Spectra of Operators in Large $N$ Tensor Models

We study the operators in the large $N$ tensor models, focusing mostly on the fermionic quantum mechanics with $O(N)^3$ symmetry which may be either global or gauged. In the model with global symmetry we study the spectra of bilinear operators, which are in either the symmetric traceless or the antisymmetric representation of one of the $O(N)$ groups. In the symmetric traceless case, the spectrum of scaling dimensions is the same as in the SYK model with real fermions; it includes the $h=2$ zero-mode. For the operators anti-symmetric in the two indices, the scaling dimensions are the same as in the additional sector found in the complex tensor and SYK models; the lowest $h=0$ eigenvalue corresponds to the conserved $O(N)$ charges. A class of singlet operators may be constructed from contracted combinations of $m$ symmetric traceless or antisymmetric two-particle operators. Their two-point functions receive contributions from $m$ melonic ladders. Such multiple ladders are a new phenomenon in the tensor model, which does not seem to be present in the SYK model. The more typical $2k$-particle operators do not receive any ladder corrections and have quantized large $N$ scaling dimensions $k/2$. We construct pictorial representations of various singlet operators with low $k$. For larger $k$ we use available techniques to count the operators and show that their number grows as $2^k k!$. As a consequence, the theory has a Hagedorn phase transition at the temperature which approaches zero in the large $N$ limit. We also study the large $N$ spectrum of low-lying operators in the Gurau-Witten model, which has $O(N)^6$ symmetry. We argue that it corresponds to one of the generalized SYK models constructed by Gross and Rosenhaus. Our paper also includes studies of the invariants in large $N$ tensor integrals with various symmetries.


Introduction and Summary
Models where the degrees of freedom are tensors of rank r > 2 offer the possibility of large N limits dominated by the so-called melon diagrams, if the interactions are chosen appropriately [1][2][3][4][5][6][7][8][9][10][11]. In models where the tensor indices are distinguishable, so that the symmetry group is O(N ) r for example, the proofs of melonic limits have been available for several years. 1 During the recent months, interest in the melonic large N tensor models has been boosted by their connections [17,18] with the Sachdev-Ye-Kitaev model [19][20][21][22] and its generalizations [23], as well as by connections with the large N matrix models [24]. In particular, the Schwinger-Dyson equations which determine the scaling dimensions of a class and U (N ) 2 × O(N ) symmetries, as well as the symmetric traceless and fully antisymmetric rank-3 tensors under a single O(N ) group.
Beyond classifying the invariant operators, it is important to determine their infrared scaling dimensions. We begin work on this in section 5 and point out that there is a large class of 2k-particle operators whose large N scaling dimensions are simply additive, i.e. k/2. This is because the melonic ladders contribute only to 1/N corrections. However, although less generic, there are operators whose dimensions are not simply quantized. While the Regge trajectory operators studied in [18,22,23,[25][26][27] receive single ladder contributions, there are operators whose two-point functions have multi-ladder contributions. Since a ladder may contain an h = 2 zero-mode, the m-ladder diagram seems to produce a low-temperature enhancement by (βJ) m . This may be an important physical effect in the melonic tensor models, whose detailed analysis we leave for the future.
Besides our analysis of the spectra of O(N ) 3 symmetric models, we make some comments about the O(N ) 6 symmetric Gurau-Witten model [17]. Some features of its spectrum are identical to those in the q = 4, f = 4 Gross-Rosenhaus flavored generalization [23] of the SYK model. The connections of the Gurau-Witten model with this Gross-Rosenhaus model have been also noted using combinatorial analysis in [47].
After this paper was completed, we became aware of the interesting paper [48], which has some overlap with our results. This model is a somewhat simplified version of the O(N ) 6 symmetric Gurau-Witten model [17]. Both are in the class of 3-tensor models which possess a "melonic" large N limit where J = gN 3/2 is held fixed [1][2][3][4][5][6][7][8][9][10][11]. The large N model is nearly conformal in the IR [19,22]; for example, the two-point function is T (ψ abc (t 1 )ψ a b c (t 2 )) = −δ aa δ bb δ cc 1 4πg 2 N 3 1/4 sgn(t 1 − t 2 ) |t 1 − t 2 | 1/2 . Each O(N ) group includes parity transformations (axis reflections) P a 0 : for a given a 0 , P a 0 sends ψ a 0 bc → −ψ a 0 bc for all b, c and leaves all ψ a 1 bc , a 1 = a 0 invariant. In a physical language, these are "big" gauge transformations and operators should be invariant under them. Therefore we can build operators using ψ abc and the delta symbol δ aa only. In the case of SO(N ) gauge group one can use the fully antisymmetric tensor a 1 ...a N as well; it is invariant under SO(N ), but changes its sign under the parity transformations. Because of this, there are additional "long" operators containing at least N fields, like The difference between gauging O(N ) and SO(N ) becomes negligible in the large N limit.
Let us define three operations which permute pairs of the O(N ) symmetry groups (and thus interchange indices in the tensor field), while also reversing the direction of time, Each of these transformations preserves the equations of motion for the ψ abc field, The Hamiltonian, including a quantum shift due to (2.2), changes sign under each of the transformations s ab , s bc , s ac (this is discussed in section 4).
This means that these transformations are unitary: they preserve e iHt . In contrast, the usual time reversal transformation is anti-unitary because it also requires complex conjugation The O(N ) 3 invariant operators form representations under the permutation group S 3 , which acts on the three O(N ) symmetry groups (it contains the elements s ab , s bc and s ac ).
For example, H is in the degree 1 "sign representation" of S 3 : it changes sign under any pair interchange, but preserves its sign under a cyclic permutation.
It is also interesting to study the spectrum of eigenstates of the Hamiltonian for small values of N ; first steps on this were made in [49][50][51]. When gauging the O(N ) 3 symmetry one needs to worry about the Z 2 anomaly, which affects the gauged O(N ) quantum mechanics with an odd number of flavors of real fermions in the fundamental representation [52,53].
Since for each of the three O(N ) groups we find N 2 flavors of fundamental fermions, the gauged model is consistent for even N , but is anomalous for odd N . 4 This means that, for odd N , the spectrum does not contain states which are invariant under O(N ) 3 (for N = 3 this can be seen via an explicit diagonalization of the Hamiltonian (2.12) [49]). 4 We are grateful to E. Witten for pointing this out to us.

