arXiv subjects

Dmitry Galakhov

Publications and source records attributed to Dmitry Galakhov.

At least 19 recordsLinked to original sources

arXiv2026

Two roles of Alexander in two Kashaev phases

The crucial feature of resurgence theory is the ambiguity of non-perturbative behavior, reflected either in the different choices of integration contours or in the existence of several solutions to Ward identities. This is well illustrated by considering exactly solvable models, of which the prominent example is Chern-Simons theory. Its important chapter, which should have a direct generalization to arbitrary Yang-Mills, is the consideration of Wilson averages in the double-scaling limit of large representation and small coupling. For historical reasons, we call it a Kashaev limit. It possesses a natural interpretation in terms of quasiclassical/WKB approximation, which is, however, somewhat peculiar and thus sheds new light on the old story. The crucial point is the appearance of Alexander polynomials $Δ$ in two seemingly opposite roles: the classical $A$-polynomials have common roots with $Δ$, while Jones polynomials tend to $Δ^{-1}$ in the perturbative expansion. The consistency is provided by the peculiar form of the quantum $A$-polynomial, and the resolution of the puzzle is the co-existence of two different branches (phases) in the quasiclassical limit -- with non-trivial and with vanishing classical actions. The first leads to classical $A$-polynomials and hyperbolic volumes, the second -- to inverse Alexanders.

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arXiv2026

Shading A-polynomials via huge representations of $U_q(\mathfrak{su}_N)$

Classical A-polynomials $A(\ell,m)$ define constraints on coordinates $\ell$ and $m$ in $SL(2,\mathbb{C})$ (a complexification of $SU(2)$) character varieties associated to knot complements $S^3\setminus K$. Quantum A-polynomials $\hat A(\hat \ell,\hat m)$ are difference operators annihilating Jones polynomials believed to represent wave functions of 3d Chern-Simons theory with gauge group $SU(2)$ on a toroidal pipe surrounding the knot $K$ strand -- a boundary of the knot complements $S^3\setminus K$. We suggest a construction of classical shaded A-polynomials $A_a(\ell_b,m_c)$ associated to Lie groups $SU(N)$. We exploit a formalism of Clebsh-Gordan (CG) chords, where indices $a$, $b$, $c$ run over $1,\ldots,N-1$. CG chords have a natural interpretation in terms of 2d CFTs of WZW type, or, alternatively, in terms of quantum group $U_q(\mathfrak{su}_N)$. In the case of $\mathfrak{su}_2$ CG chords could be associated to Reeb chords in a knot contact homology (KCH) framework. KCH suggests its own analogue of A-polynomials known as augmentation polynomials allowed to have extra spurious roots in principle. Yet the CG chord formalism could be easily extended to arbitrary $\mathfrak{su}_N$ allowing us to generalize the construction of A(ugmentation)-polynomials to arbitrary $\mathfrak{su}_N$ and arbitrary representation as well. Primarily we aim at classical A-polynomials by considering a double scaling limit when $q=e^{\hbar}$, $\hbar\to 0$ and the representations are huge, in particular, highest weight vector components $w_i\to \infty$ so that $\hbar w_i\sim m_i$ remain finite. Still we expect the presented techniques would be helpful in deriving quantum A-polynomials for arbitrary Lie (super)algebras $\mathfrak{g}$. Also we discuss explicit examples of A-polynomials for knots $3_1$, $4_1$ and $5_1$ for $\mathfrak{g}=\mathfrak{su}_3$.

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arXiv2026

Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs

Topological quantum field theory (TQFT) is a powerful tool to describe homologies, which normally involve complexes and a variety of maps/morphisms, what makes a functional integration approach with a sum over a single kind of maps seemingly problematic. In TQFT this problem is overcame by exploiting the rich set of zero modes of BRST operators, which appear sufficient to describe complexes. We explain what this approach looks like for the important class of Khovanov-Rozansky (KR) cohomologies, which categorify the observables (Wilson lines or knot polynomials) in 3d Chern-Simons theory. We develop a construction of odd differential operators, associated with all link diagrams, including tangles with open ends. These operators become nilpotent only for diagram with no external legs, but even for open tangles one can develop a factorization formalism, which preserve Reidemeister/topological invariance -- the symmetry of the problem. This technique seems much more ``physical'' than conventional language of homological algebra and should have many applications to various problems beyond Chern-Simons theory. We also hope that this language will provide efficient algorithms, and finally allow to computerize the calculation of KR cohomologies -- for closed diagrams and for open tangles.

