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Regular version of the site
Of all publications in the section: 32
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Article
Takebe T., Teo L. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2006. Vol. 2. No. 072. P. 1-30.

We define the coupled modified KP hierarchy and its dispersionless limit. This integrable hierarchy is a generalization of the ''half'' of the Toda lattice hierarchy as well as an extension of the mKP hierarchy. The solutions are parametrized by a fibered flag manifold. The dispersionless counterpart interpolates several versions of dispersionless mKP hierarchy.

Added: Aug 13, 2014
Article
Takebe T. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2014. Vol. 10. P. 1-13.

We show that N-variable reduction of the dispersionless BKP hierarchy is described by a Loewner type equation for the quadrant.

Added: Mar 15, 2014
Article
Gusein-Zade S. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2020. Vol. 16. No. 051. P. 1-15.

P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct mirror symmetric Calabi–Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. In a previous paper, we explained that this construction should work only under a special condition on the permutation group called parity condition (PC). Here we prove that, if the permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign.

Added: Oct 27, 2020
Article
Тарасов В. О., Uvarov F. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2020. Vol. 16. P. 035.

We consider the Knizhnik-Zamolodchikov and dynamical operators, both differential and difference, in the context of the (gl(k), gl(n))-duality for the space of polynomials in kn anticommuting variables. We show that the Knizhnik-Zamolodchikov and dynamical operators naturally exchange under the duality.

Added: Oct 27, 2020
Article
Burman Y. M., Berenstein A. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2009. Vol. 5. No. 57. P. 1-18.

Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.

Added: Oct 12, 2012
Article
Khoroshkin S. M., Пакуляк С. З. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2008. Vol. 4. No. 081. P. 23.
Added: Oct 15, 2012
Article
Levin A., Olshanetsky M., Smirnov A. V. et al. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2012. Vol. 8. No. 095.
We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint G-bundles of different topological types over complex curves Sigma(g,n) of genus g with n marked points. The bundles are defined by their characteristic classes - elements of H-2 (Sigma(g,n), Z(G)), where Z (G) is a center of the simple complex Lie group G. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness.
Added: Feb 7, 2013
Article
Nirov Khazret S., Razumov A. V. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2017. Vol. 13. No. 043. P. 1-31.

We discuss highest ℓ-weight representations of quantum loop algebras and the corresponding functional relations between integrability objects. In particular, we compare the prefundamental and q-oscillator representations of the positive Borel subalgebras of the quantum group Uq(L(sll+1)) for arbitrary values of l. Our article has partially the nature of a short review, but it also contains new results. These are the expressions for the L-operators, and the exact relationship between different representations, as a byproduct resulting in certain conclusions about functional relations.

Added: Jan 29, 2018
Article
Loktev S., Natanzon S. M. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2011. Vol. 7. No. 70. P. 1-15.

We show that any complex (respectively real) representation of finite group naturally generates a open-closed (respectively Klein) topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in this theory with the Frobenius-Schur indicator on the representation. We relate any complex simple Klein TFT to a real division ring.

Added: Oct 12, 2012
Article
Rybnikov L. G., Chervov A., Falqui G. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2009. Vol. 5. No. 29. P. 17.
We consider the XXX homogeneous Gaudin system with N sites, both in classical and the quantum case. In particular we show that a suitable limiting procedure for letting the poles of its Lax matrix collide can be used to define new families of Liouville integrals(in the classical case) and new “Gaudin” algebras (in the quantum case). We will especially treat the case of total collisions, that gives rise to (a generalization of) the so called Bending flows of Kapovich and Millson. Some aspects of multi-Poisson geometry will be addressed(in the classical case). We will make use of properties of “Manin matrices” to provide explicit generators of the Gaudin Algebras in the quantum case.
Added: Oct 12, 2012
Article
Spiridonov V., Magadov K. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2018. Vol. 14. No. 121. P. 1-13.

We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.  

Added: Nov 14, 2018
Article
Akhtar M., Coates T., Galkin S. et al. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2012. Vol. 8. No. 094. P. 1-707.

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P, or in terms of piecewise-linear transformations acting on the dual polytope P* (much like cluster transformations). Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f. Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period.

Added: Sep 14, 2013
Article
Zotov A., Levin A., Olshanetsky M. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2009. Vol. 5. No. 065. P. 1-22.
Modifications of bundles over complex curves is an operation that allows one to construct a new bundle from a given one. Modifications can change a topological type of bundle. We describe the topological type in terms of the characteristic classes of the bundle.Being applied to the Higgs bundles modifications establish an equivalence between different classical integrable systems. Following Kapustin and Witten we define the modifications in terms of monopole solutions of the Bogomolny equation. We find the Dirac monopole solution in the case R × (elliptic curve). This solution is a three-dimensional generalization of the Kronecker series. We give two representations for this solution and derive a functional equation for it generalizing the Kronecker results. We use it to define Abelian modifications for bundles of arbitrary rank. We also describe non-Abelian modifications in terms of thetafunctions with characteristic.
Added: Oct 15, 2012
Article
Hutsalyuk A., Liashyk A., Pakuliak S. et al. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2016. Vol. 12. No. 099. P. 1-22.

We study gl(2|1) symmetric integrable models solvable by the nested algebraic Bethe ansatz. Using explicit formulas for the Bethe vectors we derive the actions of the monodromy matrix entries onto these vectors. We show that the result of these actions is a finite linear combination of Bethe vectors. The obtained formulas open a way for studying scalar products of Bethe vectors.  

Added: Nov 17, 2016
Article
Victor A. Vassiliev. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2020. Vol. 16. P. 1-21.

Local diffusion of strictly hyperbolic higher-order PDE's with constant coefficients at all simple singularities of corresponding wavefronts can be explained and recognized by only two local geometrical features of these wavefronts. We radically disprove the obvious conjecture extending this fact to arbitrary singularities: namely, we present examples of diffusion at all non-simple singularity classes of generic wavefronts in odd-dimensional spaces, which are not reducible to diffusion at simple singular points.

Added: Mar 21, 2020
Article
Kirillov A. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2016. Vol. 12.

We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.

Added: May 5, 2016
Article
Ayano T., Nakayashiki A. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2013. Vol. 9. No. 046.
Added: Nov 2, 2016
Article
Деркачев С. Э., Манашов А. Н. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2020. Vol. 16. No. 003. P. 1-20.

It was observed recently that relations between matrix elements of certain operators in the SL(2,R) spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with SL(2,C) symmetry group and L2(C) as a local Hilbert space give rise to a new type of Γ-function integrals. In this work we present a direct calculation of two such integrals. We also analyse properties of these integrals and show that they comprise the star-triangle relations recently discussed in the literature. It is also shown that in the quasi-classical limit these integral identities are reduced to the duality relations for Dotsenko–Fateev integrals.

Added: Nov 12, 2020
Article
Ayano T. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2016. Vol. 12. No. 086.
Added: Nov 2, 2016
Article
Gavrylenko P., Iorgov N., Lisovyy O. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2018. Vol. 14. P. 1-27.

We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlevé VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for $c=N-1$.

Added: Nov 22, 2018
Article
Buryak A., Rossi P. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2018. Vol. 14. No. 120. P. 1-7.

In this note we present a simple Lax description of the hierarchy of the intermediate long wave equation (ILW hierarchy). Although the linear inverse scattering problem for the ILW equation itself was well known, here we give an explicit expression for all higher flows and their Hamiltonian structure in terms of a single Lax difference-differential operator.

Added: Sep 27, 2020
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