Composite Operators and Schwinger-Dyson Equations
The scaling dimensions of a class of bilinear operators may be extracted from the 4-point and factorizing it in the channel where t 1 → t 2 and t 3 → t 4 . A class of melonic ladder graphs appears in this channel in the large N limit; it may be summed by means of a Schwinger-Dyson equation. The singlet bilinear operators form a "Regge trajectory." Their scaling dimensions are the same as in the SYK model [19,22], and they have been extensively analyzed in the literature [23,[25][26][27]. The dimensions are determined by the equation and the first few solutions are h = 2, 3.77, 5.68, . . .. As pointed out in [18], the model also contains a multitude of multi-particle singlet operators. As we will see, some special combinations of the multi-particle operators are related by the equations of motion to the operators (3.2), but most multi-particle operators are genuinely new.
Interestingly, there are also certain non-singlet operators which are renormalized by the melonic ladder diagrams. This can be seen, for example, from the 4-point function factorized in the channel t 1 → t 2 and t 3 → t 4 . As shown in figure 1, all the melonic ladders again make non-vanishing contributions in the large N limit. Here we find two classes of non-singlet bilinear operators: those symmetric and traceless in a 1 and a 2 , and those anti- The first few solutions of this equation are h = 0, 2.65, 4.58, . . ., and each one appears with multiplicity 3 2 N (N − 1). The spectrum includes the special h = 0 mode corresponding here to the n = 0 operators, which are the O(N ) 3 charges (2.6).
The 4-point function (3.4) may also be factorized in the channel t 1 → t 3 and t 2 → t 4 . 5 We are grateful to Shiraz Minwalla for very useful discussions on this; see the paper [48].

This leads to the spectrum of operators
We can see from figure 2 that the ladder contribution to this operator are subleading in 1/N : the rightmost diagram is of ladder type and is ∼ g 2 N 3 , which is suppressed by a power of N relative to the other two diagrams. Therefore the large N scaling dimensions of these operators are 1/2 + m.

