2005/10/17 by Andrew J. Hanson, Ji-Ping Sha
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Particle physics theoretical and experimental studies #math-ph #math.MP #msc:30E20 #msc:57M50 #msc:83E30
paper · pdf · doi:10.1088/0305-4470/39/10/017
published as J. Phys. A: Math. Gen. 39 (2006) 2509-2537 · 43 pages and 44 figures
arxiv created 2005/10/17 · openalex publication_date 2006/02/22 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.