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A. Morozov

Publications and source records attributed to A. Morozov.

At least 19 recordsLinked to original sources
arXiv2026

Determinantal representation of colored HOMFLY for double braids

Starting from the known differential expansions, we express the HOMFLY-PT polynomials of twist and antiparallel double-braid knots in rectangular representations \([r^s]\) as determinants of \(r\times r\) and \(s\times s\) matrices. We first give an elementary derivation for the figure-eight knot and then re-sum the KNTZ formulas for twist and double-braid evolution. The resulting matrices are constructed from single-hook evolution coefficients and explicit representation-dependent weights. These formulas provide a starting point for investigating possible extensions of knot polynomials to KP/Toda \(\tau\)-functions. Extensions to non-rectangular representations and to more general knot families remain open problems.

hep-th↗
arXiv2026

Non-symmetric triads and Baker-Akhiezer functions

The symmetric Macdonald polynomial at peculiar values of parameter $t=q^{-m}$, $m\in\mathbb{Z}_{\ge 0}$ is naturally split into non-symmetric parts, which are the (quasi)polynomial Baker-Akhiezer (BA) functions. One may think this is due to symmetricity, and one just picks up this way non-symmetric parts already containing all the information. However, we demonstrate that, in the case of {\bf non-symmetric} Macdonald polynomials, it still works, though each single BA function splits into $N!$ distinct (quasi)polynomial BA functions. The sum of these functions gives rise to the universal solution of the eigenstate problem for the Cherednik Hamiltonians. Extending to arbitrary values of $t$ is also immediate giving rise to counterparts of the Noumi-Shiraishi power series. Altogether, this power series and its reductions to non-symmetric Macdonald polynomials and to BA functions form a non-symmetric triad. There are $N!$ different branches of the non-symmetric triad, each branch being split into $N!$ distinct triads, and of these $(N!)^2$ triads $N!(N-1)!$ are independent. We describe in detail the simplest $N=2$ case.

hep-th↗
arXiv2026

Cherednik integrable system: eigenfunctions at generic eigenvalues

Symmetric Macdonald polynomials of $N$ variables provide eigenfunctions of the $N$-body trigonometric Ruijsenaars-Schneider integrable system at particular eigenvalues. In order to construct eigenfunctions with arbitrary eigenvalues, M. Noumi and J. Shiraishi used a recursion in $N$ (branching rule) for the symmetric Macdonald polynomials and analytically continued them. This generated a power series, which is a part of triad (universal solution). In the present paper, we demonstrate that a similar procedure is available for another integrable system, $N$-body Cherednik integrable system inspired by the DAHA of type $A$, which has non-symmetric Macdonald polynomials as its polynomial eigenfunctions. However, in this system, the generic eigenfunction is more complicated: it is not just a simple power series as in the Noumi-Shiraishi case, but has an involved structure with $N!$ branches, each of them being a sum over the Weyl chambers of power series of the Noumi-Shiraishi type. As an illustration, we also provide explicit formulas for particular cases.

hep-th↗
arXiv2026

Towards Equations for String Amplitudes

Generic Feynman integrals are widely studied as solutions of Picard-Fuchs equations on moduli spaces of their parameters, and this calls for consideration of this phenomenon at a more basic level - of string amplitudes which are integrals over true non-singular module space of Riemann surfaces and their various generalizations. The main puzzle here is that a single string amplitude involves mane different particle diagrams, corresponding to different parts of the same moduli space, but different particle diagrams are usually believed to satisfy different equations, not unified into a common entity. We begin investigation of this problem, starting from Koba-Nielsen diagrams. While there is nothing interesting at this level for particles, the tree-level open bosonic string amplitudes satisfy non-trivial linear difference equations in kinematic variables. Moreover, the integration-by-parts on moduli space, standing behind Picard-Fuchs equations for particle loops, for strings are operative already at the tree level. We construct a complete system of such equations for arbitrary n-point tree amplitudes, with the number of independent relations matching the kinematic parameters. In variance with the particle case equations are difference ones rather than differential. The low-energy limit $\alpha \to 0$ smoothly recovers the algebraic QFT structure.

