CFT approach to constraint operators for (β-deformed) hermitian one-matrix models

We explain how the spectral curve can be extracted from the $$ \mathcal{W} $$ W -representation of a matrix model. It emerges from the part of the $$ \mathcal{W} $$ W -operator, which is linear in time-variables. A possibility of extracting the spectral curve in this way is important because there are models where matrix integrals are not yet available, and still they possess all their important features. We apply this reasoning to the family of WLZZ models and discuss additional peculiarities which appear for the non-negative value of the family parameter n, when the model depends on additional couplings (dual times). In this case, the relation between topological and 1/N expansions is broken. On the other hand, all the WLZZ partition functions are τ-functions of the Toda lattice hierarchy, and these models also celebrate the superintegrability properties.

Section: Model With M =mentioning

confidence: 99%

Spectral curves and W-representations of matrix models

Mironov

Морозов

2023

“…From the commutation relations (3.1) it is clear that operators Fn complemented by Ŵ0 generate the whole "positive" part of the w 1+∞ algebra that correspond to Ôn,k with k < 0. These operators are associated to what was called the positive branch of the WLZZ models [25][26][27]. In order to come to the negative WLZZ branch, one has to make [1] the transformation of any operator Ô(z) in accordance with Ô(z) −→ −z −1 Ô(z −1 )z.…”

Section: Integer Raysmentioning

confidence: 99%

Commutative families in W∞, integrable many-body systems and hypergeometric τ-functions

Mironov,

Mishnyakov,

Morozov

et al. 2023

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 arXiv:2303.05273, 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.

Commutative families in DIM algebra, integrable many-body systems and q, t matrix models

Mironov,

Morozov,

Popolitov

2024

We extend our consideration of commutative subalgebras (rays) in different representations of the W1+∞ algebra to the elliptic Hall algebra (or, equivalently, to the Ding-Iohara-Miki (DIM) algebra $$ {U}_{q,t}\left({\hat{\hat{\mathfrak{gl}}}}_1\right) $$ U q , t gl ̂ ̂ 1 ). Its advantage is that it possesses the Miki automorphism, which makes all commutative rays equivalent. Integrable systems associated with these rays become finite-difference and, apart from the trigonometric Ruijsenaars system not too much familiar. We concentrate on the simplest many-body and Fock representations, and derive explicit formulas for all generators of the elliptic Hall algebra en,m. In the one-body representation, they differ just by normalization from $$ {z}^n{q}^{m\hat{D}} $$ z n q m D ̂ of the W1+∞ Lie algebra, and, in the N -body case, they are non-trivially generalized to monomials of the Cherednik operators with action restricted to symmetric polynomials. In the Fock representation, the resulting operators are expressed through auxiliary polynomials of n variables, which define weights in the residues formulas. We also discuss q, t-deformation of matrix models associated with constructed commutative subalgebras.