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Of all publications in the section: 39
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Article
Takebe T. Letters in Mathematical Physics. 2002. Vol. 59. P. 157-172.
Article
Buryak A., Shadrin S. Letters in Mathematical Physics. 2010. Vol. 93. No. 3. P. 243-252.

In this note we use the formalism of multi-KP hierarchies in order to give some general formulas for infinitesimal deformations of solutions of the Darboux-Egoroff system. As an application, we explain how Shramchenko's deformations of Frobenius manifold structures on Hurwitz spaces fit into the general formalism of Givental-van de Leur twisted loop group action on the space of semi-simple Frobenius manifolds.

Article
Natanzon S. M., Orlov Aleksandr Yur'evich. Letters in Mathematical Physics. 2017. Vol. 107. P. 1065-1109.

We consider $d$-fold branch covering of the real projective plane $RP^2$ and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz numbers. In paticular we get the $RP^2$ analogue of the both $CP^1$ generating functions proposed by A.Okounkov and by Goulden-Jackson. Other examples are Hurwitz numbers weighted by Hall-Littlewood and by Macdonald polynomials. We also consider integrals of tau functions which generate projective Hurwitz numbers.

Article
Bershtein M., Feigin B. L., Litvinov A. Letters in Mathematical Physics. 2016. Vol. 106. No. 1. P. 29-56.

We study the conformal vertex algebras which naturally arise in relation to the Nakajima–Yoshioka blow-up equations.

Article
Aleksei Ilin, Rybnikov L. G. Letters in Mathematical Physics. 2018. Vol. 108. No. 4. P. 1083-1107.

We study degenerations of Bethe subalgebras B(C) in the Yangian Y(gln), where C is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parameterizes all possible degenerations, is the Deligne–Mumford moduli space of stable rational curves M0,n+2¯. All subalgebras corresponding to the points of M0,n+2¯are free and maximal commutative. We describe explicitly the “simplest” degenerations and show that every degeneration is the composition of the simplest ones. The Deligne–Mumford spaceM0,n+2¯ generalizes to other root systems as some De Concini–Procesi resolution of some toric variety. We state a conjecture generalizing our results to Bethe subalgebras in the Yangian of arbitrary simple Lie algebra in terms of this De Concini–Procesi resolution.

Article
Willwacher T., Zivkovic M., Khoroshkin A. Letters in Mathematical Physics. 2017. Vol. 107. No. 10. P. 1781-1797.

We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many nonzero hairy graph cohomology classes out of (known) non-hairy classes by studying the cancellations in those sequences. This provide a first glimpse at the tentative global structure of the hairy graph cohomology.

Article
Buryak A., Rossi P. Letters in Mathematical Physics. 2016. Vol. 106. No. 3. P. 289-317.

In this paper we define a quantization of the Double Ramification Hierarchies using intersection numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. We provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with the first descendant of the unit of the cohomological field theory only. We study various examples which provide, in very explicit form, new (1+1)-dimensional integrable quantum field theories whose classical limits are well-known integrable hierarchies such as KdV, Intermediate Long Wave, Extended Toda, etc. Finally we prove polynomiality in the ramification multiplicities of the integral of any tautological class over the double ramification cycle.

Article
Gurevich D., Saponov P. A., Talalaev D. Letters in Mathematical Physics. 2018. Vol. 108. P. 2303-2314.

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skewinvertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).

Article
Gurevich D., Talalaev D. Letters in Mathematical Physics. 2018. Vol. 108. No. 10. P. 2303-2314.

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skew-invertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).

Article
Tarasov V., Uvarov F. Letters in Mathematical Physics. 2020. No. 110. P. 3375-3400.