Construction of O(N ) 3 invariant operators
In this section we study the spectrum of O(N ) 3 invariant operators. Since a time derivative may be removed using the equations of motion (2.11), we may write the operators in a form where no derivatives are present. The bilinear singlet operator, ψ abc ψ abc , vanishes classically by the Fermi statistics, while at the quantum level taking into account (2.2), it is a C-number.
The first non-trivial operators appear at the quartic level and are shown in figure 4 (from here on we will not be careful about the quantum corrections to operators). Figure 4: All the four-particle operators, the tetrahedron and the three pillows, with the index contractions shown explicitly.
On the left is the "tetrahedron operator" O tetra , which is proportional to the Hamiltonian (2.12): One can check that and also that s ab O tetra = −O tetra and s ac O tetra = −O tetra . Thus, the tetrahedron operator O tetra is in the degree 1 "sign representation" of S 3 : it changes sign under any pair interchange, but preserves its sign under a cyclic permutation.
The three additional operators in figure 4, which we denote as O pillow and O pillow , are the "pillow" operators in the terminology of [6,10]; they contain double lines between a pair of vertices. For example, for O Under the S 3 the three pillow operators decompose into the trivial representation of degree If we iterate the use of the equation of motion (2.11), then all derivatives in an operator may be traded for extra ψ-fields. Thus, a complete basis of operators may be constructed by multiplying some number 2k of ψ-fields and contracting all indices. In this approach, there is a unique operator with k = 2(n + 1) which is equal to the Regge trajectory operator ψ abc ∂ 2n+1 t ψ abc . For n = 0 this operator is O tetra , which is proportional to the Hamiltonian; for n = 1 it will be constructed explicitly in section 4.1.
may be written as This may be seen by cutting the diagram for this operator in figure 5 along the vertical symmetry axis. To show that also vanishes, we may permute the first two ψ-fields to write it as One may wonder if the vanishing extends to the 10-particle operators. We have checked that the operators shown in figure 6 all vanish; this is due to the reflection symmetry present for these operators. For example, the left operator in figure 6 vanishes because it may be written as (ψ 5 ) abc (ψ 5 ) abc , which may be seen by cutting the diagram along the vertical symmetry axis. We note that Similarly, by cutting the third diagram in figure 6 along its vertical symmetry axis, we see that the corresponding operator may be written as ( vanishes as well. This argument extends to all the reflection symmetric (4n + 2)-particle diagrams.
However, not all 10-particle operators vanish. For example, the operators shown in figure   7 do not have a reflection symmetry, and we have checked that they do not vanish.
Let us note that each gauge invariant operator, where all the indices are contracted, corresponds to a vacuum Feynman diagram in the theory with three scalar fields and interaction λϕ 1 ϕ 2 ϕ 3 (the three different propagators correspond to the lines of three different colors in our figures). In the theory of bosonic tensors φ abc , the number of operators made out of 2k Figure 7: Some non-vanishing ten-particle operators.
fields is precisely the number of distinct Feynman diagrams appearing at order λ 2k , which grows as k!2 k . In the fermionic model, some of the operators vanish by the Fermi statistics, while others due to the gauge constraint. Nevertheless, we will find that the factorial growth holds also in the fermionic model.

Eight-particle operators
In this section we explicitly construct all the eight-particle operators without bubble (double Their pictorial representations are shown in the first column of figure 9. Using the equations of motion, we may write them as It follows that which up to a total derivative equals the Regge trajectory operator ψ abc ∂ 3 t ψ abc . It follows that (s ab , s ac , s bc ) :  Similarly, we may write down the three operators which correspond to the second column in figure 9 (the first of these operators,Õ 1 , was written down in [18]): Via the equations of motion, these operators are related to the bilinear operators defined in These relations will be used in the next section.

