Thermalization in Large-N CFTs
aa r X i v : . [ h e p - t h ] F e b Thermalization in Large- N CFTs
Robin Karlsson, Andrei Parnachev and Petar Tadi´c
School of Mathematics, Trinity College Dublin, Dublin 2, Ireland
Abstract In d -dimensional CFTs with a large number of degrees of freedom an important set ofoperators consists of the stress tensor and its products, multi stress tensors. Thermalizationof such operators, the equality between their expectation values in heavy states and at finitetemperature, is equivalent to a universal behavior of their OPE coefficients with a pair ofidentical heavy operators. We verify this behavior in a number of examples which includeholographic and free CFTs and provide a bootstrap argument for the general case. In afree CFT we check the thermalization of multi stress tensor operators directly and alsoconfirm 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 bydisentangling the contributions of other light operators. Unlike multi stress tensors, theselight operators violate the Eigenstate Thermalization Hypothesis and do not thermalize.February 2021 karlsson, parnachev, tadicp @ maths.tcd.ie ontents
1. Introduction and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Thermalization and universality . . . . . . . . . . . . . . . . . . . . . . . . . 53. OPE coefficients in the free adjoint scalar model . . . . . . . . . . . . . . . . . . 103.1. Stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2. Double-stress tensor with minimal twist . . . . . . . . . . . . . . . . . . . . 123.3. Double-stress tensor with minimal twist and spin s = 6 . . . . . . . . . . . . . 133.4. Minimal-twist multi stress tensors . . . . . . . . . . . . . . . . . . . . . . 133.5. Double-stress tensors with non-minimal twist . . . . . . . . . . . . . . . . . 154. Thermal one-point functions in the free adjoint scalar model . . . . . . . . . . . . 164.1. Stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2. Double-stress tensor with minimal twist . . . . . . . . . . . . . . . . . . . . 174.3. Minimal-twist multi stress tensors . . . . . . . . . . . . . . . . . . . . . . 174.4. Double-stress tensors with non-minimal twist . . . . . . . . . . . . . . . . . 184.5. Triple-stress tensors with non-minimal twist . . . . . . . . . . . . . . . . . . 195. Thermal two-point function and block decomposition . . . . . . . . . . . . . . . . 215.1. Thermal two-point function of a single trace scalar operator . . . . . . . . . . . 215.2. CFT data of scalar operators with dimensions two and four . . . . . . . . . . . 235.3. CFT data of single-trace operator with twist two and spin four . . . . . . . . . . 255.4. CFT data of double-trace operators with twist and spin equal to four . . . . . . . 266. Comparison with the eigenstate thermalization hypothesis . . . . . . . . . . . . . 277. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Appendix A. OPE coefficients from Wick contractions . . . . . . . . . . . . . . . . 33Appendix B. Subleading twist double-stress tensors . . . . . . . . . . . . . . . . . . 38Appendix C. Single trace operator with dimension ∆ ∼ C T . . . . . . . . . . . . . . 40Appendix D. Stress tensor thermal one-point function . . . . . . . . . . . . . . . . . 43Appendix E. Dimension-six spin-four single trace operator . . . . . . . . . . . . . . . 46Appendix F. Thermal one-point functions of multi-trace operators in the large- N limit . . 47Appendix G. Free boson in two dimensions . . . . . . . . . . . . . . . . . . . . . 48G.1. Review free boson in two dimensions . . . . . . . . . . . . . . . . . . . . . 49G.2. Thermal two-point function of quasi-primary operator . . . . . . . . . . . . . 50G.3. Quasi-primaries, OPE coefficients, and thermal one-point functions . . . . . . . 51G.4. Free adjoint scalar model in two dimensions . . . . . . . . . . . . . . . . . . 54Appendix H. Vector model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Appendix I. Factorization of thermal correlators . . . . . . . . . . . . . . . . . . . 61 . Introduction and summary Holography [1-3] provides us with a useful tool to study d -dimensional CFTs at largecentral charge C T , especially when combined with modern CFT techniques (see e.g. [4-6]for reviews). One of the basic objects in this setup is a Witten diagram with a singlegraviton exchange which contributes to four-point functions. It can be decomposed intothe conformal blocks of the stress-tensor and of the double-trace operators made out ofexternal fields [7].When a pair of the external operators denoted by O H is taken to be heavy, with theconformal dimension ∆ H ∼ C T , and the other pair denoted by O L stays light, the resultingheavy-heavy-light-light (HHLL) correlator describes a light probe interacting with a heavystate. In this case, operators which are comprised out of many stress tensors (multi stresstensor operators) contribute, together with the multi-trace operators involving O L . As wereview below, the OPE coefficients of the scalar operators with a (unit-normalized) multistress tensor operator T kτ,s , which contains k stress tensors and has twist τ and spin s ,scale like λ O ∆ O ∆ T kτ,s ∼ ∆ k /C k/ T for large ∆.The contribution of a given multi stress tensor operator to the HHLL four-pointfunction hO H O L O L O H i can be compared to the contribution of the same operator to thecorresponding two-point function at finite temperature β − , hO L O L i β . In this paper weargue that they are the same in generic large- C T CFTs. As we explain later, this meansthat OPE coefficients of T kτ,s with the two heavy operators O H , hO H T kτ,s O H i , are equalto their finite temperature expectation values, h T kτ,s i β . The relation between the inversetemperature β and the conformal dimension ∆ H is set by considering the stress tensor( k = 1 , τ = d − , s = 2), but the equality between the thermal expectation values andthe OPE coefficients for all other multi stress tensor operators is a nontrivial statement.We call it “the thermalization of the stress tensor sector” . It is directly related to theEigenstate Thermalization Hypothesis (ETH) [23-27], as we review below. Hence, we arguethat all multi stress tensor operators in the large- C T CFTs satisfy the ETH. In d = 2 theETH and thermalization have been studied in e.g. [28-56]. See [8-22] for some previous work on finite temperature conformal field theories in d > We show this explicitly for certain primary heavy operators O H in free CFTs. We also observethat other light operators do not satisfy the thermalization property that the stress tensor sectorenjoys. d > T kτ,s . The heavy state we consider iscreated by a scalar operator O H with dimension ∆ H ∼ C T and by thermalization of amulti stress tensor operator we mean hO H | T kτ,s |O H i (cid:12)(cid:12)(cid:12) ∆ kHCk/ T = λ O H O H T kτ,s (cid:12)(cid:12)(cid:12) ∆ kHCk/ T = h T kτ,s i β , (1 . |O H i on the sphere of unit radius is created by the operator O H , λ O H O H T kτ,s are the OPE coefficients of T kτ,s in the O H × O H OPE and | ∆ kH /C k/ T meanswe keep only leading terms that scale like ∆ kH /C k/ T ∼ C k/ T . In (1.1) h T kτ,s i β is the one-point function on the sphere at finite temperature β − . Note that the OPE coefficientsinvolving the stress tensor are fixed by the Ward identity, and hence eq. (1.1) for the Here we are suppressing the tensor structure. Note that all terms scale like C k/ T which isconsistent with T kτ,s being unit-normalized. β − and ∆ H . By the large- C T factorization , the thermal one-point functions of multi stress tensors can be relatedto the thermal one-point function of the stress tensor itself. Explicitly, h T kτ,s i β = c kτ,s ( h T d − , i β ) k = c kτ,s ( λ O H O H T d − , ) k , (1 . c kτ,s are theory-independent coefficients that appear because of the index structure in h T kτ,s i 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 coefficientsis universal, i.e. it does not depend on the theory . We provide a bootstrap argument forthis universality in all large- C T theories. Also note that (1.2) is written for multi-traceoperators T kτ,s which do not contain derivatives, but the presence of derivatives does notaffect the statement of universality.In Section 3, we check the universality by computing a number of the multi stress ten-sor OPE coefficients in a free SU ( N ) adjoint scalar theory in d = 4 dimensions. We com-pare the leading ∆ H behavior in the free theory with results from holography/bootstrapand find perfect agreement in all cases listed below. After fixing the coefficients for thestress tensor case in Section 3.1, we look at the first nontrivial case, T , in Section 3.2.Section 3.3 is devoted to the double stress tensor with two derivatives, T , . This is anoperator whose finite temperature expectation value vanishes in the large volume limit (onthe plane), but is finite on the sphere. In Section 3.4 we consider minimal twist multi stresstensors of the type T k k, k . Section 3.5 is devoted to multi stress tensors with non-minimaltwist, T , and T , .In Section 4, we verify that (1.1) holds in the free adjoint scalar theory for a varietyof operators. In this section we again consider d = 4, but in addition, take the infinitevolume limit. This is for technical reasons – it is easier to compute a finite temperatureexpectation value on the plane than on the sphere. We spell out the index structure in See [69] for a general discussion of large- N factorization and [70,71] and [10] for the discussionin the context of gauge theories and CFTs respectively. The factorization holds in adjoint modelsin the ’t Hooft limit at finite temperature, but there are counterexamples, like e.g. a directproduct of low- C T CFTs. However the factorization of multi stress tensors would still apply inthese models. This amounts to the large- C T factorization of correlators hO H | T µν . . . T αβ |O H i in heavystates. d = 4. By decomposing the correlator into thermal blocks we read off theproduct of thermal one-point functions and the OPE coefficients for several operators oflow dimension and observe agreement with the results of Sections 3 and 4. Due to thepresence of multiple operators with the same dimension and spin, we have to solve a mixingproblem to find which operators contribute to the thermal two-point function.In Section 6 we explain the relation between our results and the Eigenstate Ther-malization Hypothesis. We observe that unlike multi stress tensors, other light operatorsexplicitly violate the Eigenstate Thermalization Hypothesis and do not thermalize. Weend with a discussion in Section 7.Appendices A, B, and C contain explicit calculations of OPE coefficients while in Ap-pendices D and E thermal one-point functions are calculated. In Appendix F we reviewthe statement that the thermal one-point functions of multi-trace operators with deriva-tives vanish on S × R d − . In appendix G we study a free scalar in two dimensions andcalculate thermal two-point functions of certain quasi-primary operators. In AppendixH we consider a free scalar vector model in four dimensions. Appendix I discusses thefactorization of multi-trace operators in the large volume limit.