We consider actions of the current Lie algebras $\mathfrak{gl}_{n}[t]$ and $\mathfrak{gl}_{k}[t]$ on the space of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1},\dots ,z_{k})$ and $\bar{\alpha}=(\alpha_{1},\dots ,\alpha_{n})$, respectively. We show that the images of the Bethe algebras $\mathcal{B}^{\langle n\rangle}_{\bar{\alpha}}\subset U(\mathfrak{gl}_{n}[t])$ and $\mathcal{B}^{\langle k\rangle}_{\bar{z}}\subset U(\mathfrak{gl}_{k}[t])$ under these actions coincide. To prove the statement, we use the Bethe ansatz description of eigenvalues of the actions of the Bethe algebras via spaces of quasi-exponentials and establish an explicit correspondence between these spaces for the actions of $\mathcal{B}^{\langle n\rangle}_{\bar{\alpha}}$ and $\mathcal{B}^{\langle k\rangle}_{\bar{k}}$.

Article
Buryak A. Letters in Mathematical Physics. 2015. Vol. 105. No. 10. P. 1427-1448.

In a recent paper R. Pandharipande, J. Solomon and R. Tessler initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. The authors conjectured KdV and Virasoro type equations that completely determine all intersection numbers. In this paper we study these equations in detail. In particular, we prove that the KdV and the Virasoro type equations for the intersection numbers on the moduli space of Riemann surfaces with boundary are equivalent.

Article
Feigin B. L., Feigin E., Jimbo M. et al. Letters in Mathematical Physics. 2009. Vol. 88. No. 1-3. P. 39-77.

We use the Whittaker vectors and the Drinfeld Casimir element to show that eigenfunctions of the difference Toda Hamiltonian can be expressed via fermionic formulas. Motivated by the combinatorics of the fermionic formulas we use the representation theory of the quantum groups to prove a number of identities for the coefficients of the eigenfunctions.

Article
Malikov F., Feigin B. L. Letters in Mathematical Physics. 1994. Vol. 31. No. 4. P. 315-325.
Article
Dunin-Barkowski P., Shadrin S., Spitz L. Letters in Mathematical Physics. 2013. Vol. 103. No. 5. P. 533-557.

Inversion symmetry is a very non-trivial discrete symmetry of Frobenius manifolds. It was obtained by Dubrovin from one of the elementary Schlesinger transformations of a special ODE associated to a Frobenius manifold. In this paper, we review the Givental group action on Frobenius manifolds in terms of Feynman graphs and obtain an interpretation of the inversion symmetry in terms of the action of the Givental group. We also consider the implication of this interpretation of the inversion symmetry for the Schlesinger transformations and for the Hamiltonians of the associated principle hierarch

Article
Pyatov P. N., Gurevich D., Saponov P. A. Letters in Mathematical Physics. 1997. Vol. 41. P. 255-264.
We discuss how properties of Hecke symmetry (i.e., Hecke type R-matrix) influ- ence the algebraic structure of the corresponding Reflection Equation (RE) algebra. Analogues of the Newton relations and Cayley-Hamilton theorem for the matrix of generators of the RE algebra related to a finite rank even Hecke symmetry are derived.
Article
Gavrylenko P., Iorgov N., Lisovyy O. Letters in Mathematical Physics. 2020. Vol. 110. No. 2. P. 327-364.

We construct the general solution of a class of Fuchsian systems of rank N as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of WN-algebra with central charge c = N − 1. The simplest example is given by the tau function of the FujiSuzuki-Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respect to intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the minimal set of matrix elements of vertex operators of the WN-algebra for generic central charge and prove several properties of semi-degenerate vertex operators and conformal blocks for c = N − 1.

Article
Rybnikov L. G., Chervov A., Falqui G. Letters in Mathematical Physics. 2010. Vol. 91. No. 1. P. 129-150.
Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U(g) of a semisimple Lie algebra g. This family is parameterized by collections of pairwise distinct complex numbers z1; : : : ; zn . We obtain some new commutative subalgebras in U(g)­n as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the Hamiltonians of bending °ows and to the Gelfand{Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.
Article
Varchenko A., Schechtman V., Feigin B. L. Letters in Mathematical Physics. 1990. Vol. 20. No. 4. P. 291-297.