Scaling Dimensions of Multi-Particle Operators
We have seen that the tensor models admit a variety of singlet operators. In this section we discuss their scaling dimensions.
ladder contributions in the large N limit, we expect a large class of m-particle operators to Figure 10: Diagrammatics for the "typical' operators whose IR dimensions are quantized. Each line denotes a dressed propagator. a) The melonic diagrams that contribute to the operator two-point functions in the large N limit. b) The ladder diagrams which do not contribute in the large N limit.
have the quantized dimensions: 7 This is the dimension of an operator which is not renormalized by ladder diagrams because every pair of tensors have at most one index in common. This situation is illustrated in figure 10: the dominant contribution comes from the two operators contracted using the IR two-point function (2.3), and the ladder insertions are suppressed by 1/N . We find that this applies to most of the 17 eight-particle operators shown in figure 9. The exceptions are operators O i andÕ i , defined in (4.10), (4.14), and shown in columns 1 and 2. For example, each of the operatorsÕ i in column 2 is renormalized by two ladders, as we discuss below.
Thus, the m/4 rule does not apply to all operators: it is violated for the operators whose two-point functions receive the melonic ladder contributions in the large N limit. One class of such singlet operators is the Regge trajectory we have discussed before: After applying the equation of motion (2.11), which schematically may be represented as we may represent the Regge trajectory operators in terms of multi-particle operators without derivatives. For example, the n = 0 operator is equivalent to the 4-particle "tetrahedron" operator O tetra , while the n = 1 operator is equivalent to , as well as the analogous operators O , are shown in figure 15.  rule (since the charges are conserved, we a priori expect their scaling dimension to be zero).
In fact, any operator whose diagram contains a bubble subdiagram (i.e. two tensors with a double index contraction) is renormalized by a ladder, and there are as many ladders as there are bubbles. For example, a pillow operator contains two bubbles and is renormalized by two ladders.
Moreover, if we take an operator diagram renormalized by multiple ladders and change one vertex in the diagram from ψ to ∂ t ψ (blue to white vertex), it will still be renormalized by the same number of ladders. With derivatives we can convert a pillow operator into the second operator in fig. 8. It is easy to check that this operator is renormalized by two ladders. Since each of the ladders contains the h = 2 zero-mode in its spectrum, and a zero-mode produces a low-temperature enhancement by a factor of βJ [26], we expect the double-ladder to produce an effect of order (βJ) 2 . The multi-ladder enhancements by (βJ) n seem to be a new effect in the tensor model, which clearly needs to be studied in more detail. understanding of the operators renormalized by multiple ladders and to study their lowtemperature contributions. We hope to address these questions elsewhere.

Some Scaling Dimensions in the Gurau-Witten Model
Let us now consider the O(N ) 6 symmetric quantum mechanical model [17]. as the basis, then the kernel is a 2 × 2 symmetric matrix with zeros on the diagonal; hence, the two eigenvalues are equal and opposite. To fix the normalization, we note that the two functions g ± (h) are proportional tog(h), which is given in (3.7). Therefore, g + (h) =g(h) and g − (h) = −g(h).  We may also study the bilinear singlet operators like For n = 0 this operator vanishes after the use of equations of motion, but it is non-trivial for n = 1, 2, . . .. To calculate the scaling dimensions of these operators using the S-D equations we note that the kernel is the SYK kernel, The solutions to this equation are shown in figure 16. 8 There is a series of solutions that lie slightly below 2n + 3 2 , for n = 1, 2, 3, . . . and approach it at large n. In other words, they lie slightly below the naive dimensions of operators O n − . For n = 1 the numerical value is 3.39, which is close to 3.5. There is also an exact solution with h = 1, whose interpretation is not completely clear.
where J ijkl are random couplings. The operators which are analogous to O n − are The n = 0 operator vanishes by the equation of motion for any value of J ijkl , which appears to explain the decoupling of the h = 1 mode.

Counting singlet operators in d = 1
In this section we proceed to do the singlet operator counting in the O(N ) 3 quantum mechanics more systematically. We employ the technique used in [39,40] to find the partition function and free energy of gauge theory. In our case, we will see that the free energy diverges wildly, but nevertheless this procedure allows to count the operators in the gauged or ungauged fermionic and scalar theories.
We work in the one-dimensional spacetime with fields living in the tri-fundamental rep- 3 , in the limit of N → ∞. We will mainly address the case of the free tensor model, which describes the UV fixed point, but also make comments about the IR theory. The partition function may be written in the form: To find Z s.s. explicitly, we use an elegant formula from [40]: Here m belongs to the set of square-free integers Ω = {2, 3, 5, 6, 7, 10, 11, 13, . . . }: Our goal in this section is to find the single-sum partition function for the scalar and fermionic tensor models. The partition function for the scalar theory in the UV with one group can be found as [37][38][39]: and for the fermionic theory it is: with M in the symmetry group and χ(M ) being the character of the desired representation.
In our case, we substitute: and take χ(M ) = tr M .
The single-letter partition functions for scalars and (Majorana) fermions correspondingly are as follows: To find Z, we will need the integrals of characters of O(N ) [40]: dM l tr M l a l = l l odd, a l even (2l) a l /2 1 √ π Γ a l 2 + 1 2 , l even a l /2 k=0 a l 2k (2l) k 1 √ π Γ k + 1 2 .

