Many-body integrable systems implied by WLZZ models

Abstract

We provide some details about the recently discovered integrable systems implied by commutativity of $W$ operators along the rays on the plane of roots of $w_{\infty }$-algebra. The simplest system of this type is the rational Calogero model, other systems escaped attention in the past. Existence of these systems is intimately tied to the very interesting WLZZ matrix models, which are now under intensive study.

arXiv comment8 pages

References in corpus (11)

• Collective fields, Calogero-Sutherland model and generalized matrix models
• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• Instantons and Merons in Matrix Models
• Algebra of differential operators associated with Young diagrams
• On Equivalence of Topological and Quantum 2d Gravity
• M-Theory of Matrix Models
• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• Cut-and-Join operator representation for Kontsevich-Witten tau-function
• Superintegrability summary
• On KP-integrable skew Hurwitz $\tau$-functions and their $\beta$-deformations
• Superintegrability of Calogero-Moser systems associated with the cyclic quiver

Cited by in corpus (20)

• Commutative subalgebras from Serre relations
• $(q,t)$-deformed (skew) Hurwitz $\tau$-functions
• Commutative families in $W_{\infty }$, integrable many-body systems and hypergeometric $\tau$-functions
• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• 3-Schurs from explicit representation of Yangian $Y({\widehat{\mathfrak{gl}}}_1)$. Levels 1-5
• On the status of DELL systems
• Algorithms for representations of quiver Yangian algebras
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• On character expansion and Gaussian regularization of Itzykson-Zuber measure
• Towards the theory of Yangians
• Generalized $\tilde{W}$ algebras
• Higher order constraints for the ($\beta$-deformed) Hermitian matrix models
• $W_{1+\infty }$ flows and multi-component hierarchy (KP case)
• Phases in WLZZ Matrix Models
• CMM formula as superintegrability property of unitary model
• Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations