W-representations of the fermionic matrix and Aristotelian tensor models

We show that the fermionic matrix model can be realized by $W$-representation. We construct the Virasoro constraints with higher algebraic structures, where the constraint operators obey the Witt algebra and null 3-algebra. The remarkable feature is that the character expansion of the partition function can be easily derived from such Virasoro constraints. It is a $\tau$-function of the KP hierarchy. We construct the fermionic Aristotelian tensor model and give its $W$-representation. Moreover, we analyze the fermionic red tensor model and present the $W$-representation and character expansion of the partition function.


Introduction
Matrix models provide a rich set of approaches to physical systems. Much interest has been attributed to their remarkable properties. As is well known, the partition function of some matrix models, such as the Gaussian Hermitian and complex matrix models [1]- [4], Kontsevich matrix model [5], can be presented as the forms of W -representations and character expansions of Schur polynomials. There is the superintegrability for these matrix models, which means that the average of a properly chosen symmetric function is proportional to ratios of symmetric functions on a proper locus, i.e., < character >∼ character. For the cases of β-and q, tdeformed matrix models, the character expansions of the partition functions are given by the Jack and Macdonald polynomials, respectively [6]- [8]. The W -representation gives a dual expression for partition function through differentiation rather than integration. The W -representations in terms of characters were analyzed in [3,4]. The constraints for matrix models enable us to effectively analyze the structures of matrix models. There have already been many works along this direction. Recently the remarkable works are devoted to analyze the character expansion of the Gaussian Hermitian model by using the Virasoro constraints [9,10].
Matrix models are associated with discretized random surfaces and 2D quantum gravity.
Tensor models are the generalizations of matrix models from matrices to tensor, which are originally introduced to describe the higher dimensional quantum gravity [11]- [13]. A few methods which allow one to connect calculations in the tensor models to those in the matrix models were introduced in [14]- [16]. The W -representations and character expansions of the partition functions have been extended to (rainbow) tensor models [17]- [20].
The fermionic matrix models involve matrices with anticommuting elements [21]- [25]. A well known fermionic one-matrix model is given by [25] where ψ andψ are independent complex Grassmann-valued N × N matrices, the integration measure in (1)  The fermionic matrix models are also generalized to the fermionic tensor cases [26,27]. It seems that the fermionic tensor model rather than the bosonic tensor model is more relevant to the problem of understanding holography. The gauge invariants of fermionic tensor model can be constructed by means of the representation theory. There are the different counting formulas for the number of gauge invariant operators in bosonic and fermionic tensor models [26].
Already a considerable amount is known about the W -representations of matrix and tensor models. Recently the corresponding analysis has been carried out for the supereigenvalue models [28,29]. The aim of the present paper is to make an attempt of studying W -representations of fermionic matrix and tensor models. More precisely, we focus on the fermionic matrix and Aristotelian tensor models and present their W -representations.
This paper is organized as follows. In section 2, we analyze the fermionic matrix model and give its W -representation. Furthermore, we derive the character expansion of the partition function and compact expressions of the correlators from the Virasoro constraints. In section 3, we construct the fermionic Aristotelian tensor model and give its W -representation. In section 4, we give W -representation of the fermionic red tensor model. Moreover we show that there is the character expansion for the partition function. We end this paper with the conclusions in section 5.
where all traces are normalized as trA = 1 The following Virasoro constraints can be derived from (2) by implementing invariance under the infinitesimal shifts ψ → ψ + ψ(ψψ) n (n ≥ 0),ψ →ψ, where The constraint operators (4) obey the Witt algebra However, straightforward calculation shows that they do not yield the closed higher algebraic structures.
Let us now take the change of variables given by By requiring that the partition function is invariant under the infinitesimal transformations (6), it leads to the constraint where the operatorsD andŴ are given bŷ To explore the properties ofD andŴ , we rewrite (2) as where Z (s) and l n=1 tr(ψψ) kn are the correlators defined by l n=1 tr(ψψ) kn = dψdψ l n=1 tr(ψψ) kn exp(N 2 trψψ) dψdψ exp(N 2 trψψ) .
Through the operatorsD andŴ acting on Z (s) DZ (s) Ŵ Z (s) we see thatD andŴ are the operators preserving and increasing the grading, respectively. The commutator ofD withŴ is In terms of the operatorŴ , the partition function (2) can be realized by the W -representation which is the dual expression of (2) through differentiation rather than integration.
Let us write the higher power ofŴ aŝ where P i 1 ,··· ,i l j 1 ,··· ,j k are the expansion coefficients. Then it is not difficult to give the correlators from the W -representation (15) where m = i 1 + · · · + i l , τ denotes all distinct permutations of (i 1 , · · · , i l ) and λ (i 1 ,··· ,i l ) is the number of τ with respect to (i 1 , · · · , i l ).
Here the chiral transformation is the analog of the reflection of the hermitian matrix φ → −φ in the Gaussian Hermitian matrix model, which makes all the odd moments vanishing, i.e., tr(φφ) 2k+1 = 0.

