Thermalization in Large-N CFTs

In d-dimensional CFTs with a large number of degrees of freedom an important set of operators consists of the stress tensor and its products, multi stress tensors. Thermalization of such operators, the equality between their expectation values in heavy states and at finite temperature, is equivalent to a universal behavior of their OPE coefficients with a pair of identical heavy operators. We verify this behavior in a number of examples which include holographic and free CFTs and provide a bootstrap argument for the general case. In a free CFT we check the thermalization of multi stress tensor operators directly and also confirm the equality between the contributions of multi stress tensors to heavy-heavy-light-light correlators and to the corresponding thermal light-light two-point functions by disentangling the contributions of other light operators. Unlike multi stress tensors, these light operators violate the Eigenstate Thermalization Hypothesis and do not thermalize.


Introduction and summary
Holography [1][2][3] provides us with a useful tool to study d-dimensional CFTs at large central charge C T , especially when combined with modern CFT techniques (see e.g. [4][5][6] for reviews). One of the basic objects in this setup is a Witten diagram with a single graviton exchange which contributes to four-point functions. It can be decomposed into the conformal blocks of the stress-tensor and of the double-trace operators made out of external fields [7].
When a pair of the external operators denoted by O H is taken to be heavy, with the conformal dimension ∆ H ∼ C T , and the other pair denoted by O L stays light, the resulting heavy-heavy-light-light (HHLL) correlator describes a light probe interacting with a heavy state. In this case, operators which are comprised out of many stress tensors (multi stress tensor operators) contribute, together with the multi-trace operators involving O L . As we review below, the OPE coefficients of the scalar operators with a (unit-normalized) multi stress tensor operator T k τ,s , which contains k stress tensors and has twist τ and spin s, The contribution of a given multi stress tensor operator to the HHLL four-point argue that they are the same in generic large-C T CFTs. As we explain later, this means that OPE coefficients of T k τ,s with the two heavy operators O H , O H T k τ,s O H , are equal to their finite temperature expectation values, T k τ,s β . The relation between the inverse temperature β and the conformal dimension ∆ H is set by considering the stress tensor (k = 1, τ = d − 2, s = 2), but the equality between the thermal expectation values and the OPE coefficients for all other multi stress tensor operators is a nontrivial statement.
Here we want to address the d > 2 case. In holographic theories CFT and bootstrap techniques provide a lot of data which indicates that the thermalization of the stress tensor sector happens [62][63][64][65][66][67][68][69][70][71][72][73]. Some of the OPE coefficients in holographic CFTs were computed using two-point functions in a black hole background [63] -these are thermal correlators according to the standard holographic dictionary. It is also worth noting that the leading ∆ behavior of the OPE coefficients in holographic models does not depend on the coefficients of the higher derivative terms in the bulk lagrangian [71] (this should not be confused with the universality of the OPE coefficients of the minimal-twist multi stress tensors [63]).
Such a universality follows from the thermalization of the stress tensor sector as we discuss below.
A natural question is whether the thermalization of the stress tensor sector is just a property of holographic CFTs or if it holds more generally. In this paper we argue for the latter scenario. We compute the OPE coefficients (and the thermal expectation values) for a number of multi stress tensor operators in a free CFT and observe thermalization as well as universality of OPE coefficients. We also provide a bootstrap argument for all CFTs with a large central charge.
The rest of the paper is organized as follows. In Section 2, we begin by considering the thermalization of multi stress tensor operators T k τ,s . The heavy state we consider is created by a scalar operator O H with dimension ∆ H ∼ C T and by thermalization of a multi stress tensor operator we mean 3 T . In (1.1) T k τ,s β is the onepoint function on the sphere at finite temperature β −1 . Note that the OPE coefficients involving the stress tensor are fixed by the Ward identity, and hence eq. (1.1) for the 3 Here we are suppressing the tensor structure. Note that all terms scale like C k/2 T which is consistent with T k τ,s being unit-normalized.
stress tensor establishes a relation between the temperature β −1 and ∆ H . By the large-C T factorization 4 , the thermal one-point functions of multi stress tensors can be related to the thermal one-point function of the stress tensor itself. Explicitly, where c k τ,s are theory-independent coefficients that appear because of the index structure in T k τ,s β . In the second equality in (1.2) we used (1.1) for the stress tensor. Note that (1.1) and (1.2) imply that the leading ∆ H behavior of the multi stress tensor OPE coefficients is universal, i.e. it does not depend on the theory 5 . We provide a bootstrap argument for this universality in all large-C T theories. Also note that (1.2) is written for multi-trace operators T k τ,s which do not contain derivatives, but the presence of derivatives does not affect the statement of universality.
In Section 3, we check the universality by computing a number of the multi stress tensor OPE coefficients in a free SU (N ) adjoint scalar theory in d = 4 dimensions. We compare the leading ∆ H behavior in the free theory with results from holography/bootstrap and find perfect agreement in all cases listed below. After fixing the coefficients for the stress tensor case in Section 3.1, we look at the first nontrivial case, T 2 4,4 in Section 3.2. Section 3.3 is devoted to the double stress tensor with two derivatives, T 2 4,6 . This is an operator whose finite temperature expectation value vanishes in the large volume limit (on the plane), but is finite on the sphere. In Section 3.4 we consider minimal twist multi stress tensors of the type T k 2k,2k . Section 3.5 is devoted to multi stress tensors with non-minimal twist, T 2 6,2 and T 2 8,0 . In Section 4, we verify that (1.1) holds in the free adjoint scalar theory for a variety of operators. In this section we again consider d = 4, but in addition, take the infinite volume limit. This is for technical reasons -it is easier to compute a finite temperature expectation value on the plane than on the sphere. We spell out the index structure in 4 See [74] for a general discussion of large-N factorization and [75,76] and [8] for the discussion in the context of gauge theories and CFTs respectively. The factorization holds in adjoint models in the 't Hooft limit at finite temperature, but there are counterexamples, like e.g. a direct product of low-C T CFTs. However the factorization of multi stress tensors would still apply in these models. 5 This amounts to the large-C T factorization of correlators O H |T µν . . . T αβ |O H in heavy states.
(1.1) in detail and go over all the examples discussed in the previous section. In addition, we discuss some triple stress tensor operators.
We continue in Section 5 by studying thermal two-point functions in the free adjoint scalar model in d = 4. By decomposing the correlator into thermal blocks we read off the product of thermal one-point functions and the OPE coefficients for several operators of low dimension and observe agreement with the results of Sections 3 and 4. Due to the presence of multiple operators with the same dimension and spin, we have to solve a mixing problem to find which operators contribute to the thermal two-point function.
In Section 6 we explain the relation between our results and the Eigenstate Thermalization Hypothesis. We observe that unlike multi stress tensors, other light operators explicitly violate the Eigenstate Thermalization Hypothesis and do not thermalize. We end with a discussion in Section 7.
Appendices A, B, and C contain explicit calculations of OPE coefficients while in Appendices D and E thermal one-point functions are calculated. In Appendix F we review the statement that the thermal one-point functions of multi-trace operators with derivatives vanish on S 1 × R d−1 . In appendix G we study a free scalar in two dimensions and calculate thermal two-point functions of certain quasi-primary operators. In Appendix H we consider a free scalar vector model in four dimensions. Appendix I discusses the factorization of multi-trace operators in the large volume limit.

Thermalization and universality
In the following we consider large-C T CFTs on a (d − 1)-dimensional sphere of radius R, which we set to unity for most of this section. As reviewed in [71], the stress tensor sector of conformal four-point functions consists of the contributions of the stress tensor and all its composites (multi stress tensors). The HHLL correlators we consider involve two heavy operators inserted at x 0 E = ±∞ and two light operators inserted on the Euclidean cylinder, with angular separation ϕ and time separation x 0 E . The correlator in a heavy state (the HHLL correlator on the cylinder) is related to the correlator on the plane by a conformal transformation where the cross-ratios (z,z) on the plane are related to the coordinates (x 0 E , ϕ) via The stress tensor sector of the HHLL correlator is given by and can be expanded in conformal blocks where τ, s, k label the twist, spin, and multiplicity of multi stress tensors. We are interested in the double scaling limit where the central charge and the dimension of O H are large, In this limit the products of the OPE coefficients which appear in (2.4) are given by We now consider these CFTs at finite temperature β −1 . To isolate the contribution of the conformal family associated with T k τ,s , we can write the thermal correlator as where is the finite temperature one-point function on the sphere of the T k τ,s operator and the dots denote contributions from other operators. In (2.8) Z(β) is the partition function and the sum runs over all operators, including descendants 6 . Note that Here and below the indices are suppressed (see e.g. [17] for the explicit form) and f k τ,s (β) ∼ C k/2 T is a theory-dependent nontrivial function of β which approaches a constant f k τ,s (0) in the large volume (β → 0) limit.
Consider the thermalization of the stress tensor sector: Note that T k τ,s is unit-normalized, so all terms in (2.10) scale like C k/2 T . Eq. (2.10) implies the equality between (2.6) and the corresponding term in (2.7). Note that the left-hand side of (2.10) is a function of the energy density while the right-hand side is a function of temperature. The relationship is fixed by considering the stress tensor case: the corresponding function f 1 d−2,2 (β) is determined by the free energy on the sphere (see Section 6).
In the following, we will first discuss the case where the multi stress operators T k τ,s do not have any derivatives inserted, and then show that the derivatives do not change the conclusions. Assuming large-C T factorization, the leading C T behavior of T k τ,s β on the sphere is determined by that of the stress tensor. Schematically, where c k τ,s are numerical coefficients, which depend on k, τ, s, but are independent of the details of the theory and the dots stand for terms subleading in C −1 T . By combining (2.11) and (2.10), one can formulate a universality condition where the last equality follows from the stress tensor Ward identity for the three-point In other words, thermalization and large-C T factorization imply that the leading ∆ k /C k/2 T behavior of the multi stress tensor OPE coefficients is completely fixed and given by (2.12) in all large-C T CFTs.
In the paragraph above we considered multi stress tensor operators that did not contain any derivatives in them. However, the story largely remains the same when the derivatives are included, as long as their number does not scale with C T . Indeed, the three-point function involving the stress-tensor with added derivatives, (2.12) still holds, provided thermalization and large-C T factorization hold on the sphere.
Note that due to conformal invariance, correlators on the sphere depend on R only through the ratio β/R. Moreover, in the large volume limit, factors of R need to drop out of (2.6) and (2.7) to have a well defined limit. To see this we use that (1 − z) → 0 and (1 −z) → 0 when R → ∞ and the conformal blocks behave as (see e.g. [6]) (2.13) this agrees with the thermal block on S 1 × R d−1 in [13]. Now from the thermalization of the stress tensor we will find in the large volume limit that and from (2.12) and (2.13) it follows that The dimension of multi stress tensors T k τ,s is given by τ + s = dk + n where n = 0, 2, . . .. Therefore, the only multi stress tensors that contribute in the large volume limit have dimensions dk. Restoring R in (2.6)-(2.7) and inserting (2.15) one finds that R drops out in the large volume limit. The correct dependence β −(τ +s) from (2.9) in the R → ∞ limit is also recovered in (2.6) using (2.15). The multi stress tensor operators that contribute in the large volume limit are therefore of the schematic form T µ 1 ν 1 T µ 2 ν 2 · · · T µ k ν k with arbitrarily many contractions and no derivatives.
In holographic theories thermalization and the Wilson line prescription for the correlator allows one to compute the universal part of the OPE coefficients (see [66,78] for explicit computations in the d = 4 case). It is also easy to check explicitly that the universality (2.12) holds for holographic theories with a Gauss-Bonnet gravitational coupling added.
While the statement was shown to be true for the leading twist OPE coefficients in [63], it was not immediately obvious for multi stress tensors of non-minimal twist. Some such OPE coefficients were computed in [63,71] [79][80][81]. The subleading contributions in 1/C T are then due to corrections to the anomalous dimension and OPE coefficients of [OT αβ ] n,l and single trace operators in the O × T µν OPE. An important example of the latter is the exchange of the single trace operator O, whose contribution is universally fixed by the stress tensor Ward identity to be (λ OT 1 d−2,2 O ) 2 ∝ ∆ 2 /C T times the conformal block. This gives a universal contribution to λ OOT 2 τ,s as was also noted in [73]. We now want to consider the case where ∆ ∼ C T and study the OPE coefficients of the double-stress tensor operators in the O × O OPE. Firstly, note that the contribution from T 2 τ,s to the four-point function expanded in the direct channel is proportional to λ OOT 2 τ,s λ T T T 2 τ,s . The OPE coefficients λ T T T 2 τ,s are fixed by the MFT and are independent of ∆ and therefore the dependence on the scaling dimension comes solely from the OPE 7 This correlator for finite ∆ was recently considered in holographic CFTs with ∆ gap ≫ 1 and ∆ ≪ ∆ gap in [73]. . This contribution is universal since it only depends on (λ OT 1 d−2,2 O ) 2 ∝ ∆ 2 /C T in the cross-channel, which is fixed by the Ward identity. The contributions from other operators O ′ to the same OPE coefficient will be denoted by λ . Note that it also follows from the stress tensor Ward identity that the only scalar primary that appears in the cross-channel is O. The operator O ′ therefore necessarily has spin s = 0.
To prove universality we need to show that λ by studying the ∆ dependence of the OPE coefficients λ OT 1 d−2,2 O ′ in the cross-channel. For operators O ′ , such that ∆ O ′ ≪ ∆, we expect that these OPE coefficients are heavily suppressed. It would be interesting to understand if one could put a general bound on the contribution of these operators in the cross-channel in any large-C T theory. On the other hand, assuming thermalization, the OPE coefficients due to operators O ′ such that ∆ O ′ ∼ ∆ have been calculated in [23]. The obtained results are in agreement with our expectation, namely, these OPE coefficients are suppressed in 1 ≪ ∆ ∝ C T limit. Additionally, in the cross-channel we have double-trace operators [OT αβ ] n,l , whose OPE is fixed by the MFT and it does not get ∆-enhanced. Note that in holographic theories with a large gap, and its contribution to multi-stress tensor OPE coefficients was studied in [73]. This is different from the regime considered in this paper where ∆ ≫ ∆ gap .
One can iteratively extend the argument given here to multi stress tensors operators (with k > 2) by considering multi stress tensors as external operators. For example, to argue the universality of λ OOT 3 τ,s one may consider OT 1 d−2,2 T 2 τ,s O . The bootstrap argument above can be applied again by using the fact that OPE coefficients λ OOT 2 τ,s are universal, and the OPE coefficients λ OT 2 τ,s O ′ are again expected to be subleading.

