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We argue that the recently discovered bilinear superintegrability http://arxiv.org/2206.02045 generalizes, in a non-trivial way, to monomial matrix models in pure phase. The structure is much richer: for the trivial core Schur functions required modifications are minor, and the only new ingredient is a certain (contour-dependent) permutation matrix; for non-trivial-core Schur functions, in both bi-linear and tri-linear averages the deformation is more complicated: averages acquire extra N-dependent factors and selection rule is less straightforward to imply. [ABSTRACT FROM AUTHOR]

We explain that the set of new integrable systems, generalizing the Calogero family and implied by the study of WLZZ models, which was described in , is only the tip of the iceberg. We provide its wide generalization and explain that it is related to commutative subalgebras (Hamiltonians) of the W1+∞ algebra. We construct many such subalgebras and explain how they look in various representations. We start from the even simpler w∞ contraction, then proceed to the one-body representation in terms of differential operators on a circle, further generalizing to matrices and in their eigenvalues, in finally to the bosonic representation in terms of time-variables. Moreover, we explain that some of the subalgebras survive the β-deformation, an intermediate step from W1+∞ to the affine Yangian. The very explicit formulas for the corresponding Hamiltonians in these cases are provided. Integrable many-body systems generalizing the rational Calogero model arise in the representation in terms of eigenvalues. Each element of W1+∞ algebra gives rise to KP/Toda τ-functions. The hidden symmetry given by the families of commuting Hamiltonians is in charge of the special, (skew) hypergeometric τ-functions among these. [ABSTRACT FROM AUTHOR]

We suggest a two-matrix model depending on three (infinite) sets of parameters which interpolates between all the models proposed in Wang et al. (Eur Phys J C 82:902, arXiv:2206.13038, 2022) and defined there through W-representations. We also discuss further generalizations of the WLZZ models, realized by W-representations associated with infinite commutative families of generators of w ∞ -algebra which are presumably related to more sophisticated multi-matrix models. Integrable properties of these generalizations are described by what we call the skew hypergeometric τ -functions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the phenomenon of superintegrability. This phenomenon is manifested in the existence of a formula for character averages, expressed through the same characters at special points and of its various generalization. In this paper we develop a method of proving such formulas from first principle from Virasoro constraints and W-representation. We apply it to prove the formula for the Jack functions averages – appropriate analogue of characters for the β -deformed Hermitian Gaussian matrix model. We also sketch the construction of W-operators from Calogero–Ruijsenaars Hamiltonians. [ABSTRACT FROM AUTHOR]

How should modern people evaluate Feynman diagrams? This question has been receiving considerable attention in recent years. While the current answer is far from being complete, one can select several attack directions under development. One of such directions is the differential equations method. We attempt to review some of its features and outline the ideas that could help establish a more general framework. [ABSTRACT FROM AUTHOR]

Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a group element , i.e. satisfying the property Δ (X) = X ⊗ X , then their generating functions obey bilinear Hirota equations and hence are named τ -functions. However, dealing with group elements is not always easy, especially for non-commutative algebras of functions, and this slows down the development of τ -function theory and the study of integrability properties of non-perturbative functional integrals. A simple way out is to use arbitrary elements of the universal enveloping algebra, and not just the group elements. Then the Hirota equations appear to interrelate a whole system of generating functions, which one may call generalized τ -functions. It was recently demonstrated that this idea can be applicable even to a somewhat sophisticated case of the quantum toroidal algebra. We consider a number of simpler examples, including ordinary and quantum groups, to explain how the method works and what kind of solutions one can obtain. [ABSTRACT FROM AUTHOR]

In this letter we continue the development of W -representations. We propose several generalizations of the known models, such as the hypergeometric Hurwitz τ -functions. We construct W -representations for multi-character expansions, which involve a generic number of sets of time variables. We propose integral representations for such kind of partition functions which are given by tensor models and multi-matrix models with multi-trace couplings. We further propose the β -deformation of the discussed W -representation for the Hurwitz case for two sets of times as well as for the multi-character case. [ABSTRACT FROM AUTHOR]

