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Vassily Gorbounov

Publications and source records attributed to Vassily Gorbounov.

At least 19 recordsLinked to original sources
arXiv2025

Generalized electrical Lie algebras

We generalize the electrical Lie algebras originally introduced by Lam and Pylyavskyy in several ways. To each Kac-Moody Lie algebra $\mathfrak{g}$ we associate two types (vertex type and edge type) of the generalized electrical algebras. The electrical Lie algebras of vertex type are always subalgebras of $\mathfrak{g}$ and are flat deformations of the nilpotent Lie subalgebra of $\mathfrak{g}$. In many cases including $sl_n$, $so_n$, and $sp_{2n}$ we find new (edge) models for our generalized electrical Lie algebras of vertex type. Finding an edge model in general is an interesting an open problem.

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arXiv2025

Quantum K--theory of Grassmannians from a Yang-Baxter algebra

In an earlier paper, two of the authors defined a $5$-vertex Yang-Baxter algebra (a Hopf algebra) which acts on the sum of the equivariant quantum K-rings of Grassmannians $\mathrm{Gr}(k;n)$, where $k$ varies from $0$ to $n$. We construct geometrically defined operators on quantum K-rings describing this action. In particular, the $R$-matrix defining the Yang-Baxter algebra corresponds to the left Weyl group action. Most importantly, we use the `quantum=classical' statement for the quantum K-theory of Grassmannians to prove an explicit geometric interpretation of the action of generators of the Yang-Baxter algebra. The diagonal entries of the monodromy matrix are given by quantum K-multiplications by explicitly defined classes, and the off-diagonal entries by certain push-pull convolutions. We use this to find a quantization of the classes of fixed points in the quantum K-rings, corresponding to the Bethe vectors of the Yang-Baxter algebra. On each of the quantum K-rings, we prove that the two Frobenius structures (one from geometry, and the other from the integrable system construction) coincide. We discuss several applications, including an action of the extended affine Weyl group on the quantum K-theory ring (extending the Seidel action), a quantum version of the localization map (which is a ring homomorphism with respect to the quantum K-product), and a graphical calculus to multiply by Hirzebruch $λ_y$ classes of the dual of the tautological quotient bundle. In an Appendix we illustrate our results in the case when $n=2$.

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arXiv2024

Symplectic geometry of electrical networks

In this paper we relate a well-known in symplectic geometry compactification of the space of symmetric bilinear forms considered as a chart of the Lagrangian Grassmannian to the specific compactifications of the space of electrical networks in the disc obtained in \cite{L}, \cite{CGS} and \cite{BGKT}. In particular, we state an explicit connection between these works and describe some of the combinatorics developed there in the language of symplectic geometry. We also show that the combinatorics of the concordance vectors forces the uniqueness of the symplectic form, such that corresponding points of the Grassmannian are isotropic. We define a notion of Lagrangian concordance which provides a construction of the compactification of the space of electrical networks in the positive part of the Lagrangian Grassmannian bypassing the construction from \cite{L}.

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arXiv2024

Electrical Networks, Lagrangian Grassmannians and Symplectic Groups

We refine the result of T. Lam \cite{L} on embedding the space $E_n$ of electrical networks on a planar graph with $n$ boundary points into the totally non-negative Grassmannian $\mathrm{Gr}_{\geq 0}(n-1,2n)$ by proving first that the image lands in $\mathrm{Gr}(n-1,V)\subset \mathrm{Gr}(n-1,2n)$ where $V\subset \mathbb{R}^{2n}$ is a certain subspace of dimension $2n-2$. The role of this reduction in the dimension of the ambient space is crucial for us. We show next that the image lands in fact inside the Lagrangian Grassmannian $\mathrm{LG}(n-1,V)\subset \mathrm{Gr}(n-1,V)$. As it is well known $\mathrm{LG}(n-1)$ can be identified with $\mathrm{Gr}(n-1,2n-2)\cap \mathbb{P} L$ where $L\subset \bigwedge^{n-1}\mathbb R^{2n-2}$ is a subspace of dimension equal to the Catalan number $C_n$, moreover it is the space of the fundamental representation of the symplectic group $Sp(2n-2)$ which corresponds to the last vertex of the Dynkin diagram. We show further that the linear relations cutting the image of $E_n$ out of $\mathrm{Gr}(n-1,2n)$ found in \cite{L} define that space $L$. This connects the combinatorial description of $E_n$ discovered in \cite{L} and representation theory of the symplectic group.

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arXiv2021

With Wronskian through the Looking Glass

In the work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the Plücker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.

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arXiv2020

Chiral differential operators via Batalin-Vilkovisky quantization

We show that the local observables of the curved beta gamma system encode the sheaf of chiral differential operators using the machinery of the book "Factorization algebras in quantum field theory", by Kevin Costello and the second author, which combines renormalization, the Batalin-Vilkovisky formalism, and factorization algebras. Our approach is in the spirit of deformation quantization via Gelfand-Kazhdan formal geometry. We begin by constructing a quantization of the beta gamma system with an n-dimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields on the target disk, and it is naturally identified with the first Pontryagin class in Gelfand-Fuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of formal vector fields by the closed 2-forms on the disk. By results in the book listed above, we then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal beta gamma vertex algebra. Next, we introduce a version of Gelfand-Kazhdan formal geometry suitable for factorization algebras, and we verify that for a complex manifold with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs on the complex manifold.