OPE coefficients in the free adjoint scalar model
In this section we consider a four-dimensional theory of a free scalar in the adjoint representation of SU (N ), see [82][83][84][85][86][87] for related work. The relation between N and the central charge C T in this theory is [88] and we consider the large-N (large-C T ) limit. The propagator for the scalar field φ i j is given by A single trace scalar operator with dimension ∆ is given by where : . . . : denotes the oscillator normal ordering and the normalization is fixed by The CFT data that we compute in this section are the OPE coefficients of multi stress tensors in the O ∆ × O ∆ OPE. Assuming we can take ∆ → ∆ H ∼ C T , the large-∆ limit of these OPE coefficients is shown to be universal. One may worry that for ∆ H ∼ C T we can no longer trust the planar expansion, but, as we show in Appendix C, the large-∆ limit of the planar result yields the correct expression even for ∆ H ∼ C T .

Stress tensor
The stress tensor operator is given by where the normalization with I µ ν (x) := δ µ ν − 2x µ x ν |x| 2 . The OPE coefficient is fixed by the stress tensor Ward identity to be It is also useful to find (3.7) using Wick contractions since an analogous calculation will be necessary for multi stress tensors. We do this explicitly in Appendix A.

Double-stress tensor with minimal twist
In this section we study the minimal-twist composite operator made out of two stress with the normalization Consider the following three-point function is given at leading order in the large-C T limit by Evaluating P (HH,LL) T 2 4,4 defined by (2.5) in the large-∆ limit 8 , we obtain where we use the following relation The result (3.12) agrees with the leading behavior of the corresponding OPE coefficients computed using holography in [63] and bootstrap in [66,68].

Double-stress tensor with minimal twist and spin s = 6
We consider double-stess tensor operator with two (uncontracted) derivatives inserted (3.14) Using the conformal algebra eq. (C.2), it is straightforward to check that this operator is primary. It is unit-normalized such that By a calculation similar to those summarized in Appendix A, we observe that the OPE coefficient of (T 2 ) µνρσηκ in the O ∆ × O ∆ OPE is given at leading order in the large-C T limit by , defined by (2.5), in the large-∆ limit, we obtain (3.17) The result (3.17) agrees with the leading behavior of the corresponding OPE coefficients computed using holography in [63] and bootstrap in [66,68].

Minimal-twist multi stress tensors
We now consider multi stress tensors T k 2k,2k . Just like the double stress tensor (k = 2), we show that these have universal OPE coefficients in the large-∆ limit for any k.
Consider the unit-normalized minimal-twist multi stress tensor operator given by The OPE coefficient of (T k ) µ 1 µ 2 ...µ 2k in the O ∆ × O ∆ OPE, in the large-C T limit is given by 9 , defined by (2.5), in the large-∆ limit. We obtain this OPE coefficient from (3.19) for k = 3, (3.20) The result (3.20) agrees with the leading behavior of the corresponding OPE coefficients computed using holography in [63] and bootstrap in [68].
Additionally, we consider the OPE coefficient P in the large-∆ limit for general k, (3.21) If we consider the limit 1 −z ≪ 1 − z ≪ 1, such that µ(1 −z)(1 − z) 3 is held fixed, only operators T k 2k,2k contribute to the heavy-heavy-light-light four-point function given by eq. (2.3). The conformal blocks of T k 2k,2k in this limit are given by and we can sum all contributions in eq. (2.4) explicitly to obtain Notice that the term in the exponential is precisely the stress-tensor conformal block in the limit 1 −z ≪ 1 − z ≪ 1 times its OPE coefficient. Therefore, the OPE coefficients (3.19) imply the exponentiation of stress-tensor conformal block. We conclude that these OPE coefficients are the same as the ones computed using holography and bootstrap in the limit of large ∆.

Double-stress tensors with non-minimal twist
So far we have shown that the minimal-twist multi stress tensor OPE coefficients are universal in the limit of large ∆. In this subsection, we extend this to show that the simplest non-minimal twist double-stress tensors also have universal OPE coefficients at large ∆.
The subleading twist double-stress tensor with twist τ = 6 is of the schematic form : T µ α T αν : and has dimension ∆ = 8 and spin s = 2. It is given by The normalization in (3.24) is again chosen such that (T 2 ) µν is unit-normalized, see Appendix B for details.
from which we read off the OPE coefficient , defined by (2.5), in the large-∆ limit, we obtain The result (3.27) agrees with the leading behavior of the corresponding OPE coefficients computed using holography in [63] and bootstrap in [71].
We further consider the scalar double-stress tensor with ∆ = 8 and spin s = 0 which is given by The three point function from which we read off the OPE coefficient The result (3.31) agrees with the leading behavior of the corresponding OPE coefficients computed using holography in [63] and bootstrap in [71].

Thermal one-point functions in the free adjoint scalar model
In this section we explicitly show that multi stress tensor operators thermalize in the free theory by calculating the thermal one-point function of some of these operators on S 1 × R 3 . One-point functions of primary symmetric traceless operators at finite temperature are fixed by symmetry up to a dimensionless coefficient b O (see e.g. [8,13]) Here e µ is a unit vector along the thermal circle.
To compare the thermal one-point functions and OPE coefficients from the previous section, we need to derive a relation between ∆ H C T and the temperature 10 β −1 . Here ∆ H ∼ N 2 refers to the scaling dimension of a heavy operator O H with OPE coefficients given by the large-∆ limit of those obtained in Section 3. One can relate the inverse temperature β to the parameter µ = 160 The energy of the state E is related to its conformal dimension ∆ via E = ∆/R. One can then use vol(S 3 ) = 2π 2 R 3 and the relation between N and C T given by (3.1), to find

Stress tensor
The thermal one-point function for the stress tensor T 1 2,2 = T µν is calculated in Appendix D where we find that b T 1 2,2 is given by Using (4.2) and (4.3) one arrives at

