Curve integral formula for the Möbius strip

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 Möbius 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.

Figures (17)

• Figure 1: $g$-vectors for $S$ being disk with $5$ marked points. There are five vectors corresponding to five curves, and the resultant five cones are colored. The headlight functions evaluated in the cones are written.
• Figure 2: List of all the curves on the Möbius strip with two marked points.
• Figure 3: Combinatorial polytopes $M_{1,2}$ obtained from (quasi-)mutations of the Möbius strip with one and two marked points.
• Figure 4: $g$-vector fans for the Möbius strip with one or two marked points on its boundary, with the 'wheel' graph ($n$-gon) as the reference triangulation.
• Figure 5: Evaluation of $g$-vectors for Möbius${}_1$ by embedding it into an annulus.