D>2 Stress-Tensor Operator Product Expansion Near a Line

PHYSICAL REVIEW D 103, L121702 (2021) Letter

d > 2 stress-tensor operator product expansion near a line

Kuo-Wei Huang Department of Physics, Boston University, Commonwealth Avenue, Boston, Massachusetts 02215, USA

(Received 1 April 2021; accepted 21 May 2021; published 10 June 2021)

We study the operator product expansion of stress tensors (TT OPE) in d>2 conformal field theories whose bulk dual is Einstein gravity. Directly from the TT OPE, we obtain, in a certain null-like limit, an 2 algebraic structure consistent with the Jacobi identity: ½Lm; Ln¼ðm − nÞLmþn þ Cmðm − 1Þδmþn;0. The dimensionless constant C is proportional to the central charge CT . Transverse integrals in the definition of Lm play a crucial role. We comment on the corresponding limiting procedure and point out a curiosity related to the central term. A connection between the d>2 near-lightcone stress-tensor conformal block and the d ¼ 2 W algebra is observed. This note is motivated by the search for a field-theoretic derivation of d>2 correlators in strong coupling critical phenomena.

DOI: 10.1103/PhysRevD.103.L121702

I. INTRODUCTION theories, we find it necessary to revisit the d>2 TT OPE structure. In this paper, we adopt the following two The operator product expansions of stress tensors (TT simplifying limits to reduce the complexity of the TT OPE) in d ¼ 2 conformal field theory (CFT) OPE: (i) Infinitely large higher-spin gap, and (ii) Null/ c 2 1 lightcone-like limit.1 Tðz ÞTðz Þ¼ þ Tðz Þþ ∂ Tðz ÞþOð∂2TÞ; – Δ > 2 1 2 2s4 s2 2 s z2 2 As shown in [3 5], the gap gap to the lightest spin single-trace primary controls the higher-order corrections s ¼ z1 − z2 ð1Þ Δ → ∞ to Einstein gravity; the limit gap then selects CFTs with an Einstein gravity bulk dual. We will focus on stress- leads to the Virasoro algebra tensor contribution to the TT OPE and suppress other 2 c 2 primary operators. On the other hand, the lightcone limit ½Lm;Ln¼ðm − nÞLmþn þ mðm − 1Þδmþn;0; has been adopted in the recent computation of the multi- I 12 1 stress-tensor OPE data in d>2 holographic CFTs [7–19]. L ¼ dzzmþ1TðzÞ: ð Þ C m 2πi 2 The near-lightcone correlator at large central charge T is independent of higher-curvature terms in the purely gravi- The Virasoro algebra is omnipresent in two-dimensional tational action [7]; however, the correlator depends on critical phenomena [1] and has enormous implications; in certain nonminimal coupling bulk interactions which are particular, the algebra provides a nonperturbative derivation suppressed at an infinite gap [9]. These results suggest that of d ¼ 2 conformal correlators. In higher dimensions, the simplest starting point is to impose the limits (i), and TT 3 the general TT OPE is contaminated by many model- (ii) on the OPE. dependent details. However, we ask the question; can one In general, the results depend on the order of limits generalize the derivation (1)–(2) to d>2 CFT in certain (i.e., limiting procedure). A related motivation of this work physical limits? is to help identify a lightcone-like limiting procedure that Over 27 years ago, Osborn and Petkos [2] computed the may be implemented to compute multi-stress-tensor OPE d>2 stress-tensor contribution to the d>2 TT OPE, but we coefficients and near-lightcone correlators in CFTs have not found any computation based on such an explicit TT OPE. A reason, presumably, is that the TT OPE is complicated. Given the recent developments of gauge/ 1 “ ” gravity correspondence and d>2 strongly coupled field We use -like to distinguish our limiting procedure from similar limits used in the literature: the null-line limit is often þ 2 defined by directly setting x ¼ x⊥ ¼ 0 in the Lorentzian sig- 0 1 Published by the American Physical Society under the terms of nature, where x ¼ x x and x⊥ denotes transverse directions. the Creative Commons Attribution 4.0 International license. 2For instance, the three-point function hTTOi is suppressed as Δ → ∞ Further distribution of this work must maintain attribution to gap [6]. 3 the author(s) and the published article’s title, journal citation, We should also assume the usual large N and large CT limits 3 and DOI. Funded by SCOAP . but we will keep CT in our expressions.