Character expansion from the Virasoro constraints
By means of the operatorsD andŴ , we may construct the Virasoro constraints where the constraint operators are given by which yield the Witt algebra (5) and null 3-algebra There is a significant difference between the Virasoro constraint operators (4) and (21). From the Virasoro constraints (20), we can derive the correlators (17) as well. There are the recursive relations of correlators from the Virasoro constraints (3). We can calculate the correlators from these recursive relations. However, it is hard to give the compact expression (17).
To achieve the character expansion of the partition function, let us rewrite (2) as where R = {R 1 , · · · , R l (R) } are the Young diagrams of the given size (number of boxes) |R| = i R i and length l (R) , χ R is the Schur polynomial, and we have used the Cauchy formula In order to show that there is the character expansion for the partition function, the key point depends on the action result ofŴ m+1 · χ R .
Since the operatorŴ acting on χ R giveŝ here (i , j ) are the coordinates of the squares added to the Young diagram R, we havê where S is skew Young diagram with |S| = m+1, n(S) is the number of standard Young tableau with shape S.
Due to (26), there are the recursive relations from (24) where |R| = m + 1, we have interchanged the indices R and S for later convenience.
The explicit form of C R is obtained from (27) with initial value C ∅ = 1 The coefficient n(R) in (28) also represents the dimension of the irreducible representation π R of symmetry group S |R| , i.e., n(R) = is the number of boxes in hook. By using the hook formulas [30,31] and where D R (N ) = χ R {p k = N } and d R = χ R {p k = δ k,1 } are respectively the dimension of representation R for the linear group GL(N ) and symmetric group S |R| divided by |R|!, we obtain the final expression of (28) Thus we reach the desired result For any partition function of the form [32] Z with the function ω R = Π (i,j)∈R f (i−j) andp k just arbitrary parameters, it is a τ -function of the KP hierarchy [33]- [36]. Since (32) coincides with (33), we conclude that the character expansion of the fermionic matrix model is indeed a τ -function of the KP hierarchy.
We note that the action ofŴ n · 1 can be written aŝ where λ = (λ 1 , · · · , λ l ) is a partition of n, σ is a permutation in the symmetry group S l , ψ R (λ) is the symmetric group character. In deriving the above action, we have used the formula p λ = By means of (34), it is easy to give another expression for C R from (32)

