W-representation of Rainbow tensor model

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.


Introduction
W -representation of matrix model which realizes partition function by acting on elementary functions with exponents of the given W -operator has attracted considerable attention. It indeed gives a dual expression for partition function through differentiation rather than integration. As the fundamental matrix models, it turned out that the Gaussian Hermitian and complex matrix models can be written as the form of the W -representations [1]- [4]. Since these W -representations can be expressed in terms of characters, the corresponding matrix models are reformulated as the sum of Schur functions over all Young diagrams. For the β-deformed Gaussian Hermitian and complex models, their W -representations still exist. The character expansions of these models can be given by the Jack polynomials. The studies of W -representations have also been devoted to the supersymmetric generalizations of matrix models, i.e., supereigenvalue models [5,6].
As the generalizations of matrix models from matrices to tensor, tensor models become very useful in the deep study of higher dimensional quantum gravity [7]- [9]. Quite recently, the operators/Feynman diagrams correspondence in quantum field theory was provided [10]. For the number of Feynman diagrams with n propagators in the rank r − 1 complex tensor model, it is equal to the number of singlet operators with 2n vertices in the rank r complex tensor model.
Tensor models are also very interesting in their own right [11]- [22]. Recently the tensorial generalization of characters [18,19] and correlators in tensor models from character calculus [20]- [22] have been analyzed. The Gaussian tensor model is a model of complex r-tensors with the Gaussian action. It can be expressed as the forms of the characters and W -representation [22], where ψ R (∆), z ∆ and χ Rm {p (m) } are respectively the character of symmetric group, the symmetry factor of Young diagram ∆ and the Schur function as a function of time-varibles p k , the operatorŴ(N 1 , · · · , N r ) is given bŷ (1.2) here the subscript m ofÔ m (N ) means that this operator acts on the variables p (m) The Aristotelian rainbow tensor model with a single complex tensor of rank 3 and the RGB (red-green-blue) symmetry is the simplest of the rainbow tensor models [23]- [25]. Recently, with the example of the Aristotelian tensor model, Itoyama et al. [23] introduced a few methods which allow one to connect calculations in the tensor models to those in the matrix models.
Well known is that the partition functions of various matrix models can be realized by the Wrepresentations, where the operators preserving and increasing the grading play a crucial role [1]. The goal of this paper is to make a step towards the W -representation of the rainbow tensor model. We present its W -representation and give the compact expression of correlators.
This paper is organized as follows. In section 2, we show that the rainbow tensor model can be realized by acting on elementary function with exponent of the given operator. The Virasoro constraints are also presented. Then we derive the compact expressions of the correlators. In sections 3 and 4, we focus on the correlators in the Aristotelian and r = 4 tensor models, respectively. In section 5, we consider the (Non-Gaussian) red tensor model. We end this paper with the conclusions in section 6.