2470-0010=2021=103(12)=L121702(7) L121702-1 Published by the American Physical Society KUO-WEI HUANG PHYS. REV. D 103, L121702 (2021) from the first principle via a Virasoro-like field-theoretic complex coordinates. We will then focus on the approach. TzzTzz OPE. The main result of this work is that we find a structure The TT OPE simplifies significantly when one focuses similar to (2) in higher dimensions. Intuitively, one may on the Tzz component. Using (3), we obtain expect that a Virasoro-like structure arises because the null- Izz;zzðsÞ like limit brings stress tensors close to a line, a picture Tzzðx ÞTzzðx Þ¼C þ Aˆ zzzz ðsÞTzzðx Þ 1 2 T 8 zz 2 reminiscent of the two-dimensional case where s π z¯ z¯ ¯ π zz zzzz λ zz 2 T ¼ − 2 T ðzÞ, T ¼ − 2 T ðz¯Þ are holomorphic and anti- þ B zzλðsÞ∂ T ðx2ÞþOð∂ TÞð6Þ holomorphic functions, respectively. Note we are not I zz;zz ¼ 4ðszÞ4 introducing a physical line defect. where s4 , This paper is organized as follows. In Sec. II, we discuss d ¼ 4 4ðszÞ2 the stress-tensor OPE structure. We focus on for Aˆ zzzz ¼ ðð2bþcÞðs⊥Þ4 zz 10 concreteness and expect that our results generalize to other CTs dimensions. In Sec. III, we consider a null-line limit and −2szsz¯ðð8a−b−3cÞðs⊥Þ2 þð6a−b−2cÞszsz¯ÞÞ; obtain a Virasoro-like commutator via the stress-tensor OPE. A d ¼ 4 single-stress-tensor-exchange derivation ð7Þ without explicitly using an algebra is discussed in sz¯ Sec. IV, where we point out a curiosity related to the zzzz ˆ zzzz B zzz ¼ A zz; ð8Þ central term. We observe a connection between the d>2 4 lightcone stress-tensor conformal block and the central- s⊥ d ¼ 2 W Bzzzz ¼ Aˆ zzzz ; ð Þ term of the algebra. zz⊥ 2 zz 9 II. STRESS-TENSOR OPE ðszÞ3 Bzzzz ¼ ðð64a þ 18b − 11cÞðs⊥Þ4 zzz¯ 9C s10 Our starting point is the stress-tensor contribution to the T d ¼ 4 TT OPE [2], − 2szsz¯ð4ð3a − b − 2cÞðs⊥Þ2 z z¯ Iμν;σρðsÞ þð26a − 4b − 9cÞs s ÞÞ: ð10Þ Tμνðx ÞTσρðx Þ¼C þ Aˆ μνσρ ðsÞTαβðx Þ 1 2 T 8 αβ 2 s We will argue that higher-order pieces, Oð∂2TÞ, are μνσρ λ αβ 2 þ B αβλðsÞ∂ T ðx2ÞþOð∂ TÞ; ð3Þ irrelevant when imposing the null-like limit considered in Sec. III. Observe that, from (7), ð8a − b − 3cÞþ s ¼ x − x ð6a − b − 2cÞ¼ 3 C where 1 2. The first term has the familiar form π2 T. While this combination is interest- N 1 1 ing, we here consider a large , large-gap condition which Iμν;σρðsÞ¼ ðIμσðsÞIνρðsÞþIμρðsÞIνσðsÞÞ − δμνδσρ; places strong constraints on the flux parameters “t2” and 2 4 “t4 – μ σ " of the energy flux escaping to null infinity [20 22], μσ μσ s s I ðsÞ¼δ − 2 : ð4Þ 30ð13a þ 4b − 3cÞ s2 t ¼ ¼ 0; 2 ð14a − 2b − 5cÞ ˆ The structures of Aμνσραβ and Bμνσραβλ are cumbersome so −15ð81a þ 32b − 20cÞ t ¼ ¼ 0: ð Þ we shall not list them here; see [2] for detailed expressions. 4 2ð14a − 2b − 5cÞ 11 As noted in [2], there are three undetermined coefficients in the TT OPE, denoted as a, b, c. The central charge CT is It is worth mentioning that two trace-anomaly central charges 4 given by become the same under these conditions. By imposing t2 ¼ t4 ¼ 0 without first requiring a strictly infinite CT,wecan π2 C ¼ ð14a − 2b − 5cÞ: ð Þ reduce three parameters to one parameter. T 3 5 III. STRESS-TENSOR OPE NEAR A LINE In the lightcone limit, the relevant contribution is the AND A VIRASORO-LIKE COMMUTATOR lightcone component of the stress tensor, Tþþ. We will mostly work in the Euclidean space and adopt the line Consider the following operator in d ¼ 4, P ðiÞ element ds2 ¼ dzdz¯ þ ðdx Þ2 where z, z¯, the I Z i¼1;2 ⊥ κ Euclidean analogue of the lightcone coordinates, are L ¼ dz¯z¯mþ1 d2x Tzzðz; z;¯ xð1Þ;xð2ÞÞ; m 2πi ⊥ ⊥ ⊥ Z Z Z l l 4 c 2 ð1Þ ð2Þ The parameter here should not be confused with the central where d x⊥ ¼ dx dx ð12Þ C ¼ c ⊥ ⊥ charge in two dimensions where T 2π2. 0 0

