# Overview on High energy String Scattering Amplitudes and Symmetries of String Theory

## TL;DR

This paper reviews symmetries and relations of high energy string scattering amplitudes in different regimes, discovering infinite linear and recurrence relations that unify and extend previous results, revealing potential spacetime symmetries.

## Contribution

It introduces new linear and recurrence relations for high energy string scattering amplitudes, unifying fixed angle and Regge regimes and suggesting underlying spacetime symmetries.

## Key findings

- Discovered infinite linear relations among fixed angle amplitudes.
- Found recurrence relations among Regge amplitudes.
- Expressed all amplitudes in terms of a single base amplitude.

## Abstract

We overview symmetries of string scattering amplitudes in the high energy limits of both the fixed angle or Gross regime (GR) and the fixed momentum transfer or Regge regime (RR). We calculated high energy string scattering amplitudes (SSA) at arbitrary mass levels for both regimes. We discovered the infinite linear relations among fixed angle string amplitudes and the infinite recurrence relations among Regge string amplitudes. The linear relations we obtained in the GR corrected the saddle point calculations by Gross, Gross and Mende. In addition, for the high energy closed string scatterings, our results differ from theirs by an oscillating prefactor which was crucial to recover the KLT relation valid for all energies. We showed that all the high energy string amplitudes can be solved by the linear or recurrence relations so that all the string amplitudes can be expressed in terms of a single string amplitude. We further found that, at each mass level, the ratios among the fixed angle amplitudes can be extracted from the Regge string scattering amplitudes. Finally, we review the recent developments on the discovery of infinite number of recurrence relations valid for all energies among Lauricella SSA. The symmetries or relations among SSA at various limits obtained previously can be exactly reproduced. It leads us to argue that the known SL(K+3,C) dynamical symmetry of the Lauricella function maybe crucial to probe spacetime symmetry of string theory.