W-representation of Rainbow tensor model

Bei Kang,Ke Wu,Jie Yang,Lu-Yao Wang,Wei-Zhong Zhao

doi:10.1007/jhep05(2021)228

Abstract

We analyze the rainbow tensor model and present the Virasoro constraints, where the constraint operators obey the Witt algebra and null 3-algebra. We generalize the method of W-representation in matrix model to the rainbow tensor model, where the operators preserving and increasing the grading play a crucial role. It is shown that the rainbow tensor model can be realized by acting on elementary function with exponent of the operator increasing the grading. We derive the compact expression of correlators and apply it to several models, i.e., the red tensor model, Aristotelian tensor model and r = 4 rainbow tensor model. Furthermore, we discuss the case of the non-Gaussian red tensor model and present a dual expression for partition function through differentiation.

Highlights

Here the subscript m of Om(N ) means that this operator acts on the variables p(km), the operator
We derive the compact expression of correlators and apply it to several models, i.e., the red tensor model, Aristotelian tensor model and r = 4 rainbow tensor model
W -representation is important for the understanding of matrix model, since it provides a dual formula for partition function through differentiation

Summary

W -representation of rainbow tensor model

For the rainbow model with the rank r complex tensors and with the gauge symmetry U =. Where Aji1,...,jr−1 is a tensor of rank r with one covariant and r − 1 contravariant indices, its conjugate tensor is Aij1,...,jr−1, σ is an element of the double coset Snr = Sn\Sn⊗r/Sn. and deg σ = n. We may choose some operators in (2.1) to generate a graded ring of gauge invariant operators with addition, multiplication, cut and join operations. The connected operators in this ring can generate the renormalization group (RG) completed rainbow tensor model [25]. DrZR(s) = sZR(s), WrZR(s) = μ(s + 1)ZR(s+1) It is similar with the case of the Gaussian hermitian model [1].

Correlators in the Aristotelian tensor model

Correlators in the red tensor model

Non-Gaussian red tensor model

Conclusions