Explicit Calculation of Multiloop Amplitudes in the Superstring Theory

TL;DR

This paper explicitly calculates multiloop superstring amplitudes using Ward identities, revealing complexities in the period matrix dependence on spinor structures and confirming finiteness under supersymmetry.

Contribution

It provides an explicit method for multiloop superstring amplitude calculation and identifies limitations of the naive generalization of the Belavin-Knizhnik theorem.

Findings

Superstring amplitudes depend on spinor structures and odd moduli.

Naive generalization of Belavin-Knizhnik theorem is incorrect.

Superstring theory is finite with ten-dimensional supersymmetry.

Abstract

Multiloop superstring amplitudes are calculated in the explicit form by the solution of Ward identities. A naive generalization of Belavin-Knizhnik theorem to the superstring is found to be incorrect since the period matrix turns out to be depended on the spinor structure over the terms proportional to odd moduli. These terms appear because fermions mix bosons under the two-dim. supersymmetry transformations. The closed, oriented superstring turns out to be finite, if it possesses the ten-dimensional supersymmetry, as well as the two-dimentional one. This problem needs a further study.