Unimodular transformations of the supermanifolds and the calculation of the multi-loop amplitudes in the superstring theory

The modular transformations of the (1|1) complex supermanifolds in the likeSchottky modular parameterization are discussed. It is shown that these ”supermodular” transformations depend on the spinor structure of the (1|1) complex supermanifold by terms proportional to the odd modular parameters. The above terms are calculated in the explicit form. The discussed terms are important for the study of the possible divergencies 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 modular 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. It is demonstrated in the explicit form the covariance of the above partition functions at zero odd moduli under those supermodular transformations in the Ramond sector, which turn a pair of even genus-1 spinor structures to a pair of the odd genus-1 spinor ones. The brief consideration of the cancellation of divergences is given. E-mail address: [email protected] 0