L121702-2 d>2 STRESS-TENSOR OPERATOR PRODUCT EXPANSION … PHYS. REV. D 103, L121702 (2021) Z 5 in the null-like limit z → 0, l → 0. We will determine the 2 ˆ zzzz zz lim d ðx1Þ⊥ðA zzðx1 − x2ÞT ðx2ÞÞ overall normalization factor κ later. The interpretation of sz→δ the small l limit is that we consider the stress- Tzzðx Þ ¼ fða; b; cÞ 2 þ OðδÞðÞ tensor contribution near a two-dimensional plane. We πðsz¯Þ2 15 are interested in computing the commutator ½Lm; Ln. The transverse integrals are crucial, as we will see, for fða; b; cÞ¼−52aþ10bþ19c extracting a central extension consistent with a Witt-like where 14a−2b−5c . The leading-order term is ðx Þ algebra.6 cutoff-independent and only depends on 2 ⊥ through the sz Let us first consider the c-number term which is stress tensor. To take the small limit, we may assume Tzzðx Þ controlled by the stress-tensor two-point function. After 2 is a suitable test function having a finite contribu- y ¼ z ¼ 0 performing the transverse integrations, we consider a small tion only near 2 2 , and then perform all the z sz s expansion, transverse integrations before imposing the small limit. But we find it simpler, as we did above, to take sz → δ right Z after performing the integrations over the first set of C I zz;zzðsÞ 4πC l2 7πC l C 8 4 T T T T transverse coordinates ðx1Þ⊥. It is now straightforward lim d x⊥ ¼ − pffiffiffiþ sz→δ s8 5ðsz¯Þ5 δ 16ðsz¯Þ9=2 δ 5ðsz¯Þ4 to complete the rest of the integrations, ð356þ315πÞC δ4 I I þ T þ: ð Þ κ 2 fða; b; cÞ 14400 l8 13 ½L ; L j ¼ dz¯ z¯nþ1 dz¯ m n A-term 2 2 1 2πi π Cð0Þ Cðz¯ Þ Z 2 Tzzðx Þ We would like to extract the cutoff-independent piece. We z¯mþ1 d2ðx Þ 2 × 1 2 ⊥ z¯ 2 do so by next imposing a l → 0 limit such that the first two ðs Þ terms are suppressed. The last piece of (13) and higher- κfða; b; cÞ ¼ ðm þ 1ÞL : ð Þ order terms, although divergent as l → 0, do not have a z¯ π mþn 16 CT pole and thus do not contribute to the commutator. The ðsz¯ Þ4 term shares the same form as the c-number term in the Next, we have d ¼ 2 TT OPE (1). The transverse integrals compensate Z d>2 TT 2 zzzz z zz for the additional dimensions of the OPE. lim d ðx1Þ⊥ðB zzzðsÞ∂ T ðx2ÞÞ This c-number term derivation does not require a sz→δ large-gap condition. The Cauchy’s integral formula ∂ Tzzðx Þ ¼ fða; b; cÞ z¯ 2 þ OðδÞ; ð Þ now leads to7 2πsz¯ 17 I I Z κ 2 C 2 zzzz zz nþ1 mþ1 T lim d ðx1Þ⊥ðB zz⊥ðsÞ∂⊥T ðx2ÞÞ ¼ OðδÞ: ð18Þ ½Lm; Lnj ¼ dz¯2z¯ dz¯1z¯ sz→δ CT 2πi 2 1 5ðsz¯Þ4 Cð0Þ Cðz¯2Þ C zz 2 T 2 Since we only focus on the T component in the null-like ¼ κ mðm − 1Þδmþn;0: ð14Þ 30 limit and effectively turn off other components of the stress tensor, the conservation of the stress tensor implies that we ∂ Tzz We next turn to the operator part of the TT OPE, keeping also drop z . Observe that the structures (including the explicit a, b, c parameters and imposing the conditions (11) relative coefficients) of (15) and (17), are the same as the at the end. We evaluate two-dimensional case (1). From (17), we get κfða; b; cÞ 5 ½L ; L j ¼ − ðm þ n þ 2ÞL : ð Þ One may perform a Wick-rotation to Lorentzian space and m n B-term mþn 19 impose the lightcone limit, and then Wick-rotate back to the 2π Euclidean space to carry out the z¯ integral via the residue theorem. One may also formally impose a small z limit directly Similar to the corresponding d ¼ 2 computation, we in Euclidean space, which is what we will do here. For two stress z have performed an integration by parts to evaluate the tensors, we take a small s . A similar analysis applies to the ∂ Tzz Tz¯ z¯Tz¯ z¯ sz¯ z¯ term. OPE if one instead chooses a small limit. Oð∂2TÞ 6This construction is essentially the same as the mode operator We will not include the higher-order corrections introduced in [23], but in that work the author adopts a different in the TT OPE, but, based on the pattern from (15) limiting procedure. See also [24–27] for related discussions. 7In general d, we find 8In the process of simplifying (15), we formally assume l> −1 −1 ðx2Þ⊥ > 0 to adopt the identity tan ðXÞþtan ð1=XÞ¼π=2 d 2 4CT 2 ½Lm; Lnj ¼ð−1Þ κ mðm − 1Þδmþn;0: with X>0. This, strictly speaking, means the end points of the CT Γðd þ 2Þ ðx2Þ⊥ integrals should be removed.

