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

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

2005-10-17

arxiv.org/abs/math-ph/0510064v1

J. Phys. A: Math. Gen. 39 (2006) 2509-2537

Physics

Mathematical Physics

43 pages and 44 figures

Scientific paper

10.1088/0305-4470/39/10/017

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.

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