Virasoro constraints for Kontsevich-Hurwitz partition function

Abstract

M.Kazarian and S.Lando found a 1-parametric interpolation between Kontsevich and Hurwitz partition functions, which entirely lies within the space of KP tau-functions. V.Bouchard and M.Marino suggested that this interpolation satisfies some deformed Virasoro constraints. However, they described the constraints in a somewhat sophisticated form of AMM-Eynard equations for the rather involved Lambert spectral curve. Here we present the relevant family of Virasoro constraints explicitly. They differ from the conventional continuous Virasoro constraints in the simplest possible way: by a constant shift u^2/24 of the L_{-1} operator, where u is an interpolation parameter between Kontsevich and Hurwitz models. This trivial modification of the string equation gives rise to the entire deformation which is a conjugation of the Virasoro constraints L_m -> U L_m U^{-1} and "twists" the partition function, Z_{KH}= U Z_K. The conjugation U is expressed through the previously unnoticed operators which annihilate the quasiclassical (planar) free energy of the Kontsevich model, but do not belong to the symmetry group GL(\infty) of the universal Grassmannian.

36 pages

References in corpus (27)

• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
• Topological Strings and Integrable Hierarchies
• All genus correlation functions for the hermitian 1-matrix model
• Two Dimensional QCD is a String Theory
• Hurwitz numbers and intersections on moduli spaces of curves
• Integrability and Matrix Models
• Open string amplitudes and large order behavior in topological string theory
• Hermitean matrix model free energy: Feynman graph technique for all genera
• Twists and Wilson Loops in the String Theory of Two Dimensional QCD
• Free energy topological expansion for the 2-matrix model
• Towards unified theory of $2d$ gravity
• Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model
• Conformal Matrix Models as an Alternative to Conventional Multi-Matrix Models
• Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
• Holomorphic anomaly and matrix models
• $2d$ gravity and matrix models. I. $2d$ gravity
• Character Expansion Methods for Matrix Models of Dually Weighted Graphs
• Instantons and Merons in Matrix Models
• Unitary matrix integrals in the framework of Generalized Kontsevich Model. I. Brezin-Gross-Witten Model
• Unification of All String Models with $c<1$
• Extended Seiberg-Witten Theory and Integrable Hierarchy
• Solving Virasoro Constraints in Matrix Models
• On Equivalence of Topological and Quantum 2d Gravity
• DV and WDVV
• Unified description of correlators in non-Gaussian phases of Hermitean matrix model
• From Virasoro Constraints in Kontsevich's Model to $\mathcal{W}$-constraints in 2-matrix Models
• Geometry of meromorphic functions and intersections on moduli spaces of curves

Cited by in corpus (35)

• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• On AGT relation in the case of U(3)
• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• A matrix model for simple Hurwitz numbers, and topological recursion
• Generation of Matrix Models by W-operators
• Algebra of differential operators associated with Young diagrams
• The Power of Nekrasov Functions
• BGWM as Second Constituent of Complex Matrix Model
• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• Interpolating Matrix Models for WLZZ series
• Additional symmetries of constrained CKP and BKP hierarchies
• Exact 2-point function in Hermitian matrix model
• Weighted Hurwitz numbers and topological recursion
• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra
• Matrix model partition function by a single constraint
• CFT exercises for the needs of AGT
• Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations
• Generalized string equations for double Hurwitz numbers
• Virasoro type algebraic structure hidden in the constrained discrete KP hierarchy
• Spectral curves and $W$-representations of matrix models
• Quantization of Harer-Zagier formulas
• Challenges of beta-deformation
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators
• Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity
• Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integrals
• Virasoro constraints and polynomial recursion for the linear Hodge integrals
• From $r$-Spin Intersection Numbers to Hodge Integrals
• Genus expansion of matrix models and $\hslash$ expansion of KP hierarchy
• Hurwitz numbers and integrable hierarchy of Volterra type
• Matrix model generating function for quantum weighted Hurwitz numbers
• Higher order constraints for the ($\beta$-deformed) Hermitian matrix models
• Symmetric deformed 2D/3D Hurwitz-Kontsevich model and affine Yangian of gl(1)
• A new kind of anomaly: on W-constraints for GKM