L121702-3 KUO-WEI HUANG PHYS. REV. D 103, L121702 (2021) and (17), it seems reasonable to assume that the higher- an algebra. In fact, as we will see, this derivation presents a order terms do not have a relevant pole in the null-like limit. central-term curiosity. Combining (19), (16), and (14), the result is The scalar four-point conformal correlator can be written in terms of the conformal block decomposition [31], 2 CT 2 ½Lm; Ln¼ðm − nÞLmþn þ κ mðm − 1Þδmþn;0: ð20Þ 30 hOHð∞ÞOHð1ÞOLðz; z¯ÞOLð0Þi 2π X Bðz; z;¯ τ;JÞ where κ ¼ . fða;b;cÞ ¼ c ðΔT;JÞ ð21Þ κ OPE ðzz¯ÞΔL We choose a normalization such that the noncentral ΔT ;J term has a simple coefficient. If we now impose the conditions listed in (11), we find the normalization factor where the twist of an operator is its dimension minus its 90π 9 to be κ ¼ − 181. spin, τ ¼ ΔT − J. We formally name OH the “heavy" scalar To summarize, we have described a null-line-like limit- and OL the “light” scalar although the heavy-light limit ing procedure that allows us to extract an algebraic [i.e., ΔH, CT → ∞ with ΔH=CT fixed and ΔL ∼ Oð1Þ] does structure from the TT OPE. The result (20) is strikingly not play a special role in the single-stress-tensor-exchange similar to the two-dimensional Virasoro algebra. We do not computation. We adopt this notation as an example which is use holographic duality here, but it would be nice to find a useful to compare with the literature that discusses multi- potential connection to the AdS=CFT computation dis- stress-tensor contributions to the heavy-light correlator. cussed some time ago [28,29] where a higher-dimensional The Ward identity fixes the stress-tensor OPE coefficient to ΔHΔL generalization to the Brown-Henneaux symmetry [30] was be c ð4; 2Þ¼ 4 in the convention of (3). The con- OPE 9π CT identified in a certain infinite momentum frame. Most formal block is12 likely, whether or not there is a Virosoro-like structure at 10 infinity depends on boundary conditions. zz¯ τþ2J τ−2 τþ2J τþ2J Bðz;z;¯ τ;JÞ¼ z 2 z¯ 2 F ; ;τþ2J;z It would certainly be of great interest to extend the two- z−z¯ 2 1 2 2 dimensional CFT analysis to higher dimensions in the null/ lightcone-like limit, where one expects to find relatively τ−2 τ−2 × F1 ; ;τ−2;¯z −ðz↔z¯Þ : ð22Þ robust structures. By first focusing on a special class of 2 2 2 higher-dimensional CFTs with an Einstein gravity dual, we would like to know if there is an effective algebraic In the limit z → 0, the stress-tensor contribution in d ¼ 4 derivation of the multi-stress-tensor OPE coefficients and reads13 conformal correlators. Considering perturbative corrections Δ due to a large but finite higher-spin gap could be interesting limððzz¯Þ L hOHð∞ÞOHð1ÞOLðz; z¯ÞOLð0ÞijTÞ z→0 as well. 1 Δ Δ ¼ H L z¯3 F ð3; 3; 6; z¯Þz þ Oðz2Þ 9π4 C 2 1 IV. A d = 4 SINGLE-STRESS-TENSOR-EXCHANGE T 10 Δ Δ 3ðz¯ − 2Þz¯ − ð6 þðz¯ − 6Þz¯Þ ð1 − z¯Þ DERIVATION ¼ H L ln z 4 2 3π CT z¯ We conclude this paper by presenting some observations, þ Oðz2Þ: ð Þ which hopefully shed light on more general cases. In the 23 following, we point out a simple derivation of the d ¼ 4 near-lightcone conformal scalar correlator via a mode The higher-order pieces represent multi-stress-tensor con- summation.11 This derivation does not explicitly rely on tributions to the correlator. It is instructive if we temporarily forget about the algebra