W -representation of rainbow tensor model
For the rainbow model with the rank r complex tensors and with the gauge symmetry U = U (N 1 ) ⊗ · · · ⊗ U (N r ), the gauge-invariant operators of level n are given by [25] is a tensor of rank r with one covariant and r − 1 contravariant indices, its conjugate tensor isĀ i j 1 ,...,j r−1 , σ is an element of the double coset S r n = S n \S ⊗r n /S n and deg σ = n. Here the different types of indices in the fields and fields themselves are assigned with different color. We may choose some operators in (2.1) to generate a graded ring of gauge invariant operators with addition, multiplication, cut and join operations. These operators are called keystones. The connected operators in this ring can generate the renormalization group (RG) completed rainbow tensor model [25].
Let us introduce the RG-completed rainbow tensor model where µ is a constant, t and the correlators K (n 1 ) .
where ∆ and {} are respectively the cut and join operations, the actions of the cut and join operations on the gauge-invariant operators are and ∆ β 1 ,··· ,β k α and γ β σ,α are the coefficients.
From (2.5), we may deduce that the partition function (2.2) satisfies (2.10) The commutation relation betweenD r andŴ r is In terms of the operatorsD r andŴ r , we may introduce the Virasoro constraints where the constraint operators L m are given by which yield the Witt algebra and null 3-algebra Let us consider the operatorsD r andŴ r acting on Z (s) R , respectively. We havê It is similar with the case of the Gaussian hermitian model [1]. We immediately recognize that the operatorsD r andŴ r are indeed the operators preserving and increasing the grading, respectively. Thus the partition function can be realized by acting on elementary function with exponents of the operatorŴ r As done in the matrix models, we formally write the m-th power of the operatorŴ r aŝ where the coefficients P α 1 ,··· ,α i r and (P r ) α 1 ,··· ,α i β 1 ,··· ,β j are the polynomials of N 1 , · · · , N r . Substituting (2.19) into (2.18) and comparing the coefficients of t (a 1 ) α i in the expansion of (2.18) with the corresponding terms in (2.2), we finally derive the compact expression of correlators where m = a 1 + · · · + a i , τ denotes all distinct permutations of (α 1 , · · · , α i ) and λ (α 1 ,··· ,α i ) is the number of τ with respect to α 1 , · · · , α i .
Let us turn to the Virasoro constraints (2.12). It can be rewritten aŝ Since the coefficients of t (a 1 ) α i on both sides in (2.21) with i j=1 a j = m are equal, we can not only derive the correlators (2.20), but also the exact correlators For the Gaussian average of the rank r operator O (r) in the rank r complex tensor model, there is a limit relation with O (r+1) r+1 in the rank r + 1 model [10], i.e., where a 1 + · · · + a i = m, S m is the symmetric group that consists of permutations of m elements, When particularized to the special correlators (K 1 ) i r , we have Taking i = 1, 2 and 3 in (2.25), respectively, it gives

Correlators in the Aristotelian tensor model
In the Aristotelian model with the tensor A j 1 ,j 2 i of rank r = 3, the ring is generated by keystone operators [23] α, σ and β are taken from the indices of connected operators in the ring.
For the Aristotelian tensor model (3.2), the Virasoro constraint operators in (2.12) become whereD 3 is given by (2.9) with r = 3 in which the index α is an element of the double coset S 3 n = S n \S ⊗3 n /S n . Since the coefficients P τ (α 1 ),··· ,τ (α i ) in (2.20) follow from the precise expression ofŴ m , we may give the exact correlators from (2.20). In particular, Let us give the correlators K 1 K 1 and K 1 K 1 K 1 and represent these correlators graphically as follows: where the thick line depicts the Feynman propagator and each propagator gives a factor 1 µ , the red, green and blue circles represent N 1 , N 2 and N 3 , respectively. By calculatingŴ i 3 , i = 1, ..., 4, we obtain the correlators which have been derived in [25]. In the following, we give two correlators by calculatingŴ 5 (3.7)

Correlators in the r = rainbow tensor model
In the rainbow tensor model with the tensor A j 1 ,j 2 ,j 3 i of rank r = 4, the keystone operators are (4.1) The ring generated by the keystone operators contains the tree and loop operators, red circles K n = K (12···n)⊗id⊗id⊗id , green circles K n = K id⊗(12···n)⊗id⊗id and disconnected collections of the red and green circles, respectively.
In similarity with the case of Aristotelian tensor model, the tree and loop operators can also be constructed. In figures.3 and 4, we draw some tree and loop operators involving chains with both K 2 and K 2 , respectively.
where the operatorŴ 4 is given bŷ For the partition function (4.3), the Virasoro constraint operators in (2.12) are whereD 4 is given by (2.9) with r = 4 in which the index α is an element of the double coset S 4 n = S n \S ⊗4 n /S n . We may give the exact correlators from (2.20), where the coefficients P τ (α 1 ),··· ,τ (α i ) in (2.20) follow from the power ofŴ 4 (4.4). The special correlators (K 1 ) i are given by (2.22) with Let us list some correlators (2.20) by calculatingŴ i 4 , i = 1, 2, 3, and represent them graphically with the same rules as the case of Aristotelian tensor model. Noted that the thick (4colored) lines depict the Feynman propagators, and the yellow circles represent N 4 .
(i) (4.6) there is only one possible attachment of the Feynman propagators to the operator K 1 , giving the result N 4 µ . (ii) Due to too many Feynman diagrams for the correlators (4.9), we do not present them here.
We introduce the red tensor model where we denote the index i in A j 1 ,...,j r−1 i with color red, the operatorŴ is given bŷ When particularized to the rank 3 tensor case in the partition function (5.3), the operator (5.4) reduces to the result derived in Ref. [23].
For the case of the red tensor model (5.3), the Virasoro constraint operators in (2.12) are 5) and the correlators (2.20) become where m = a 1 + · · · + a i andD = ∞ a=1 at a ∂ ∂t a . Furthermore the correlators (K 1 ) i are given by (2.22) with N r = N 1 N 2 N 3 · · · N r .