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arXiv2026

Weyl Mutations in Quiver Yangians

The problem of solving non-linear equations would be considerably simplified by a possibility to convert known solutions into the new ones. This could seem an element of art, but in the context of ADHM-like equations describing quiver varieties there is a systematic approach. In this note we study moduli spaces and dualities of quiver gauge theories associated to effective dynamics of D-branes compactified on Calabi-Yau resolutions. We concentrate on a subfamily of quivers $\mathfrak{Q}_{\mathfrak{g}}$ covering Dynkin diagrams for simple Lie algebras $\mathfrak{g}$, where the respective BPS algebra is expected to be the Yangian algebra $Y(\mathfrak{g})$. For Yangians labeled by quivers their representations are described by solutions of ADHM-like equations. As quivers substitute Dynkin diagrams a generalization of the Weyl group $\mathcal{W}_{\mathfrak{g}}$ acts on the ADHM solutions. Here we work with the case $\mathfrak{g}=\mathfrak{sl}_{n+1}$ and treat this group as a group of electro-magnetic Seiberg-like dualities (we call them Weyl mutations) on the respective quiver gauge theories. We lift it to the case of higher representations associated to rectangular Young diagrams. An action of Weyl mutations on the BPS Yangian algebra is also discussed.

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arXiv2025

On geometric bases for A-polynomials II: $\mathfrak{su}_3$ and Kuberberg bracket

We continue the study of quantum A-polynomials -- equations for knot polynomials with respect to their coloring (representation-dependence) -- as the relations between different links, obtained by hanging additional ``simple'' components on the original knot. Depending on the choice of this ``decoration'', the knot polynomial is either multiplied by a number or decomposes into a sum over ``surrounding'' representations by a cabling procedure. What happens is that these two of decorations, when complicated enough, become dependent -- and this provides an equation. Remarkably it can be made independent of the representation. However, the equivalence of links is not a topological property -- it follows from the properties of $R$-matrices, and strongly depends on the choice the gauge group and particular links. The relatively well studied part of the story concerns $\mathfrak{su}_2$, where $R$-matrices can be chosen in an especially convenient Kauffman form, what makes the derivation of equations rather geometrical. To make these geometric methods somewhat simpler we suggest to use an arcade formalism/representation of the braid group to simplify decorating links universally. Here we attempt to extend this technique to the next case, $\mathfrak{su}_3$, where the Kauffman rule is substituted by a more involved Kuberberg rule, still remains more geometric than generic analysis of MOY-diagrams, needed for higher ranks. Already in this case we encounter a classification problem for possible ``decorations'' and emergence of two-lined Young diagrams in enumeration of representations.

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arXiv2025

Tunnels Under Geometries (or Instantons Know Their Algebras)

In the tight binding model with multiple degenerate vacua we might treat wave function overlaps as instanton tunnelings between different wells (vacua). An amplitude for such a tunneling process might be constructed as $\mathsf{T}_{i\to j}\sim e^{-S_{\mathrm{ inst}}}{\mathbf{v}}_j^{+}{\mathbf{v}}_i^{-}$, where there is canonical instanton action suppression, and $\mathbf{v}_i^{-}$ annihilates a particle in the $i^{\mathrm{th}}$ vacuum, whereas $\mathbf{v}_j^{+}$ creates a particle in the $j^{\mathrm{th}}$ vacuum. Adiabatic change of the wells leads to a Berry-phase evolution of the couplings, which is described by the zero-curvature Gauss-Manin connection i.e. by a quantum $R$-matrix. Zero-curvature is actually a consequence of level repulsion or topological protection, and its implication is the Yang-Baxter relation for the $R$-matrices. In the simplest case the story is pure Abelian and not very exciting. But when the model becomes more involved, incorporates supersymmetry, gauge and other symmetries, such amplitudes obtain more intricate structures. Operators $\mathbf{v}_i^{-}$, $\mathbf{v}_j^{+}$ might also evolve from ordinary Heisenberg operators into a more sophisticated algebraic object -- a ``tunneling algebra''. The result for the tunneling algebra would depend strongly on geometry of the QFT we started with, and, unfortunately, at the moment we are unable to solve the reverse engineering problem. In this note we revise few successful cases of the aforementioned correspondence: quantum algebras $U_q(\mathfrak{g})$ and affine Yangians $Y(\hat{\mathfrak{g}})$. For affine Yangians we demonstrate explicitly how instantons ``perform'' equivariant integrals over associated quiver moduli spaces appearing in the alternative geometric construction.