2. Thermalization and universality
In the following we consider large- C T CFTs on a ( d − R , which we set to unity for most of this section. As reviewed in [66], the stress tensorsector of conformal four-point functions consists of the contributions of the stress tensorand all its composites (multi stress tensors). The HHLL correlators we consider involve twoheavy operators inserted at x E = ±∞ and two light operators inserted on the Euclideancylinder, with angular separation ϕ and time separation x E . The correlator in a heavystate (the HHLL correlator on the cylinder) is related to the correlator on the plane by aconformal transformation hO H |O ( x E , ϕ ) O (0) |O H i = lim x →∞ x H ( z ¯ z ) − ∆ / hO H ( x ) O (1) O ( z, ¯ z ) O H (0) i , (2 . z, ¯ z ) on the plane are related to the coordinates ( x E , ϕ ) via z = e − x E − iϕ , ¯ z = e − x E + iϕ . (2 . G ( z, ¯ z ) = lim x →∞ x H hO H ( x ) O (1) O ( z, ¯ z ) O H (0) i (cid:12)(cid:12)(cid:12) multi stress tensors (2 . G ( z, ¯ z ) = 1[(1 − z )(1 − ¯ z )] ∆ X T kτ,s P ( HH,LL ) T kτ,s g (0 , τ,s (1 − z, − ¯ z ) , (2 . τ, s, k label the twist, spin, and multiplicity of multi stress tensors. We are interestedin the double scaling limit where the central charge and the dimension of O H are large, C T , ∆ H → ∞ with their ratio µ ∝ ∆ H /C T fixed. In this limit the products of the OPEcoefficients which appear in (2.4) are given by P ( HH,LL ) T kτ,s = (cid:18) − (cid:19) s λ OO T kτ,s λ O H O H T kτ,s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∆ HCT (cid:1) k , (2 . (cid:16) ∆ H √ C T (cid:17) k term in the OPE coefficients λ O H O H T kτ,s , butretain all terms in the OPE coefficients of the light operators λ OO T kτ,s . The contributionof the conformal family of a multi stress operator T kτ,s to the HHLL correlator is therefore hO H |O ( x E , ϕ ) O (0) |O H i| T kτ,s = P ( HH,LL ) T kτ,s g (0 , τ,s (1 − z, − ¯ z )[ √ z ¯ z (1 − z )(1 − ¯ z )] ∆ . (2 . β − . To isolate the contributionof the conformal family associated with T kτ,s , we can write the thermal correlator as hO ( x E , ϕ ) O (0) i β = 1 Z ( β ) X i e − β ∆ i hO i |O ( x E , ϕ ) O (0) |O i i = 1[ √ z ¯ z (1 − z )(1 − ¯ z )] ∆ X T kτ,s (cid:18) − (cid:19) s λ OO T kτ,s g (0 , τ,s (1 − z, − ¯ z ) h T kτ,s i β + . . . , (2 . h T kτ,s i β = 1 Z ( β ) X i e − β ∆ i λ O i O i T kτ,s (2 . T kτ,s operator and the dotsdenote contributions from other operators. In (2.8) Z ( β ) is the partition function and thesum runs over all operators, including descendants . Note that h T kτ,s i β = β − ( τ + s ) f kτ,s ( β ) . (2 . f kτ,s ( β ) ∼ C k/ T is a theory-dependent nontrivial function of β which approaches a constant f kτ,s (0)in the large volume ( β →
0) limit.Consider the thermalization of the stress tensor sector: hO H | T kτ,s |O H i (cid:12)(cid:12)(cid:12) ∆ kHCk/ T = λ O H O H T kτ,s (cid:12)(cid:12)(cid:12) ∆ kHCk/ T = h T kτ,s i β . (2 . T kτ,s is unit-normalized, so all terms in (2.10) scale like C k/ T . Eq. (2.10) impliesthe equality between (2.6) and the corresponding term in (2.7). Note that the left-handside of (2.10) is a function of the energy density while the right-hand side is a functionof temperature. The relationship is fixed by considering the stress tensor case: the corre-sponding function f d − , ( β ) is determined by the free energy on the sphere (see Section6). In the following, we will first discuss the case where the multi stress operators T kτ,s donot have any derivatives inserted, and then show that the derivatives do not change theconclusions. Assuming large- C T factorization, the leading C T behavior of h T kτ,s i β on thesphere is determined by that of the stress tensor. Schematically, h T kτ,s i β = c kτ,s ( h T d − , i β ) k + . . . , (2 . c kτ,s are numerical coefficients, which depend on k, τ, s , but are independent of thedetails of the theory and the dots stand for terms subleading in C − T . By combining (2.11)and (2.10), one can formulate a universality condition λ O H O H T kτ,s (cid:12)(cid:12)(cid:12) ∆ kHCk/ T = c kτ,s ( λ O H O H T d − , ) k = c kτ,s (cid:18) d − d (cid:19) k ∆ kH C k T , (2 . The corresponding conformal blocks can be obtained in the usual way by applying thequadratic conformal Casimir and solving the resulting differential equation [72]. λ O H O H T d − , ( T d − , here is unit-normalized). In other words, ther-malization and large- C T factorization imply that the leading ∆ k /C k/ T behavior of themulti 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 con-tain any derivatives in them. However, the story largely remains the same when thederivatives are included, as long as their number does not scale with C T . Indeed, thethree-point function involving the stress-tensor with added derivatives, ∂ α . . . ∂ β T µν stillbehaves like λ O H O H ∂ α ...∂ β T µν ≃ ∆ H / √ C T up to a theory-independent coefficient. Hence,(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 onlythrough the ratio β/R . Moreover, in the large volume limit, factors of R need to drop outof (2.6) and (2.7) to have a well defined limit. To see this we use that (1 − z ) → − ¯ z ) → R → ∞ and the conformal blocks behave as (see e.g. [6]) g (0 , τ,s (1 − z, − ¯ z ) ∼ N d,s [(1 − z )(1 − ¯ z )] τ + s C ( d/ − s (cid:16) (1 − z ) + (1 − ¯ z )2 p (1 − z )(1 − ¯ z ) (cid:17) ∼ N d,s | x | τ + s R τ + s C ( d/ − s (cid:16) x E | x | (cid:17) , (2 . | x | = p ( x E ) + x , C ( d/ − s ( x E | x | ) is a Gegenbauer polynomial and N d,s = s !( d/ − s .Including the factor [(1 − z )(1 − ¯ z )] − ∆ from (2.6) in (2.13) this agrees with the thermalblock on S × R d − in [13]. Now from the thermalization of the stress tensor we will findin the large volume limit that ∆ H C T ∝ (cid:16) Rβ (cid:17) d , (2 . g (0 , τ,s (1 − z, − ¯ z ) λ OO T kτ,s λ O H O H T kτ,s (cid:12)(cid:12)(cid:12) ∆ kHCkT ∝ R dk − ( τ + s ) β − dk . (2 . T kτ,s is given by τ + s = dk + n where n = 0 , , . . . .Therefore, the only multi stress tensors that contribute in the large volume limit havedimensions dk . Restoring R in (2.6)-(2.7) and inserting (2.15) one finds that R drops outin the large volume limit. The correct dependence β − ( τ + s ) from (2.9) in the R → ∞ limitis also recovered in (2.6) using (2.15). The multi stress tensor operators that contribute8n the large volume limit are therefore of the schematic form T µ ν T µ ν · · · T µ k ν k witharbitrarily many contractions and no derivatives.In holographic theories thermalization and the Wilson line prescription for the correla-tor allows one to compute the universal part of the OPE coefficients (see [61,73] for explicitcomputations 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 [58],it was not immediately obvious for multi stress tensors of non-minimal twist. Some suchOPE coefficients were computed in [58,66]. (See e.g. eqs. (5.48), (5.51), (5.52), (5.57) and(D.1)-(D.5) in [66]). Indeed, the leading ∆ k /C k/ T behavior of these OPE coefficients isindependent of the Gauss-Bonnet coupling.What about a general large- C T theory? We first consider the OPE coefficients ofdouble-stress tensors. To this end, consider the four point function hO T µν T ρσ Oi where O is a scalar operator with scaling dimension ∆. In the direct channel O × O → O ′ → T µν × T ρσ for finite ∆ and large C T , the leading contribution in the large- C T limit comesfrom the identity operator O × O → → T µν × T ρσ . The subleading contributions in thedirect-channel are due to single trace operators as well as double trace operators made outof the external operators of the schematic form T τ,s and [ OO ] n,l =: O ∂ n ∂ . . . ∂ l O :. Theexchange of the identity operator is reproduced in the cross-channel O× T µν → [ O T αβ ] n,l →O × T ρσ by mixed double-trace operators [ O T αβ ] n,l with OPE coefficients fixed by theMFT [74-76]. The subleading contributions in 1 /C T are then due to corrections to theanomalous dimension and OPE coefficients of [ O T 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 traceoperator O , whose contribution is universally fixed by the stress tensor Ward identity tobe ( λ O T d − , O ) ∝ ∆ /C T times the conformal block. This gives a universal contributionto λ OO T τ,s as was also noted in [68].We now want to consider the case where ∆ ∼ C T and study the OPE coefficients ofthe double-stress tensor operators in the O × O
OPE. Firstly, note that the contributionfrom T τ,s to the four-point function expanded in the direct channel is proportional to λ OO T τ,s λ T T T τ,s . The OPE coefficients λ T T T τ,s are fixed by the MFT and are independentof ∆ and therefore the dependence on the scaling dimension comes solely from the OPE This correlator for finite ∆ was recently considered in holographic CFTs with ∆ gap ≫ ≪ ∆ gap in [68]. λ OO T τ,s . In the cross-channel, we analyze two kinds of contributions: from theexchanged operator O and from all other operators O ′ = O . From the operator O we get auniversal contribution to the OPE coefficients in the direct channel λ OO T τ,s , that we denoteby λ (1) OO T τ,s . This contribution is universal since it only depends on ( λ O T d − , O ) ∝ ∆ /C T in the cross-channel, which is fixed by the Ward identity. The contributions from otheroperators O ′ to the same OPE coefficient will be denoted by λ (2) OO T τ,s , such that λ OO T τ,s = λ (1) OO T τ,s + λ (2) OO T τ,s . Note that it also follows from the stress tensor Ward identity thatthe only scalar primary that appears in the cross-channel is O . The operator O ′ thereforenecessarily has spin s = 0.To prove universality we need to show that λ (2) OO T τ,s ≪ ∆ /C T in limit 1 ≪ ∆ ∝ C T by studying the ∆ dependence of the OPE coefficients λ O T d − , O ′ in the cross-channel.For operators O ′ , such that ∆ O ′ ≪ ∆, we expect that these OPE coefficients are heavilysuppressed. It would be interesting to understand if one could put a general bound on thecontribution of these operators in the cross-channel in any large- C T theory. On the otherhand, assuming thermalization, the OPE coefficients due to operators O ′ such that ∆ O ′ ∼ ∆ have been calculated in [20]. The obtained results are in agreement with our expectation,namely, these OPE coefficients are suppressed in 1 ≪ ∆ ∝ C T limit. Additionally, in thecross-channel we have double-trace operators [ O T αβ ] n,l , whose OPE is fixed by the MFTand it does not get ∆-enhanced.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 OO T τ,s one may consider hO T d − , T τ,s Oi . The bootstrapargument above can be applied again by using the fact that OPE coefficients λ OO T τ,s areuniversal, and the OPE coefficients λ O T τ,s O ′ are again expected to be subleading.
3. OPE coefficients in the free adjoint scalar model
In this section we consider a four-dimensional theory of a free scalar in the adjointrepresentation of SU ( N ), see [77-82] for related work. The relation between N and thecentral charge C T in this theory is [83] C T = 43 ( N − , (3 . N (large- C T ) limit. The propagator for the scalar field φ ij isgiven by h φ ij ( x ) φ kl ( y ) i = (cid:18) δ il δ kj − N δ ij δ kl (cid:19) | x − y | . (3 . O ∆ ( x ) = 1 √ ∆ N ∆2 : T r ( φ ∆ ) : ( x ) , (3 . . . . : denotes the oscillator normal ordering and the normalization is fixed by hO ∆ ( x ) O ∆ ( y ) i = 1 | x − y | . (3 . O ∆ × O ∆ OPE. Assuming we can take ∆ → ∆ H ∼ C T , the large-∆ limit ofthese OPE coefficients is shown to be universal. One may worry that for ∆ H ∼ C T we canno longer trust the planar expansion, but, as we show in Appendix C, the large-∆ limit ofthe planar result yields the correct expression even for ∆ H ∼ C T . The stress tensor operator is given by T µν ( x ) = 13 √ C T : T r (cid:18) ∂ µ φ∂ ν φ − φ∂ µ ∂ ν φ − (trace) (cid:19) : ( x ) , (3 . h T µν ( x ) T ρσ (0) i = 1 | x | (cid:16) I ( µρ ( x ) I ν ) σ ( x ) − (traces) (cid:17) , (3 . I µν ( x ) := δ µν − x µ x ν | x | . The OPE coefficient is fixed by the stress tensor Ward identityto be λ O ∆ O ∆ T , = − √ C T . (3 . .2. Double-stress tensor with minimal twist In this section we study the minimal-twist composite operator made out of two stresstensors ( T ) µνρσ ( x ) = 1 √ T ( µν T ρσ ) : ( x ) − (traces) , (3 . h ( T ) µνρσ ( x )( T ) κλδω (0) i = 1 | x | (cid:16) I ( µκ I ν λ I ρδ I σ ) ω − (traces) (cid:17) . (3 . hO ∆ ( x ) O ∆ ( x )( T ) µνρσ ( x ) i = λ O ∆ O ∆ T , | x | − | x | | x | ( Z µ Z ν Z ρ Z σ − (traces)) , (3 . Z µ = x µ | x | − x µ | x | . It is shown in Appendix A that the OPE coefficient λ O ∆ O ∆ T , is given at leading order in the large- C T limit by λ O ∆ O ∆ T , = 8 √ − C T . (3 . P ( HH,LL ) T , defined by (2.5) in the large-∆ limit , we obtain P ( HH,LL ) T , = (cid:18) − (cid:19) λ O H O H T , λ O ∆ O ∆ T , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∆ HCT (cid:1) = 881 ∆ H C T (cid:0) ∆ + O (∆) (cid:1) = µ (cid:18) ∆ O (∆) (cid:19) , (3 . µ = 1603 ∆ H C T . (3 . By the large-∆ limit, we strictly speaking mean 1 ≪ ∆ ≪ C T . However in this paper weoften extrapolate this to the ∆ ∼ C T regime. .3. Double-stress tensor with minimal twist and spin s = 6We consider double-stess tensor operator with two (uncontracted) derivatives inserted( T ) µνρσηκ ( x ) = 12 √
182 : (cid:18) T ( µν ∂ ρ ∂ σ T ηκ ) ( x ) − (cid:0) ∂ ( ρ T µν (cid:1) (cid:0) ∂ σ T ηκ ) (cid:1) ( x ) − (traces) (cid:19) : . (3 . h ( T ) µνρσηκ ( x )( T ) αβγδξǫ (0) i = 1 | x | (cid:16) I ( µα I ν β I ργ I σδ I ηξ I κ ) ǫ − (traces) (cid:17) . (3 . T ) µνρσηκ in the O ∆ × O ∆ OPE is given at leading order in the large- C T limit by λ O ∆ O ∆ T , = 83 r
291 ∆(∆ − C T . (3 . P ( HH,LL ) T , , defined by (2.5), in the large-∆ limit, we obtain P ( HH,LL ) T , = (cid:18) − (cid:19) λ O H O H T , λ O ∆ O ∆ T , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∆ HCT (cid:1) = 2819 ∆ H C T (cid:0) ∆ + O (∆) (cid:1) = µ (cid:18) ∆ O (∆) (cid:19) . (3 . We now consider multi stress tensors T k k, k . 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( T k ) µ µ ...µ k ( x ) = 1 √ k ! : T ( µ µ T µ µ · · · T µ k − µ k ) : ( x ) − (traces) . (3 . T k ) µ µ ...µ k in the O ∆ × O ∆ OPE, in the large- C T limit is givenby λ O ∆ O ∆ T k k, k = (cid:18) − (cid:19) k √ k ! C k/ T Γ(∆ + 1)Γ(∆ − k + 1) . (3 . See Appendix A for detailed computations of similar OPE coefficients. P ( HH,LL ) T , , defined by (2.5), in the large-∆ limit. We obtain this OPEcoefficient from (3.19) for k = 3, P ( HH,LL ) T , = (cid:18) − (cid:19) λ O H O H T , λ O ∆ O ∆ T , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∆ HCT (cid:1) = 322187 ∆ H C T (cid:0) ∆ + O (cid:0) ∆ (cid:1)(cid:1) = µ (cid:18) ∆ O (∆ ) (cid:19) . (3 . P ( HH,LL ) T k k, k in the large-∆ limit for general k , P ( HH,LL ) T k k, k = (cid:18) − (cid:19) k λ O H O H T k k, k λ O ∆ O ∆ T k k, k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∆ HCT (cid:1) k = 1 k ! (cid:18) (cid:19) k ∆ kH C kT (cid:0) ∆ k + O (cid:0) ∆ k − (cid:1)(cid:1) = µ k (cid:18) ∆ k k k ! + O (∆ k − ) (cid:19) . (3 . − ¯ z ≪ − z ≪
1, such that µ (1 − ¯ z )(1 − z ) is held fixed,only operators T k k, k contribute to the heavy-heavy-light-light four-point function givenby eq. (2.3). The conformal blocks of T k k, k in this limit are given by g (0 , k, k (1 − z, − ¯ z ) ≈ (1 − ¯ z ) k (1 − z ) k , (3 . G ( z, ¯ z ) ≈ − z )(1 − ¯ z )) ∆ e µ ∆120 (1 − ¯ z )(1 − z ) . (3 . − ¯ z ≪ − z ≪ .5. Double-stress tensors with non-minimal twist So far we have shown that the minimal-twist multi stress tensor OPE coefficients areuniversal in the limit of large ∆. In this subsection, we extend this to show that thesimplest non-minimal twist double-stress tensors also have universal OPE coefficients atlarge ∆.The subleading twist double-stress tensor with twist τ = 6 is of the schematic form: T µα T αν : and has dimension ∆ = 8 and spin s = 2. It is given by( T ) µν ( x ) = 1 √ T µα T αν : ( x ) − (trace) . (3 . T ) µν is unit-normalized, see Ap-pendix B for details.The OPE coefficient of ( T ) µν in the O ∆ × O ∆ OPE is found from the three-pointfunction in the large- C T limit, for details see Appendix B, hO ∆ ( x ) O ∆ ( x )( T ) µν ( x ) i = 4 √ − C T Z µ Z ν − (trace) | x | − | x | | x | , (3 . λ O ∆ O ∆ T , = 4 √ − C T . (3 . P ( HH,LL ) T , , defined by (2.5), in the large-∆ limit, we obtain P ( HH,LL ) T , = (cid:18) − (cid:19) λ O H O H T , λ O ∆ O ∆ T , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∆ HCT (cid:1) = 881 ∆ H C T (cid:0) ∆ + O (∆) (cid:1) = µ (cid:18) ∆ O (∆) (cid:19) . (3 . s = 0 whichis given by ( T )( x ) = 13 √ T µν T µν : ( x ) . (3 . hO ∆ ( x ) O ∆ ( x )( T )( x ) i is found in Appendix B to be hO ∆ ( x ) O ∆ ( x )( T )( x ) i = 2 √ − C T | x | − | x | | x | , (3 . λ O ∆ O ∆ T , = 2 √ − C T . (3 . P ( HH,LL ) T , in the large-∆ limit P ( HH,LL ) T , = λ O H O H T , λ O ∆ O ∆ T , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∆ HCT (cid:1) = 881 ∆ H C T (cid:0) ∆ + O (∆) (cid:1) = µ (cid:18) ∆ O (∆) (cid:19) . (3 .