hep-th↗
arXiv2026

More on Kashaev limits of the quantum $A$-polynomials

"Colored" knot polynomials satisfy difference equation w.r.t. the highest weights of the underlying representation -- which in the case of symmetrically colored Jones are named "quantum $A$-polynomials". In the double scaling quasiclassical (Kashaev) limit, when representation size $r\sim \hbar^{-1}$, there are different phases -- in one of them the classical action vanishes and in another one it is a deformation of hyperbolic volume (of a knot complement in $S^3$). This corresponds to a splitting of the non-homogeneous version of the quantum $A$-polynomial into two pieces, which we illustrate by more examples than just a figure-eight knot $4_1$ in the original paper. From the point of view of quasiclassics, hyperbolic volume is just an integration constant, which is not fully determined by the $A$-polynomial equation -- and actually remains ambiguous in this formalism. As a byproduct, we expect that classical $A$-polynomial at $L=1$ becomes proportional to Alexander: $A^{\cal K}(1,M)\sim \Delta^{\cal K}(M)$ -- this seems true, but $A$ should be consistent with the polynomiality of {\it non-homogeneous quantum} ${\cal A}$-polynomial, what sometime implies that it is not minimal.

hep-th↗
arXiv2026

Khovanov complexes for bipartite links

Recently, for a limited class for bipartite links, the complicated Khovanov-Rozansky matrix factorization technique was reduced to an analogue of elementary Kauffman-Khovanov cycle calculus for an arbitrary $N$. In this note, we demonstrate the consistency of such reduction with the computation of the bipartite Khovanov polynomials for $N=2$. Namely, we explain how the Kauffman-Khovanov $2^2$-hypercube is shrinked to the bipartite 3-hypercube.

hep-th↗
arXiv2026

Reductions in Khovanov-Rozansky operator formalism

Sophisticated Khovanov-Rozansky (KhR) description of knot invariants in the fundamental representation can be reformulated in terms of bicomplex with a simple physical meaning. Namely, the counterintuitive matrix factorization is substituted by simple operators $D$, locally constructed for every MOY resolution of a link diagram, which becomes nilpotent when the diagram has no external lines. Operators for different resolutions are related by equally simple conjugations $\chi^{(\pm)}$. The KhR procedure then splits in two steps - defining ``vertical'' cohomologies of $D$, which are associated with particular resolutions and will be put at vertices of the hypercube, and conjugations $\chi^{(\pm)}$, that define morphisms along its edges. As usual, standard combinations of morphisms are nilpotent, and one can define ``horizontal'' cohomologies - which are then combined into Poincar\'e polynomial, called KhR polynomial in application to links. This construction remains global in the sense that resulting cohomologies depend on the entire link diagram, but all its building blocks, including the operators and morphisms are local in the sense that they are defined for its particular vertices. Sometimes, this allows simple local reductions, allowing to eliminate or change particular vertices or sets of those. Along with the obvious case of Reidemeister equivalencies this happens also for antiparallel-lock tangles, what is responsible for simplification of bipartite calculus. In the $N=2$ and arbitrary $N$ bipartite cases, one can also provide global reductions transferring the local construction of the KhR double-complex to the global construction of the Khovanov(-like) single-complex.

hep-th↗
arXiv2026

On the geometric algebras of the Ising model

We revisit the classical transfer matrix solution of the one- and two-dimensional Ising model from the perspective of Clifford and conformal geometric algebras. Building on Kaufman's spinor formulation, we show that all elements entering the solution, including the transfer matrix, its eigenvectors, and the quasiparticle excitations, admit a natural and unified interpretation as elements of an appropriate conformal Clifford algebra. In particular, the transfer matrix can be viewed as a dilation generated by a conformal bivector, while its eigenvectors correspond to null combinations of Clifford generators, closely paralleling the emergence of Majorana fermionic degrees of freedom. In the two-dimensional case, the standard eigenvalue equation for the row-to-row transfer matrix is reinterpreted as a dispersion relation for quasiparticle excitations, exposing the connection between the Ising model and a theory of free Majorana fermions. While all the explicit exact results recovered are well known, this geometric reformulation provides a unified algebraic framework which is compact and physically interpretable. Specifically, this clarifies the role of scale transformations, fermionic modes, and duality in the Ising model. We believe this approach offers a useful pedagogical complement to more conventional fermionic, Grassmann, or field theoretic treatments.

