We study the Mana and Magic for quantum states. They have a standard definition through the Clifford group, which is finite and thus classically computable. We introduce a modified Mana and Magic, which keep their main property of classical computability, while making other states classically computable. We also apply these new definitions to the studies of knot states of 2-strand knots.

Within the context of wavefunctions of integrable many-body systems, rational multivariable Baker-Akhiezer (BA) functions were introduced by O. Chalykh, M. Feigin and A. Veselov and, in the case of the trigonometric Ruijsenaars-Schneider system, can be associated with a reduction of the Macdonald symmetric polynomials at t = q(-m) with integer partition labels substituted by arbitrary complex numbers. A parallel attempt to describe wavefunctions of the bispectral trigonometric Ruijsenaars-Schneider problem was made by M. Noumi and J. Shiraishi who proposed a power series that reduces to the Macdonald polynomials at particular values of parameters. It turns out that this power series also reduces to the BA functions at t = q(-m), as we demonstrate in this letter. This makes the Macdonald polynomials, the BA functions and the Noumi-Shiraishi (NS) series a closely tied triad of objects, which have very different definitions, but are straightforwardly related with each other. In particular, theory of the BA functions provides a nice system of simple linear equations, while the NS functions provide a nice way to represent the multivariable BA function explicitly with arbitrary number of variables.

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 T_i→ j∼e^-S_instv_j^+v_i^- , where there is canonical instanton action suppression, and v_i^- annihilates a particle in the ith vacuum, whereas v_j^+ creates a particle in the jth 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 v_i^- , 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 Uq( 𝔤 ) and affine Yangians Y( 𝔤̂ ). For affine Yangians we demonstrate explicitly how instantons “perform” equivariant integrals over associated quiver moduli spaces appearing in the alternative geometric construction.

We continue the development of a position space approach to equations for Feynman multi-loop integrals. The key idea of the approach is that unintegrated products of Greens functions in position space are still loop integral in momentum space. The natural place to start are the famous banana diagrams, which we explore in this paper. In position space, these are just products of $n$ propagators. Firstly, we explain that these functions satisfy an equation of order $2^n$. These should be compared with Picard-Fuchs equations derived for the momentum space integral. We find that the Fourier transform of the position space operator contains the Picard-Fuchs one as a rightmost factor. The order of these operators is a special issue, especially since the order in momentum space is governed by degree in $x$ in position space. For the generic mass case this factorization pattern is complicated and it seems like the order of the Fourier transformed position space operators is much bigger than that of the Picard-Fuchs. Furthermore, one may ask what happens if after factorization we take the Picard-Fuchs operators back into position space. We discover that the result is again factorized, with the rightmost factor being the original position space equation. We demonstrate how this works in examples and discuss implications for more sophisticated Feynman integrals.

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 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, 𝔰𝔲_3 , where the Kauffman rule is substituted by a more involved Kuperberg 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.

Abstract 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 $$\mathfrak {su}_2$$ 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, $$\mathfrak {su}_3$$ su 3 , where the Kauffman rule is substituted by a more involved Kuperberg 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.

We propose an extension of the position space approach to Feynman integrals from the banana family to generic Feynman diagrams. Our approach is based on omitting integration in position space and then writing differential equations for the products of propagators defined for any graph. We employ the so-called 'bananization', where we start with simple Feynman graphs and further substitute each edge with a multiple one. We explain how the previously developed theory of banana diagrams can be used to describe what happens to the differential equations on Feynman diagrams after this transformation. Our approach works for generic (large enough) dimension and masses. We expect that after Fourier transform our equations should be related to the Picard-Fuchs equations. Therefore, we describe the challenges of the Fourier transform that arise in our approach.

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.

We prove that normalized colored Alexander polynomial (the A→1 limit of colored HOMFLY-PT polynomial) evaluated for one-hook (L-shape) representation R possesses scaling property: it is equal to the fundamental Alexander polynomial with the substitution q→q|R|. The proof is simple and direct use of Reshetikhin-Turaev formalism to get all required R-matrices.

We present a review of the relations between various equations for maximal cut banana Feynman diagrams, i.e. integrals with propagators substituted with δ-functions. We consider both equal and generic masses. There are three types of equation to consider: those in coordinate space, their Fourier transform and Picard-Fuchs equations originating from the parametric representation. First we review the properties of the corresponding differential operators themselves, mainly their factorization properties at the equal mass locus and their form at special values of the dimension. Then we study the relation between the Fourier transform of the coordinate space equations and the Picard-Fuchs equations and show that they are related by factorization as well. The equations in question are the counterparts of the Virasoro constraints in the much-better studied theory of eigenvalue matrix models and are the first step towards building a full-fledged theory of Feynman integrals, which will reveal their hidden integrable structure.

