More on superstring chiral measures. (English) Zbl 1207.81132
Summary: In this paper we study the expressions of the superstring chiral measures for \(g\leq 5\). In genus three and four we obtain certain new equivalent expressions for the measures which are functions of higher powers of theta constants. For \(g=3\) we show that the measures can be written in terms of fourth power of theta constants and for \(g=4\) in terms of squares of theta constants. In both cases the forms \(\Xi_8^{(g)}[0^{(g)}]\) appearing in the expression of the measures are defined on the whole Siegel upper half space. Instead, for \(g=5\) we find a form \(\Xi_8^{(5)}[0^{(5)}]\) which is a polynomial in the classical theta constants, well defined on the Siegel upper half space and satisfying some suitable constraints on the moduli space of curves (and not on the whole Siegel upper half space) that could be a candidate for the genus five superstring measure. Moreover, we discuss the problem of the uniqueness of this form in genus five. We also determine the dimension of certain spaces of modular forms and reinterpret the vanishing of the cosmological constant in terms of group representations.
MSC:
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 46T12 | Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds |
| 14K25 | Theta functions and abelian varieties |
| 14H15 | Families, moduli of curves (analytic) |
| 20C15 | Ordinary representations and characters |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
Keywords:
superstrings; amplitudes; modular forms; finite geometry; group representations; theta constantsReferences:
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