Double-stress tensor with minimal twist
In this section we calculate the thermal one-point function of the double-stress tensor operator with τ = 4 and spin s = 4. The operator is written explicitly in (3.8). The leading contribution to the thermal one-point function of (T 2 ) µνρσ follows from the large-N factorization and is given by (4.5) Using the relation (4.2) and the OPE coefficient (3.11), we observe the thermalization of

Minimal-twist multi stress tensors
Consider now multi stress tensors T k 2k,2k with twist τ = 2k and spin s = 2k. We show that these operators thermalize for any k by calculating their thermal one-point functions: where the leading behavior of b T k 2k,2k follows from the large-N factorization: Eqs. (4.2) and (4.8) may be combined to yield

Double-stress tensors with non-minimal twist
The subleading twist double-stress tensor is of the schematic form : T µ α T αν : and has twist τ = 6 and spin s = 2. The explicit form can be found in (3.24). The leading term in the thermal one-point function is given by Taking the large-∆ limit of the OPE coefficient in (3.26) and substituting (4.2), we observe thermalization, We further consider the scalar double-stress tensor with τ = 8 and s = 0 which is given by (3.28). The thermal one-point function for this operator is where the factor of 3 4 in the first line comes from the index contractions. Hence, (4.14) Using (4.14), (3.30) and (4.2), we again observe thermalization,

Triple-stress tensors with non-minimal twist
We consider the triple stress tensors with τ = 8, s = 4 and τ = 10, s = 2. The unit-normalized triple stress tensor with τ = 8 can be written as where |α| denotes that index α is excluded from the symmetrization. The thermal onepoint function follows from large-N factorization The OPE coefficient of the operator with same quantum numbers (∆ = 12, s = 4) is calculated holographically and is given by (D.1) in [71]. In the large-∆ limit it can be written as where we use the relation between central charge C T and N given by (3.1). Using (4.2) one can obtain We also consider the triple stress tensors with quantum numbers ∆ = 12 and s = 2.
There are two linearly independent such operators that schematically can be written as : T αβ T αβ T µν : and : T µα T αβ T βν :. We write the following linear combinations of these The thermal one-point functions of these operators, obtained by large-N factorization, in the large-N limit are given by  The holographic OPE coefficient of the operator with the same quantum numbers (∆ = 12, s = 2), with external scalar operators is given by (5.57) in [71]. In the large-∆ limit it can be written as Again, using (4.2), one can confirm that this operator thermalizes (4.29)

Thermal two-point function and block decomposition
In this section we study the thermal two-point function O ∆ O ∆ β and decompose it in thermal blocks. We determine the contributions of a few low-lying operators, including the stress tensor T

Thermal two-point function of a single trace scalar operator
The correlator at finite temperature β −1 in the free theory can be calculated by Wick contractions using the propagators on S 1 × R 3 . Explicitly, the two-point function at finite temperature is given by 11 The dots in (5.1) contain contributions due to further self-contractions which will not be important below 12 . Taking the β → ∞ limit of (5.1) we can read off the decomposition of the two-point function in terms of thermal conformal blocks on S 1 × R 3 with coordinates . 11 Here and below we assume that ∆ > 4. We further drop the disconnected term O ∆ 2 β ∼ N 2 . 12 These terms will be proportional to β −2ag (x 0 E , |x|) ∆−a , with a ≥ 4. When decomposed into thermal blocks, these will not affect the operators with dimension ∆ < 8 or ∆ = 8 with non-zero spin s = 0.
Following [13], if |x| = (x 0 E ) 2 + x 2 ≤ β the two-point function can be evaluated using the OPE: Using (4.1) together with (5.3), the two-point function on S 1 × R 3 can be organized in the following way [13]: where we sum over primary operators O τ,s , with twist τ and spin s, appearing in the OPE s (x 0 E /|x|) is a Gegenbauer polynomial which, together with a factor of |x| −2∆+τ −s , forms a thermal conformal block in d = 4 dimensions and the coefficients a O τ,s are given by From the expansion (5.6), we can read off the coefficients a τ ′ ,s ′ : we sum over all operators with twist τ ′ and spin s ′ : For future reference, expanding Note that due to the mixing of operators with the same twist and spin, a τ,s generically contains the contribution from multiple operators. In the following section we calculate the OPE coefficients and thermal one-point functions of operators which are not multi stress tensors but contribute to (5.7) and (5.8).

CFT data of scalar operators with dimensions two and four
We explicitly calculate the thermal one-point contractions. This is done to find which operators contribute to the thermal two-point function and to resolve a mixing problem.
For τ ′ = 2 there is only one such operator, the single trace operator The three-point function is calculated in Appendix A, in the large-N limit, and it is given to leading order in 1/N . To calculate the thermal onepoint function ∝ T r(φ 2 ) β , we include self-contractions, i.e. contractions of fundamental fields within the same composite operator separated by a distance mβ along the thermal circle for m = 0 and integer. Explicitly, the one-point function of O 2 is given by The contribution to the thermal two-point function a O 2 is found using (5.10) and (5.12) This agrees with (5.7) which was obtained from the thermal two-point function.
We now continue with scalar operators of twist four. There are two such linearly independent operators appearing in the O ∆ × O ∆ OPE. In order to construct an orthonormal basis, consider the following single and double trace operators: (5.14) We further construct the operatorÕ 4 that has vanishing overlap with O 4,DT (x) as follows: Note that even though the second term in (5.15) is suppressed by 1/N , it can still contribute to the thermal two-point function due to the scaling of OPE coefficients and one-point The one-point function and the OPE coefficient for O 4 is found analogously to that of O 2 in the large-N limit Consider now the double trace operator given in (5.14). The one-point function factorizes in the large-N limit: Likewise, the OPE coefficient can be computed in the large-N limit (see Appendix A) Consider now the thermal one-point function ofÕ 4 in (5.17) which agrees with (5.7).

CFT data of single-trace operator with twist two and spin four
The primary single trace operator Ξ = O 2,4 with twist τ = 2 and spin s = 4 is given by The relative coefficients follow from requiring that the operator is a primary, see Appendix E for details.
The thermal one-point function of this operator is found from Wick contractions in the large-N limit to be and therefore the OPE coefficient Now, it is easy to check that which agrees with a 2,4 in (5.8).

