1. Commutative families in $W_\infty$, integrable many-body systems and hypergeometric $τ$-functions
- Author
-
Mironov, A., Mishnyakov, V., Morozov, A., and Popolitov, A.
- Subjects
High Energy Physics - Theory (hep-th) ,FOS: Physical sciences ,Mathematical Physics (math-ph) - Abstract
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 $W_{1+\infty}$ algebra. We construct many such subalgebras and explain how they look in various representations. We start from the even simpler $w_\infty$ 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 $W_{1+\infty}$ 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 $W_{1+\infty}$ 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., 43 pages
- Published
- 2023
- Full Text
- View/download PDF