(7.11)
In the next chapter, we first find partition functions for both the fermionic and scalar d = 1 models without the constraint that the charges (2.6) vanish. Then, to find the partition function for the operators in the gauged model, we subtract the contribution from the operators containing O(N ) charge, or a "bubble" subdiagram (2.6) (see fig. 3). Such operators should vanish in the gauged version of quantum mechanics.

Fermions
The single-letter partition function for real fermions z F,d is not well defined in one dimension.
This reflects the divergence of the partition function (and hence free energy). To regularize it, we formally proceed in (1 + 2 ) dimension and neglect all the terms proportional to in the single-letter partition function; in other words, we simply take: We can justify this choice as follows. The single-letter partition function counts all local operators containing one field ψ abc with any number of derivatives. In our case, the only such operator is ψ abc : since ∂ t ψ abc vanishes by equations of motion in the free theory, all the operators with higher derivatives will vanish too.
In other words, in the fermionic case we are counting only the operators made of fermions without derivatives. We can think of this as operator counting in a d = 0 model (for a review see [2]), but with the Fermi statistics imposed.
Computing Z and using (7.7), (7.11), we find to first several orders in x: From this we can find the single-sum partition function, which counts connected operators: Z F s.s. = 4x 4 + 60x 8 + 116x 10 + 2802x 12 + 24324x 14 + 396196x 16 + . . . . (7.14) The order 2k in x 2k gives the number of fermions in the operator. So we see there are four four-fermion operators: one tetrahedron and three differently colored pillows (see figure 4).  Figure 17: Logarithm of the number of allowed (2k)-particle fermionic operators as a function of k. We see that the number of operators grows like ∼ k!2 k .
Note that, although we employed a gauged theory to count these operators, the pillows and other operators containing O(N ) charges are still present. At the sixth order, there are no operators because of the Fermi statistics as we noticed before, but at order 8 there are 60 operators.
The number of 2k-particle operators grows roughly as (see fig. 17): To count operators in the gauged model where the vanishing of O(N ) charges (2.6) is imposed, we have to disregard the operators containing their insertions, i.e. the "bubble" subgraphs. In order to do that, we subtract the operators having the same quantum numbers as a bubble in the exponent of (7.7). Each O(N ) charge (2.6) is antisymmetric in its two indices, which means that it lives in the representation (N ⊗ N ) antisym with the character: The bubble is a bosonic operator and its conformal dimension in the UV is 2 . Bringing it all together, we find that the partition function for operators in the gauge theory is: The single-sum partition function for the gauge theory then is as follows: s.s. = x 4 + 17x 8 + 24x 10 + 617x 12 + 4887x 14 + 82466x 16 + . . . . (7.18) We see that at the fourth order we are left with one operator; namely, the tetrahedron. At the eighth order we see 17 operators, as we already found in section 4.1 via explicit construction (see fig. 9) We have computed the single-sum partition function up to order 30, and the result matches the same factorial growth as in the model where the O(N ) 3 symmetry is not gauged (see fig. 18).
Finally, let us comment on the IR theory, where we believe there is similarly rapid growth of the number of operators as a function of the conformal dimension. Since for the majority of 2k-particle operators the large N IR dimension is h = k/2, in view of the result (7. 15) we expect that the number of operators of dimension h to grow as Γ(2h + 1), up to an exponential prefactor.