CFT data of double-trace operators with twist and spin equal to four
To find the full contribution to the thermal two-point function from the operators with τ = 4 and s = 4 we need to take into account the contribution of all operators with these quantum numbers and solve a mixing problem. In addition to the double-stress tensor operator with these quantum numbers, the other double trace primary operator which contributes is given by where the operator is unit-normalized. Notice that this is the double trace operator obtained by taking the normal ordered product of two single trace operators, the scalar operator with dimension 2 and the single trace spin-4 operator with dimension 6. There are more double trace operators with these quantum numbers which are, however, not simply products of single trace operators. These do not contribute to the thermal two-point function to leading order in 1 N 2 (see appendix F). Note that it follows from large-N factorization that the overlap of this operator with (T 2 ) µνρσ is suppressed by powers of 1 N ; since both of these are double trace operators and obey the scaling (5.18), to study the thermal two-point function to leading order in N 2 , one can therefore neglect this overlap. where we used the thermal one-point functions for each single trace operator given by (5.11) and (5.25). The OPE coefficient is calculated in Appendix A, Using the thermal one point function and the OPE coefficient in (5.30) and (5.31) respectively, it is found that it the operator O DT µνρσ gives the following contribution to the thermal two point function: The total contribution from T 2 4,4 together with that of O DT 4,4 , using (3.11), (4.5) and (5.32), is This agrees with a 4,4 in (5.8).

Comparison with the eigenstate thermalization hypothesis
In this section we discuss the relation of our results to the eigenstate thermalization hypothesis (ETH). We argue that the stress tensor sector of the free SU (N ) adjoint scalar theory in d = 4 satisfies the ETH to leading order in C T ∼ N 2 ≫ 1. We explain the equivalence of the micro-canonical and canonical ensemble when ∆ H ∼ C T in large-C T theories. In this regime, the diagonal part of the ETH is (up to exponentially suppressed terms which we do not consider), equivalent to thermalization. Note that in two dimensions the Virasoro descendants of the identity satisfy the ETH (see e.g. [42] for a recent discussion).
We begin by showing the equivalence between the micro-canonical and the canonical [9,11,12,14,89] for a similar discussion at infinite volume as well as [47] in the two-dimensional case. The expectation value in the micro-canonical ensemble of an operator O, which we take to be a scalar for simplicity, at where we sum over states |Õ with energy (E, E + δE) and N (E) is the number of states in this interval. On the other hand, consider the partition function at inverse temperature β given by where we sum over all states in the theory. In the second line in (6.2) we have approximated the sum of delta-functions by a continuous function ρ(∆). Expectation values in the canonical ensemble is then computed by 13 Consider the partition function in (6.2) with a free energy F = −β −1 log Z(β). By an inverse Laplace transform of (6.2) we find the density of states 13 It was argued in [14] that the existence of the thermodynamic limit implies that we only need to sum over operators with low spin.
For ∆ H ∼ C T and a large free energy 14 F ∼ C T , we can evaluate (6.4) using a saddlepoint approximation with the saddle at β given by Consider now the thermal expectation value in (6.3), multiplying both sides by Z(β) and doing an inverse Laplace transform evaluated at ∆ H ∼ C T we find For F ∼ C T ≫ 1 we again use a saddlepoint approximation to evaluate (6.6) with the saddle at β determined by (6.5), assuming O β ′ does not grow exponentially with C T . The RHS of (6.6) is therefore the thermal expectation value O β multiplied by the saddlepoint approximation of the density of states in (6.4). It then follows that with β determined by (6.5). In particular, in the infinite volume limit R → ∞, the free energy is given by 15 . Inserting (6.8) in (6.5) we find [9] We can use (6.7) to see the thermalization of the stress tensor. The free energy is related to the expectation value of the stress tensor T (can) µν [5] T (can) 00 (6.11) 14 We consider a CFT in a high temperature phase. 15 Here we denote the canonically normalized stress tensor by T  This shows the thermalization of the stress tensor in heavy states where F ∼ ∆ H ∼ C T in large-C T theories. Note that this follows from (6.7) since we can replace the microcanonical expectation value at E = ∆ H /R, on the LHS, with the expectation value in any single heavy state with dimension ∆ H due to the Ward identity, independent of the heavy state. Put differently, the stress tensor satisfies the ETH as we will review below.
We now consider the eigenstate thermalization hypothesis for CFTs at finite temperature on the sphere S d−1 of radius R. The diagonal part of the ETH is given by The operator O τ,s on the other hand can have non-zero spin. 16 . In (6.13), e S(E) is the density of states at energy E = ∆ H /R. As shown in (6.7), in the limit ∆ H ∼ C T ≫ 1, the micro-canonical ensemble is equivalent to the canonical ensemble at inverse temperature β determined by (6.5). It then follows from (6.13) that the diagonal part of the ETH can written in terms of OPE coefficients and thermal one-point functions: where f O τ,s also appears in (2.9). This is equivalent to the statement of thermalization discussed in the rest of the paper.
In this paper we observed that the multi stress tensor operators satisfy (6.14). One can also ask if (6.14) holds for any operator in the specific heavy state we considered. By comparing eqs. (5.10) and (5.11) using (4.2), one can check that operator O 2 = 1 √ 2N : T r(φ 2 ) : does not satisfy (6.14). Since this is a free theory, it is not a surprise that the ETH is not satisfied by all operators in the spectrum which is seen explicitly in this case. 16 The tensor structure in (6.13) is suppressed.