4. Thermal one-point functions in the free adjoint scalar model
In this section we explicitly show that multi stress tensor operators thermalize in thefree theory by calculating the thermal one-point function of some of these operators on S × R . One-point functions of primary symmetric traceless operators at finite temperatureare fixed by symmetry up to a dimensionless coefficient b O (see e.g. [10,13]) hO µ ··· µ s O i β = b O β ∆ O (cid:16) e µ · · · e µ s O − (traces) (cid:17) . (4 . e µ is a unit vector along the thermal circle.To compare the thermal one-point functions and OPE coefficients from the previoussection, we need to derive a relation between ∆ H C T and the temperature β − . Here ∆ H ∼ N refers to the scaling dimension of a heavy operator O H with OPE coefficients given bythe large-∆ limit of those obtained in Section 3. One can relate the inverse temperature β to the parameter µ = H C T using the Stefan-Boltzmann’s law E/ vol( S ) = N π / β .The energy of the state E is related to its conformal dimension ∆ via E = ∆ /R . One canthen use vol( S ) = 2 π R and the relation between N and C T given by (3.1), to find µ = 1603 ∆ H C T = 1603 E RC T = 83 (cid:16) πRβ (cid:17) . (4 . See also Section 6 and Appendix D for alternative derivations. .1. Stress tensor The thermal one-point function for the stress tensor T , = T µν is calculated in Ap-pendix D where we find that b T , is given by b T , = − π N √ . (4 . b T , β − = λ O H O H T , . (4 . In this section we calculate the thermal one-point function of the double-stress tensoroperator with τ = 4 and spin s = 4. The operator is written explicitly in (3.8). Theleading contribution to the thermal one-point function of ( T ) µνρσ follows from the large- N factorization and is given by h ( T ) µνρσ i β = 1 √ h T ( µν i β h T ρσ ) i β − (traces)= 2 √ π N β ( e µ e ν e ρ e σ − (traces)) . (4 . b T , β − = λ O H O H T , (cid:12)(cid:12)(cid:12) ∆2 HCT . (4 . Consider now multi stress tensors T k k, k with twist τ = 2 k and spin s = 2 k . We showthat these operators thermalize for any k by calculating their thermal one-point functions: h ( T k ) µ µ ...µ k i β = b T k k, k β k ( e µ e µ · · · e µ k − (traces)) , (4 . b T k k, k follows from the large- N factorization: b T k k, k = 1 √ k ! ( b T , ) k = ( − ) k N k π k k √ k ! . (4 . b T k k, k β − k = λ O H O H T k k, k (cid:12)(cid:12)(cid:12) ∆ kHCk/ T . (4 . The subleading twist double-stress tensor is of the schematic form : T µα T αν : and hastwist τ = 6 and spin s = 2. The explicit form can be found in (3.24). The leading term inthe thermal one-point function is given by h ( T ) µν i β = 1 √ h T µα i β h T ν α i β − (trace)= b T , √ β ( e µ e ν − δ µν )= √ N π β ( e µ e ν − δ µν ) , (4 . b T , = √ N π . (4 . b T , β − = λ O H O H T , (cid:12)(cid:12)(cid:12) ∆2 HCT . (4 . τ = 8 and s = 0 which isgiven by (3.28). The thermal one-point function for this operator is h ( T ) i β = 13 √ h T µν i β h T µν i β = 13 √ b T , β − = π N √ β , (4 . in the first line comes from the index contractions. Hence, b T , = π N √ . (4 . b T , β − = λ O H O H T , (cid:12)(cid:12)(cid:12) ∆2 HCT . (4 . .5. Triple-stress tensors with non-minimal twist We consider the triple stress tensors with τ = 8 , s = 4 and τ = 10 , s = 2. Theunit-normalized triple stress tensor with τ = 8 can be written as( T ) µνρσ ( x ) = 1 √ (cid:0) : T ( µν T ρ | α | T ασ ) : ( x ) − (traces) (cid:1) , (4 . | α | denotes that index α is excluded from the symmetrization. The thermal one-point function follows from large- N factorization h ( T ) µνρσ i β = 1 √ (cid:0) h T ( µν i β h T ρ | α | i β h T ασ ) i β − (traces) (cid:1) = 12 √ b T , β ( e µ e ν e ρ e σ − (traces))= − π N β ( e µ e ν e ρ e σ − (traces)) , (4 . b T , = − π N . (4 . s = 4)is calculated holographically and is given by (D.1) in [66]. In the large-∆ limit it can bewritten as P ( HH,LL ) T , = (cid:18) − (cid:19) λ O ∆ O ∆ T , λ O H O H T , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∆ HCT (cid:1) = 642187 ∆ H ∆ C T + O (∆ ) . (4 . λ O ∆ O ∆ T , in the large-∆ limit λ O ∆ O ∆ T , = − √ C T / + O (∆ ) = − N + O (∆ ) , (4 . C T and N given by (3.1). Using (4.2)one can obtain b T , β − = λ O H O H T , (cid:12)(cid:12)(cid:12) ∆3 HC / T . (4 . s = 2.There are two linearly independent such operators that schematically can be written as19 T αβ T αβ T µν : and : T µα T αβ T βν :. We write the following linear combinations of theseoperators( T ) µν ( x ) = 110 √ (cid:0) : T αβ T αβ T µν : ( x ) + 4 : T µα T αβ T βν : ( x ) − (trace) (cid:1) , (4 . T ) µν ( x ) = 720 (cid:18) : T αβ T αβ T µν : ( x ) −
127 : T µα T αβ T βν : ( x ) − (trace) (cid:19) . (4 . T ) µν and ( ˜ T ) µν are unit-normalized and their overlap vanishes in the large- N limit h ( T ) µν ( x )( ˜ T ) ρσ ( y ) i = O (1 /N ) . (4 . N factorization, inthe large- N limit are given by h ( T ) µν i β = − r N π β ( e µ e ν − (trace)) , h ( ˜ T ) µν i β = O ( N ) , (4 . b T , = − r N π ,b ˜ T , = 0 . (4 . s = 2), with external scalar operators is given by (5.57) in [66]. In the large-∆limit it can be written as P ( HH,LL ) T , = (cid:18) − (cid:19) λ O ∆ O ∆ T , λ O H O H T , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ∆ HCT (cid:1) = 32729 ∆ H ∆ C T + O (∆ ) . (4 . λ O ∆ O ∆ T , : λ O ∆ O ∆ T , = − √
227 ∆ C / T + O (∆ ) = − √ √ N + O (∆ ) . (4 . b T , β − = λ O H O H T , (cid:12)(cid:12)(cid:12) ∆3 HC / T . (4 . . Thermal two-point function and block decomposition In this section we study the thermal two-point function hO ∆ O ∆ i β and decompose it inthermal blocks. We determine the contributions of a few low-lying operators, including thestress tensor T , and the double stress tensor T , . They exactly match the correspondingOPE coefficients and thermal expectation values computed in previous sections. Due tothe presence of multiple operators with equal scaling dimension and spin, there is a mixingproblem which we solve explicitly in a few cases. Related appendices include AppendixF, where we review the statement that the thermal one-point functions of multi-traceoperators with derivatives vanish on S × R d − and Appendix G, where we consider two-dimensional thermal two-point functions. In Appendix H we do a similar analysis for thevector model in four dimensions. The correlator at finite temperature β − in the free theory can be calculated by Wickcontractions using the propagators on S × R . Explicitly, the two-point function at finitetemperature is given by hO ∆ ( x ) O ∆ (0) i β = ˜ g ( x E , | x | ) ∆ + π ∆(∆ − β ˜ g ( x E , | x | ) ∆ − + . . . , (5 . g ( x E , | x | ) = ∞ X m = −∞ x E + mβ ) + x = π β | x | h Coth (cid:16) πβ ( | x | − ix E ) (cid:17) + Coth (cid:16) πβ ( | x | + ix E ) (cid:17)i . (5 . . Taking the β → ∞ limit of (5.1) we can read off the decomposition ofthe two-point function in terms of thermal conformal blocks on S × R with coordinates x = ( x E , x ). Here and below we assume that ∆ >
4. We further drop the disconnected term hO ∆ i β ∼ N . These terms will be proportional to β − a ˜ g ( x E , | x | ) ∆ − a , with a ≥
4. When decomposed intothermal blocks, these will not affect the operators with dimension ∆ < s = 0. | x | = p ( x E ) + x ≤ β the two-point function can be evaluatedusing the OPE: hO ∆ ( x ) O ∆ (0) i β = X O λ O ∆ O ∆ O | x | τ − x µ · · · x µ s O hO µ ··· µ s O i β , (5 . λ O ∆ O ∆ O is the OPE coefficient, τ and s O is the twist and spin of O , respectively.Using (4.1) together with (5.3), the two-point function on S × R can be organized in thefollowing way [13]: hO ∆ ( x ) O ∆ (0) i β = X O τ,s ∈O ∆ ×O ∆ a O τ,s β ∆ O | x | − τ + s C (1) s (cid:18) x E | x | (cid:19) , (5 . O τ,s , with twist τ and spin s , appearing in the OPE O ∆ × O ∆ ∼ O τ,s + . . . . In (5.4) C (1) s ( x E / | x | ) is a Gegenbauer polynomial which, togetherwith a factor of | x | − τ − s , forms a thermal conformal block in d = 4 dimensions and thecoefficients a O τ,s are given by a O τ,s = (cid:18) (cid:19) s λ O ∆ O ∆ O τ,s b O τ,s . (5 . β → ∞ one finds: hO ∆ ( x ) O ∆ (0) i β = 1 | x | h π ∆3 β | x | + π ∆90 β | x | (3 x (5∆ −
9) + (15∆ − x E ) ) + O ( β − ) i . (5 . a τ ′ ,s ′ := P O τ ′ ,s ′ a O τ ′ ,s ′ wherewe sum over all operators with twist τ ′ and spin s ′ : a , = π ∆3 ,a , = π ∆(3∆ − ,a , = π ∆45 . (5 . O ( β ) one finds a , = 2 π ∆945 ,a , = π ∆(∆ − . (5 . a τ,s genericallycontains the contribution from multiple operators. In the following section we calculatethe OPE coefficients and thermal one-point functions of operators which are not multistress tensors but contribute to (5.7) and (5.8).22 .2. CFT data of scalar operators with dimensions two and four We explicitly calculate the thermal one-point functions hOi β = b O β − ∆ O and OPEcoefficients λ O ∆ O ∆ O for scalar operators O with twist τ ′ = 2 and τ ′ = 4 using Wickcontractions. This is done to find which operators contribute to the thermal two-pointfunction and to resolve a mixing problem.For τ ′ = 2 there is only one such operator, the single trace operator O ( x ) = √ N : T r ( φ ) : ( x ) given in (3.3). The OPE coefficient is found by considering the three-pointcorrelator hO ∆ ( x ) O ∆ ( x ) O ( x ) i = λ O ∆ O ∆ O | x | − | x | | x | . (5 . N limit, and it is givenby hO ∆ ( x ) O ∆ ( x ) O ( x ) i = √ N | x | − | x | | x | , (5 . λ O ∆ O ∆ O = √ N to leading order in 1 /N . To calculate the thermal one-point function ∝ h T r ( φ ) i β , we include self-contractions, i.e. contractions of fundamentalfields within the same composite operator separated by a distance mβ along the thermalcircle for m = 0 and integer. Explicitly, the one-point function of O is given by hO ( x ) i β = 1 √ N X m =0 N ( mβ ) = π N √ β , (5 . b O = π N √ . (5 . a O is found using (5.10) and (5.12) a , = b O λ O ∆ O ∆ O = π ∆3 . (5 . O ∆ × O ∆ OPE. In order to construct an orthonormalbasis, consider the following single and double trace operators: O ( x ) = 12 N : T r ( φ ) : ( x ) , O , DT ( x ) = 12 √ N : T r ( φ ) T r ( φ ) : ( x ) . (5 . O that has vanishing overlap with O , DT ( x ) as follows:˜ O = N h O − c O O , DT O , DT i , (5 . N a normalization constant and c O O , DT is the overlap defined by hO ( x ) O , DT ( y ) i = c O O , DT | x − y | . (5 . c O O , DT = √ N and N = √ in the large- N limit, and the scalardimension four operator orthogonal to the double trace operator O , DT is therefore˜ O = 1 √ h O − √ N O , DT i . (5 . /N , it can still contributeto the thermal two-point function due to the scaling of OPE coefficients and one-pointfunction of a k -trace operator O ( k ) : b O ( k ) ∼ N k ,λ O ∆ O ∆ O ( k ) ∼ N k , (5 . N → ∞ .The one-point function and the OPE coefficient for O is found analogously to thatof O in the large- N limit b O = π N ,λ O ∆ O ∆ O = 2∆ N . (5 . N limit: hO , DT ( x ) i β = 1 √ hO ( x ) i β ) = π N √ β . (5 . N limit (see Appendix A) λ O ∆ O ∆ O , DT = √ − N . (5 . O in (5.17) h ˜ O i β = 1 √ β h b O − √ N b O , DT i = O ( N − ) , (5 . N − , it follows that the only scalar operator with dimension four contributingto the thermal two-point function is the double trace operator O , DT . From the OPEcoefficient and thermal one-point function of this double trace operator, using (5.20) and(5.21), we find the following contribution to the thermal two-point function a , = π ∆(3∆ − , (5 . The primary single trace operator Ξ = O , with twist τ = 2 and spin s = 4 is givenby Ξ µνρσ ( x ) = 196 √ N : T r (cid:0) φ ( ∂ µ ∂ ν ∂ ρ ∂ σ φ ) − ∂ ( µ φ )( ∂ ν ∂ ρ ∂ σ ) φ )+ 18( ∂ ( µ ∂ ν φ )( ∂ ρ ∂ σ ) φ ) − (traces) (cid:1) : ( x ) . (5 . N limit to be h Ξ µνρσ i β = 8 π N √ β ( e µ e ν e ρ e σ − (traces)) . (5 . O ∆ × O ∆ OPE can again be calculated using Wickcontractions similarly to how it was done for T , in Appendix A. By explicit calculationone finds hO ∆ ( x ) O ∆ ( x )Ξ µνρσ ( x ) i = 4∆ √ N Z µ Z ν Z ρ Z σ − (traces) | x | − | x | | x | , (5 . λ O ∆ O ∆ O , is given by λ O ∆ O ∆ O , = 4∆ √ N . (5 . λ O ∆ O ∆ O , b O , = 2 π ∆945 , (5 . a , in (5.8). 25 .4. 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 operatorswith τ = 4 and s = 4 we need to take into account the contribution of all operators withthese quantum numbers and solve a mixing problem. In addition to the double-stresstensor operator with these quantum numbers, the other double trace primary operatorwhich contributes is given by O DT µνρσ ( x ) = 196 √ N : T r ( φ ) (cid:16) T r ( φ∂ µ ∂ ν ∂ ρ ∂ σ φ ) − T r ( ∂ ( µ φ∂ ν ∂ ρ ∂ σ ) φ )+18 T r ( ∂ ( µ ∂ ν φ∂ ρ ∂ σ ) φ )( x ) − (traces) (cid:17) : ( x ) , (5 . N (see appendix F).Note that it follows from large- N factorization that the overlap of this operator with( T ) µνρσ is suppressed by powers of N ; since both of these are double trace operators andobey the scaling (5.18), to study the thermal two-point function to leading order in N ,one can therefore neglect this overlap.The thermal one-point function of O DT µνρσ follows from the large- N factorization andwe find that b O DT4 , = r
