Bilinear equations for the strict KP hierarchy

In this paper we show first of all that for solutions of the strict KP hierarchy it is sufficient to work in a standard setting. Further we discuss a minimal realization of the hierarchy and present the scaling invariance of the Lax equations of the hierarchy.

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Genus expansion of matrix models and ћ expansion of KP hierarchy

Journal of High Energy Physics ◽

10.1007/jhep12(2020)038 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

A. Andreev ◽

A. Popolitov ◽

A. Sleptsov ◽

A. Zhabin

Keyword(s):

Matrix Models ◽

Matrix Model ◽

Special Interest ◽

Hermitian Matrix ◽

Geometric Meaning ◽

Kp Hierarchy ◽

Hurwitz Numbers ◽

Kp Equations ◽

Genus Expansion ◽

Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

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BPS Quivers of Five-Dimensional SCFTs, Topological Strings and q-Painlevé Equations

Annales Henri Poincaré ◽

10.1007/s00023-021-01034-3 ◽

2021 ◽

Author(s):

Giulio Bonelli ◽

Fabrizio Del Monte ◽

Alessandro Tanzini

Keyword(s):

Topological String ◽

Cluster Algebra ◽

Painlevé Equations ◽

Partition Functions ◽

Quantum Field Theories ◽

Particle Spectrum ◽

Yang Mills ◽

Painleve Equations ◽

Rank One ◽

Bilinear Equations

AbstractWe study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $$\tau $$ τ -functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU(2) pure super Yang–Mills and $$N_f=2$$ N f = 2 on a circle.

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Intertwining operator and integrable hierarchies from topological strings

Journal of High Energy Physics ◽

10.1007/jhep05(2021)216 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Jean-Emile Bourgine

Keyword(s):

Topological Strings ◽

Vertex Operator ◽

Integrable Hierarchies ◽

Kp Hierarchy ◽

Tau Function ◽

Time Deformation ◽

Toroidal Algebra ◽

Crystal Model ◽

Melting Crystal ◽

Definition Of

Abstract In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum W1+∞ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner — or vertex operator — of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $$ \mathfrak{gl} $$ gl (1) (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.

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