cond-mat.stat-mech↗
arXiv2026

Experimental Scaling of Diffraction Efficiency in Laser-Induced Plasma Gratings

We demonstrate efficient diffraction of intense ultrashort laser pulses using optical-field-ionization-induced plasma-neutral gratings formed by spatially structured ionization of a neutral molecular gas in the interference field of two femtosecond pump pulses. The transient refractive index modulation of the plasma structure persists for at least 10 picoseconds and is used to diffract intense femtosecond signal pulses into the 1st order of diffraction with an average efficiency of up to 35$\%$. Plasma gratings are shown to provide stable diffraction at signal laser intensities greater than $ 10^{14}\text{ W/cm}^2$, exceeding the damage thresholds of conventional solid-state optics by more than two orders of magnitude, continuously for hours at a 10-Hz repetition rate. The experimental diffraction efficiency scales with the grating aperture allowing for a larger millimeter-scale plasma optic, increases with the pump energy and electron density, and reaches a maximum at a specific grating length in agreement with the coupled-mode theory for periodic media. These results demonstrate the scalability, tunability, and high damage threshold of transmissive plasma-based photonic structures, opening new prospects for controlling multi-petawatt laser beams.

physics.optics↗
arXiv2026

A new Uncertainty Principle in Machine Learning

Many scientific problems in the context of machine learning can be reduced to the search of polynomial answers in appropriate variables. The Hevisidization of arbitrary polynomial is actually provided by one-and-the same two-layer expression. What prevents the use of this simple idea is the fatal degeneracy of the Heaviside and sigmoid expansions, which traps the steepest-descent evolution at the bottom of canyons, close to the starting point, but far from the desired true minimum. This problem is unavoidable and can be formulated as a peculiar uncertainty principle -- the sharper the minimum, the smoother the canyons. It is a direct analogue of the usual one, which is the pertinent property of the more familiar Fourier expansion. Standard machine learning software fights with this problem empirically, for example, by testing evolutions, originated at randomly distributed starting points and then selecting the best one. Surprisingly or not, phenomena and problems, encountered in ML application to science are pure scientific and belong to physics, not to computer science. On the other hand, they sound slightly different and shed new light on the well-known phenomena -- for example, extend the uncertainty principle from Fourier and, later, wavelet analysis to a new peculiar class of nearly singular sigmoid functions.

cs.LG↗
arXiv2026

Generating twisted Cherednik eigenfunctions

Hamiltonians ${\cal H}^{a}_k$ of new integrable systems associated with the integer rays $(1,a)$ (commutative subalgebras) of Ding-Iohara-Miki (DIM) algebra in the $N$-body representation are closely related to commuting twisted Cherednik Hamiltonians $\mathfrak{C}_i^{(a)}$, ${\cal H}^{a}_k = \sum_{i=1}^N (\mathfrak{C}_i^{(a)})^k$. Moreover, symmetric combinations of eigenfunctions in the twisted Cherednik system were recently shown to produce the DIM Hamiltonian eigenstates. We explicitly construct these twisted Cherednik eigenfunctions recurrently by action of some (creation and permutation) operations. It resembles of a far-going generalization of Kirillov-Noumi operators, but exact relation remains to be specified.

hep-th↗
arXiv2026

Correlators in the theory of Integral Discriminants

Integral discriminants provide a simple and fundamental model for non-Gaussian integrals, associated with homogeneous polynomials of degree r in n variables. We argue that, in this context, the study of correlators is equally if not more important. In this paper, we study a natural class of correlators in this model -- the invariant correlators. We suggest a general method to compute invariant correlators using differential operators that act on the partition function. This method allows to compute general invariant correlators in terms of the fundamental invariants. Moreover, in some cases the correlators appear to be simply polynomials in the invariants. This could be an interesting manifestation of superintegrability phenomenon in the theory of integral discriminants.

hep-th↗
arXiv2026

Averages of Exponentials from the point of view of Superintegrability

We calculate Gaussian averages of arbitrary exponentials of the matrix variable $X$ with the help of superintegrability, which provides explicit expressions for Schur averages. As in the simpler cases the answer is expressed in terms of Laguerre polynomials, but in a somewhat sophisticated way. It involves triangular sum over partitions, with simple exponential factor and a complicated polynomial prefactor. Some ingredients of the formula are not found in full generality and there is still a room for further work.