In the recent study of Virasoro action on characters, we discovered that it gets especially simple for peculiar linear combinations of the Virasoro operators: particular harmonics of w ^ -operators. In this letter, we demonstrate that even more is true: a singlew-constraint is sufficient to uniquely specify the partition functions provided one assumes that it is a power series in time-variables. This substitutes the previous specifications in terms of two requirements: either a string equation imposed on the KP/Toda τ -function or a pair of Virasoro generators. This mysterious single-entry definition holds for a variety of theories, including Hermitian and complex matrix models, and also matrix models with external matrix: the unitary and cubic Kontsevich models. In these cases, it is equivalent to W-representation and is closely related to super integrability. However, a similar single equation that completely determines the partition function exists also in the case of the generalized Kontsevich model (GKM) with potential of higher degree, when the constraint algebra is a larger W-algebra, and neither W-representation, nor superintegrability are understood well enough. [ABSTRACT FROM AUTHOR]

We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of N and generalizes the previously known "tug-the-hook" symmetry of the colored Alexander polynomial (Mishnyakov et al. in Annales Henri Poincaré, 2021. https://doi.org/10.1007/s00023-020-00980-8, arXiv:2001.10596). As we show, the symmetry has a superalgebra origin, which we discuss qualitatively. Our main focus are the constraints that such a property imposes on the general group-theoretical structure, namely the sl (N) weight system, arising in the perturbative expansion of the invariant. Finally, we demonstrate its tight relation to the eigenvalue conjecture. [ABSTRACT FROM AUTHOR]

Relation between the Virasoro constraints and KP integrability (determinant formulas) for matrix models is a lasting mystery. We elaborate on the claim that the situation is improved when integrability is enhanced to superintegrability, i.e., to explicit formulas for Gaussian averages of characters. In this case, the Virasoro constraints are equivalent to simple recursive formulas, which have appropriate combinations of characters as their solutions. Moreover, one can easily separate dependence on the size of matrix, and deduce superintegrability from the Virasoro constraints. We describe one of the ways to do so for the Gaussian Hermitian matrix model. The result is a spectacularly elegant reformulation of Virasoro constraints as identities for the Schur functions evaluated at appropriate loci in the space of time-variables. [ABSTRACT FROM AUTHOR]

We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum sl N invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry imposes on the group theoretic structure of the loop expansion and provide solutions to those constraints. The symmetry is a powerful tool for research on polynomial knot invariants and in the end we suggest several possible applications of the symmetry. [ABSTRACT FROM AUTHOR]

We demonstrate that commutativity of numerous one-dimensional subalgebras in W 1 + ∞ algebra, i.e. the existence of many non-trivial integrable systems described in recent arXiv:2303.05273 follows from the subset of relations in algebra known as Serre relations. No other relations are needed for commutativity. The Serre relations survive the deformation to the affine Yangian Y ( gl ˆ 1) , hence the commutative subalgebras do as well. A special case of the Yangian parameters corresponds to the β -deformation. The preservation of Serre relations can be thought of a selection rule for proper systems of commuting β -deformed Hamiltonians. On the contrary, commutativity in the extended family associated with "rational (non-integer) rays" is not reduced to the Serre relations, and uses also other relations in the W 1 + ∞ algebra. Thus their β -deformation is less straightforward. [ABSTRACT FROM AUTHOR]

The answers for Feynman diagrams satisfy various kinds of differential equations – which is not a surprise, because they are defined as Gaussian correlators, possessing a vast variety of Ward identities and superintegrability properties. We study these equations in the simplest example of banana diagrams. They contain any number of loops, but can be efficiently handled in position rather than momentum representation, where loop integrals do not show up. We derive equations for the case of scalar fields, explain their origins and drastic simplification at coincident masses. To further simplify the story we do not consider coincident points, i.e. ignore delta-function contributions and ultraviolet divergences for the most part. The equations in this case reduce to homogeneous and have as many solutions as there are different Green functions – 2 n + 1 for n loops in quadratic theory, what reduces to just n + 1 for coincident masses, i.e. for a single field. We comment on the recovery of the delta-functions directly from the homogeneous equations and also compare our result with momentum space formulas known in the literature. [ABSTRACT FROM AUTHOR]