Vogel's universality implies a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters α,β,γ, which are homogeneous coordinates of Vogel's plane. Actually this is true (if at all) only for a piece of representation theory captured by knot/Chern-Simons theory, where some irreducible representations are often undistinguishable and combined into new “universally-irreducible" entities (uirreps). We consider from this point of view the recently discovered Macdonald deformation of quantum dimensions, for which a kind of universality holds for the ADE series. The claim is that universal are not Macdonald dimensions themselves, but their products with Littlewood-Richardson coefficients, which themselves are functions of q and t in Macdonald theory. These products are precisely what arises in knot/refined Chern-Simons theory. Actually, we consider the simplest decomposition of adjoint square into six uirreps and obtain the universal formulas for hyperpolynomials of the Hopf link and, more generally, of the torus links T[2,2n].

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.

We demonstrate the consistency of character expansion for the Itzykson-Zuber (IZ) model in terms of Schur polynomials with the old formulas for pair correlators with the IZ measure. An essential new feature of the correlators is that they are not symmetric in eigenvalues – and thus can not be expressed through Schur polynomials only. Instead, we demonstrate that an expression is possible in terms of Schur derivatives. This opens a new way to study arbitrary IZ correlators of any order in character expansion.

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.

We generalize the recently discovered planar decomposition (Kauffman bracket) for the Hoste-Ocneanu-Millett-Freyd-Lickorish-Yetter (HOMFLY) polynomials of bipartite knot/link diagrams to (anti)symmetrically colored HOMFLY polynomials. Cabling destroys planarity, but it is restored after projection to (anti)symmetric representations. This allows one to go beyond arborescent-induced calculus, which so far produced the majority of results for colored polynomials. Technicalities include combinations of projectors, and these can be handled rigorously, without any guess work—what can be also useful for other considerations, where reliable quantization was so far unavailable. We explicitly provide simple examples of calculation of the HOMFLY polynomials in symmetric representations with the use of our planar technique. These examples reveal what we call the bipartite evolution and the bipartite decomposition of squares of R matrix eigenvalues in the antiparallel channel. Published by the American Physical Society 2025

Abstract 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 $$ \textrm{Y}\left({\mathfrak{sl}}_2\right) $$ Y sl 2 , $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) $$ Y gl ̂ 1 and $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_{\left.1\right|1}\right) $$ Y gl ̂ 1 1 .

A typical example of superintegrability is provided by expression of the Hopf link hyperpolynomial in an arbitrary representation through a pair of the Macdonald polynomials at special points. In the simpler case of the Hopf link HOMFLY-PT polynomial and a pair of the Schur functions, it is a relation in the unitary matrix model. We explain that the Cherednik-Mehta-Macdonald identity for bilinear Macdonald residues with an elliptic weight function is nothing but a reformulation of these same formulas. Their lifting to arbitrary knots and links, even torus ones, remains obscure.

We extend our consideration of commutative subalgebras (rays) in different representations of the W1+∞ algebra to the elliptic Hall algebra (or, equivalently, to the Ding-Iohara-Miki (DIM) algebra U_q,t(𝔤̂̂̂𝔩̂̂̂_1) ). Its advantage is that it possesses the Miki automorphism, which makes all commutative rays equivalent. Integrable systems associated with these rays become finite-difference and, apart from the trigonometric Ruijsenaars system not too much familiar. We concentrate on the simplest many-body and Fock representations, and derive explicit formulas for all generators of the elliptic Hall algebra en,m. In the one-body representation, they differ just by normalization from z^nq^mD̂ of the W1+∞ Lie algebra, and, in the N -body case, they are non-trivially generalized to monomials of the Cherednik operators with action restricted to symmetric polynomials. In the Fock representation, the resulting operators are expressed through auxiliary polynomials of n variables, which define weights in the residues formulas. We also discuss q, t-deformation of matrix models associated with constructed commutative subalgebras.

A typical example of superintegrability is provided by expression of the Hopf link hyperpolynomial in an arbitrary representation through a pair of the Macdonald polynomials at special points. In the simpler case of the Hopf link HOMFLY-PT polynomial and a pair of the Schur functions, it is a relation in the unitary matrix model. We explain that the Cherednik-Mehta-Macdonald (CMM) identity for bilinear Macdonald residues with an elliptic weight function is nothing but a reformulation of these same formulas. Their lifting to arbitrary knots and links, even torus ones remains obscure.

A detailed review of the $p,q$-duality for Calogero system and its generalizations is given. For the first time, we present some of elliptic-trigonometric Hamiltonians dual to the elliptic Ruijsenaars Hamiltonians (i.e. trigonometric-elliptic ones), and explain their relations to the bi-elliptic Koroteev-Shakirov (KS) model. The most interesting self-dual double-elliptic (DELL) system remains a mystery, but we provide a clearer formulation of the problem and describe the steps that are still to be done.