On the domain of string perturbation theory

Abstract

For a large class of effectively closed surfaces, it is shown that the only divergences in string scattering amplitudes at each order in perturbation theory are those associated with the coincidence of vertex operators and the boundary of moduli space. This class includes all closed surfaces of finite genus, and infinite-genus surfaces which can be uniformised by a group of Schottky type. While the computation is done explicitly for bosonic strings in their ground state, it can also be extended to excited states and to superstrings. The properties of these amplitudes lead to a definition of the domain of perturbation theory as the set of effectively closed surfaces. The implications of the restriction to effectively closed surfaces on the behaviour of the perturbation series are discussed.

Divergences in the moduli space integral and accumulating handles in the infinite-genus limit

“…Even in the intrinsic metric, which expands areas by a factor of g, the handles would still be accumulating to a point. This example confirms the theorem proven earlier [24] relating surfaces that can be uniformized by groups of Schottky type and those in the class O G . For the manifolds considered in this paper, the length of the closed geodesics are bounded below and the surfaces of infinite genus could be expanded to be manifolds of infinite extent with boundary.…”

“…The growth of the bounds for the integrals over each range of values of |K n | and |ξ 1n −ξ 2n |, n=1,...,g, reflects the geometrical characteristics of the string worldsheet. The class of effectively closed surfaces with handles of thickness diminishing at a rate of 1 n q , q > 1 2 [24], would be excluded by the cut-off. It might appear that configurations of isometric circles on the extended complex plane with the square of the radii |γ n | −2 decreasing as O( 1 n ) would be consistent with a genus-independent cut-off on the length of closed geodesics.…”

On the absence of large‐order divergences in superstring theory

“…The last component of this expression absorbs all except four of the integrals over the Grassmann variables. Examples of graviton scattering show that the ϑ 21 , ϑ 2g ,θ 21 ,θ 2g integrals are non-zero because the Green function on the super-Riemann surface depends on all of the elements of the super-Schottky group [10,13]. The first term in the integral over |H n | produces a dependence on the fixed-point distance of |ξ 1n − ξ 2n | −1 , whereas the dominant contribution to the integral with Grassmann variables is given by…”

“…The relevant constraints are those listed for the multipliers and ordinary fixed points. Moreover, conditions such as − 1 2 ≤ (Re τ ) mn ≤ 1 2 and (Im τ ) 1n ≥ 0 reduce the integrals by an exponential function of the genus [21]. Other exponential factors arise from the angular integrations of the arguments of K n and ξ 1n − ξ 2n , the integrations over ξ 2n and the primitive-element products.…”

“…Study of the large-genus limit in bosonic string theory shows that the requirement of a finite-size interaction region restricts the surfaces to be either closed or effectively closed infinite-genus surfaces [21]. Both types of surfaces always can be constructed by placing handles on spheres, which decrease in size to zero when there are an infinite number, thereby breaking the bound on the minimum length of closed geodesics.…”