We extend the old formalism of cut-and-join operators in the theory of Hurwitz τ -functions to description of a wide family of KP-integrable skew Hurwitz τ -functions, which include, in particular, the newly discovered interpolating WLZZ models. Recently, the simplest of them was related to a superintegrable two-matrix model with two potentials and one external matrix field. Now we provide detailed proofs, and a generalization to a multi-matrix representation, and propose the β -deformation of the matrix model as well. The general interpolating WLZZ model is generated by a W -representation given by a sum of operators from a one-parametric commutative sub-family (a commutative subalgebra of w ∞). Different commutative families are related by cut-and-join rotations. Two of these sub-families ('vertical' and '45-degree') turn out to be nothing but the trigonometric and rational Calogero-Sutherland Hamiltonians, the 'horizontal' family is represented by simple derivatives. Other families require an additional analysis. [ABSTRACT FROM AUTHOR]

Painlevè equation for conformal blocks is a combined corollary of integrability and Ward identities, which can be explicitly revealed in the matrix model realization of AGT relations. We demonstrate this in some detail, both for q -Painlevè equations for the q -Virasoro conformal block, or AGT dual gauge theory in 5 d , and for ordinary Painlevè equations, or AGT dual gauge theory in 4 d. Especially interesting is the continuous limit from 5 d to 4 d and its description at the level of equations for eight τ -functions. Half of these equations are governed by integrability and another half by Ward identities. [ABSTRACT FROM AUTHOR]

We reconsider the simple matrix model description of Hurwitz numbers proposed by S. Natanzon and A. Orlov, which uses the superintegrability property of the complex matrix model, and discuss a way of its possible supersymmetric extension to approach spin Hurwitz numbers. [ABSTRACT FROM AUTHOR]

W -representation is a miraculous possibility to define a non-perturbative (exact) partition function as an exponential action of somehow integrated Ward identities on unity. It is well known for numerous eigenvalue matrix models, when the relevant operators are of a kind of W -operators: for the Hermitian matrix model with the Virasoro constraints, it is a W 3 -like operator, and so on. We extend this statement to the monomial generalized Kontsevich models (GKM), where the new feature is appearance of an ordered P-exponential for the set of non-commuting operators of different gradings. [ABSTRACT FROM AUTHOR]

In this paper we study the group theoretic structures of colored HOMFLY polynomials in a specific limit. The group structures arise in the perturbative expansion of S U (N) Chern-Simons Wilson loops, while the limit is N → 0. The result of the paper is twofold. First, we explain the emergence of Kadomsev-Petviashvily (KP) τ -functions. This result is an extension of what we did in [1] , where a symbolic correspondence between KP equations and group factors was established. In this paper we prove that integrability of the colored Alexander polynomial is due to it's relation to soliton τ -functions. Mainly, the colored Alexander polynomial is embedded in the action of the KP generating function on the soliton τ -function. Secondly, we use this correspondence to provide a rather simple combinatoric description of the group factors in term of Young diagrams, which is otherwise described in terms of chord diagrams, where no simple description is known. This is a first step providing an explicit description of the group theoretic data of Wilson loops, which would effectively reduce them to a purely topological quantity, mainly to a collection of Vassiliev invariants. [ABSTRACT FROM AUTHOR]

We follow the general recipe for constructing commutative families of W -operators, which provides Hurwitz-like expansions in symmetric functions (Macdonald polynomials), in order to obtain a difference operator example that gives rise to a (q , t) -deformation of the earlier studied models. As before, a key role is played by an appropriate deformation of the cut-and-join rotation operator. We outline its expression both in terms of generators of the quantum toroidal algebra and in terms of the Macdonald difference operators. [ABSTRACT FROM AUTHOR]

We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: A R K ( q ) = A [ 1 ] K ( q | R | ) for all 1-hook Young diagrams R . Via the Kontsevich construction, it is reformulated as a system of linear equations. It appears that the solutions of this system induce the KP equations in the Hirota form. The Alexander polynomial is a specialization of the HOMFLY polynomial, and it is a kind of a dual to the double scaling limit, which gives the special polynomial, in the sense that, while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials provide the equations of this hierarchy. This gives a new connection with integrable properties of knot polynomials and puts an interesting question about the way the KP hierarchy is encoded in the full HOMFLY polynomial. [ABSTRACT FROM AUTHOR]