A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6

TL;DR

This paper introduces a geometric and integral representation for the dual five-point function B5, revealing complex polyhedral structures and symmetries that generalize properties of the Euler Beta function to higher dimensions.

Contribution

It provides a contour integral representation for B5 involving a genus five surface and uncovers symmetric embeddings with polyhedral structures, extending classical complex analysis methods.

Findings

Polyhedral structure of the five-crosscap surface with 12 pentagonal faces

A Pochhammer-like contour integral representation for B5

Symmetric embedding of genus four surface with 24 pentagonal faces

Abstract

The invention of the "dual resonance model" N-point functions BN motivated the development of current string theory. The simplest of these models, the four-point function B4, is the classical Euler Beta function. Many standard methods of complex analysis in a single variable have been applied to elucidate the properties of the Euler Beta function, leading, for example, to analytic continuation formulas such as the contour-integral representation obtained by Pochhammer in 1890. Here we explore the geometry underlying the dual five-point function B5, the simplest generalization of the Euler Beta function. Analyzing the B5 integrand leads to a polyhedral structure for the five-crosscap surface, embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120 in PGL(6). We find a Pochhammer-like representation for B5 that is a contour integral along a surface of genus five. The symmetric embedding of the five-crosscap surface in RP5 is doubly covered by a symmetric embedding of the surface of genus four in R6 that has a polyhedral structure with 24 pentagonal faces and a symmetry group of order 240 in O(6). The methods appear generalizable to all N, and the resulting structures seem to be related to associahedra in arbitrary dimensions.