9 and instead adopt the following operator, An overall rescaling of the mode operator Lm should not affect a I scalar correlator computation. But one might wonder if the “right” dz¯ κ ¼ − 90π ¼ − π ˜ T mþ2 zz ⊥ proportionality constant should instead be 180 2.Ifwe Lm ¼ lim z¯T T ðzT; z¯T;x ¼ 0Þ: ð24Þ formally adopt free-theory values of a, b, c [2], we notice that zT →δ 2πi π 18π κ ¼ − 2 for both a fermion and a Uð1Þ gauge field, but κ ¼ − 37 for π x⊥ ¼ 0 z → δ a scalar. In fact, κ ¼ − 2 is true only under the condition Notice we directly set in this definition. The T 4a þ 2b − c ¼ 0, which holds for both a free fermion and a represents the null-line limit for the stress tensor. The Uð1Þ 4a þ 2b − c ¼ − 1 gauge field, but a free scalar has 9π6.(In notation “m þ 2” will result in slightly more symmetric d ¼ 2, on the other hand, 4a þ 2b − c ¼ 0 for both a free scalar expressions in the following computation. The mode and a free fermion.) 10I thank Gary Gibbons for related remarks. 11 12 1 J The derivation presented here is simpler than previous work Our convention differs by an overall factor of ð− 2Þ from the [23] and we can avoid an arbitrary parameter introduced in that convention used in Dolan and Osborn [31]. paper. 13One can also choose z¯ → 0 as the lightcone limit.

