The two-loop superstring five-point amplitude and S-duality

TL;DR

This paper computes the low-energy limit of two-loop five-point superstring amplitudes using the pure spinor formalism, revealing their relation to S-duality and modular forms, and providing insights into amplitude ratios for different states.

Contribution

It presents the first calculation of the two-loop five-point amplitude in superstring theory and links the results to S-duality through modular functions.

Findings

The five-graviton amplitude ratio matches the modular function $E_{5/2}$.

For R-symmetry violating states, the amplitude ratio includes a factor of -3/5.

The overall coefficient of the amplitude is determined from first principles.

Abstract

The low-energy limit of the massless two-loop five-point amplitudes for both type IIA and type IIB superstrings is computed with the pure spinor formalism and its overall coefficient determined from first principles. For the type IIB theory, the five-graviton amplitude is found to be proportional to its tree-level counterpart at the corresponding order in $\alpha'$. Their ratio ties in with expectations based on S-duality since it matches the same modular function $E_{5/2}$ which relates the two-loop and tree-level four-graviton amplitudes. For R-symmetry violating states, the ratio between tree-level and two-loop amplitudes at the same $\alpha'$-order carries an additional factor of $-3/5$. Its S-duality origin can be traced back to a modular form derived from $E_{5/2}$.