On the absence of large‐order divergences in superstring theory

Abstract

The genus‐dependence of multi‐loop superstring amplitudes is estimated at large orders in perturbation theory using the super‐Schottky group parameterization of supermoduli space. Restriction of the integration region to a subset of supermoduli space and a single fundamental domain of the super‐modular group suggests an exponential dependence on the genus. Upper bounds for these estimates are obtained for arbitrary N‐point superstring scattering amplitudes and are shown to be consistent with exact results obtained for special type II string amplitudes for orbifold or Calabi‐Yau compactifications. The genus‐dependence is then obtained by considering the effect of the remaining contribution to the superstring amplitudes after the coefficients of the formally divergent parts of the integrals vanish as a result of a sum over spin structures. The introduction of supersymmetry therefore leads to the elimination of large‐order divergences in string perturbation theory, a result which is based only on the supersymmetric generalization of the Polyakov measure and not the gauge group of the string model.

Genus dependence of superstring amplitudes

“…and it has been demonstrated that the products Q 0 j1 K 1=2 j and Q 0 j1 ÿ K 1=2 j are exponential functional of the genus [5], the sum (4.1) is bounded by the product of an exponential and…”

“…While a minimum-length condition is satisfied by the categories of isometric circles [4] i 0 g 1ÿ2q 0 jK n j the sizes of the handles decrease to zero even in the intrinsic metric in the following ranges for the parameters [5] 0 n q 00 jK n j 0 0 n q 00 ; q 00 > 1; n 1; 2; 3; . .…”

A resolution of the integration region problem for the supermoduli space integral

“…13 Given that the first two sectors are chosen to be the Neveu-Schwarz and Ramond sectors, the remaining spin structures can be obtained from modular transformations of S 1 and S 2 . While it has not been established whether there exists a set of 2 g − 2 transformations which generates the 2 2g spin structures without any overlaps, it has been found that there is a set of 3 g − 2 g − 1 modular transformations {ρ r }, with S 1 = S ′ 1 ,…”

“…13 Although there is a quotient by a factorial for each value of q, the values of q for isometric circles can be chosen such that the maximum contribution would have an exponential dependence of the genus, since there is a factor of g g arising from the fixed point integrals, except from the first category defined in Sec. 2.…”

Univalent functions and Diff(S 1)/S 1

“…This exponential growth resembles the upper bound for the scattering amplitudes of superstring theory [12,13]. A proof of this conjecture requires an analysis of the maximum magnitude of the coefficients for convex and starlike functions on the superdisk.…”