L121702-4 d>2 STRESS-TENSOR OPERATOR PRODUCT EXPANSION … PHYS. REV. D 103, L121702 (2021)

˜ ˜ † operator (24) is essentially the same as the lightray operator N m ¼hLmLmi which does not contain transverse integrals [24,26,27,32]. I I dz¯2T dz¯1T We here use the Euclidean signature with complex coor- ¼ lim sz→δ 2πi 2πi dinates z; z¯. Similar to the d ¼ 2 case, we may expect that Cð0Þ Cðz¯2T Þ mþ2 −mþ2 zz zz the stress-tensor-exchange contribution can be computed × ðz¯1TÞ ðz¯2TÞ hT ðz1T; z¯1TÞT ðz2T; z¯2TÞi via the following mode summation, I I mþ2 −mþ2 dz¯2T dz¯1T 4CTðz¯1TÞ ðz¯2TÞ ¼ lim sz→δ 2πi 2πi ðz¯ − z¯ Þ6ðz − z Þ2 Cð0Þ Cðz¯2T Þ 1T 2T 1T 2T X∞ ˜ † ˜ hOHð∞ÞOHð1ÞLmihLmOLðz; z¯ÞOLð0Þi C ðm þ 2Þðm þ 1Þmðm − 1Þðm − 2Þ VT ¼ lim ¼ T : ð Þ z→0 hO ð∞ÞO ð1ÞiN hO ðz; z¯ÞO ð0Þi 2 29 m¼m H H m L L 30 δ ð25Þ The UV-cutoff dependencies cancel out in the final mode summation and we obtain exactly the d ¼ 4 stress-tensor- ˜ ˜ † where the normalization factor is N m ¼hLmLmi. We will exchange structure (23), find m ¼ 3. Using the three-point function 10 Δ Δ X∞ ðm − 1Þðm − 2Þz¯m V ¼ H L z T 3π4 C mðm þ 1Þðm þ 2Þ T m¼3 c XμXν δμν hTμνðx ÞO ðx ÞO ðx Þi ¼ TOO − 1 Δ Δ 1 Δ 2 Δ 3 4 4 2Δ−4 2 ¼ H L z¯3 F ð3; 3; 6; z¯Þz: ð Þ x12x13x23 X 4 4 2 1 30 9π CT ð26Þ This computation does not require a large gap. Xμ ¼ xμ =x2 − xμ =x2 c ¼ − 2Δ It is peculiar that we are able to reproduce the d ¼ 4 with 12 12 13 13 and TOO 3π2, we first obtain near-lightcone correlator, including the correct OPE coefficient, via a mode summation. The final result is finite and cutoff independent. Although the above single-stress-tensor ˜ hLmOLðz; z¯ÞOLð0Þi computation does not rely on knowing an algebra, we lim lim zT →δ z→0 hOLðz; z¯ÞOLð0Þi would like to ask why such a derivation exists. Recall that, I 2Δ dz¯ z¯3z in two-dimensions, a similar derivation exists because of L T mþ2 d>2 ¼ − lim lim z¯T the Virasoro symmetry. Given the above computa- 3π2 z →δ z→0 2πi z¯3 z ðz¯ − z¯Þ3ðz − zÞ T Cðz¯Þ T T T T tion, one may speculate that a certain symmetry emerges 2Δ ðm − 1Þðm − 2Þz¯m near the lightcone. An underlining algebra would provide a ¼ − L z 2 lim lim 3π zT →δ z→0 2zTðzT − zÞ precise interpretation of the modes counting in (30). Since Δ ðm − 1Þðm − 2Þz¯m we have extracted a Virasoro-like commutator from the ¼ − L z; ð Þ 2 2 27 stress-tensor OPE (20), it seems natural to link the 3π δ correlator computation to the algebra. However, we find a curiosity related to the central term, where we introduce a short-distance cutoff δ. We shall find or more generally, to the limiting procedure. In the above that the final four-point scalar correlator is independent of correlator computation, we emphasize that we take the UV cutoff. It is important to adopt a proper order of x⊥ → 0 before imposing z → 0.14 The resulting “central” m limits. term hasH the following -dependence (in the notation of ˜ † ˜ ˜ mþ2 On the other hand, by taking Lm ¼ L−m, we find Lm ∼ dz¯z¯ T),