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arXiv2025

Super-Hamiltonians for super-Macdonald polynomials

The Macdonald finite-difference Hamiltonian is lifted to a super-generalization. In addition to canonical bosonic time variables $p_k$ new Grassmann time variables $θ_k$ are introduced, and the Hamiltonian is represented as a differential operator acting on a space of functions of both types of variables $p_k$ and $θ_k$. Eigenfunctions for this Hamiltonian are a suitable generalization of Macdonald polynomials to super-Macdonald polynomials discussed earlier in the literature. Peculiarities of the construction in comparison to the canonical bosonic case are discussed.

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arXiv2024

On geometric bases for quantum A-polynomials of knots

A simple geometric way is suggested to derive the Ward identities in the Chern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots. In quasi-classical limit it is closely related to the well publicized augmentation theory and contact geometry. Quantization allows to present it in much simpler terms, what could make these techniques available to a broader audience. To avoid overloading of the presentation, only the case of the colored Jones polynomial for the trefoil knot is considered, though various generalizations are straightforward. Restriction to solely Jones polynomials (rather than full HOMFLY-PT) is related to a serious simplification, provided by the use of Kauffman calculus. Going beyond looks realistic, however it remains a problem, both challenging and promising.

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arXiv2024

Supersymmetric polynomials and algebro-combinatorial duality

In this note we develop a systematic combinatorial definition for constructed earlier supersymmetric polynomial families. These polynomial families generalize canonical Schur, Jack and Macdonald families so that the new polynomials depend on odd Grassmann variables as well. Members of these families are labeled by respective modifications of Young diagrams. We show that the super-Macdonald polynomials form a representation of a super-algebra analog $\mathsf{T}(\widehat{\mathfrak{gl}}_{1|1})$ of Ding-Ioahara-Miki (quantum toroidal) algebra, emerging as a BPS algebra of D-branes on a conifold. A supersymmetric modification for Young tableaux and Kostka numbers are also discussed.

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arXiv2024

Macdonald polynomials for super-partitions

We introduce generalization of famous Macdonald polynomials for the case of super-Young diagrams that contain half-boxes on the equal footing with full boxes. These super-Macdonald polynomials are polynomials of extended set of variables: usual $p_k$ variables are accompanied by anti-commuting Grassmann variables $θ_k$. Starting from recently defined super-Schur polynomials and exploiting orthogonality relations with triangular decompositions we are able to fully determine super-Macdonald polynomials. These new polynomials have similar properties to canonical Macdonald polynomials -- they respect two different orderings in the set of (super)-Young diagrams simultaneously.

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arXiv2024

Algorithms for representations of quiver Yangian algebras

In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm's implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of $\mathsf{Y}(\mathfrak{sl}_2)$, $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1})$ and $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$.

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arXiv2024

Wall-Crossing Effects on Quiver BPS Algebras

BPS states in supersymmetric theories can admit additional algebro-geometric structures in their spectra, described as quiver Yangian algebras. Equivariant fixed points on the quiver variety are interpreted as vectors populating a representation module, and matrix elements for the generators are then defined as Duistermaat-Heckman integrals in the vicinity of these points. The well-known wall-crossing phenomena are that the fixed point spectrum establishes a dependence on the stability (Fayet-Illiopolous) parameters $ζ$, jumping abruptly across the walls of marginal stability, which divide the $ζ$-space into a collection of stability chambers -- ``phases'' of the theory. The standard construction of the quiver Yangian algebra relies heavily on the molten crystal model, valid in a sole cyclic chamber where all the $ζ$-parameters have the same sign. We propose to lift this restriction and investigate the effects of the wall-crossing phenomena on the quiver Yangian algebra and its representations -- starting with the example of affine super-Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. In addition to the molten crystal construction more general atomic structures appear, in other non-cyclic phases (chambers of the $ζ$-space). We call them glasses and also divide in a few different classes. For some of the new phases we manage to associate an algebraic structure again as a representation of the same affine Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This observation supports an earlier conjecture that the BPS algebraic structures can be considered as new wall-crossing invariants.

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arXiv2024

Charging solid partitions

Solid partitions are the 4D generalization of the plane partitions in 3D and Young diagrams in 2D, and they can be visualized as stacking of 4D unit-size boxes in the positive corner of a 4D room. Physically, solid partitions arise naturally as 4D molten crystals that count equivariant D-brane BPS states on the simplest toric Calabi-Yau fourfold, $\mathbb{C}^4$, generalizing the 3D statement that plane partitions count equivariant D-brane BPS states on $\mathbb{C}^3$. In the construction of BPS algebras for toric Calabi-Yau threefolds, the so-called charge function on the 3D molten crystal is an important ingredient -- it is the generating function for the eigenvalues of an infinite tower of Cartan elements of the algebra. In this paper, we derive the charge function for solid partitions. Compared to the 3D case, the new feature is the appearance of contributions from certain 4-box and 5-box clusters, which will make the construction of the corresponding BPS algebra much more complicated than in the 3D.

