A Relation between One-Loop Amplitudes of Closed and Open Strings (One-Loop KLT Relation)

TL;DR

This paper extends the KLT relations from tree-level to one-loop string amplitudes, expressing closed string amplitudes as sums over open string subamplitudes, with implications for gauge-gravity duality and field theory limits.

Contribution

It generalizes the KLT relations to one-loop amplitudes, incorporating complex structure moduli and loop momenta, applicable to both massless and massive states with or without supersymmetry.

Findings

Closed string one-loop amplitudes expressed as sums over open string subamplitudes.

Relations hold for both massless and massive states, with or without supersymmetry.

Implications for one-loop gauge-gravity relations and color kinematics duality.

Abstract

We express one-loop closed string amplitudes as weighted sums over squares of open string one-loop subamplitudes. These findings generalize - subject to final complex structure modulus integration - the celebrated tree-level relationships known as Kawai-Lewellen-Tye (KLT) relations to higher loops and can be applied for both the massless and massive case - with or without supersymmetry. As a consequence in the field-theory limit our relations capitalize solid one-loop gauge-gravity relations including loop-level color kinematics duality. In particular, in gravitational one-loop amplitudes a graviton is traded for two gluons just like at tree-level. Our results are derived on the underlying string world-sheet torus by splitting each complex string coordinate into a pair of two real cylinder coordinates by means of an analytic continuation on the elliptic curve. This manipulation involves the loop momentum, which mediates between the two one-loop open string cylinder amplitudes.