Abstract :
The modular transformations of the (1∥1) complex supermanifolds in the Schottky-like modular parameterization are discussed. It is shown that these “supermodular” transformations depend on the spinor structure of the (1∥1) complex supermanifold through terms proportional to the odd modular parameters. The above terms are calculated in explicit form. The discussed terms are important for the study of possible divergences in the Ramond-Neveu-Schwarz superstring theory. In addition, they are necessary to calculate the dependence on the odd moduli of the fundamental domain in the moduli space. The supermodular transformations of the multi-loop superstring partition functions calculated by the solution of the Ward identities are studied. In the present paper, it is shown that the above Ward identities are covariant under the supermodular transformations. Hence the considered partition functions necessarily possess the covariance under the supermodular transformations discussed. The covariance of the partition functions at zero odd moduli under those supermodular transformations in the Ramond sector, which turn a pair of even genus-1 spinor structures into a pair of odd genus-1 spinor structures is explicitly demonstrated. A brief consideration of the cancellation of divergences is given.
