AI 中文总结
本文用超对称性阐释佐藤1972年提出的无穷阶微分算子方法，结合OSp(1|2n)等超群，研究黎曼θ函数的模性相关结构，获超对称生成元的特殊对易关系及BGG型分解结果。
AI 中文摘要
1972年，M.佐藤提出一种方法，仅通过模变量相关的特定无穷阶微分算子（DOI）刻画θ零值这类形式，以证明其模性。DOI是导数的无穷级数，其衰减速度足够快，可作为层态射作用于全纯函数。该方法经Kashiwara、Kawai、Takei和Yoshida等多位学者发展。本文利用超对称性对该方法进行阐释，超对称性为DOI提供了自然来源：任意奇性超对称生成元的朴素指数均为DOI。亏格为n的黎曼θ函数的情形由超群OSp(1|2n)（及其双重复盖）控制，作用于西格尔平面的自然超加厚空间。当n=2时，该空间对应三维N=1超共形群，相关结构恰好是自由无质量标量超多重态（结合拉普拉斯方程与狄拉克方程）；当n>2时，得到拉格朗日格拉斯曼流形（如同任意埃尔米特对称空间）上广义共形结构的超扩张。此处另一有趣特征是，“壳上”（即运动方程的解空间中）作用的奇性超对称生成元满足偶性海森堡对易关系。这些运动方程升级为微分算子复形，对应osp(1|2n)的超外尔表示的自然BGG型分解。
英文摘要
In 1972, M. Sato proposed an approach to proving modularity of forms like Thetanullwerte by characterizing them via certain differential operators of infinite order (DOI) in the modular variable(s) alone. A DOI is an infinite series in derivatives decreasing so fast that it acts on holomorphic functions by a sheaf morphism. This approach was developed by several authors including Kashiwara, Kawai, Takei and Yoshida. We give an interpretation of this approach using supersymmetry which provides a natural source of DOIs: the naive exponential of any odd supersymmetry generator is a DOI. The case of the Riemann theta function of genus n is governed by the supergroup OSp(1|2n) (and its metaplectic cover) acting on a natural super-thickening of the Siegel plane. For n=2 this is the 3-dimensional N=1 superconformal group and the structure at hand is precisely the free massless scalar supermultiplet (combining the Laplace and Dirac equations). For n>2 we get a super-extension of the generalized conformal structure existing on the Lagrangian Grassmannian as on any Hermitian symmetric space. An additional interesting feature here is that the odd supersymmetry generators acting ``on-shell'' (i.e., in the space of solutions of the equations of motion) satisfy even-style Heisenberg commutation relations. These equations of motion upgrade to a complex of differential operators corresponding to a natural BGG-type resolution of the super-Weil representation of osp(1|2n).
Comments126 pages, 1 figure