I hO ð∞ÞO ð1ÞL˜ † i 2Δ dz¯ ðz¯ Þ−mþ2 TypeA∶ CTðm þ 2Þðm þ 1Þmðm − 1Þðm − 2Þδmþn;0: ð31Þ H H m ¼ H T T 2 lim 3 hOHð∞ÞOHð1Þi 3π zT →δ Cð1Þ 2πi ðz¯ − 1Þ ðz − 1Þ T T Such an m-dependence is quite different from the central 2Δ ðm − 1Þðm − 2Þ ¼ H term in the Virasoro-like commutator, which has the 2 lim 3π zT →δ 2ðzT − 1Þ structure, Δ ¼ − H ðm − 1Þðm − 2Þ: ð Þ 2 28 3π Type B∶ CTðm þ 1Þmðm − 1Þδmþn;0: ð32Þ

The normalization factor can be computed using the stress- 14Using this order of limits, one can include transverse tensor two-point function, integrals but the correlator result is unchanged.

L121702-5 KUO-WEI HUANG PHYS. REV. D 103, L121702 (2021) P z → 0 3 As shown above, in this case, we take before where Λm ¼ nðLm−nLnÞ − 10 ðm þ 3Þðm þ 2ÞLm. Wm is ⊥ imposing x → 0. The Type B central term has the familiar the Laurent modes of a spin-three primary current. Note that form fixed by the Jacobi identity, but the correlator closure of the W3 algebra requires first knowing the operator derivation suggests that the Type A structure plays a Lm that satisfies the Virasoro algebra. In general (even) d, nontrivial role in recovering the scalar correlator. The ∼C mðm2 − 1Þðm2 − we find the Type A structure is T H Type A structure, however, is incompatible with the Witt 2 d 2 ˜ ðmþdÞ 4Þðm − ð Þ Þ in the notation of Lm ∼ dz¯z¯ 2 T. algebra. 2 ˜ ˜ To our knowledge, a connection between d>2 CFT Ideally, we would like to also compute ½Lm; Ln via the near-lightcone correlators and the W (-like) symmetry has d ¼ 4 TT OPE, but we find that the commutator compu- 15 L˜ TT not been mentioned before. This central-term curiosity tation using m requires a higher-order term in the needs to be better understood. Perhaps exploring more OPE. Such a computation will not be included in this paper. general structures involving multi-stress-tensor exchanges On the other hand, we observe that, up to an overall in d>2 CFTs can help clarify its algebraic underpinnings. coefficient, the d ¼ 4 Type A structure (31) is identical to W d ¼ 2 the central term of the 3 algebra in CFTs [33] (see ACKNOWLEDGMENTS [34] for a review), I am grateful to L. Fitzpatrick, J. Maldacena, H. Osborn, c ½W ;W ¼ ðm þ 2Þðm þ 1Þmðm − 1Þðm − 2Þδ and A. Parnachev for helpful comments. This work was m n 360 mþn;0 supported in part by the U.S. Department of Energy Office 1 þðm − nÞ ðm þ n þ 3Þðm þ n þ 2Þ of Science No. DE-SC0015845 and the Simons 15 Collaboration on the Nonperturbative Bootstrap. 1 − ðm þ 2Þðn þ 2Þ L 15 6 mþn In the context of supersymmetric CFTs in 3, 4, and 6 dimensions (with at least 8 real supercharges), it was shown in 16 þ ðm − nÞΛ ; ð Þ [35] that there is a subsector of the theory which is described by a 22 þ 5c mþn 33 d ¼ 2 chiral algebra. In [36], it was argued that, in d ¼ 6 maximally supersymmetric CFTs, the algebra is the WN algebra. ½L ;W ¼ð2m − nÞW ; ð Þ (See also [37] for an alternative viewpoint.) These results use very m n mþn 34 different methods and apply to a distinct class of CFTs.

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