Bosons
We can also count the allowed operators in the scalar theory. Proceeding in the same fashion, we define single-letter partition function in (1 + 2 ) dimensions as follows: where − 1 2 + is the dimension of the scalar field. The partition function is: The single-sum partition function, which includes the operators with bubble insertions, is: In the second order we have operators φ abc φ abc , φ abc ∂ t φ abc , and ∂ t φ abc ∂ t φ abc . In the fourth order, we find the pillows and tetrahedra with various insertions of ∂ t . This partition function also diverges at → 0 and displays the factorial growth of the number of operators with their order.
To count operators in the gauged theory, we once again have to take care of the subgraphs corresponding to the gauge group charge. For a scalar theory, the gauge charge operator is: This operator lives in the adjoint representation, just like the gauge field. Its dimension is 2 = − 1 2 + + 1 2 + . The character of the adjoint representation is: Taking all this into account, we write the partition function as: To the first six orders, the partition function reads as: The single-sum partition function, which counts only the operators with connected diagrams, is as follows: The first term in this expression corresponds to the operators φ abc φ abc , φ abc ∂ t φ abc , and ∂ t φ abc ∂ t φ abc (the second of these operators is a total derivative; such descendant operators are included in the counting). The number 11 in the third term corresponds to all the six-particle graphs discussed in Section 4. Now the number of operators containing a string of 2k scalars is approximately Compared to the fermionic case 7.15 we have an additional factor of 2 2k . As we will see in the next section, for d = 0 the leading asymptotic for the number of operators is the same for scalars and fermions. Therefore, the factor 2 2k comes from distributing the time derivatives ∂ t among 2k fields. Since in the free theory ∂ 2 t φ abc = 0, each of the 2k fields may be acted on by one or no derivatives. This indeed contributes a factor of 2 2k .

Counting the Invariants in d = 0
Here we use methods similar to those in the previous section to discuss the counting of invariants in the d = 0 model which is simply an integral over the tensor. The construction and counting of such invariants, which are made out of products of tensors with all indices contracted, has been addressed in [2,[42][43][44][45][46]. These papers primarily discuss the complex bosonic rank-r tensor models which possess U (N ) r symmetry. We will first consider the bosonic rank-3 tensor model with O(N ) 3 symmetry and perform the counting using the methods developed in [39,40]. The model of a real fermionic tensor ψ abc does not work in d = 0: since the O(N ) 3 invariant ψ abc ψ abc vanishes, it is impossible to write down a Gaussian integral. One can write down models of complex fermionic tensors in d = 0, but we won't study them here. We will address the bosonic rank-3 symmetric traceless and antisymmetric tensors in subsection 8.1, and the bosonic complex tensors with U (N ) 3  The single-letter partition function counts all the invariants containing one field. In our case the only such operator is φ abc , so the single-letter partition function is: The invariants in this case are given by the diagrams with 2k vertices and three edges of different colors meeting at each vertex. Thus, the invariants are isomorphic to the Feynman diagrams in the theory of three scalar fields with interaction ϕ 1 ϕ 2 ϕ 3 . Every edge of the diagram is assigned one of the three colors, and every vertex joins the edges of three different colors. This is a non-trivial condition; for example, one-particle reducible graphs cannot be colored in this way. We consider different colorings of the diagrams as different invariants, so each topology can enter multiple times if there are several distinct ways to color it. Using (7.7), we find the full partition function: where we have used the character of a tri-fundamental representation (7.8). Taking this integral and using (7.11), we find in the first several orders: This partition function counts all the invariants, including the disconnected ones. To remove the latter, we compute the single-sum partition function using (7.4): Z 0 s.s. = x 2 + 4x 4 + 11x 6 + 60x 8 + 318x 10 + 2806x 12 + 29359x 14 + . . . . The number of invariants made out of 2k fields grows asymptotically as (see fig.19): We can find this asymptotic from an analytic estimate. The key observation is that the integral (7.11) grows factorially as (a l /2)! for large a l , while only as a power l a l /2 for large l. Besides, for large a l there is no difference in the leading order between odd and even l.
Therefore, the leading contribution to x 2k will come simply from the m = 1 term: Since the dominant term originates only from m = 1 term, the same estimate is valid for the fermions.

