## $$c=I$$ conformal field theories on Riemann surfaces.(English)Zbl 0649.32019

The commented paper studies the class of Gaussian $$c=I$$ conformal field theories which consists of the torus models describing a free massless scalar field $$\phi(z,\bar z)$$ compactified on the cirl $${\mathbb{R}}/2\pi R{\mathbb{Z}}$$ with the action $$({1/2}\pi) \int d^ 2z\partial \phi {\bar \partial}\phi$$ and $${\mathbb{Z}}_ 2$$ orbifold models which are obtained from the corresponding torus models identifying $$\phi$$ with $$-\phi$$. The main purpose of the paper is to investigate the structure of these theories on higher genus Riemann surfaces.
The authors analyse the partition functions on arbitrary Riemann surfaces and their behaviour under the action of modular group. The correlation functions are obtained through factorization of the partition function at the boundary of moduli space. To analyze the $$Z_ 2$$ orbifold models the authors use the theory of double coverings of Riemann surfaces and theta- functions defined on Prym varieties. The equivalence of the $$R=\sqrt{2}$$ orbifold and $$R=(1/2)\sqrt{2}$$ torus models gives a physical interpretation of nontrivial identities between theta-functions known as Schottky relations. It appears that in contrast with toroidal models the orbifold partition functions are sensitive to the Torelli group. The operator formalism on Riemann surfaces for both torus and orbifold models is constructed and the relation with the $$\tau$$-functions of the KP and BKP hierarchy is discussed.
Reviewer: V.Enol’skij

### MSC:

 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 35Q99 Partial differential equations of mathematical physics and other areas of application 81T08 Constructive quantum field theory 30F99 Riemann surfaces
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