Non-Gaussian red tensor model
The correlators in the matrix models including supereigenvalue models have attracted considerable attention. Much interest has also been attributed to the non-Gaussian cases [26]- [30].
But so far, no investigation has been made for the non-Gaussian tensor models. The red tensor model is a simple rainbow tensor model. In the previous section, we have presented its W -representation and the correlators. Let us now focus on the non-Gaussian red tensor model t n K n ). (5.8) In this model, the keystone operator is K 2 (5.2), the tree and loop operators are the same with the case of the red tensor model.
The operators yield the Witt algebra (2.14), but the null 3-algebra (2.15) does not hold.
Let us take the deformation δA = ∞ m=0 (m + p)t m+p ∂K m+1 ∂Ā of the integration variable in the integral (5.8), we obtain On the other hand, there are the additional constraints for the partition function (5.8) Combining (5.14) and (5.11), we have By expanding the exponential term in (5.8), we rewrite the partition function as t n 1 t n 2 K n 1 K n 2 N G + · · · ), (5.16) where Z (s) l=0 n 1 +···+n l =s 1 l! t n 1 t n 2 · · · t n l K n 1 K n 2 · · · K n l N G , (5.17) the correlators are defined by .

(5.19)
The operatorsD andŴ acting on Z (5.20) Thus the partition function (5.8) can be expressed as We observe that (5.21) is similar with the case of non-Gaussian matrix model [30]. It shows that the dual expression for the non-Gaussian red tensor model (5.8) through differentiation can also be formulated.
Since the usual W -representation of (5.8) fails, we can not present the compact expression of correlators here. In principle, we can give the correlators from (5.21). Let us list some correlators as follows: ∂T q Z N G {T }, (q = 1, · · · , p − 1). (5.22)

Conclusions
W -representation is important for the understanding of matrix model, since it provides a dual formula for partition function through differentiation. We have investigated the W -representation of the rainbow tensor model with the rank r complex tensor in this paper. By the given deformation of the integration variable in the integral, we derived the desired operatorsD r and W r which preserve and increase the grading, respectively. It was shown that the rainbow tensor model can be realized by acting on elementary function with exponent of the operatorŴ r . In terms of the operators preserving and increasing the grading, we can construct the Virasoro constraints for the rainbow tensor model, where the constraint operators obey the Witt algebra and null 3-algebra. An interesting aspect of these Virasoro constraints is that the compact expression of correlators can be derived from them. As examples, we have applied above results to analyze the red tensor model, Aristotelian tensor model and r = 4 rainbow tensor model in detail and presented the corresponding correlators in these models.
We have also considered the non-Gaussian red tensor model and presented the Virasoro constraints, where the constraint operators obey the Witt algebra, however the null 3-algebra does not hold. We showed that the partition function can be expressed as the infinite sum of the operators (D −1Ŵ ) k acting on the given function. Namely, a dual form for partition function through differentiation can be formulated. SinceŴ is not the desired operator increasing the grading, it causes the usual W -representation of the non-Gaussian red tensor model to fail here. For this reason, the dual expression (5.21) through differentiation can be regarded as the generalized W -representation. In terms of the operatorsD andŴ, we can not construct the Virasoro constraints such that the constraint operators obey the Witt algebra and null 3algebra. It should be noted that we can calculate the correlators from (5.21). However, the compact expression of correlators can not be derived. We have presented some correlators. How to represent these correlators graphically still deserves further study. Furthermore, further study should be done to investigate the W -representations of the non-Gaussian and fermionic tensor models.