Piecewise flat metrics and bosonic strings

Natalia Amberg (ITEP, Moscow)

Tuesday 25th October, 2011 15:00-16:00 Mathematics Building, room 515


In this talk we will discuss modification of bosonic string for piecewise flat metrics. Standard approach to bosonic string developed by Polyakov (Belavin and Knizhnik) is based on integration over infinite space of Rimenian metrics on the surface. Semidirect product of the diffeomorphism group and of the conformal group acts on this space and the integral reduced to a finite-dimensional integral over moduli space of algebraic curves. In this talk I will rewrite Polyakov integral for piecewise flat metrics. Surfaces are glued from flat triangles and embedding of piecewise flat surface is defined by embeddings of the vertexes. Integrals are finite-dimensional for each triangulation. I will write Polyakov action and measures on parameter space. Analog of the diffeomorphism group is discrete group of Whitehead's moves. This group is similar to modular one. For torus glued from two triangles this is $PSL(2,\mathbb Z)$. I will show that the integral reduces to the integral over moduli space and discuss examples with small number of triangles.

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