Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator

Abstract

We revisit the tree-level closed superstring amplitude and identify its α′-expansion as series with single-valued multiple zeta values as coefficients. The latter represent a subclass of multiple zeta values originating from single-valued multiple polylogarithms at unity. Moreover, the α′-expansion of the closed superstring amplitude can be cast into the same algebraic form as the open superstring amplitude: the closed superstring amplitude is essentially the single-valued version of the open superstring amplitude. This fact points to a deeper connection between gauge and gravity amplitudes than is implied by Kawai–Lewellen–Tye relations. Furthermore, we argue that the Deligne associator carries the relevant information on the closed superstring amplitude. In particular, we give an explicit representation of the Deligne associator in terms of Gamma functions modulo squares of commutators of the underlying Lie algebra. This form of the associator can be interpreted as the four-point closed superstring amplitude.