simulations. We focus on the case where the target space is invariant under global rotations, in Euclidian signature, corresponding to global Lorentz transformations, in Lorentzian signature. This can describe target space supersymmetry. In this case a cross--term, that is a total derivative for abelian isometries, or when the fields are holomorphic functions of their arguments, can lead to obstructions. We study its effects and find that, in two dimensions, it cannot lead to the appearance of the holomorphic anomaly, in any event, when fluctuations are taken into account, because continuous symmetries can't be broken in two dimensions.