Evaluating one-loop string amplitudes

Abstract

We evaluate one-loop open-string amplitudes at finite \alpha'α′ for the first time. Our method involves a deformation of the integration contour over the modular parameter \tauτ to a fractal contour introduced by Rademacher in the context of analytic number theory. This procedure leads to explicit and practical formulas for the one-loop four-point amplitudes in type-I superstring theory, amenable to numerical evaluation. We plot the amplitudes as a function of the Mandelstam invariants ss and tt and directly verify long-standing conjectures about their behaviour at high energies.

Tropological sigma models

“…It should be intriguing to analyze how this construction withstands the process of tropicalization. 7 Tropical geometry has already made its appearance in the study of equilibrium string amplitudes, first in the α ′ → 0 limit [65] and more recently at general α ′ [66,67]. 8 See remarks in section 3.1 of [69].…”