235 4 π N , (5 . λ O ∆ O ∆ O DT4 , = r
235 4∆(∆ − N . (5 . O DT µνρσ gives the following contribution to the thermaltwo point function: a O DT4 , = (cid:18) (cid:19) b O DT4 , λ O ∆ O ∆ O DT4 , = 2 π ∆(∆ − . (5 . T , together with that of O DT4 , , using (3.11), (4.5) and(5.32), is a , = ( a T , + a O DT4 , ) = π ∆(∆ − . (5 . a , in (5.8). 26 . Comparison with the eigenstate thermalization hypothesis In this section we discuss the relation of our results to the eigenstate thermalizationhypothesis (ETH). We argue that the stress tensor sector of the free SU ( N ) adjoint scalartheory in d = 4 satisfies the ETH to leading order in C T ∼ N ≫
1. We explain theequivalence 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 suppressedterms which we do not consider), equivalent to thermalization. Note that in two dimen-sions the Virasoro descendants of the identity satisfy the ETH (see e.g. [38] for a recentdiscussion).We begin by showing the equivalence between the micro-canonical and the canonicalensemble on S β × S d − when ∆ H ∼ C T ≫
1. See [11,14] for a similar discussion at infinitevolume as well as [43] 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 energy E = ∆ H /R is given by hOi (micro) E = 1 N ( E ) X ˜ O h ˜ O|O| ˜ Oi , (6 . | ˜ Oi with energy ( E, E + δE ) and N ( E ) is the number of statesin this interval. On the other hand, consider the partition function at inverse temperature β given by Z ( β ) = X ˜ O e − β ˜∆ R = Z d ˜∆ ρ ( ˜∆) e − β ˜∆ R , (6 . ρ ( ˜∆). Expectation values in thecanonical ensemble is then computed by hOi β = Z ( β ) − Z d ˜∆ ρ ( ˜∆) hOi (micro) E e − β ˜∆ R . (6 . F = − β − log Z ( β ). By aninverse Laplace transform of (6.2) we find the density of states ρ (∆ H ) = 12 πiR Z dβ ′ e β ′ ( ∆ HR − F ( β ′ )) . (6 . It was argued in [14] that the existence of the thermodynamic limit implies that we onlyneed to sum over operators with low spin. H ∼ C T and a large free energy F ∼ C T , we can evaluate (6.4) using a saddlepointapproximation with the saddle at β given by∆ H R = ∂ β ′ ( β ′ F ) | β . (6 . Z ( β ) anddoing an inverse Laplace transform evaluated at ∆ H ∼ C T we find ρ (∆ H ) hOi (micro)∆ H /R = 12 πiR Z dβ ′ hOi β ′ e β ′ ( ∆ HR − F ( β ′ )) . (6 . F ∼ C T ≫ β determined by (6.5), assuming hOi β ′ does not grow exponentially with C T . TheRHS of (6.6) is therefore the thermal expectation value hOi β multiplied by the saddlepointapproximation of the density of states in (6.4). It then follows that hOi (micro)∆ H /R ≈ hOi β , (6 . β determined by (6.5). In particular, in the infinite volume limit R → ∞ , the freeenergy is given by F = b T (can) µν S d R d − dβ d , (6 . S d = V ol ( S d − ) = 2 π d / Γ( d ). Inserting (6.8) in (6.5) we find [11] βR = − ( d − b T (can) µν S d d ∆ H ! d . (6 . T (can) µν [5] h T (can)00 i β = 1 S d R d − ∂ β ( − βF ( β )) . (6 . T (can)00 in a heavy state |O H i is fixed by theWard identity to be hO H | T (can)00 |O H i = − ∆ H S d R d . (6 . We consider a CFT in a high temperature phase. Here we denote the canonically normalized stress tensor by T (can) µν , whose two-point functionis given by h T (can) µν ( x ) T (can) ρσ ( y ) i = C T S d ( I µ ( ρ I ν σ ) − (trace)). S d R d − ) − and comparing with (6.10)-(6.11) we find that hO H | T (can)00 |O H i = h T (can)00 i β . (6 . F ∼ ∆ H ∼ C T in large- C T theories. Note that this follows from (6.7) since we can replace the micro-canonical expectation value at E = ∆ H /R , on the LHS, with the expectation value in anysingle heavy state with dimension ∆ H due to the Ward identity, independent of the heavystate. Put differently, the stress tensor satisfies the ETH as we will review below.We now consider the eigenstate thermalization hypothesis for CFTs at finite temper-ature on the sphere S d − of radius R . The diagonal part of the ETH is given by hO H |O τ,s |O H i = hO τ,s i (micro) E + O (cid:16) e − S ( E ) (cid:17) , (6 . O H and O τ,s are local primary operators and hO τ,s i (micro) E is the expectation valueof O τ,s in the micro-canonical ensemble at energy E = ∆ H R . Here we assume that theoperator O H is a heavy scalar operator with large conformal dimension ∆ H ∝ C T ≫ O τ,s on the other hand can have non-zero spin. . In (6.13), e S ( E ) is thedensity of states at energy E = ∆ H /R . As shown in (6.7), in the limit ∆ H ∼ C T ≫
1, themicro-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 canwritten in terms of OPE coefficients and thermal one-point functions: λ O H O H O τ,s R τ + s = b O τ,s f O τ,s ( β/R ) β τ + s + O (cid:16) e − S ( E ) (cid:17) , (6 . f O τ,s also appears in (2.9). This is equivalent to the statement of thermalizationdiscussed in the rest of the paper.In this paper we observed that the multi stress tensor operators satisfy (6.14). Onecan also ask if (6.14) holds for any operator in the specific heavy state we considered. Bycomparing eqs. (5.10) and (5.11) using (4.2), one can check that operator O = √ N : T r ( φ ) : does not satisfy (6.14). Since this is a free theory, it is not a surprise that theETH is not satisfied by all operators in the spectrum which is seen explicitly in this case. The tensor structure in (6.13) is suppressed. . Discussion In this paper we argued that multi stress tensor operators T kτ,s in CFTs with a largecentral charge C T thermalize: their expectation values in heavy states are the same astheir thermal expectation values. This is equivalent to the statement that multi stresstensor operators in higher-dimensional CFTs satisfy the diagonal part of the ETH in thethermodynamic limit. The analogous statement in the d = 2 case is that the Virasorodescendants of the identity satisfy the ETH condition in the large- C T limit.We observed that the operator O = √ N : T r ( φ ) : does not satisfy the ETH.This is seen by comparing eqs. (5.10) and (5.11) using (4.2). While this operator doesnot thermalize in the heavy states we considered, the OPE coefficient averaged over alloperators with ∆ H ∼ C T is expected to be proportional to the thermal one-point function.The averaged OPE coefficients should therefore scale like ∼ √ ∆ H compared to λ O H O H O ∼ ∆ H / √ C T for the heavy states we considered. It would be interesting to exhibit heavyoperators that produce the former scaling.We provided a bootstrap argument in favor of the thermalization of multi stress ten-sor operators. One should be able to refine it to give an explicit form for leading be-havior of the multi stress tensor OPE coefficients – we leave it for future work. Theholographic/bootstrap OPE coefficients for the leading twist double stress tensor opera-tors can be found in e.g. [61] – they are nontrivial functions of the spin. As explained in[61,63], the leading ∆ behavior of the minimal-twist double- and triple-stress tensor OPEcoefficients is consistent with the exponentiation of the near lightcone stress tensor confor-mal block. One can go beyond the leading twist multi stress tensors. In holographic HHLLcorrelators each term of the type (∆ µ ) k ∼ (∆∆ H /C T ) k comes from the exponentiation ofthe stress-tensor block – this follows from the Wilson line calculation of the correlator inthe AdS-Schwarzschild background [84,61,73].In this paper we argue that this behavior is universal, and is not just confined toholographic theories. Hence, one can formulate another statement equivalent to the ther-malization of multi stress tensor operators. Namely, scalar correlators of pairwise identicaloperators of dimensions ∆ , in large- C T theories in the limit ∆ , ≫
1, ∆ ∆ /C T fixedare given by the exponentials of the stress tensor conformal block . This is similar towhat happens in two-dimensional CFTs. See [29] for previous work on the eikonalization of the multi stress tensor OPE coefficientsat large spin. gap ) corrections to the multi stress tensor OPE coefficients wereconsidered. In particular, for double stress tensors, such corrections behave like ∆ / ∆ gap which is clearly at odds with the universality statement. Of course, the results of [68]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 coefficientsbeyond leading order in ∆. In particular, in [73], it was shown that the bootstrap resultfor the HHLL correlator exactly matches the holographic Wilson line calculation (in thedouble scaling limit where only the minimal twist multi stress tensor operators contribute).This corresponds to including terms beyond the exponential of the stress tensor block –one 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 resultin [73] is given by an elliptic integral – is it applicable beyond holography?In [61] terms subleading in ∆ were shown to be important for the computation ofthe phase shift. The simplest nontrivial case in two spacetime dimensions is the operatorΛ which is a level four Virasoro descendant of the identity (see e.g.