hep-th↗
arXiv2026

Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems

We discuss interrelations between eigenfunctions of the Hamiltonians associated with the commutative (integer ray) subalgebras of the Ding-Iohara-Miki algebra and those of the twisted Cherednik system. In the case of $t=q^{-m}$ with natural $m$, eigenfunctions of the first system of Hamiltonians are the twisted Baker-Akhiezer functions (BAFs) introduced by O. Chalykh, while eigenfunctions of the twisted Cherednik Hamiltonians are twisted non-symmetric Macdonald polynomials. Actually, the twisted Cherednik ground state is symmetric and coincides with a peculiar symmetric BAF. We lift this correspondence to excited states, and claim that both Cherednik eigenfunctions and BAF's can be combined to produce symmetric functions, which coincide with each other and are eigenfunctions of the both DIM Hamiltonians and power sums of the twisted Cherednik Hamiltonians at once. This reflects the correspondence between the DIM algebra and the spherical DAHA explicitly.

hep-th↗
arXiv2026

Nested ansatz method for Baker-Akhiezer functions

We explain that the logic behind the derivation of the Noumi-Shiraishi function can be applied directly to the Baker-Akhiezer function (BAF). This amounts to changing an ansatz for BAF to a nested one, where the BAF of N + 1 variables is recursively expressed as a sum over BAFs of N variables. This may be seen as a generalization of symmetrization trick from [1], but for the generally non-symmetric BAF. We demonstrate that, for usual non-twisted (a = 1) BAFs, this method correctly reproduces the Noumi-Shiraishi formula directly from linear equations, resolving the ambiguity related to non-simple roots. For the first non-trivial twisted case (N = 3, a = 2) this method also fixes this ambiguity, moreover, answers for the first few layers of coefficients are in the form of direct quantization of [1].

hep-th↗
arXiv2026

Twisted Cherednik spectrum as a $q,t$-deformation

The common eigenfunctions of the twisted Cherednik operators can be first analyzed in the limit of $q\longrightarrow 1$. Then, the polynomial eigenfunctions form a simple set originating from the symmetric ground state of non-vanishing degree and excitations over it, described by non-symmetric polynomials of higher degrees and enumerated by weak compositions. This pattern is inherited by the full spectrum at $q\neq 1$, which can be considered as a deformation. The whole story looks like a typical NP problem: the Cherednik equations are difficult to solve, but easy to check the solution once it is somehow found.

hep-th↗
arXiv2025

Twisted Cherednik systems and non-symmetric Macdonald polynomials

We consider eigenfunctions of many-body system Hamiltonians associated with generalized (a-twisted) Cherednik operators used in construction of other Hamiltonians: those arising from commutative subalgebras of the Ding-Iohara-Miki (DIM) algebra. The simplest example of these eigenfunctions is provided by non-symmetric Macdonald polynomials, while generally they are constructed basing on the ground state eigenfunction coinciding with the twisted Baker-Akhiezer function being a peculiar (symmetric) eigenfunction of the DIM Hamiltonians. Moreover, the eigenfunctions admit an expansion with universal coefficients so that the dependence on the twist $a$ is hidden only in these ground state eigenfunctions, and we suggest a general formula that allows one to construct these eigenfunctions from non-symmetric Macdonald polynomials. This gives a new twist in theory of integrable systems, which usually puts an accent on symmetric polynomials, and provides a new dimension to the {\it triad} made from the symmetric Macdonald polynomials, untwisted Baker-Akhiezer functions and Noumi-Shiraishi series.

hep-th↗
arXiv2025

5D AGT conjecture for circular quivers

The best way to represent generic conformal blocks is provided by the free-field formalism, where they acquire a form of multiple Dotsenko-Fateev-like integrals of the screening operators. Degenerate conformal blocks can be described by the same integrals with special choice of parameters. Integrals satisfy various recurrent relations, which for the special choice of parameters reduce to closed equations. This setting is widely used in explaining the AGT relation, because similar integral representations exist also for Nekrasov functions. We extend this approach to the case of q-Virasoro conformal blocks on elliptic surface -- generic and degenerate. For the generic case, we check equivalence with instanton partition function of a 5d circular quiver gauge theory. For the degenerate case, we check equivalence with partition function of a defect in the same theory, also known as the Shiraishi function. We find agreement in both cases. This opens a way to re-derive the sophisticated equation for the Shiraishi function as the equation for the corresponding integral, what seems straightforward, but remains technically involved and is left for the future.

hep-th↗