W -representation of the fermionic Aristotelian tensor model
The Aristotelian tensor model with a single complex tensor of rank 3 and the RGB (red-greenblue) symmetry is the simplest of the rainbow tensor models [14]. It can be realized by Wrepresentation. In the previous section, we have analyzed the fermionic matrix model. Let us now turn to the fermionic generalization of Aristotelian tensor model.
We introduce the gauge-invariant operators of level n where the gauge symmetry is U (N 1 )⊗U (N 2 )⊗U (N 3 ),Ψ i j 1 ,j 2 and Ψ j 1 ,j 2 i are the fermionic tensors of rank 3 with one covariant and two contravariant indices which are assigned with different color, σ is an element of the double coset S 3 n = S n \S ⊗3 n /S n . The fermions are described by Grassmann variables. There is a natural action of S 3 n on U (N 1 ) ⊗ U (N 2 ) ⊗ U (N 3 ) defined as follows: where ξ is a permutation in S 3 n . In similarity with the Aristotelian tensor model, we may introduce the cut operation ∆ F and join operation { } F on the gauge-invariant operators and where ∆ β 1 ,··· ,β k α and γ β σ,α are coefficients.
It is known that the keystone operators play an important role in the Aristotelian tensor model [14,15]. They may generate a graded ring of gauge invariant operators with addition, multiplication, cut and join operations. The ring contains the so-called tree and loop operators.
When the operator belongs to the sub-ring generated only by the join operation, this operator is the tree operator, otherwise, it is the loop operator.
For the case of fermionic Aristotelian tensor model, the ring is generated by the following keystone operators In terms of the keystones operators, connected tree and loop operators in the ring, we introduce the fermionic Aristotelian tensor model where the measure is induced by the norm δΨδΨ = δΨ i and the correlators K (n 1 ) σ 2 · · · K (n l ) σ l are defined by Considering the deformation δΨ = and N = N 1 N 2 N 3 , the permutations α, σ and β are taken from indices of connected operators in the ring.
The commutator ofD withW is Straightforward calculation of the operatorsD andW acting on Z (s) It indicates that the operatorsD andW are also the operators preserving and increasing the grading, respectively. Thus the partition function can be realized by acting on elementary function with exponents of the operatorW We may also introduce the Virasoro constraints where the constraint operatorsL m are given bỹ which yield the Witt algebra (5) and null 3-algebra (22).

W -representation and character expansion of the fermionic red tensor model
The red tensor model was constructed in [14]. It is indeed equivalent to a rectangular matrix model. In order to generalize the red tensor model to the fermionic case, let us consider the rank r fermionic tensors and introduce the gauge-invariant operators where the gauge symmetry is U (N 1 ) ⊗ · · · ⊗ U (N r ), we take σ = (12 · · · n) ⊗ id ⊗ · · · ⊗ id to be the simplest element of the double coset S r n = S n \ S ⊗r n /S n . It is clear that K n are the connected operators.
We introduce the fermionic red tensor model where N r = N 1 N 2 N 3 · · · N r .
Following the same procedure as in the previous section, we may obtain the corresponding operators preserving and increasing the gradinḡ In deriving the above operators, we have considered the deformation δΨ = Similarly, the partition function (58) can be realized by the W -representation Moreover, there are the Virasoro constraints where the Virasoro constraint operators areL m =W m (W −D) which yield the Witt algebra (5) and null 3-algebra (22) as well.
Following the same procedure as in the fermionic matrix model, we may derive the character expansion of (58) from the Virasoro constraints (62) where It also coincides with the τ -function (33) of the KP hierarchy.
Moreover there is the compact expression of correlators where P τ (α 1 ),··· ,τ (α i ) are the coefficients of the term t (a 1 ) τ (α i ) inW m . The first several exact correlators are K 1 = 1,

Conclusions
We have analyzed the fermionic matrix model with complex Grassmann-valued N × N matrices.
Similar to the Gaussian hermitian model case [1], there exist the operatorsD andŴ which preserve and increase the grading, respectively. Thus the partition function can be realized by the W -representation. The compact expression of correlators has been presented. In terms of the operators preserving and increasing the grading, we may construct the Virasoro constraints such that the constraint operators obey the Witt algebra and null 3-algebra. The remarkable feature is that the character expansion of the partition function can be easily derived from such Virasoro constraints. It is known that the superintegrability looks like a property < character >∼ character [4]. The result of character expansion of the fermionic matrix model shows that it is a τ -function of the KP hierarchy.
We have given the fermionic Aristotelian and red tensor models which can also be realized by the W -representations. Since there exist the operators preserving and increasing the grading in these models, similarly, we may construct the desired Virasoro constraint operators which lead to the higher algebraic structures. The compact expressions of correlators in these models can be derived from such Virasoro constraints. For the fermionic Aristotelian tensor model, it should be noted that we have to add the infinite set of time variables in the partition function.
As the case of fermionic matrix model, there is one set of time variables in the fermionic red tensor model. Thus similar character expansion of the partition function can be derived from the Virasoro constraints with higher algebraic structures, which is a τ -function of the KP hierarchy as well. Finally, it should be pointed out that the Virasoro constraints with higher algebraic structures are also applicable to the character expansions of the Gaussian hermitian matrix, complex matrix and red tensor models. For further research, it would be interesting to study the cases of β and q, t-deformed models