Symmetric traceless and antisymmetric tensors
Let us also discuss the counting of invariants in models with a single O(N ) symmetry, where we will consider the tensors which are either symmetric traceless or fully antisymmetric.
Such models with the tetrahedral interactions were recently studied in [12], where evidence was provided that they have melonic large N limits. The full partition function is where for the 3-index symmetric traceless representation the character in the large N limit For the fully antisymmetric representation the character is In the symmetric traceless case, the partition function is found to be Extracting the single-sum expression, we find Z + s.s. = x 2 + 2x 4 + 6x 6 + 20x 8 + 91x 10 + 509x 12 + . . . .
where we used the integrals (7.11).

Complex 3-Tensors
Let us now consider the complex 3-tensors with U (N ) 3 or U (N ) 2 × O(N ) symmetries. The latter symmetry is particularly interesting because it is preserved by the tetrahedral interac- This means that there are interacting melonic theories with the U (N ) 2 × O(N ) symmetry [6,9,18].
In the U (N ) 3 case we have the fields φ abc andφ abc , which are in the tri-fundamental representations N × N × N andN ×N ×N respectively. The partition function reads: ) .

(8.15)
It is straightforward to compute it using the following large N result [40]: For the scalar we take z S,0 (x) = x and find Z U (N ) 3 = 1 + x 2 + 4x 4 + 11x 6 + 43x 8 + 161x 10 + . . . . (8.17) This expansion matches the results obtained in [46] using group-theoretic methods. Extracting from Z the single-sum partition function, we find The coefficient 3 of x 4 is in agreement with the fact that the tetrahedron invariant is not allowed by the U (N ) 3 symmetry. Only the 3 pillow invariants are allowed, and their form is The asymptotic number of operators can be estimated as follows. As in the O(N ) case, the integral (8.16) grows factorially in a l and only as a power in l. It means that the term with m = 1 again dominates. Besides, to get a non-zero answer we need to extract the term with an equal number of χ(M i ) andχ(M i ). Therefore, Using the same method as in the U (N ) 3 case, the asymptotic growth can be found to be n U (N ) 2 ×O(N ) 2k ∼ 2 k k! (8.25)

The Hagedorn Transition
The special features of the thermodynamics of free theories where the fields are tensors of rank r ≥ 3 under some global symmetry group were recently studied in [40]. It was found that the Hagedorn temperature vanishes in the large N limit as ∼ 1/ log N [40]. In this section we show that this also applies to the models with O(N ) 3 symmetry studied in this paper.
An essential feature of the large N tensor models is that the low temperature expansion of the partition function has the approximate structure k 2 k k!x 2k , where − ln x is proportional to β. This power series is divergent and non-Borel summable; therefore, strictly speaking the partition function is not defined for any finite temperature. To illustrate the basic points, we study the large N behavior of the integral (8.2) in a standard fashion (it will be convenient to assume that N is even). First of all, for large N there should be no difference between Index r labels different SO(N ) r groups and i, j = 1, . . . , N/2 go over rotation angles. Also we have introduced a single-letter partition function z(x) to work in more generality. The above equation is valid for scalars, while for fermions we need to include the factor (−1) m+1 in front of z(x m ). However, we will see in a moment that for the Hagedorn transition only m = 1 term is relevant. Therefore, our main results will be applicable for both cases.  There are three saddle-point equations. One of them is: z(x m ) sin(mα i 1 ) j 2 ,j 3 cos(mα j 2 2 ) cos(mα j 3 3 ) = 0 . (9. 3) The other two can be obtained by cyclic permutations of α i 1 , α i 2 , α i 3 . Introducing density functions: The saddle-point equation can be rewritten as: π −π dα 1 ρ 1 (α 1 ) cot z(x m ) sin(mα 1 )ρ m 2 ρ m 3 = 0 , (9.5) where ρ m r = π −π dαρ r (α) cos(mα) . (9.6) It is natural to assume that because of the cyclic symmetry ρ 1 = ρ 2 = ρ 3 = ρ(α). Moreover, we will assume that ρ is an even function: ρ(α) = ρ(−α). With these assumptions the saddle-point equation reads as: This is exactly the saddle-point equation studied in [40], with their 6N replaced by our 2N .
They have found that there is Hagedorn transition: for low temperatures when N z(x) < More details can be found in [40] and [39].
For example, we can study the fermions in d = 1 + 2 . According to eq.