Physics Letters B 802 (2020) 135237
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Complete solution to Gaussian tensor model and its integrable
properties
H. Itoyama
a
, A. Mironov
b,c,d,∗
, A. Morozov
e,c,d
a
Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP) and Department of Mathematics and Physics, Graduate School of Science, Osaka City
University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
b
I.E. Tamm Theory Department, Lebedev Physics Institute, Leninsky prospect, 53, Moscow 119991, Russia
c
ITEP, B. Cheremushkinskaya, 25, Moscow, 117259, Russia
d
Institute for Information Transmission Problems, Bolshoy Karetny per. 19, build. 1, Moscow 127051, Russia
e
MIPT, Dolgoprudny, 141701, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
25 October 2019
Received
in revised form 10 January 2020
Accepted
14 January 2020
Available
online 20 January 2020
Editor:
M. Cveti
ˇ
c
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.
© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
For Gaussian measures, it is possible to find an explicit full basis of gauge invariant observables, which have factorized averages and can
be written in the form of explicit rational functions of matrix size [1]. This means that matrix models are not just integrable, i.e. expressed
through distinguished, still transcendental τ -functions [2–4], but super-integrable like particles moving in especially nice potentials, say,
in oscillator or Coulomb ones. This special basis is actually formed by “characters”, which are the Schur or Macdonald polynomials [5] (in
the case of q, t-deformed models [6]). A similar property persists for logarithmic (hypergeometric) measures [7] when Selberg integrals
convert the generalized Macdonald polynomials [8]into factorized Nekrasov functions [9], this fact is used in the conformal matrix model
[10] proof [11]of the AGT relations [12].
In
this letter, we explain what happens in still another generalization: from matrices to tensors [13–17]. As explained in [18], in this
case of Gaussian measure, there is a large kernel, still the rainbow tensor models [19]remainsuper-integrable in the sense that beyond
the kernel one can still find an explicit basis with nicely factorized and explicitly calculable averages.
This
poses a further question of what super-integrability in the above sense implies for the ordinary integrability. The latter is usually
seen at two levels: as an infinite set of linear Ward identities (Virasoro-like constraints) [20] and as a set of bilinear Hirota-like equations
[21]. In the both cases, one needs an additional input, an appropriately defined generating function [22]of averages, which is then
identified as a τ -function subject to an additional string/Painleve constraint (this peculiar class is called “matrix-model τ -functions) [2,3,
23–26].
A natural choice of partition function for the correlators of characters are Cauchy sums with weights which are also characters,
in physical literature they are also known as Ooguri-Vafa partition functions (since they were used in the widely known paper [27]).
We demonstrate that, in the tensor case, these Ooguri-Vafa partition functions have rich, still limited integrability properties, which can
stimulate a new attention to [22] and a search for a somewhat better prescription for making the generating functions.
*
Corresponding author.
E-mail
address: mironov@lpi.ru (A. Mironov).
https://doi.org/10.1016/j.physletb.2020.135237
0370-2693/
© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.