Discussion
In this paper we argued that multi stress tensor operators T k τ,s in CFTs with a large central charge C T thermalize: their expectation values in heavy states are the same as their thermal expectation values. This is equivalent to the statement that multi stress tensor operators in higher-dimensional CFTs satisfy the diagonal part of the ETH in the thermodynamic limit. The analogous statement in the d = 2 case is that the Virasoro descendants of the identity satisfy the ETH condition in the large-C T limit.
We observed that the operator O 2 = 1 √ 2N : T r(φ 2 ) : does not satisfy the ETH. This is seen by comparing eqs. (5.10) and (5.11) using (4.2). While this operator does not thermalize in the heavy states we considered, the OPE coefficient averaged over all operators with ∆ H ∼ C T is expected to be proportional to the thermal one-point function.
The averaged OPE coefficients should therefore scale like for the heavy states we considered. It would be interesting to exhibit heavy operators that produce the former scaling.
We provided a bootstrap argument in favor of the thermalization of multi stress tensor operators. One should be able to refine it to give an explicit form for leading behavior of the multi stress tensor OPE coefficients -we leave it for future work. The holographic/bootstrap OPE coefficients for the leading twist double stress tensor operators can be found in e.g. [66] -they are nontrivial functions of the spin. As explained in [66,68], the leading ∆ behavior of the minimal-twist double-and triple-stress tensor OPE coefficients is consistent with the exponentiation of the near lightcone stress tensor conformal block. One can go beyond the leading twist multi stress tensors. In holographic HHLL correlators each term of the type (∆µ) k ∼ (∆∆ H /C T ) k comes from the exponentiation of the stress-tensor block -this follows from the Wilson line calculation of the correlator in the AdS-Schwarzschild background [90,66,78].
In this paper we argue that this behavior is universal, and is not just confined to holographic theories. Hence, one can formulate another statement equivalent to the thermalization of multi stress tensor operators. Namely, scalar correlators of pairwise identical operators of dimensions ∆ 1,2 in large-C T theories in the limit ∆ 1,2 ≫ 1, ∆ 1 ∆ 2 /C T fixed are given by the exponentials of the stress tensor conformal block 17 . This is similar to what happens in two-dimensional CFTs.
17 See [33] for previous work on the eikonalization of the multi stress tensor OPE coefficients at large spin.
Note that the universality of the OPE coefficients is naively in tension with the results of [73], where finite gap (∆ gap ) corrections to the multi stress tensor OPE coefficients were considered. In particular, for double stress tensors, such corrections behave like ∆ 3 /∆ gap which is clearly at odds with the universality statement. Of course, the results of [73] are obtained in the limit ∆ ≪ ∆ gap , while in this paper we consider the opposite regime ∆ ≫ ∆ gap .
One may also wonder what happens with the universality of the OPE coefficients beyond leading order in ∆. In particular, in [78], it was shown that the bootstrap result for the HHLL correlator exactly matches the holographic Wilson line calculation (in the double scaling limit where only the minimal twist multi stress tensor operators contribute).
This corresponds to including terms beyond the exponential of the stress tensor blockone needs to compute the HHLL correlator, take a logarithm of the result, divide by ∆, and then take the large-∆ limit. The result is sensitive to terms subleading in the large-∆ limit of the multi stress tensor OPE coefficients. In four spacetime dimensions the result in [78] is given by an elliptic integral -is it applicable beyond holography?
In [66] terms subleading in ∆ were shown to be important for the computation of the phase shift. The simplest nontrivial case in two spacetime dimensions is the operator Λ 4 which is a level four Virasoro descendant of the identity (see e.g. [91]). One could also get it by using the CFT normal ordering and imposing the quasi-primary condition [92].
Consider now the case of minimal twist (twist four) operators in four dimensions. How do we determine the analog of Λ 4 ? There is no Virasoro algebra now.
Presumably, one can reconstruct the analog of Λ 4 in four spacetime dimensions by considering a CFT normal ordered product of stress tensors, and adding a single trace term to ensure that the resulting operator is a primary and is orthogonal to the stress tensor itself. Note that the CFT normal ordering differs from the oscillator normal ordering in a free theory by the addition of a single trace operator, as reviewed in Appendix G. This procedure can then be generalized to other multi-trace operators. We leave it for future work.
Comparing the two results with the help of (4.2) one observes that such operators do not thermalize in the free theory for generic ∆. The situation is more nontrivial in holography where we do not know the large µ behavior of the OPE coefficients 19 . As pointed out in [63], the contribution of double-trace operators to thermal two-point functions is different from that of multi stress tensors. The latter is only sensitive to the behavior of the metric near the boundary, but the former knows about the full black hole metric. This seems to indicate that the thermalization of the double trace operators in holographic theories is also unlikely 20 .
It is a natural question how generic are the heavy states for which the stress tensor sector thermalizes. The results of our paper seem to suggest that such thermalization is more generic than the thermalization of other light operators 21 . Other interesting questions include generalizations to the case of finite but large central charge and to non-conformal quantum field theories. 18 This scaling is obtained by computing the OPE coefficient C T and extrapolating it to the ∆ H ∼ C T regime. 19 Note that the large-N scaling in holography is different. Both the OPE coefficients and the thermal expectation values behave like C 0 T as opposed to C T ∼ N 2 . 20 A simple way to decouple such operators is to take the large-∆ limit. 21 A closely related question of finding "typical" states where the stress tensor sector thermalizes in the large volume limit in d = 2 was recently discussed in [58]. There it was observed that such states are Virasoro descendants when the central charge is finite. To begin with, since we consider a large-N matrix theory, it is convenient to use the double-line notation for fundamental field propagators. In Fig. 1     Continuing with Fig. 6, the first contraction gives a factor of 2∆, the second contraction ∆ and the third one a factor of 2(∆ − 1). What remains is then fixed by imposing that the diagram is planar. The type A diagrams in Fig. 6 therefore further contributes

Consider now the stress tensor OPE coffiecient
and the three-point function O ∆ O ∆ T µν : From the definition of T µν in (A.5) it is clear that the only term that contributes to term x 13µ x 13ν comes from the second term in (A.5) that is of the form ∝ T r(φ∂ µ ∂ ν φ). Up to the derivatives, the diagram will look like those visualised in Fig. 3. The number of diagrams is half of that given in (A.2) since we restrict to terms proportional to x 13µ x 13ν : from which we reproduce (3.7).
Now we want to find the OPE coefficient λ O ∆ O ∆ T 2 4,4 for the double-stress tensor T 2 4,4 . This is done similarly to the way the stress tensor OPE coefficient was found. First, the operator (T 2 ) µνρσ was given in (3.8) to be Consider the term in (A.9) proportional to x 13µ x 13ν x 13ρ x 13σ . This will be due to the term in (T 2 ) µνρσ of the form T r(φ∂ (µ ∂ ν φ)T r(φ∂ ρ ∂ σ) φ). Using this we find that , come from diagrams of the form given in Fig. 6. Since we are considering the term proportional x 13µ x 13ν x 13ρ x 13σ , the number of such diagrams are reduced compared to scalar double trace operator. Instead the first contraction, (1) in Fig. 6, give a factor of ∆, the second contraction, (2), a factor of (∆ − 1), the third contraction (3) gives a further factor ∆ after which everything is fixed by imposing that the diagram is planar. We therefore find that .11) and inserting this in (A.10) gives and therefore reproduces (3.11).
Similar to the double-stress tensor, consider the dimension-eight spin-four double trace (A.14) By again considering terms in (A.14) proportional to x 13µ x 13ν x 13ρ x 13σ we find that each term in (A.13) will contribute planar diagram of the type in Fig. 5, while only the term ∼ T r(φ∂ 4 φ) also give a contribution of the type in Fig. 6. Considering first the terms coming from the diagram in Fig. 5, one finds that this contribution vanishes. The remaining contribution to the term (A.14) proportional to x 13µ x 13ν x 13ρ x 13σ comes from the first term in (A.13) and the planar diagram pictured in Fig. 6; there are 2∆ 2 (∆ − 1) contractions giving such a planar diagram leading to 15) where the 384 in the numerator come from the derivatives. This gives the OPE coefficient:

Appendix B. Subleading twist double-stress tensors
In this Appendix we study the subleading twist double-stress tensors, both with dimension 8 and spin s = 0, 2 denoted (T 2 ) and (T 2 ) µν respectively. The calculations needed to find the OPE coefficient in the O ∆ × O ∆ OPE are reviewed as well as the normalization of (T 2 ) µν .
The (T 2 ) µν was defined in (3.24) which we repeat here: The operator (T 2 ) µν can be seen to be unit-normalized to leading order in N : Using the two-point function of the stress tensor in (3.6) and I µ α I α ρ = δ µ ρ one finds from which it is seen that (T 2 ) µν is unit-normalised.

