Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory

Abstract

We define cut-and-join operator in Hurwitz theory for merging of two branching points of arbitrary type. These operators have two alternative descriptions:(i) they have the GL characters as eigenfunctions and the symmetric-group characters as eigenvalues; (ii) they can be represented as differential operators of the $W$-type (in particular, acting on the time-variables in the Hurwitz-Kontsevich tau-function). The operators have the simplest form if expressed in terms of the matrix Miwa-variables. They form an important commutative associative algebra, a Universal Hurwitz Algebra, generalizing all group algebra centers of particular symmetric groups which are used in description of the Universal Hurwitz numbers of particular orders. This algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear form on the linear space of Young diagrams, evaluated at the product of all diagrams, which characterize particular ramification points of the covering.

21 pages

References in corpus (11)

• Seiberg-Witten Theory and Random Partitions
• Generation of Matrix Models by W-operators
• Seiberg-Witten theory and matrix models
• Symplectic invariants, Virasoro constraints and Givental decomposition
• Virasoro constraints for Kontsevich-Hurwitz partition function
• Hodge integrals, Hurwitz numbers, and Symmetric Groups
• Hurwitz numbers, matrix models and enumerative geometry
• KP hierarchy for Hodge integrals
• From matrix models' topological expansion to topological string theories: counting surfaces with algebraic geometry
• Universal Algebras of Hurwitz Numbers
• Disk single Hurwitz numbers

Cited by in corpus (121)

• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• On AGT relation in the case of U(3)
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• W-symmetry, topological vertex and affine Yangian
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Algebra of differential operators associated with Young diagrams
• On KP-integrable Hurwitz functions
• The Power of Nekrasov Functions
• BGWM as Second Constituent of Complex Matrix Model
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• On Combinatorial Expansions of Conformal Blocks
• On the complete perturbative solution of one-matrix models
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Correlators in tensor models from character calculus
• Ward identities and combinatorics of rainbow tensor models
• Integrability of Hurwitz Partition Functions. I. Summary
• Colored HOMFLY polynomials for the pretzel knots and links
• Character expansion for HOMFLY polynomials. I. Integrability and difference equations
• Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations
• Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths
• Unitary Integrals and Related Matrix Models
• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• On universal knot polynomials
• Superintegrability summary
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Towards effective topological field theory for knots
• On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• On colored HOMFLY polynomials for twist knots
• Tabulating knot polynomials for arborescent knots
• Knot polynomials in the first non-symmetric representation
• Knot invariants from Virasoro related representation and pretzel knots
• Sum rules for characters from character-preservation property of matrix models
• Cut and join operator ring in Aristotelian tensor model
• On 3d extensions of AGT relation
• Interpolating Matrix Models for WLZZ series
• On KP-integrable skew Hurwitz $\tau$-functions and their $\beta$-deformations
• Generalized Jack polynomials and the AGT relations for the $SU(3)$ group
• Genus expansion of HOMFLY polynomials
• Towards matrix model representation of HOMFLY polynomials
• Equations on knot polynomials and 3d/5d duality
• Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials
• Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial
• On genus expansion of superpolynomials
• Many-body integrable systems implied by WLZZ models
• Torus HOMFLY as the Hall-Littlewood Polynomials
• From Kronecker to tableau pseudo-characters in tensor models
• Cut-and-join structure and integrability for spin Hurwitz numbers
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• CFT approach to constraint operators for ($\beta$-deformed) hermitian one-matrix models
• CFT exercises for the needs of AGT
• Around spin Hurwitz numbers
• Complex matrix model duality
• Complete solution to Gaussian tensor model and its integrable properties
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• Hopf link invariants and integrable hierarchies
• Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations
• $(q,t)$-deformed (skew) Hurwitz $\tau$-functions
• Fermionic approach to weighted Hurwitz numbers and topological recursion
• An analogue of Schur functions for the plane partitions
• Commutative families in $W_{\infty }$, integrable many-body systems and hypergeometric $\tau$-functions
• Kerov functions revisited
• Bilinear character correlators in superintegrable theory
• Distinguishing Mutant Knots
• On Products of Random Matrices
• On Factorization of Generalized Macdonald Polynomials
• Checks of integrality properties in topological strings
• Matching branches of non-perturbative conformal block at its singularity divisor
• On Hopf-induced deformation of topological locus
• Quantization of Harer-Zagier formulas
• Tensorial generalization of characters
• Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}({\widehat{\mathfrak{gl}}}_{1|1})$
• Challenges of beta-deformation
• A Hurwitz theory avatar of open-closed strings
• Connection between cut-and-join and Casimir operators
• Colored Alexander polynomials and KP hierarchy
• 3-Schurs from explicit representation of Yangian $Y({\widehat{\mathfrak{gl}}}_1)$. Levels 1-5
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• Hurwitz numbers from Feynman diagrams
• From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators
• On the status of DELL systems
• A new symmetry of the colored Alexander polynomial
• On Hamiltonians for Kerov functions
• Colored HOMFLY-PT polynomials of quasi-alternating $3$-braid knots
• On generalized Macdonald polynomials
• Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity
• Spin Hurwitz theory and Miwa transform for the Schur Q-functions
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• Universal Algebras of Hurwitz Numbers
• Faces of matrix models
• Hook variables: cut-and-join operators and $\tau$-functions
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Colored HOMFLY-PT for hybrid weaving knot ${\hat{W}}_3(m,n)$
• Macdonald polynomials for super-partitions
• Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems
• Cardy-Frobenius extension of algebra of cut-and-join operators
• Hurwitz numbers and matrix integrals labeled with chord diagrams
• On character expansion and Gaussian regularization of Itzykson-Zuber measure
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• Matrix integrals and Hurwitz numbers
• Asymptotic Hurwitz numbers
• Towards the theory of Yangians
• From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity
• Generalized $\tilde{W}$ algebras
• Averaging method in combinatorics of symmetric polynomials
• q-Casimir and q-cut-and-join operators related to Reflection Equation Algebras
• Matrix model and dimensions at hypercube vertices
• Notes about KP/BKP correspondence
• Genus expansion of matrix models and $\hslash$ expansion of $B$KP hierarchy
• Integrals of tau functions
• Phases in WLZZ Matrix Models
• Super-Hamiltonians for super-Macdonald polynomials
• CMM formula as superintegrability property of unitary model
• A Formula about W-operator and Its Application to Hurwitz Number
• Genus Expanded Cut-and-Join operators and generalized Hurwtiz numbers
• Decomplexification of the Capelli identities and holomorphic factorization
• Shifted genus expanded ${\mathcal{W}}_{\infty }$ algebra and shifted Hurwtiz numbers
• Degree of W-operator and Noncrossing Partition
• A new kind of anomaly: on W-constraints for GKM
• W-Operator and Differential Equation for 3-Hurwitz Number