Arrangements and Frobenius like structures

2012/10/14 by Alexander Varchenko, Varchenko, Alexander
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #math.AG #math.CO #nlin.SI

paper · pdf · doi:10.48550/arxiv.1210.3802

AmsLaTeX, 55 pages, misprints corrected, a references added, abstract and introduction extended

arxiv created 2014/09/19 · arxiv updated 2014/09/22

Abstract

We consider a family of generic weighted arrangements of n hyperplanes in \C^k and show that the Gauss-Manin connection for the associated hypergeometric integrals, the contravariant form on the space of singular vectors, and the algebra of functions on the critical set of the master function define a Frobenius like structure on the base of the family. As a result of this construction we show that the matrix elements of the linear operators of the Gauss-Manin connection are given by the 2k+1-st derivatives of a single function on the base of the family, the function called the potential of second kind, see formula (6.46).

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