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arXiv2019

Electrical varieties as vertex integrable statistical models

We propose a new approach to studying electrical networks interpreting the Ohm law as the operator which solves certain Local Yang-Baxter equation. Using this operator and the medial graph of the electrical network we define a vertex integrable statistical model and its boundary partition function. This gives an equivalent description of electrical networks We show that in the important case of an electrical network on the standard graph introduced by Curtis, Ingerman and Morrow, the response matrix of an electrical network, its most important feature, and the boundary partition function of our statistical model can be recovered from each other. Defining the electrical varieties in the usual way we compare them to the theory of the Lusztig varieties developed by Berenstein, Fomin and Zelevinsky. In our picture the former turns out to be a deformation of the latter. We describe how our approach produces new interesting mathematical structures on the electrical varieties. Our results should be compared to the earlier work started in Lam and Pylyavskyy on the connection between the Lusztig varieties and the electrical varieties.

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arXiv2018

Yang-Baxter algebras as convolution algebras: The Grassmannian case

We present a simple but explicit example of a recent development which connects quantum integrable models with Schubert calculus: there is a purely geometric construction of solutions to the Yang-Baxter equation and their associated Yang-Baxter algebras which play a central role in quantum integrable systems and exactly solvable lattice models in statistical physics. We consider the degenerate five-vertex limit of the asymmetric six-vertex model and identify its associated Yang-Baxter algebra as convolution algebra arising from the equivariant Schubert calculus of Grassmannians. We show how our method can be used to construct (Schur algebra type) quotients of the current algebra $\mathfrak{gl}_2[t]$ acting on the tensor product of copies of its evaluation representation $\mathbb{C}^2[t]$. Finally we connect it with the COHA for the $A_1$-quiver.

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arXiv2017

Quantum Integrability and Generalised Quantum Schubert Calculus

We introduce and study a new mathematical structure in the generalised (quantum) cohomology theory for Grassmannians. Namely, we relate the Schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six-vertex model. Our approach offers a new perspective on already established and well-studied special cases, for example equivariant K-theory, and in addition allows us to formulate a conjecture on the so-far unknown case of quantum equivariant K-theory.

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arXiv2015

The Berkovits Complex and Semi-free Extensions of Koszul Algebras

In his extension of W. Siegel's ideas on string quantization, N. Berkovits made several observations which deserve further study and development. Indeed, interesting accounts of this work have already appeared in the mathematical literature and in a different guise due to Avramov. In this paper we bridge between these three approaches, by providing a complex that is useful in the calculation of some homologies.

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arXiv2014

Equivariant quantum cohomology and Yang-Baxter algebras

There are two intriguing statements regarding the quantum cohomology of partial flag varieties. The first one relates quantum cohomology to the affinisation of Lie algebras and the homology of the affine Grassmannian, the second one connects it with the geometry of quiver varieties. The connection with the affine Grassmannian was first discussed in unpublished work of Peterson and subsequently proved by Lam and Shimozono. The second development is based on recent works of Nekrasov, Shatashvili and of Maulik, Okounkov relating the quantum cohomology of Nakajima varieties with integrable systems and quantum groups. In this article we explore for the simplest case, the Grassmannian, the relation between the two approaches. We extend the definition of the integrable systems called vicious and osculating walkers to the equivariant setting and show that these models have simple expressions in a particular representation of the affine nil-Hecke ring. We compare this representation with the one introduced by Kostant and Kumar and later used by Peterson in his approach to Schubert calculus. We reveal an underlying quantum group structure in terms of Yang-Baxter algebras and relate them to Schur-Weyl duality. We also derive new combinatorial results for equivariant Gromov-Witten invariants such as an explicit determinant formula.

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arXiv2013

Some remarks on Landau-Ginzburg potentials for odd-dimensional quadrics

We study the possibility of constructing a Frobenius manifold for the standard Landau-Ginzburg model of odd-dimensional quadrics $Q_{2n+1}$ and matching it with the Frobenius manifold attached to the quantum cohomology of these quadrics. Namely, we show that the initial conditions of the quantum cohomology Frobenius manifold of the quadric can be obtained from the suitably modified standard Landau-Ginzburg model.

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arXiv2011

Schubert calculus and singularity theory

Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces it has to be redesigned when applied to other generalized cohomology theories such as the equivariant, the quantum cohomology, K-theory, and cobordism. All this cohomology theories are different deformations of the ordinary cohomology. In this note we show that there is in some sense the universal deformation of Schubert calculus which produces the above mentioned by specialization of the appropriate parameters. We build on the work of Lerche Vafa and Warner. The main conjecture there was that the classical cohomology of a hermitian symmetric homogeneous manifold is a Jacobi ring of an appropriate potential. We extend this conjecture and provide a simple proof. Namely we show that the cohomology of the hermitian symmetric space is a Jacobi ring of a certain potential and the equivariant and the quantum cohomology and K-theory is a Jacobi ring of a particular deformation of this potential. This suggests to study the most general deformations of the Frobenius algebra of cohomology of these manifolds by considering the versal deformation of the appropriate potential. The structure of the Jacobi ring of such potential is a subject of well developed singularity theory. This gives a potentially new way to look at the classical, the equivariant, the quantum and other flavors of Schubert calculus.

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arXiv2008

Homotopy Gerstenhaber Structures and Vertex Algebras

We provide a simple construction of a Gerstenhaber-infinity algebra structure on a class of vertex algebras V, which lifts the strict Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications: the construction of a sheaf of Gerstenhaber-infinity algebras on a Calabi-Yau manifold extending the multiplication and bracket of functions and vector fields, and of a Lie-infinity structure related to the bracket of Courant.

math.QA↗
arXiv2006

Mirror symmetry formulae for the elliptic genus of complete intersections

In this paper we calculate the elliptic genus of certain complete intersections in products of projective spaces. We show that it is equal to the elliptic genus of the Landau-Ginzburg models that are, according to Hori and Vafa, mirror partners of these complete intersections. This provides additional evidence of the validity of their construction.

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