## Ïåòåðáóðãñêèé ñåìèíàð ïî êâàíòîâîé òåîðèè ïîëÿ

### Äîêëàäû    â   2010   ãîäó

• 23 June
A. A. Belavin (Inst. Teor. Fiz., Moskwa)
"TWO DIMENSIONAL GRAVITY IN MATRIX MODEL, TOPOLOGICAL AND LIOUVILLE APPROACHES "
Three different approaches to 2d Gravity will be discussed. The first one is the continuous approach, in which the theory is defined through the functional integral over the Riemannian metric with appropriate gauge fixing. The choice of the conformal gauge leads to quantum Liouville theory and for that reason this approach is often called the Liouville Gravity. The second one is the discrete approach, based on the idea of approximating the fluctuating 2d geometry by an ensemble of planar graphs, so that the continuous theory is recovered in the scaling limit where the planar graphs of very large size dominate. The discrete approach is usually refereed to as the Matrix Models, since technically the ensemble of the graphs is usually generated by the perturbative expansion of the integral over $N\times N$ matrices. The third approach is 2d topological gravity. Witten built axiomatics of this theory by studying intersection theory on the moduli space of Riemann surfaces. It will be shown in what sense all these approaches are equivalent.

• 27 May
V.Tarasov (PDMI)
"Bethe subalgebras of the group algebra of the symmetric group "
I will introduce a family of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra of the symmetric group. Bethe subalgebras are deformations of the Gelfand-Zetlin subalgebra. I will describe properties of Bethe subalgebras.

• 1 April
E. K. Sklyanin (York, UK)
"Bispectrality and separation of variables in multiparticle hypergeometric system "
Hypergeometric functions depending on two sets of parameters are known to possess the property of bispectrality: they satisfy two different systems of differential/difference equations in one set of parameters the other plays the role of spectral parameter and vice versa. Examples we choosen include the open Toda lattice and Calogero - Sutherland system.

• 25 March
P. N. Mnev
"Towards (perturbative) topological quantum field theory with boundary"
We will discuss the project of quantizing AKSZ (Alexandrov-Kontsevich-Schwarz-Zaboronsky) sigma models on manifolds with boundary, in a way that is consistent with gluing. The motivation here is drawn from the example of one-dimensional Chern-Simons theory on an interval.

• 4 ìàðòà
Ë.Ä.Ôàääååâ
"3 + 1 separation of variables in new approach to the Einstein theory of gravity"