(B.5)
And lastly J (3) µνρσ :  We further need to make (B.4)-(B.6) traceless in the pairs (µ, ν) and (ρ, σ) and therefore define I (i) µνρσ as Explicitly we find that Consider now the scalar operator (T 2 ) defined by as follows whereÔ is either : T r(φ 2 ) : or, more importantly, minimal-twist multi stress tensors with any spin. Moreover, note that the LHS in (C.2) is in principle exact in C T ∼ N 2 while the RHS is obtained by keeping only planar diagrams with ∆ ≪ C T and then setting ∆ = ∆ H in the end.
The propagator for the field φ was given in (3.2) by where we introduced the notation φ i = φ(x i ) and dropped the |x ij | −2 coming from (C.3).
The position dependence is easily restored in the end. Now it is seen that the RHS of (C.4) is proportional to the two-point function 23 of O H and we therefore find that which is exact to all orders in C T . Including the normalization factor of O H in (C.1) and By comparing (C.6) with (5.10) we find that 23 Up to the position dependence.
Note that in (C.6) the normalization of O H cancels the contribution from non-planar diagrams in limit ∆ H ∼ C T . For ∆ = 2 in (3.3), it is trivial to compute the normalization exact in N to get the correction to λ O ∆ O ∆ O 2 in (C.6).
Consider now the stress tensor operator defined in (3.5) and the three-point function . This is fixed by the Ward identity but is an instructive example before considering more general multi stress tensors. In the same way as the OPE coefficient was found in the O ∆ × O ∆ OPE, due to the tensor structure being fixed by conformal symmetry, we consider the term proportional to x µ 13 x ν 13 in the three-point function. This comes from the − 1 T r(φ∂ µ ∂ ν φ) term in the stress tensor when ∂ µ ∂ ν φ is contracted with one of the ∆ H number of φ(x 1 ) fields. Doing this contraction we therefore see that where the factor 8 comes from the derivatives and we again suppress the spacetime dependence. The RHS of (C.8) is also proportional to the normalization constant of O H .
Including the normalization factor of the stress tensor in This agrees with (3.7).
We now want to show that is true for minimal-twist multi stress tensors with any spin.
For simplicity, consider the double-stress tensor with spin 4 defined in (3.8) Similarly to the calculation of the three-point function with the stress tensor, we can ob- by considering the term proportional to x µ 13 x ν 13 x ρ 13 x σ 13 . This will be due to the term when contracting ∂ µ ∂ ν φ with some φ(x 1 ) and likewise contracting ∂ ρ ∂ σ φ with some other φ(x 1 ). The number of such contractions is given by ∆ H (∆ H − 1) and we find that ) :: T r(φ ∆ H 2 ) : , where the factor of 8 2 again is due to acting with the derivatives and note that the position of the φ 3 fields in in the last line is not important. It is again seen that the RHS of (C.11) is proportional to the normalization constant of O H . Including the normalization in (3.8) and (C.1) we find the three-point function O H O H (T 2 ) µνρσ and read off the OPE coefficient: which is seen to agree with (3.11) when setting ∆ H = ∆. Note that the corrections in (C.12) are solely due to corrections in the normalization of T 2 4,4 and therefore 4,4 to all orders in C T . These arguments generalize straightforwardly to minimaltwist multi stress tensor with any spin such that the results are the same as those obtained in the planar limit for ∆ ≪ C 2 T in Section 3 by setting ∆ H = ∆. The only correction in C T is then due to the normalization of the multi stress tensor.
The same argument applies to any scalar primary multi-trace operator O ∆ , without any derivatives, with OPE coefficients given by (C.6), (C.9) and (C.12).

Appendix D. Stress tensor thermal one-point function
In order to calculate thermal one-point functions in the free adjoint scalar model we use the fact that the thermal correlation function is related to the zero-temperature case by summing over images. Consider now the thermal one-point function of the stress tensor.
Generally, the one-point function of a spin-s symmetric traceless operator with dimension . . e µ s − (traces)), (D.1) where e µ 1 is a unit-vector along the thermal circle. Consider first the canonically normalized stress tensor given by T . In order to find the one-point function, use the following: and To get the thermal correlator, we use (D.2) and (D.3) with x, y along the thermal circle separated by a distance mβ, with m integer, and sum over m = 0. The relevant terms for calculating the one-point functions in terms of fundamental fields are therefore where we note that only the first term in each equation in (D.4) contribute to the non-trace term in (D.1).
We therefore find for the stress tensor one-point function: where the 2ζ(4) comes from summing over images and we therefore have This agrees with f = b T (can) µν d in eq. (2.17) in [13] for (N 2 − 1) free scalar fields. This also agrees with a 2,2 = π 4 ∆ 45 found from the two-point thermal correlator using: . This is simply related to the one-point function for the unit-normalized stress tensor by (to leading order where the RHS is found by a conformal transformation to the plane with Z µ = where we reintroduced the radius of the sphere R, The thermal one-point function of an operator O τ,s , with twist τ and spin s, on S 1 ×S 3 is fixed by conformal symmetry [13] O τ,s ( where f O τ,s (0) = 1 and e µ = (1, 0, 0, 0).
We assume thermalization of the stress tensor in the heavy state: where T µν (x) β is the thermal one-point function at inverse temperature β evaluated on S 1 × S 3 , with R being the radius of S 3 . Using (D.10)-(D.12) we find Using (D.13) for R → ∞ in the free adjoint scalar theory, together with the one- This agrees with (4.2).

Appendix E. Dimension-six spin-four single trace operator
We want to calculate the contribution of the single trace operator with τ = 2 and s = 4. The unit-normalised O 2,4 operator is given by 24 The relative coefficients are fixed by demanding that it is a primary operator [K α , Ξ µνρσ ] = 0. Explictily, this is done using the conformal algebra and the action on the fundamental field φ The relevant commutators in order to fix Ξ µνρσ are which can also be found in e.g. Appendix F in [94].
and therefore the OPE coefficient Now, it is easy to check that which agrees with a 2,4 in (5.8).
Appendix F. Thermal one-point functions of multi-trace operators in the large- Therefore we have which is simply due to large-N factorization. As ∂ 2n ∂ l O b is a descendant of O b , it is easy to explicitly show that ∂ 2n ∂ l O b β = 0 for n = 0 or l = 0, from which it follows that Similar reasoning holds for all terms in [O a O b ] n,l , so we conclude for n = 0 or l = 0 that It is easy to generalise (n and/or l non-zero) Using the canonical scaling for the OPE coefficients (5.18) it is found that these multi-trace operators give a suppressed contribution to the thermal two point function in the large-N limit: The conclusion is that these operators with n = 0 or l = 0 do not contribute to the thermal two-point functions to leading order in N . Note that for n = l = 0, the operator . . O a k : and it does contribute to the thermal 2pt function since From (F.5) it is seen that multi stress tensor operators of the schematic form [T k ] n,l with either n or l, or both, being non-zero will not contribute to the thermal correlator to leading order in N on the plane.

