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Abstract: In this paper we study the expressions of the superstring chiral measures for$g\leq 5$. We obtain certain new expressions which are functions of higherpowers of theta constants. For $g=3$ we show that the measures can be writtenin terms of fourth power of theta constants and for $g=4$ in terms of squaresof theta constants. In both cases the forms $\Xi 8^{g}0^{g}$ appearing inthe expression of the measures are defined on the whole Siegel upper halfspace. Instead, for $g=5$ we find a form $\Xi 8^{5}0^{5}$ which is apolynomial in the classical theta constants, well defined on the Siegel upperhalf space and satisfying some suitable constraints on the moduli space ofcurves and not on the whole Siegel upper half space that could be a candidatefor the genus five superstring measure. Moreover, we discuss the problem of theuniqueness of this form in genus five. We also determine the dimension ofcertain spaces of modular forms and reinterpret the vanishing of thecosmological constant in terms of group representations.

Author: Francesco Dalla Piazza



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