Supersymmetric Hyperbolic $\sigma$-models and Decay of Correlations in Two Dimensions

In this paper we study a family of nonlinear $\sigma$-models in which the target space is the super manifold $H^{2|2N}$. These models generalize Zirnbauer's $H^{2|2}$ nonlinear $\sigma$-model which has a number of special features for which we find analogs in the general case. For example, by supersymmetric localization, the partition function of the $H^{2|2}$ model is a constants independent of the coupling constants. Here we show that for the $H^{2|2N}$ model, the partition function is a multivariate polynomial of degree $N-1$, increasing in each variable. We use these facts to provide estimates on the Fourier and Laplace transforms of the '$t$-field' when these models are specialized to $\mathbb{Z}^2$. From the bounds, we conclude the $t$-field exhibits polynomial decay of correlations and has fluctuations which are at least those of a massless free field.