Complete solution to Gaussian tensor model and its integrable properties

Abstract

Similarly to the complex matrix model, the rainbow tensor models are superintegrable in the sense that arbitrary Gaussian correlators are explicitly expressed through the Clebsh-Gordan coefficients. We introduce associated (Ooguri-Vafa type) partition functions and describe their W -representations. We also discuss their integrability properties, which can be further improved by better adjusting the way the partition function is defined. This is a new avatar of the old unresolved problem with non-Abelian integrability concerning a clever choice of the partition function. This is a part of the long-standing problem to define a non-Abelian lift of integrability from the fundamental to generic representation families of arbitrary Lie algebras.

Correlators in the Gaussian and chiral supereigenvalue models in the Neveu-Schwarz sector

“…Finally, it should be mentioned that the partition function for (deformed) Gaussian matrix models have the explicit character expansions [19]- [21], [33,34,41]. The character JHEP11(2020)119 expansions imply that the expectation values of characters are again characters, which is considered as the superintegrability property of the models [42].…”

“…Moreover, the corresponding Ramond super quantum curves were analyzed and the underlying Ramond super spectral curves were identified. It is well-known that the W -representations can be used to realize partition functions of various matrix models, such as the Gaussian Hermitian and complex matrix models [18]- [21], the Kontsevich matrix model [22] and the generalized Brezin-Gross-Witten model [23]. Namely, by acting on elementary functions with exponents of the given W -operators, we can give the corresponding partition functions of the matrix models.…”

Superintegrability of Kontsevich matrix model

“…The results of this section for the complex matrix model can be straightforwardly extended to the Gaussian tensor model in background fields. To this end, we follow [36,37] and again apply the technique of [8].…”

“…In fact, the labeling is reduced to a double coset S m \S ⊗r m /S m [38][39][40][41][42], but we do not need these details here. There is also a distinguished set of operators called generalized characters in [36,37] that are defined as…”

Genus expansion of matrix models and ћ expansion of KP hierarchy

“…Tensor models often have the same properties as matrix models, e.g. Feynman diagrams expansion and integrability [14][15][16]. Finally, ( , )-deformed matrix models have found their use in localization computations in supersymmetric gauge theories.…”