A contour integral representation for the dual five-point function and a symmetry of the genus-4 surface in R

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 formulae such as the contour-integral representation obtained by Pochhammer in 1890. However, the precise features of the expected multiplecomplex-variable generalizations to BN have not been systematically studied. Here we explore the geometry underlying the dual five-point function B5, the simplest generalization of the Euler Beta function. The original integrand defining B5 leads to a polyhedral structure for the five-crosscap surface, embedded in RP, 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 5 in CP#4CP2. The symmetric embedding of the five-crosscap surface in RP is doubly covered by a corresponding symmetric embedding of the surface of genus 4 in S ⊂ R that has a polyhedral structure with 24 pentagonal faces and a symmetry group of order 240 in O(6). These symmetries enable the construction of elegant visualizations of these surfaces. The key idea of this paper is to realize that the compactification of the set of five-point cross-ratios forms a smooth real algebraic subvariety that is the five-crosscap surface in RP. It is in the complexification of this surface that we construct the contour integral representation for B5. Our methods are generalizable in principle to higher dimensions, and therefore should be of interest for further study.