The two D6R4 type invariants and their higher order generalisation

TL;DR

This paper classifies two classes of D6R4 invariants in maximal supergravity, explores their supersymmetry constraints, and links their threshold functions to Eisenstein series, providing insights into string theory couplings.

Contribution

It identifies two distinct classes of D6R4 invariants, derives their supersymmetry constraints, and relates their threshold functions to Eisenstein series, extending understanding of supergravity invariants.

Findings

Second class includes a coupling generalizing to 1/8 BPS protected couplings.

Threshold functions satisfy homogeneous differential equations for all k>0.

Exact D6R4 threshold function is a sum of an Eisenstein series and an inhomogeneous solution.

Abstract

We show that there are two distinct classes of D6R4 type supersymmetry invariants in maximal supergravity. The second class includes a coupling in F2D4R4 that generalises to 1/8 BPS protected F2kD4R4 couplings. We work out the supersymmetry constraints on the corresponding threshold functions, and argue that the functions in the second class satisfy to homogeneous differential equations for arbitrary k>0, such that the corresponding exact threshold functions in type II string theory should be proportional to Eisenstein series, which we identify. This analysis explains in particular that the exact D6R4 threshold function is the sum of an Eisenstein function and a solution to an inhomogeneous Poisson equation in string theory.