[85]). One could alsoget it by using the CFT normal ordering and imposing the quasi-primary condition [86].Consider now the case of minimal twist (twist four) operators in four dimensions. How dowe determine the analog of Λ ? There is no Virasoro algebra now.Presumably, one can reconstruct the analog of Λ in four spacetime dimensions byconsidering a CFT normal ordered product of stress tensors, and adding a single trace termto ensure that the resulting operator is a primary and is orthogonal to the stress tensoritself. Note that the CFT normal ordering differs from the oscillator normal ordering ina free theory by the addition of a single trace operator, as reviewed in Appendix G. Thisprocedure can then be generalized to other multi-trace operators. We leave it for futurework.It is also helpful to imagine what happens in a theory like N = 4 Super Yang-Mills,where there is a marginal line connecting the weak and the strong coupling (the latteradmits a holographic description). Presumably, as the coupling is turned on, only oneoperator remains light (with dimension eight and spin four), while others get anomalousdimensions. It would be interesting to see this explicitly even to the leading nontrivial31rder in the ’t Hooft coupling. It would also be interesting to see how the correspondingOPE coefficient interpolates between its free and strong coupling values.Using crossing symmetry, we argued that the universality of multi stress tensor OPEcoefficients is related to the OPE coefficients λ O H T µν O ′ , with O ′ = O H being either heavyor light, present in the cross-channel expansion. Such OPE coefficients with at least oneoperator being heavy were recently studied in [20,87]. It would be interesting to furtherstudy the connection of our results to this work.Another interesting question concerns the fate of the double trace operators of theschematic form [ O ∆ O ∆ ] n,l . Consider the d = 4 case in the large volume limit and n, l = 0,for simplicity. We expect that the corresponding OPE coefficients in the free theory behavelike λ O H O H [ O ∆ O ∆ ] , ∝ ∆ H /C T ∝ C T µ , while their thermal one-point functions behavelike h [ O ∆ O ∆ ] , i β ∝ C T β − . Comparing the two results with the help of (4.2) oneobserves that such operators do not thermalize in the free theory for generic ∆. Thesituation is more nontrivial in holography where we do not know the large µ behavior ofthe OPE coefficients . As pointed out in [58], the contribution of double-trace operatorsto thermal two-point functions is different from that of multi stress tensors. The latteris only sensitive to the behavior of the metric near the boundary, but the former knowsabout the full black hole metric. This seems to indicate that the thermalization of thedouble trace operators in holographic theories is also unlikely .It is a natural question how generic are the heavy states for which the stress tensorsector thermalizes. The results of our paper seem to suggest that such thermalization ismore generic than the thermalization of other light operators . Other interesting questionsinclude generalizations to the case of finite but large central charge and to non-conformalquantum field theories. This scaling is obtained by computing the OPE coefficient λ O H O H [ O ∆ O ∆ ] , for 1 ≪ ∆ H ≪ C T and extrapolating it to the ∆ H ∼ C T regime. Note that the large- N scaling in holography is different. Both the OPE coefficients and thethermal expectation values behave like C T as opposed to C T ∼ N . A simple way to decouple such operators is to take the large-∆ limit. A closely related question of finding “typical” states where the stress tensor sector thermal-izes in the large volume limit in d = 2 was recently discussed in [54]. There it was observed thatsuch states are Virasoro descendants when the central charge is finite. cknowledgments : We thank Aleksandar Bukva, Ilija Buri´c, Saˇso Grozdanov, ManuelaKulaxizi, Eric Perlmutter and Larry Yaffe for discussions, correspondence and commentson the draft. The work of R.K. and A.P. is supported in part by an Irish Research CouncilLaureate Award. The work of P.T. is supported in part by an Ussher Fellowship Award. Appendix A. OPE coefficients from Wick contractions
In this appendix we go through the calculations needed for finding the OPE coefficientsof various operators using Wick contractions. This mainly amounts to counting the numberof contractions leading to a planar diagram. For simplicity, the figures are shown forexternal operators with ∆ = 4 while we write down the result for general ∆ as this isneeded for the main body of the paper.To begin with, since we consider a large- N matrix theory, it is convenient to use thedouble-line notation for fundamental field propagators. In Fig. 1 the two-point function h : T r ( φ ) :: T r ( φ ) : i is visualised. Fig. 1:
The two-point function h : T r ( φ ) :: T r ( φ ) : i before any contractions. In Fig. 2, the planar diagram is shown for ∆ = 4 and there are ∆ number of suchcontractions giving a planar diagram P h : T r ( φ ∆ ):: T r ( φ ∆ ): i = ∆ , (A.1)where the P h ... i denotes the number of planar diagrams for h ... i .33 ig. 2: The two-point function h : T r ( φ ) :: T r ( φ ) : i completely contracted. Fig. 3:
The three-point function h : T r ( φ ) :: T r ( φ ) :: T r ( φ ) : i completelycontracted. We further need the OPE coefficient λ O ∆ O ∆ O . This is shown in Fig. 3 for ∆ = 4 andthere are 2∆ possibilities for step (1), ∆ number of possibilites for step (2) after whicheverything is fixed assuming that the diagram is planar. This gives P h : T r ( φ ∆ ):: T r ( φ ∆ ):: T r ( φ ): i = 2∆ . (A.2)34n Fig. 4 the three-point function h : T r ( φ ∆ ) :: T r ( φ ∆ ) :: T r ( φ ) : i for ∆ = 4 is shown.For the first contraction (1) there are 2∆ possibilites, for the second contraction there are∆ and for step (3) there are two possibilites. This gives overall P h : T r ( φ ∆ ):: T r ( φ ∆ ):: T r ( φ ): i = 4∆ . (A.3) Fig. 4:
The three-point function h : T r ( φ ) :: T r ( φ ) :: T r ( φ ) : i completelycontracted. In Fig. 5 and Fig. 6, the three-point function h : T r ( φ ) :: T r ( φ ) : T r ( φ ) T r ( φ ) : i isshown. The reason for there being two different types of diagrams is because each traceterm in the double trace operator : T r ( φ ) T r ( φ ) : can either be contracted with the same: T r ( φ ) : (Fig. 5, type B), or to both (Fig. 6, type A).Consider first the type of diagrams in Fig. 5. For the first contraction there are 2∆such terms and the second contraction gives another factor of 2. Contraction (3) and (4)contributes factors of ∆ and 2 respectively. What remains is equivalent to the two-pointfunction h : T r ( φ ∆ − ) :: T r ( φ ∆ − ) : i which further give a factor of (∆ −
2) and thereforethere are 8∆ (∆ −
2) contractions of type B in Fig. 5.Continuing with Fig. 6, the first contraction gives a factor of 2∆, the second contrac-tion ∆ and the third one a factor of 2(∆ − (∆ −
1) planar diagrams to h : T r ( φ ∆ ) :: T r ( φ ∆ ) : T r ( φ ) T r ( φ ) : i . It is thereforefound that P h : T r ( φ ∆ ):: T r ( φ ∆ ): T r ( φ ) T r ( φ ): i = 4∆ (3∆ − . (A.4)35 ig. 5: The three-point function h : T r ( φ ) :: T r ( φ ) :: T r ( φ ) T r ( φ ) : i . There aretwo such types of contractions that give planar diagrams, here it shown when each: T r ( φ ) : connect to a separate : T r ( φ ) :. Fig. 6:
The three-point function h : T r ( φ ) :: T r ( φ ) :: T r ( φ ) T r ( φ ) : i . There aretwo such types of contractions that give planar diagrams, here it shown when each: T r ( φ ) : connect to both : T r ( φ ) : operators. Consider now the stress tensor OPE coffiecient λ O ∆ O ∆ T µν where T µν ( x ) = 12 √ N : T r (cid:18) ∂ µ φ∂ ν φ − φ∂ µ ∂ ν φ − (trace) (cid:19) : ( x ) (A.5)and the three-point function hO ∆ O ∆ T µν i : hO ∆ ( x ) O ∆ ( x ) T µν ( x ) i = λ O ∆ O ∆ T µν Z µ Z ν − traces | x | − | x | | x | , (A.6)36here Z µ = x µ | x | − x µ | x | . From the definition of T µν in (A.5) it is clear that the onlyterm that contributes to term x µ x ν comes from the second term in (A.5) that is of theform ∝ T r ( φ∂ µ ∂ ν φ ). Up to the derivatives, the diagram will look like those visualised inFig. 3. The number of diagrams is half of that given in (A.2) since we restrict to termsproportional to x µ x ν : P hO ∆ O ∆ T µν i| x µx ν = ∆ , (A.7)from which we reproduce (3.7).Now we want to find the OPE coefficient λ O ∆ O ∆ T , for the double-stress tensor T , .This is done similarly to the way the stress tensor OPE coefficient was found. First, theoperator ( T ) µνρσ was given in (3.8) to be( T ) µνρσ ( x ) = 1 √ T ( µν T ρσ ) : ( x ) − (traces) (A.8)and the three-point function hO ∆ O ∆ ( T ) µνρσ i is fixed by conformal symmetry to be hO ∆ ( x ) O ∆ ( x )( T ) µνρσ ( x ) i = λ O ∆ O ∆ T , | x | − | x | | x | ( Z µ Z ν Z ρ Z σ − (traces)) . (A.9)Consider the term in (A.9) proportional to x µ x ν x ρ x σ . This will be due to the termin ( T ) µνρσ of the form T r ( φ∂ ( µ ∂ ν φ ) T r ( φ∂ ρ ∂ σ ) φ ). Using this we find that hO ∆ ( x ) O ∆ ( x )( T ) µνρσ ( x ) i| x µ x ν x ρ x σ = 1∆ N ∆ √ (cid:18) − √ N (cid:19) N ∆ × P hO ∆ O ∆ T , i| x µx ν x ρx σ | x | − | x | | x | . (A.10)The number of contractions giving a planar diagram, P hO ∆ O ∆ T , i| x µx ν x ρx σ , comefrom diagrams of the form given in Fig. 6. Since we are considering the term proportional x µ x ν x ρ x σ , the number of such diagrams are reduced compared to scalar doubletrace operator. Instead the first contraction, (1) in Fig. 6, give a factor of ∆, the secondcontraction, (2), a factor of (∆ − P hO ∆ O ∆ T , i| x µx ν x ρx σ = ∆ (∆ − , (A.11)and inserting this in (A.10) gives λ O ∆ O ∆ T , = 2 √ − N , (A.12)37nd therefore reproduces (3.11).Similar to the double-stress tensor, consider the dimension-eight spin-four double traceoperator O DT µνρσ ( x ) = 196 √ N : T r ( φ ) (cid:16) T r ( φ∂ µ ∂ ν ∂ ρ ∂ σ φ ) − T r ( ∂ ( µ φ∂ ν ∂ ρ ∂ σ ) φ )+18 T r ( ∂ ( µ ∂ ν φ∂ ρ ∂ σ ) φ )( x ) − (traces) (cid:17) : ( x ) . (A.13)The three-point function hO ∆ ( x ) O ∆ ( x ) O DT µνρσ ( x ) i is given by hO ∆ ( x ) O ∆ ( x ) O DT µνρσ ( x ) i = λ O ∆ O ∆ O DT µνρσ | x | − | x | | x | ( Z µ Z ν Z ρ Z σ − (traces)) . (A.14)By again considering terms in (A.14) proportional to x µ x ν x ρ x σ we find that eachterm in (A.13) will contribute planar diagram of the type in Fig. 5, while only the term ∼ T r ( φ∂ φ ) also give a contribution of the type in Fig. 6. Considering first the termscoming from the diagram in Fig. 5, one finds that this contribution vanishes. The remainingcontribution to the term (A.14) proportional to x µ x ν x ρ x σ comes from the first termin (A.13) and the planar diagram pictured in Fig. 6; there are 2∆ (∆ −
1) contractionsgiving such a planar diagram leading to hO ∆ ( x ) O ∆ ( x ) O DT µνρσ ( x ) i| x µ x ν x ρ x σ = 1∆ N ∆ √ N N ∆ × (∆ − | x | − | x | | x | , (A.15)where the 384 in the numerator come from the derivatives. This gives the OPE coefficient: λ O ∆ O ∆ O DT µνρσ = r
235 4∆(∆ − N + O ( N − ) . (A.16) Appendix B. Subleading twist double-stress tensors