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arXiv2024

Simple Representations of BPS Algebras: the case of $Y(\widehat{\mathfrak{gl}}_2)$

BPS algebras are the symmetries of a wide class of brane-inspired models. They are closely related to Yangians -- the peculiar and somewhat sophisticated limit of DIM algebras. Still they possess some simple and explicit representations. We explain here that for $Y(\widehat{\mathfrak{gl}}_r)$ these representations are related to Uglov polynomials, whose families are also labeled by natural $r$. They arise in the limit $\hbar\longrightarrow 0$ from Macdonald polynomials, and generalize the well-known Jack polynomials ($β$-deformation of Schur functions), associated with $r=1$. For $r=2$ they approximate Macdonald polynomials with the accuracy $O(\hbar^2)$, so that they are eigenfunctions of {\it two} immediately available commuting operators, arising from the $\hbar$-expansion of the first Macdonald Hamiltonian. These operators have a clear structure, which is easily generalizable, -- what provides a technically simple way to build an explicit representation of Yangian $Y(\widehat{\mathfrak{gl}}_2)$, where $U^{(2)}$ are associated with the states $|λ\rangle$, parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables $p_{2n+1}$ can be expressed through mutually commuting operators from Yangian, however even time-variables $p_{2n}$ are inexpressible. Implications to higher $r$ become now straightforward, yet we describe them only in a sketchy way.

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arXiv2023

Towards the theory of Yangians

We review the main ideas underlying the emerging theory of Yangians -- the new type of hidden symmetry in string-inspired models. Their classification by quivers is a far-going generalization of simple Lie algebras classification by Dynkin diagrams. However, this is still a kind of project, while a more constructive approach goes through toric Calabi-Yau spaces, related supersymmetric systems and the Duistermaat-Heckmann/equivariant integrals between the fixed points in the ADHM-like moduli spaces. These fixed points are classified by crystals (Young-type diagrams) and Yangian generators describe ``instanton'' transitions between them. Detailed examples will be presented elsewhere.

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arXiv2023

Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$

We explicitly construct cut-and-join operators and their eigenfunctions -- the Super-Schur functions -- for the case of the affine super-Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This is the simplest non-trivial (semi-Fock) representation, where eigenfunctions are labeled by the superanalogue of 2d Young diagrams, and depend on the supertime variables $(p_k,θ_k)$. The action of other generators on diagrams is described by the analogue of the Pieri rule. As well we present generalizations of the hook formula for the measure on super-Young diagrams and of the Cauchy formula. Also a discussion of string theory origins for these relations is provided.

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arXiv2023

BPS States Meet Generalized Cohomology

In this note we review a construction of a BPS Hilbert space in an effective supersymmetric quiver theory with 4 supercharges. We argue abstractly that this space contains elements of an equivariant generalized cohomology theory $E_G^{*}(-)$ of the quiver representation moduli space giving concretely Dolbeault cohomology, K-theory or elliptic cohomology depending on the spacial slice is compactified to a point, a circle or a torus respectively, and something more amorphous in other cases. Furthermore BPS instantons -- basic contributors to interface defects or a Berry connection -- induce a BPS algebra on the BPS Hilbert spaces representing Fourier-Mukai transforms on the quiver representation moduli spaces descending to an algebra over $E_G^{*}(-)$ as its representation. In the cases when the quiver describes a toric Calabi-Yau three-fold (CY${}_3$) the algebra is a respective generalization of the quiver BPS Yangian algebra discussed in the literature, in more general cases it is given by an abstract generalized cohomological Hall algebra.

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arXiv2022

On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry

We concentrate on a treatment of a Higgs-Coulomb duality as an absence of manifest phase transition between ordered and disordered phases of 2d $\mathcal{N}=(2,2)$ theories. We consider these examples of QFTs in the Schrödinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated categories of D-brane boundary conditions. As a result of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are equivalent, this resembles an analog of homological mirror symmetry. Following construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one is able to introduce interface defects in these theories and associate them to D-brane parallel transport functors. We concentrate on surveying simple examples, analytic when possible calculations, numerical estimates and simple physical picture behind curtains of geometric objects. Categorification of hypergeometric series analytic continuation is derived as an Atiyah flop of the conifold. Finally we arrive to an interpretation of the braid group action on the derived category of coherent sheaves on cotangent bundles to flag varieties as a categorification of Berry connection on the Fayet-Illiopolous parameter space of a sigma-model with a quiver variety target space.

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