Curve integral formula for the M\"obius strip

TL;DR

This paper extends the curve integral formula to non-orientable surfaces like the M"obius strip, connecting string amplitudes with Feynman diagrams through a geometric and algebraic framework.

Contribution

It introduces a method to compute scattering amplitudes on non-orientable surfaces using quasi-cluster algebras and surface embeddings, generalizing previous orientable surface techniques.

Findings

Successfully derived Feynman diagrams from superstring amplitudes with M"obius topology.

Constructed surface Symanzik polynomials for higher genus non-orientable surfaces.

Extended the curve integral formula to arbitrary non-orientable surfaces.

Abstract

The scattering amplitudes for colored scalars can be calculated using the so-called curve integral formula, relying on simple combinatorics. It introduces a set of global Schwinger parameters for all Feynman diagrams that contribute to an amplitude. We extend this construction to non-orientable surfaces by making use of the quasi-cluster algebras defined for non-orientable surfaces. We embed the non-orientable surface in a doubled orientable surface, and project the appropriate features onto the non-orientable surface. The curve integral formula can also be thought of as the high-tension limit of an appropriate string amplitude. As a check of our construction, we take a superstring amplitude with the M\"obius strip topology and take its field theory limit to obtain the same Feynman diagrams as in the corresponding curve integral. Our construction can be generalized to arbitrary higher genus non-orientable surfaces. To illustrate this, we list the possible curves and their dual momenta for a two-loop non-orientable surface, and construct the surface Symanzik polynomials using the surface generalization of spanning trees.