Appendix G. Free boson in two dimensions
In this appendix we discuss free scalars in two dimensions. We first consider a single  25 We check this explicitly up to the O(1/β 4 ). 26 Deviation from the Virasoro vacuum block in the Regge limit of four-point HHLL correlator is observed in [95] as well.

G.1. Review free boson in two dimensions
We consider single free boson φ(z) in two dimensions. The stress tensor can be written in terms of Virasoro modes as This stress tensor is unit-normalized The fundamental field can be expressed as Laurent series where oscillators α n obey the following algebra which is properly defined when h is a positive integer. Its asymptotic in-state is given by One can check that this operator is a quasi-primary but not a Virasoro primary.
The thermal two-point function of this operator for even h is given by This expression is obtained by writing all possible Wick contractions between fundamental fields ∂φ, including those that belong to same operator O h , that we call self-contractions.
Fundamental fields are separated along the thermal circle in all Wick contractions. Factors The sum over n comes from doing n self-contractions within each of the external operators. i.e. it represents the disconnected contribution.
Since the state O h is quasi-primary, it transforms properly only with respect to the global conformal transformation. These are just the Möbius transformations in twodimensional spacetime z → az+b cz+d , with ad − bc = 1. On the other hand, the usual way to calculate the thermal two-point function of primary operators in two dimensions is to do a conformal transformation from the plane to the cylinder with radius β, z → β 2π log(z). This transformation is clearly not one of the Möbius transformations and that is why we can not use this method to compute the thermal two-point functions of quasi-primary operators.
Expanding (G.11) for T = 1 β → 0 one finds This is obtained by the Wick contractions of fundamental fields in the stress tensor, that are separated along the thermal circle. The same result can be obtained by the transform of the stress tensor from the plane to the cylinder using the Schwarzian derivative.
We define the OPE coefficient of unit-normalized operator O, with quantum numbers 2h. This OPE coefficient is fixed by the Ward identity. Now, it follows which reproduces the second term on the RHS of (G.13).
We are now interested in the contributions of quasi-primary operators with quantum numbers (4, 0). There are only two linearly independent operators with these quantum numbers given by 27  where k i > 0. It is straightforward to check It follows that α 4 −1 |0 is already quasi-primary and one can make only one more as We compute the OPE coefficients of : T T : and Λ 4 with two O h operators. We express all states in terms of free theory oscillators and use algebra (G.4) to find Now, we evaluate the thermal one-point functions of Λ 4 andΛ 4 . From (3.4) in [58] we have * T 2 * β = 3π 4 20β 4 , (G.30) 28 These states are not unit-normalized.
which is the thermal one-point function of the first term on the RHS of (G. 19). The second term can be written as − 3 . It is clear that it will not affect the thermal one-point function of Λ 4 (z), as ∂ 2 T β = 0. Therefore, from (G.19), we have Now, it follows which is the third therm at the RHS of (G.13). On the other hand, we can evaluate the

G.4. Free adjoint scalar model in two dimensions
In this subsection we study a large-c theory. Consider the free adjoint SU (N ) scalar in 2d with The thermal two point of the quasi-primary operator O h = 1 √ hN h : T r((∂φ) h ) : follows immediately from the result in four dimensions upon replacing the propagator of fundamental fields. We find that Consider first the normalized stress tensor which is given by with c = N 2 so that T (z)T (0) = 1 z 4 . By calculating the OPE coefficient with O h and the thermal one-point function of T , one finds that these are the same as those for the scalar T r(φ 2 ) operator in four dimensions so that T β = π 2 N 3 √ N , and the product reproduces the weight two term in (G.38):  Consider the state * T T * (0)|0 , which is given in terms of oscillator modes by * T T * (0)|0 = 1 Now T r(α m −1 )|0 is a quasi-primary while T r(α −3 α −1 )|0 is not. One way to make it a quasi-primary is to simply remove the second term in (G.45) and then we get a quasiprimary state which is just : T T : |0 . Another option is to remove a descendant of the stress tensor to construct |Λ 4 . To do the latter we need to remove the descendant of the stress tensor with weight 4 given by ∂ 2 T Acting on the vacuum we find Consider now L 1 = √ 2 N (T r(α −1 α 2 ) + T r(α −2 α 3 + . . .)) which acts as L 1 T r(α 2 −2 )|0 = 4 √ 2 N T r(α −1 α −2 )|0 and as L 1 T r(α −3 α −1 )|0 = 3 √ 2 N T r(α −1 α −2 )|0 . We can therefore construct a quasi-primary state annihilated by L 1 : T r(α −3 α −1 )|0 − 3 4 T r(α 2 −2 )|0 . The quasiprimary |Λ 4 is then given by: where the term ∝ 1 N O (2) β is subleading since it is single trace. We find that 55) which agrees with the weight 4 term in (G. 38).
Note that it is explicitly seen that one can write Λ 4 either as * T T * +(desc. of T) or as : T T : + 1 N O ST with O ST a quasi-primary single trace operator. In this case the single trace operator which one needs to add to : T T : to get Λ 4 can be written as a sum of descendants O (2) ∝ ∂ 2 T − 5 √ 2 T r(∂ 2 φ∂ 2 φ). Explicitly, we have β 2∆ sin ∆ πz β sin ∆ πz β = 1 (zz) ∆ 1 + π 2 ∆(z 2 +z 2 ) 6β 2 + π 4 ∆(5∆ + 2) 360β 4 (z 4 +z 4 ) + π 4 ∆ 2 36β 4 z 2z2 + O 1 β 6 . (G.58) One can easily check that the OPE coefficients of stress tensor T and its anti-holomorphic partnerT with O ∆ are given by while their thermal one-point function are given by It is easy to check that terms proportional to β −2 in (G.58) are contributions of T andT operators T β λ O ∆ O ∆ T z 2 + T β λ O ∆ O ∆Tz 2 = π 2 ∆(z 2 +z 2 ) 6β 2 . (G.61) We compute the OPE coefficient of operators Λ 4 , defined by (G. 48), and its antiholomorphic partnerΛ 4 with O ∆ and obtain which agrees with (C.26) in [66]. Its thermal one-point function (which is the same as Λ 4 β ) is given by (G.54). Another operator that contributes to thermal two-point function where N (∆) is a normalization constant which to leading order in N is given by The thermal two-point function is given by (H.5) The unit-normalized stress tensor is given by where C T = 4 3 N . The OPE coefficient of the stress tensor is again found by Wick contractions to be in agreement with the stress tensor Ward identity. The double-stress tensor is given by : We therefore see the that the thermal one-point function of the double-trace operator factorizes on the plane. We expect a similar argument to hold for multi stress tensors. 29 Note that this is not true for all operators but is in line with the fact that a 4,4 is unaffected by the second term in (H.3).