Supersymmetric Hyperbolic $$\sigma $$ σ -Models and Bounds on Correlations in Two Dimensions

In this paper we study a family of nonlinear $$\sigma $$ -models in which the target space is the super manifold $$\mathbb {H}^{2|2N}$$ . These models generalize Zirnbauer’s $$\mathbb {H}^{2|2}$$ nonlinear $$\sigma $$ -model (Zirnbauer in Commun Math Phys 141(3):503–522, 1991). The latter model has a number of special features which aid in its analysis: through a remarkable technique from symplectic geometry colloquial known as supersymmetric localization, the partition function of the $$\mathbb {H}^{2|2}$$ model is equal to one independent of the coupling constants. Our main technical observation is to generalize this fact to $$\mathbb {H}^{2|2N}$$ models as follows: the partition function is a multivariate polynomial of degree $$n=N-1$$ , increasing in each variable. As an application, these facts provide estimates on the Fourier and Laplace transforms of the ’t-field’ when we specialize to $$\mathbb {Z}^2$$ . We show that this field has fluctuations which are at least those of a massless free field. In addition we show that small fractional moments of $$e^{t_v-t_0}$$ decay at least polynomially fast in the distance of v to 0.