In this Appendix we study the subleading twist double-stress tensors, both with di-mension 8 and spin s = 0 , T ) and ( T ) µν respectively. The calculations neededto find the OPE coefficient in the O ∆ × O ∆ OPE are reviewed as well as the normalizationof ( T ) µν .The ( T ) µν was defined in (3.24) which we repeat here:( T ) µν ( x ) = 1 √ T µα T αν : ( x ) − δ µν √ T β α T αβ : ( x ) . (B.1)38he operator ( T ) µν can be seen to be unit-normalized to leading order in N : h ( T ) µν ( x )( T ) ρσ ( x ) i = 1 √ h T µα ( x ) T ρβ ( x ) ih T ν α ( x ) T β σ i + ( ρ ←→ σ ) − (traces) + O ( N − ) . (B.2)Using the two-point function of the stress tensor in (3.6) and I µα I αρ = δ µρ one finds h ( T ) µν ( x )( T ) ρσ ( x ) i = 1 | x | (cid:16) I ( µρ I ν ) σ − (traces) (cid:17) , (B.3)from which it is seen that ( T ) µν is unit-normalised.We now want to find the OPE coefficient of ( T ) µν in the O ∆ × O ∆ OPE. It can befound from the basic objects I (1) µνρσ , I (2) µνρσ and I (3) µνρσ which we calculate below.We first consider a similar quantity J (1) µνρσ : J (1) µνρσ = h : T r ( φ ∆ ) : ( x ) : T r ( φ ∆ ) : ( x ) :: T r ( ∂ µ φ∂ ν φ ) T r ( ∂ ρ φ∂ σ φ ) : ( x ) i = 2 N ∆ | x | | x | | x | − × h (2∆) (∆ − x µ x ν x ρ x σ + x µ x ν x ρ x σ )+∆ (∆ − x µ x ν ( x ρ x σ + x ρ x σ ) + x µ x ν ( x ρ x σ + x ρ x σ )) i . (B.4)Definining X µν = | x | ( − δ µν + 4 x µ x ν | x | ) we then study J (2) µνρσ : J (2) µνρσ = h : T r ( φ ∆ ) : ( x ) : T r ( φ ∆ ) : ( x ) :: T r ( φ∂ µ ∂ ν φ ) T r ( φ∂ ρ ∂ σ φ ) : ( x ) i = N ∆ | x | − h ∆ (∆ − (cid:16) X µν | x | X ρσ | x | + X µν | x | X ρσ | x | (cid:17) +((2∆) (∆ − X µν | x | X ρσ | x | +(13) ←→ (23) i . (B.5)And lastly J (3) µνρσ : J (3) µνρσ = h : T r ( φ ∆ ) : ( x ) : T r ( φ ∆ ) : ( x ) :: T r ( φ∂ µ ∂ ν φ ) T r ( ∂ ρ φ∂ σ φ ) : ( x ) i = N ∆ | x | − h ((2∆) (∆ − X µν | x | x ρ x σ | x | ++ ∆ (∆ − X µν | x | x ρ x σ + x ρ x σ | x | | x | + (13) ←→ (23) i . (B.6)39e further need to make (B.4)-(B.6) traceless in the pairs ( µ, ν ) and ( ρ, σ ) and thereforedefine I ( i ) µνρσ as I ( i ) µνρσ = J ( i ) µνρσ − δ µν J ( i ) ααρσ − δ ρσ J ( i ) µναα + δ µν δ ρσ J ( i ) ααγ γ . (B.7)From (B.4)-(B.6), the three-point function hO ∆ ( x ) O ∆ ( x )( T ) µν ( x ) i is given by hO ∆ ( x ) O ∆ ( x )( T ) µν i = 112 √ N ∆+2 ( I (1)( µ | αα | ν ) − I (3)( µ | αα | ν ) + 14 I (2)( µ | αα | ν ) − (trace)) . (B.8)Explicitly we find that hO ∆ ( x ) O ∆ ( x )( T ) µν ( x ) i = √ − N Z µ Z ν − (trace) | x | − | x | | x | + O ( N − ) . (B.9)Consider now the scalar operator ( T ) defined by( T )( x ) = 136 √ N : T µν T µν : ( x ) . (B.10)The three-point function hO ∆ ( x ) O ∆ ( x )( T )( x ) i can be found using I ( i ) defined in (B.7)as follows hO ∆ ( x ) O ∆ ( x )( T )( x ) i = 136 √ N ( I (1) µν µν − I (3) µν µν + 14 I (2) µν µν ) + O ( N − )= ∆(∆ − √ N | x | − | x | | x | + O ( N − ) . (B.11) Appendix C. Single trace operator with dimension ∆ ∼ C T In this appendix we study the single trace scalar operator O ∆ H given by O H ( x ) = 1 p N ∆ H : T r ( φ ∆ H ) : ( x ) , (C.1)with ∆ H ∼ C T and N ∆ H a normalization constant . When calculating the normalizationconstant N ∆ H as well as the three-point functions hO H ( x ) O H ( x ) O ( x ) i , non-planar Mixing with other operators with ∆ ∼ C T is not important for this discussion. H and therefore invalidates the naiveplanar expansion. The goal of this appendix is to show that hO H ( x ) O H ( x ) ˆ O ( x ) i = hO ∆ ( x ) O ∆ ( x ) ˆ O ( x ) i| ∆=∆ H , (C.2)where ˆ O is either : T r ( φ ) : or, more importantly, minimal-twist multi stress tensors withany spin. Moreover, note that the LHS in (C.2) is in principle exact in C T ∼ N while theRHS 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 h φ ij ( x ) φ kl ( y ) i = (cid:18) δ il δ kj − N δ ij δ kl (cid:19) | x − y | . (C.3)Consider now the three-point function h : T r ( φ ∆ H ) : ( x ) : T r ( φ ∆ H ) : ( x ) : T r ( φ ) : ( x ) i .Due to the normal ordering, one φ field in : T r ( φ ) : ( x ) need to be contracted with: T r ( φ ∆ H ) : ( x ) : and the other one with : T r ( φ ∆ H ) : ( x ) :. Note that for this contractionthe second term in (C.3) give a contribution proportional to T r ( φ ( x )) = 0. It is thereforeseen that h : T r ( φ ∆ H ) :: T r ( φ ∆ H ) :: T r ( φ ) : i = 2∆ H h : T r ( φ φ ∆ H − ) :: T r ( φ ∆ H ) : i , (C.4)where we introduced the notation φ i = φ ( x i ) and dropped the | x ij | − 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 of O H and we therefore find that h : T r ( φ ∆ H ) :: T r ( φ ∆ H ) :: T r ( φ ) : i = 2∆ H N ∆ H , (C.5)which is exact to all orders in C T . Including the normalization factor of O H in (C.1) and O from (3.3) we find that hO H ( x ) O H ( x ) O ( x ) i = √ H N | x | H − | x | | x | + O ( N − ) . (C.6)By comparing (C.6) with (5.10) we find that λ O H O H O = λ O ∆ O ∆ O | ∆=∆ H . (C.7) Up to the position dependence. O H cancels the contribution from non-planardiagrams in limit ∆ H ∼ C T . For ∆ = 2 in (3.3), it is trivial to compute the normalizationexact in N to get the correction to λ O ∆ O ∆ O in (C.6).Consider now the stress tensor operator defined in (3.5) and the three-point function hO H ( x ) O H ( x ) T µν ( x ) i . This is fixed by the Ward identity but is an instructive examplebefore considering more general multi stress tensors. In the same way as the OPE coefficientwas found in the O ∆ × O ∆ OPE, due to the tensor structure being fixed by conformalsymmetry, we consider the term proportional to x µ x ν in the three-point function. Thiscomes from the − √ C T T r ( φ∂ µ ∂ ν φ ) term in the stress tensor when ∂ µ ∂ ν φ is contractedwith one of the ∆ H number of φ ( x ) fields. Doing this contraction we therefore see that h : T r ( φ ∆ H ) :: T r ( φ ∆ H ) :: T r ( φ ∂ µ ∂ ν φ ) : i| x µ x ν = 8∆ H h : T r ( φ φ ∆ H − ) :: T r ( φ ∆ H ) : i , (C.8)where the factor 8 comes from the derivatives and we again suppress the spacetime de-pendence. The RHS of (C.8) is also proportional to the normalization constant of O H .Including the normalization factor of the stress tensor in (3.5) and that of O H in (C.1),the three-point function hO H O H T µν i can be obtained from (C.8) from which we read offthe OPE coefficient λ O H O H T µν = − H √ C T . (C.9)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)( T ) µνρσ ( x ) = 1 √ T ( µν T ρσ ) : ( x ) − (traces) . (C.10)Similarly to the calculation of the three-point function with the stress tensor, we can ob-tain the three-point function hO H ( x ) O H ( x )( T ) µνρσ ( x ) i by considering the term pro-portional to x µ x ν x ρ x σ . This will be due to the term √ C T T r ( φ∂ µ ∂ ν φ ) T r ( φ∂ ρ ∂ σ φ )when contracting ∂ µ ∂ ν φ with some φ ( x ) and likewise contracting ∂ ρ ∂ σ φ with some other φ ( x ). The number of such contractions is given by ∆ H (∆ H −
1) and we find that h : T r ( φ ∆ H ) :: T r ( φ ∆ H ) : : T r ( φ ∂ µ ∂ ν φ ) T r ( φ ∂ ρ ∂ σ φ ) : i| x µ x ν x ρ x σ = 8 ∆ H (∆ H − h : T r ( φ φ ∆ H − ) :: T r ( φ ∆ H ) : i , (C.11)42here the factor of 8 again is due to acting with the derivatives and note that the positionof the φ fields in in the last line is not important. It is again seen that the RHS of (C.11) isproportional to the normalization constant of O H . Including the normalization in (3.8) and(C.1) we find the three-point function hO H O H ( T ) µνρσ i and read off the OPE coefficient: λ O H O H T , = 8 √ H (∆ H − C T + O ( C − / T ) . (C.12)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 , and therefore λ O H O H T , = λ O ∆ O ∆ T , to all orders in C T . These arguments generalize straightforwardly to minimal-twist multi stress tensor with any spin such that the results are the same as those obtainedin the planar limit for ∆ ≪ C 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 ∆ , withoutany 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 weuse the fact that the thermal correlation function is related to the zero-temperature caseby 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∆ O on S × R d − is given by [13] hO µ ...µ s ( x ) i β = b O β ∆ O ( e µ . . . e µ s − (traces)) , (D.1)where e µ is a unit-vector along the thermal circle. Consider first the canonically normal-ized stress tensor given by T (can) µν = S d ( T r ( ∂ µ φ∂ µ φ ) − T r ( φ∂ µ ∂ ν φ ) − (traces)). In orderto find the one-point function, use the following: h T r ( ∂ ( x ) µ φ ( x ) ∂ ( y ) ν φ ( y )) i = 2( N − | x − y | ( δ µν − y − x ) µ ( y − x ) ν | x − y | ) (D.2)and h T r ( ∂ ( x ) µ ∂ ( x ) ν φ ( x ) φ ( y )) i = 2( N − | x − y | ( − δ µν + 4( y − x ) µ ( y − x ) ν | x − y | ) . (D.3)43o get the thermal correlator, we use (D.2) and (D.3) with x, y along the thermal circleseparated by a distance mβ , with m integer, and sum over m = 0. The relevant terms forcalculating the one-point functions in terms of fundamental fields are therefore h T r ( ∂ µ φ∂ ν φ ) i β,m = − N − mβ ) e µ e ν + 2( N − mβ ) δ µν , h T r ( ∂ µ ∂ ν φφ ) i β,m = 8( N − mβ ) e µ e ν − N − mβ ) δ µν , (D.4)where we note that only the first term in each equation in (D.4) contribute to the non-traceterm in (D.1).We therefore find for the stress tensor one-point function: h T (can) µν i β = 13 S d ( h T r ( ∂ µ φ∂ ν φ ) i β − h T r ( ∂ µ ∂ ν φφ ) i β − trace)= − N − S d ζ (4) β ( e µ e ν − (trace)) , (D.5)where the 2 ζ (4) comes from summing over images and we therefore have b T (can) µν = − N − S d ζ (4) = − π S d ( N − . (D.6)This agrees with f = b T (can) µν d in eq. (2.17) in [13] for ( N −
1) free scalar fields. This alsoagrees with a , = π ∆45 found from the two-point thermal correlator using: a , = π ∆45 = (cid:18) (cid:19) λ O ∆ O ∆ T (can) b T (can) µν C T S d , (D.7)using λ O ∆ O ∆ T (can) = − S d in this normalization and C T = ( N − N ) b T µν = b T (can) µν √ C T S d ≈ − π N √ . (D.8)Let us now consider the thermalization of the stress tensor, keeping all the indexstructures. To compare the thermal two-point function with the heavy-heavy-light-lightcorrelator, we want to relate the dimension of the heavy operator, ∆ H , to the inverse44emperature β . Consider the expectation value of the stress tensor in a heavy state createdby O H on the cylinder R × S hO H | T µν ( x E, , ˆ n ) |O H i cyl = lim x →∞ | x | H | x | λ O H O H T µν Z µ Z ν − δ µν Z ρ Z ρ | x | H − | x | | x | , (D.9)where the RHS is found by a conformal transformation to the plane with Z µ = (cid:16) x µ | x | + x µ | x | (cid:17) . When x = 0 and x → ∞ , it is seen that Z µ = − x µ | x | and (D.9)only depends on ˆ x µ = x µ | x | = ˆ r , where ˆ r is a radial unit vector. In radial quantization itfollows that hO H | T µν ( x E, , ˆ n ) |O H i cyl = λ O H O H T µν R (ˆ e µ ˆ e ν − δ µν ) (D.10)where we reintroduced the radius of the sphere R , λ O H O H T µν is the OPE coefficient of T µν in the O H × O H OPE and ˆ e µ = (1 , , , O τ,s , with twist τ and spin s , on S × S is fixed by conformal symmetry [13] hO τ,s ( x ) i β = b O τ,s f O τ,s ( βR ) β τ + s ( e µ · · · e µ s − (traces)) , (D.11)where f O τ,s (0) = 1 and e µ = (1 , , , hO H | T µν ( x ) |O H i = h T µν ( x ) i β (D.12)where h T µν ( x ) i β is the thermal one-point function at inverse temperature β evaluated on S × S , with R being the radius of S . Using (D.10)-(D.12) we find λ O H O H T µν R = b T µν f T µν ( βR ) β . (D.13)Using (D.13) for R → ∞ in the free adjoint scalar theory, together with the one-point function b T µν = − π N √ and the OPE coefficient λ O H O H T µν = − H √ C T , one finds thefollowing relation between µ = H C T and the inverse temperature β : µ = 83 (cid:18) πRβ (cid:19) . (D.14)This agrees with (4.2). 45 ppendix 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 , operator is given by Ξ µνρσ ( x ) = 196 √ N : T r (cid:0) φ ( ∂ µ ∂ ν ∂ ρ ∂ σ φ ) − ∂ ( µ φ )( ∂ ν ∂ ρ ∂ σ ) φ )+ 18( ∂ ( µ ∂ ν φ )( ∂ ρ ∂ σ ) φ ) − (traces) (cid:1) : ( x ) . (E.1)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[ K µ , P ν ] = 2 i ( η µν D − M µν ) , [ M µν , P ρ ] = − i ( η ρµ P ν − η ρν P µ ) , (E.2)and the action on the fundamental field φP µ φ (0) = − i∂ µ φ (0) ,Dφ (0) = iφ (0) . (E.3)The relevant commutators in order to fix Ξ µνρσ are[ K α , P µ φ ] = − η αµ φ, [ K α , P µ P ν φ ] = − η αµ P ν φ − η αν P µ φ + 2 η µν P α φ, [ K α , P µ P ν P ρ φ ] = − η αµ P ν P ρ φ − η αν P µ P ρ φ − η αρ P ν P µ φ + 2 η µν P ρ P α φ + 2 η ρν P µ P α φ + 2 η µρ P ν P α φ, [ K α , P µ P ν P ρ P σ φ ] = − η αµ P ν P ρ P σ φ − η αν P µ P ρ P σ φ − η αρ P ν P µ P σ φ − η ασ P ν P ρ P µ φ + 2 η µν P ρ P σ P α φ + 2 η µρ P ν P σ P α φ + 2 η µσ P ρ P ν P α φ + 2 η νρ P µ P σ P α φ + 2 η νσ P µ P ρ P α φ + 2 η ρσ P µ P ν P α φ, (E.4)which can also be found in e.g. Appendix F in [88].The thermal one-point function of this operator is found from Wick contractions tobe h Ξ µνρσ i β = 8( πT ) N √
