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In this paper we study a family of nonlinear $\sigma$-models in which thetarget space is the super manifold $H^{2|2N}$. These models generalizeZirnbauer's $H^{2|2}$ nonlinear $\sigma$-model which has a number of specialfeatures for which we find analogs in the general case. For example, bysupersymmetric localization, the partition function of the $H^{2|2}$ model is aconstants 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 onthe Fourier and Laplace transforms of the '$t$-field' when these models arespecialized to $\mathbb{Z}^2$. From the bounds, we conclude the $t$-fieldexhibits polynomial decay of correlations and has fluctuations which are atleast those of a massless free field.

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