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Supersymmetric Hyperbolic σ-models and Decay of Correlations in Two Dimensions

In this paper we study a family of nonlinear σ-models in which the target space is the super manifold H . These models generalize Zirnbauer’s H nonlinear σ-model [Zir91] 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 model is a constants independent of the coupling constants. Here we show that for the H 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 Z. 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.