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.

在本文中，我们研究了一系列非线性\(\sigma \) -模型，其中目标空间是超流形\(\mathbb {H}^{2|2N}\)。这些模型概括了 Zirnbauer 的\(\mathbb {H}^{2|2}\)非线性\(\sigma \) 模型（Zirnbauer in Commun Math Phys 141(3):503–522, 1991）。后一个模型具有许多有助于其分析的特殊特征：通过辛几何口语中称为超对称定位的一项非凡技术，\(\mathbb {H}^{2|2}\)模型的配分函数为等于 1，与耦合常数无关。我们的主要技术观察是将这个事实推广到\(\mathbb {H}^{2|2N}\)模型如下：配分函数是一个多元多项式\(n=N-1\)，在每个变量中增加。作为一个应用，当我们专门研究\(\mathbb {Z}^2\)时，这些事实提供了对“ t 场”的傅立叶和拉普拉斯变换的估计。我们表明该场的波动至少是无质量自由场的波动。此外，我们表明\(e^{t_v-t_0}\) 的小分数矩在v到 0的距离内至少以多项式快速衰减。