35 ( e µ e ν e ρ e σ − (traces)) . (E.5) We denote this operator either as O , or Ξ µνρσ depending whether we want to explicitly listthe indices or not. O ∆ ( x ) = √ ∆ N ∆ : T r (cid:0) φ ∆ (cid:1) : ( x ) canagain be calculated using Wick contractions similarly to how it was done for T µνρσ inAppendix A. By explicit calculation one finds hO ∆ ( x ) O ∆ ( x )Ξ µνρσ ( x ) i = 4∆ √ N Z µ Z ν Z ρ Z σ − (traces) | x | − | x | | x | , (E.6)and therefore the OPE coefficient λ O ∆ O ∆ O , is given by λ O ∆ O ∆ O , = 4∆ √ N . (E.7)Now, it is easy to check that 12 λ O ∆ O ∆ O , b O , = 2 π ∆945 , (E.8)which agrees with a , in (5.8). Appendix F. Thermal one-point functions of multi-trace operators in the large- N limit In (5.33), it was shown that a , was due to double trace operators which were normalordered products of single trace operators without any derivatives. There are, however,other double trace operators that have the same quantum numbers and are schematicallyrepresented as [ O a O b ] n,l . Concretely, the double trace operators with twist and spin fourbesides ( T ) µνρσ and ( O DT ) µνρσ are [ O O ] , and [ O T µν ] , . We argue that the thermalone-point functions of these operators are subleading in the large- N limit when evaluatedon the plane.Consider the thermal one-point function of a double trace operator [ O a O b ] n,l = O a ∂ n ∂ l O b + . . . , where O a and O b are single trace primary operators and dots repre-sent terms where derivatives acts on O a as well, in order to make [ O a O b ] n,l a primaryoperator. The term in the thermal one-point function that behaves as N k ( N for doubletrace operators) comes from contracting the fundamental field within each trace separately.Therefore we have hO a ∂ n ∂ l O b i β ≈ hO a i β h ∂ n ∂ l O b i β + O (1) , (F.1)which is simply due to large- N factorization. As ∂ n ∂ l O b is a descendant of O b , it is easyto explicitly show that h ∂ n ∂ l O b i β = 0 for n = 0 or l = 0, from which it follows that hO a ∂ n ∂ l O b i β = O (1) . (F.2)47imilar reasoning holds for all terms in [ O a O b ] n,l , so we conclude for n = 0 or l = 0 that h [ O a O b ] n,l i β = O (1) . (F.3)It is easy to generalise ( n and/or l non-zero) h [ O a . . . O a k ] n,l i β = O ( N k − ) . (F.4)Using the canonical scaling for the OPE coefficients (5.18) it is found that these multi-traceoperators give a suppressed contribution to the thermal two point function in the large- N limit: λ O ∆ O ∆ [ O a ... O ak ] n,l h [ O a . . . O a k ] n,l i β = O (cid:16) N (cid:17) . (F.5)The conclusion is that these operators with n = 0 or l = 0 do not contribute to thethermal two-point functions to leading order in N . Note that for n = l = 0, the operatoris just : O a O a . . . O a k : and it does contribute to the thermal 2pt function since λ O ∆ O ∆ [ O a ... O ak ] n =0 ,l =0 h [ O a . . . O a k ] n =0 ,l =0 i β = O (1) . (F.6)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 toleading 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 singlescalar and then the case of the SU ( N ) adjoint scalar. We compute two-point functionsof a particular class of quasi-primary operators at finite temperature 1 /β . These two-point functions are not determined by the conformal symmetry, because the quasi-primaryoperators do not transform covariantly from the plane to the cylinder. They transformcovariantly only with respect to the global conformal transformations. The only operatorsthat have the non-zero thermal one-point functions are the Virasoro descendants of thevacuum and therefore, only these operators contribute to the thermal two-point functionof the quasi-primary operators . Virasoro descendants of the vacuum have different OPEcoefficients with external quasi-primary operators compared with the case when primaryexternal operators are considered. We check this explicitly up to the O (1 /β ). Deviation from the Virasoro vacuum block in the Regge limit of four-point HHLL correlatoris observed in [89] as well. .1. Review free boson in two dimensions We consider single free boson φ ( z ) in two dimensions. The stress tensor can be writtenin terms of Virasoro modes as T ( z ) = √ X n z − n − L n . (G.1)This stress tensor is unit-normalized h T ( z ) T ( w ) i = 1( z − w ) . (G.2)The fundamental field can be expressed as Laurent series ∂φ ( z ) = + ∞ X n = −∞ z − n − α n , (G.3)where oscillators α n obey the following algebra[ α n , α m ] = nδ n + m, . (G.4)They act on the vacuum as α n | i = 0 , n ≥ . (G.5)The two-point function of the fundamental fields is given by h ∂φ ( z ) ∂φ ( w ) i = 1( z − w ) . (G.6)The unit-normalized stress tensor can be expressed in terms of the fundamental fieldas T ( z ) = 1 √ ∂φ∂φ : ( z ) = 1 √ X m,n z − m − n − : α m α n : , (G.7)where : ab : denotes product of operators a and b with the corresponding free theoryoscillators being normally ordered such that the operators annihilating the vacuum areput at the rightmost position. Then, it follows L n = 12 X m : α n − m α m := 12 X m ≥ α n − m α m + X m< α m α n − m . (G.8)49 .2. Thermal two-point function of quasi-primary operator We are interested in computing the thermal two-point function of quasi-primary op-erators at temperature 1 /β . Quasi-primary operators O ( z ) are defined as [ L , O ( z )] = 0,or equivalently, in therms of their asymptotic in-states O (0) | i = |Oi , as L |Oi = 0. Wedenote the quantum numbers of quasi-primary operators that correspond to eigenvaluesof L and ¯ L by ( h, ¯ h ). We consider the following unit-normalized quasi-primary operatorwith quantum numbers ( h, O h ( z ) = 1 √ h ! : ( ∂φ ) h : ( z ) = 1 √ h ! X m ,m ,...,m h z − P hi =1 m i − h : α m . . . α m h : , (G.9)which is properly defined when h is a positive integer. Its asymptotic in-state is given by |O h i = O h (0) | i = 1 √ h ! ( α − ) h | i . (G.10)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 hO h ( z ) O h (0) i β = ( h − X n =0 h !4 n ( h − n )! (cid:16) ζ (2) β (cid:17) n ∞ X m = −∞ z + mβ ) ! h − n + 2 h π Γ (cid:0) − h (cid:1) Γ( h + 1) (cid:16) ζ (2) β (cid:17) h . (G.11)This expression is obtained by writing all possible Wick contractions between fundamentalfields ∂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 (cid:16) ζ (2) β (cid:17) are due to the self-contractions, ∞ X m = −∞ ,m =0 β m = (cid:18) ζ (2) β (cid:19) . (G.12)The sum over n comes from doing n self-contractions within each of the external operators.Term h !4 n ( h − n )! counts the number of Wick contractions with n self-contractions for eachexternal operator, including 1 / √ h ! normalization factors. The term in the second line of(G.11) is due to the case when we take n = h/ O h is quasi-primary, it transforms properly only with respect tothe global conformal transformation. These are just the M¨obius transformations in two-dimensional spacetime z → az + bcz + d , with ad − bc = 1. On the other hand, the usual way tocalculate the thermal two-point function of primary operators in two dimensions is to doa conformal transformation from the plane to the cylinder with radius β , z → β π log( z ).This transformation is clearly not one of the M¨obius transformations and that is why wecan not use this method to compute the thermal two-point functions of quasi-primaryoperators.Expanding (G.11) for T = β → z h hO h ( z ) O h (0) i β = 1 + h πz ) β + h ( h − )12 ( πz ) β + O (cid:18) β (cid:19) . (G.13) G.3. Quasi-primaries, OPE coefficients, and thermal one-point functions
In expansion (G.13), terms O ( z h ) are due to the quasi-primary operator with quan-tum numbers ( h ,
0) in the operator product expansion O h × O h . Identity in the expansionis due to the identity operator. We show that the second term on the RHS is due to thestress tensor. The quantum numbers of stress tensor T ( z ) are (2 , h T i β = 1 √ ∞ X m = −∞ ,m =0 β m = π √ β . (G.14)This is obtained by the Wick contractions of fundamental fields in the stress tensor, thatare separated along the thermal circle. The same result can be obtained by the transformof 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( h O , O h operators as hO h ( z ) O h ( z ) O ( z ) i = λ O h O h O ( z − z ) h O ( z − z ) h O ( z − z ) h − h O . (G.15)Next, we evaluate its OPE coefficient of the stress tensor with O h by doing the Wickcontractions between fundamental fields hO h ( z ) O h ( z ) T ( z ) i = √ h z − z ) ( z − z ) ( z − z ) h − , (G.16)51herefore λ O h O h T = √ h . This OPE coefficient is fixed by the Ward identity. Now, itfollows z λ O h O h T h T i β = h πz ) β , (G.17)which reproduces the second term on the RHS of (G.13).We are now interested in the contributions of quasi-primary operators with quantumnumbers (4 , : T T : ( z ) = 1 √
24 : ( ∂φ ) : ( z ) = 1 √ X a,b,c,d z − a − b − c − d − : α a α b α c α d : , (G.18)Λ ( z ) = r ∞ X m,n = −∞ z − m − n − ∗ L m L n ∗ − ∞ X m = −∞ z − m − ( m + 2)( m + 3) L m ! , (G.19)where ∗ ab ∗ denotes the product where the relevant Virasoro generators are normally or-dered. It should be noted that the operator Λ ( z ) is Virasoro descendant of unity, while: T T : ( z ) is not. The relevant asymptotic in-states are given by | : T T : i =: T T : (0) | i = 1 √
24 ( α − ) | i , | Λ i = Λ (0) | i = r (cid:18) L − − L − (cid:19) | i . (G.20)In terms of oscillators, | Λ i state can be represented as | Λ i = r (cid:18)
14 ( α − ) + 25 α − α − −
310 ( α − ) (cid:19) | i . (G.21)From eqs. (G.20) and (G.21) one can see that | : T T : i and | Λ i are the only quasi-primarystates with quantum numbers (4 , α − | i , α − α − | i , α − | i , α − α − | i , α − | i , (G.22)because L N Y i =1 α − k i ! | i = N X i =1 k i ! N Y i =1 α − k i ! | i , (G.23) Both of them are unit-normalized. k i >
0. It is straightforward to check L α − | i = 4 α − | i ,L α − α − | i = 3 α − α − | i ,L α − | i = 4 α − α − | i ,L α − α − | i = 2 α − | i ,L α − | i = 0 . (G.24)It follows that α − | i is already quasi-primary and one can make only one more as α − α − | i − α − α − | i . | : T T : i and | Λ i are just the linear combination of these twostates with overall normalization.Now, one can calculate the overlap of | : T T : i and | Λ i states as h | Λ (0) : T T : (0) | i = √ . (G.25)The state orthogonal to | Λ i can be written as | ˜Λ i = 32 : T T : (0) − √
53 Λ (0) ! | i . (G.26)Using (G.20) and (G.21), it can be written in terms of free theory oscillators.We compute the OPE coefficients of : T T : and Λ with two O h operators. We expressall states in terms of free theory oscillators and use algebra (G.4) to find λ O h O h : T T : = hO h |O h (1) | : T T : i = √ h ( h − , (G.27) λ O h O h Λ = hO h |O h (1) | Λ i = r h (cid:18) h − (cid:19) , (G.28) λ O h O h ˜Λ = hO h |O h (1) | ˜Λ i = 2 √ h ( h − . (G.29)Now, we evaluate the thermal one-point functions of Λ and ˜Λ . From (3.4) in [54]we have h∗ T ∗i β = 3 π β , (G.30) These states are not unit-normalized. − P ∞ m = −∞ z − m − ( m + 2)( m + 3) L m = − √ ∂ T ( z ) . It is clearthat it will not affect the thermal one-point function of Λ ( z ), as h ∂ T i β = 0.Therefore, from (G.19), we have h Λ i β = r h∗ T ∗i β = π √ β . (G.31)Now, it follows z h Λ i β λ O h O h Λ = π z β h (cid:18) h − (cid:19) , (G.32)which is the third therm at the RHS of (G.13). On the other hand, we can evaluate thethermal one-point function of : T T : ( z ) operator by Wick contractions of fundamentalfields separated along the thermal circle h : T T : i β = π √ β . (G.33)Using (G.26), it is straightforward to confirm that h ˜Λ i β = 0. Therefore, as we expected,operator ˜Λ does not contribute to the thermal two-point function of O h operators, eventhought it is present in the operator product expansion O h × O h .This is a general property of two-dimensional CFTs, that only the operators in theVirasoro vacuum module have non-zero expectation value on the cylinder. G.4. Free adjoint scalar model in two dimensions
