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## Free fermions and tau-functions

We review the formalism of free fermions used for construction of tau-functions of classical integrable hierarchies and give a detailed derivation of group-like properties of the normally ordered exponents, transformations between different normal orderings, the bilinear relations, the generalized Wick theorem and the bosonization rules. We also consider various examples of tau-functions and give their fermionic realization.

Correlators in topological theories are given by the values of a linear form on the products of operators from a commutative associative algebra (CAA). As a corollary, partition functions of topological theory always satisfy the generalized WDVV equations of. We consider the Hurwitz partition functions, associated in this way with the CAA of cut-and-join operators. The ordinary Hurwitz numbers for a given number of sheets in the covering provide trivial (sums of exponentials) solutions to the WDVV equations, with finite number of time-variables. The generalized Hurwitz numbers from provide a non-trivial solution with infinite number of times. The simplest solution of this type is associated with a subring, generated by the dilatation operators ˆW1 = trD = trX@/@X.

Following the approach of [Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages, arXiv:1112.3310], we construct the master T-operator for the quantum Gaudin model with twisted boundary conditions and show that it satisfies the bilinear identity and Hirota equations for the classical KP hierarchy. We also characterize the class of solutions to the KP hierarchy that correspond to eigenvalues of the master T-operator and study dynamics of their zeros as functions of the spectral parameter. This implies a remarkable connection between the quantum Gaudin model and the classical Calogero-Moser system of particles.

The construction of the master T-operator recently suggested is applied to integrable vertex models and associated quantum spin chains with trigonometric R-matrices. The master T-operator is a generating function for commuting transfer matrices of integrable vertex models depending on infinitely many parameters. At the same time it turns out to be the tau-function of an integrable hierarchy of classical soliton equations in the sense that it satisfies the the same bilinear Hirota equations. The class of solutions of the Hirota equations that correspond to eigenvalues of the master T-operator is characterized and its relation to the classical Ruijsenaars-Schneider system of particles is discussed.

We construct partition functions that are tau-functions of integrable hierarchies.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.