In this subsection we study a large- c theory. Consider the free adjoint SU ( N ) scalarin 2d with ∂φ ( z ) ab = X m z − m − ( α m ) ab (G.34)with [( α m ) ab , ( α n ) cd ] = mδ m + n (cid:16) δ ad δ cb − N δ ab δ cd (cid:17) . (G.35)The thermal two point of the quasi-primary operator O h = √ hN h : T r (( ∂φ ) h ) : follows im-mediately from the result in four dimensions upon replacing the propagator of fundamentalfields. We find that hO h ( z ) O h (0) i β = g d ( z ) h + π h ( h − β g d ( z ) h − + . . . , (G.36)54here g d ( z ) = ∞ X m = −∞ z + mβ ) = (cid:16) πβ sin( πz/β ) (cid:17) . (G.37)Expanding (G.36) for β → ∞ we find hO h ( z ) O h (0) i β = z − h h π h β z + π h (15 h − β z + O ( β − ) i . (G.38)Consider first the normalized stress tensor which is given by T = 1 √ N : T r ( ∂φ∂φ ) : , (G.39)with c = N so that h T ( z ) T (0) i = z . By calculating the OPE coefficient with O h andthe thermal one-point function of T , one finds that these are the same as those for thescalar T r ( φ ) operator in four dimensions so that h T i β = π N √ β and λ O h O h T = √ hN , andthe product reproduces the weight two term in (G.38): h T i β λ O h O h T = π h β . (G.40)Consider now ∗ T T ∗ defined by ∗ T T ∗ (0) = lim z → T ( z ) T (0) − (sing . terms) . (G.41)The OPE of the stress tensor in (G.39) can be found in the free theory by first performingWick contractions T ( z ) T (0) = 12 N : T r ( ∂φ ( z ) ∂φ ( z )) :: T r ( ∂φ (0) ∂φ (0)) :=: T T : (0) + . . . + 2 N z : T r ( ∂φ ( z ) ∂φ (0)) : + 1 z , (G.42)and expanding the second term in (G.42) for z → T ( z ) T (0) =: T T : (0) + . . . + 2 N z : T r ( ∂φ (0) ∂φ (0)) :+ 2 N z : T r ( ∂ φ (0) ∂φ (0)) : + 1 N : T r ( ∂ φ (0) ∂φ (0)) : + . . . + 1 z , (G.43)55here the dots refer to higher order terms in z . Inserting the OPE (G.43) in (G.41) wefind that ∗ T T ∗ (0) =: T T : (0) + 1 N : T r ( ∂ φ (0) ∂φ (0)) : . (G.44)Consider the state ∗ T T ∗ (0) | i , which is given in terms of oscillator modes by ∗ T T ∗ (0) | i = 12 N T r ( α − ) T r ( α − ) | i + 2 1 N T r ( α − α − ) | i . (G.45)Now T r ( α m − ) | i is a quasi-primary while T r ( α − α − ) | i is not. One way to make it aquasi-primary is to simply remove the second term in (G.45) and then we get a quasi-primary state which is just : T T : | i . Another option is to remove a descendant of thestress tensor to construct | Λ i . To do the latter we need to remove the descendant of thestress tensor with weight 4 given by ∂ T∂ T = √ N : T r ( ∂ φ∂φ ) : + √ N : T r ( ∂ φ∂ φ ) : . (G.46)Acting on the vacuum we find ∂ T (0) | i = 2 √ N T r ( α − α − ) | i + √ N T r ( α − ) | i . (G.47)Consider now L = √ N ( T r ( α − α ) + T r ( α − α + . . . )) which acts as L T r ( α − ) | i = √ N T r ( α − α − ) | i and as L T r ( α − α − ) | i = √ N T r ( α − α − ) | i . We can therefore con-struct a quasi-primary state annihilated by L : T r ( α − α − ) | i − T r ( α − ) | i . The quasi-primary | Λ i is then given by: | Λ i = 1 √ h ∗ T T ∗ (0) | i − √ N ∂ T (0) | i i . = 12 √ N h T r ( α − ) T r ( α − ) | i − T r ( α − ) | i + 85 T r ( α − α − ) | i i (G.48)There are two more weight 4 single trace quasi-primary operators given by O (1) = 12 N T r (( ∂φ ) ) O (2) = n O (2) N ( T r ( ∂ φ∂φ ) − T r ( ∂ φ∂ φ )) , = n O (2) N ( 12 ∂ T r ( ∂φ∂φ ) − T r ( ∂ φ∂ φ )) , (G.49)56here n O (2) is some N -independent normalization constant. The state | Λ i can be writtenin terms of : T T : (0) | i + a O (0) | i in the following way | Λ i = 1 √ h : T T : (0) | i + 25 N n O (2) O (2) | i i . (G.50)The OPE coefficient for : T T : is up to a normalization the same as the scalar dimension4 double trace operator in 4d and is given by hO h O h : T T : i = 1 hN h N h (3 h − N h z z z h − = 1 N h (3 h −
5) 1 z z z h − , (G.51)where 4 h (3 h −
5) come from the number of contractions giving planar diagrams. Considernow the OPE coefficient for O (2) . One finds hO h O h O (2) i = n O (2) N h hN h +1 z z z h − h ( − − h ( z + z ) −
32 2 h ( − ) z z i = 6 hn O (2) N z z z h − . (G.52)Using (G.51), (G.52) and (G.50) we find the OPE coefficient for | Λ ihO h O h Λ i = √ h (15 h − N . (G.53)Note that the h dependence matches that of the weight 4 term in the two-point function(G.38). Additionally, the OPE coefficient given by (G.53) can not be extrapolated to thelimit when h ∼ C T , as in this limit the planar expansion used for calculating (G.53) breaksdown. For this reason, we can not test the thermalization of Λ in heavy state O h H . Letus consider the thermal one-point function which is given by h Λ i β = h √ b T + O (1) i = π N √ β , (G.54)where the term ∝ N hO (2) i β is subleading since it is single trace. We find that h Λ i β λ O h O h Λ = π h (15 h − β , (G.55)which agrees with the weight 4 term in (G.38).57ote that it is explicitly seen that one can write Λ either as ∗ T T ∗ +(desc . of T) oras : T T : + N O ST with O ST a quasi-primary single trace operator. In this case the singletrace operator which one needs to add to : T T : to get Λ can be written as a sum ofdescendants O (2) ∝ ∂ T − √ T r ( ∂ φ∂ φ ). Explicitly, we have | Λ i = 1 √ h ∗ T T ∗ (0) − √ N ∂ T (0) i | i = 1 √ h : T T : (0) + 25
N n O (2) O (2) i | i . (G.56)As we saw above, using the second line in (G.56) it is straightforward to calculate correla-tion functions using Wick contractions to see that Λ gives the full weight four contributionsto the thermal two-point function for large- N theories.Now, we consider the following quasi-primary operator O ∆ ( z, ¯ z ) = √ √ ∆ N ∆ / : T r (cid:16) ( ∂φ ¯ ∂ ¯ φ ) ∆2 (cid:17) : ( z, ¯ z ) , (G.57)where we denote the anti-holomorphic part of the free field by ¯ φ = ¯ φ (¯ z ). The thermaltwo-point function of this operator, up to the terms subleading in large- N expansion, isgiven by hO ∆ ( z, ¯ z ) O ∆ (0 , i β = π β sin ∆ (cid:16) πzβ (cid:17) sin ∆ (cid:16) π ¯ zβ (cid:17) = 1( z ¯ z ) ∆ (cid:18) π ∆( z + ¯ z )6 β + π ∆(5∆ + 2)360 β ( z + ¯ z ) + π ∆ β z ¯ z + O (cid:18) β (cid:19)(cid:19) . (G.58)One can easily check that the OPE coefficients of stress tensor T and its anti-holomorphicpartner ¯ T with O ∆ are given by λ O ∆ O ∆ T = λ O ∆ O ∆ ¯ T = ∆ √ N , (G.59)while their thermal one-point function are given by h T i β = h ¯ T i β = π N √ β . (G.60)It is easy to check that terms proportional to β − in (G.58) are contributions of T and ¯ T operators h T i β λ O ∆ O ∆ T z + h ¯ T i β λ O ∆ O ∆ ¯ T ¯ z = π ∆( z + ¯ z )6 β . (G.61)58e compute the OPE coefficient of operators Λ , defined by (G.48), and its anti-holomorphic partner ¯Λ with O ∆ and obtain λ O ∆ O ∆ Λ = λ O ∆ O ∆ ¯Λ = ∆(5∆ + 2)10 √ N , (G.62)which agrees with (C.26) in [61]. Its thermal one-point function (which is the same as h ¯Λ i β ) is given by (G.54). Another operator that contributes to thermal two-point function(G.58) is : T ¯ T :. Its OPE coefficient with O ∆ and thermal one-point function are given by λ O ∆ O ∆ : T ¯ T : = ∆ N h : T ¯ T : i β = π N β . (G.63)Again, it is easy to check h Λ i β λ O ∆ O ∆ Λ z + h ¯Λ i β λ O ∆ O ∆ ¯Λ ¯ z + h : T ¯ T : i β λ O ∆ O ∆ : T ¯ T : z ¯ z == π ∆(5∆ + 2)360 β ( z + ¯ z ) + π ∆ β z ¯ z , (G.64)which matches with the corresponding terms in (G.58).The OPE coefficients λ O ∆ O ∆ Λ , λ O ∆ O ∆ ¯Λ , and λ O ∆ O ∆ : T ¯ T : can be extrapolated to thelimit ∆ ∼ N , by the same logic as in Appendix C. Then, we can explicitly check thethermalization property of Λ , ¯Λ , and : T ¯ T :. To establish a relation between the inversetemperature β and the conformal dimension ∆ H of heavy state O H = O ∆ ∼ N , we assumethe thermalization of stress tensor h T i β = λ O H O H Λ , (G.65)which implies ∆ H N = π β . (G.66)Using this relation, it is easy to show h Λ i β = λ O H O H Λ (cid:12)(cid:12)(cid:12) ∆2 HN , h ¯Λ i β = λ O H O H ¯Λ (cid:12)(cid:12)(cid:12) ∆2 HN , h : T ¯ T : i β = λ O H O H : T ¯ T : (cid:12)(cid:12)(cid:12) ∆2 HN . (G.67)This means that operators Λ , ¯Λ , and : T ¯ T : thermalize in the quasi-primary state O H similarly to the thermalization in a Virasoro primary states in large- c theory, that wasanalyzed in [38]. 59 ppendix H. Vector model In this section we study the free scalar vector model at large- N . Consider the scalaroperator O ∆ = 1 p N (∆) : ( ϕ i ϕ i ) ∆2 : ( x ) , (H.1)where N (∆) is a normalization constant which to leading order in N is given by N (∆) ≈ (∆)!! N ∆2 . (H.2)The thermal two-point function is given by hO ∆ ( x ) O ∆ (0) i β = ˜ g ( x E , | x | ) ∆ + (cid:16) ∆2 (cid:17)
1∆ ˜ g ( x E , | x | ) ∆ − + . . . , (H.3)where ˜ g ( x E , | x | ) = ∞ X m = −∞ x E + mβ ) + x = π β | x | h Coth (cid:16) πβ ( | x | − ix E ) (cid:17) + Coth (cid:16) πβ ( | x | + ix E ) (cid:17)i . (H.4)The thermal a τ,J coefficients a , and a , are the same as in the adjoint model (this is sosince the second term in (H.3) does not affect these): a , = π ∆45 ,a , = π ∆(∆ − . (H.5)The unit-normalized stress tensor is given by T µν ( x ) = 13 √ C T : (cid:18) ∂ µ ϕ i ∂ ν ϕ i − ϕ i ∂ µ ∂ ν ϕ i − (trace) (cid:19) : ( x ) , (H.6)where C T = N . The OPE coefficient of the stress tensor is again found by Wick contrac-tions to be λ O ∆ O ∆ T µν = − √ C T , (H.7)in agreement with the stress tensor Ward identity. The double-stress tensor is given by T µνρσ = 1 √ T ( µν T ρσ ) : − (traces) , (H.8)60nd the OPE coefficient is calculated precisely as for the adjoint model and we find λ O ∆ O ∆ T , = 8 √ C T ∆(∆ − . (H.9)There is another double-trace operator with twist 4 and spin 4 and takes the same form: O O , : as for the adjoint model O DT µνρσ ( x ) = 196 √ N : ϕ i ϕ i (cid:16) ϕ j ∂ µ ∂ ν ∂ ρ ∂ σ ϕ j − ∂ ( µ ϕ j ∂ ν ∂ ρ ∂ σ ) ϕ j +18 ∂ ( µ ∂ ν ϕ j ∂ ρ ∂ σ ) ϕ j − (traces) (cid:17) : ( x ) . (H.10)The OPE coefficient and the thermal one-point function yields the same result as for thecorresponding operator in the adjoint model . It then follows that the a , extracted from(H.3) is reproduced by the sum of the double stress tensor and (H.10). Appendix I. Factorization of thermal correlators
In this appendix we argue for the factorization of thermal expectation values of multi-trace operators in large- C T theories on S × R d − . Consider the thermal two-point functionof a scalar operator O with dimension ∆: hO ( x ) O (0) i β = hOi β hOi β + hO ( x ) O (0) i β,c , (I.1)where the second term consist of the connected part of the correlator. Note that thedisconnected term in (I.1) is independent of the position x . On the other hand we canevaluate (I.1) using the OPE on the plane which takes the form O ( x ) O (0) = 1 | x | + X n,l λ OO [ OO ] n,l x n + l [ OO ] n,l + . . . , (I.2)when written in terms of primaries and the dots refer to terms surpressed in the large- C T limit. Note that λ OO [ OO ] n,l are the MFT OPE coefficient which are of order 1. The termin (I.2) that is independent of x is due to the n = l = 0 term in (I.2) and inserting theOPE on the LHS of (I.2), we find that λ OO [ OO ] , h [ OO ] , i β = hOi β . 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