W (1+ Infinity) and the Celestial Sphere

w1+∞ and the Celestial Sphere

Andrew Strominger

Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA

Abstract

It is shown that the inﬁnite tower of tree-level soft graviton symmetries in asymptotically ﬂat 4D quantum gravity can be organized into a single chiral 2D Kac-Moody symmetry based

on the wedge algebra of w1+∞. The inﬁnite towers of soft photon or gluon symmetries also

transform irreducibly under w1+∞. arXiv:2105.14346v1 [hep-th] 29 May 2021 Contents

1 Introduction 1

2 Gravity 2

3 Gauge theory 4

4 Speculations 5

1 Introduction

Quantum gravity and quantum field theory in an asymptotically flat 4D spacetime exhibit an infinite number of symmetries whose Ward identities are soft theorems [1–3]. A complete characterization of these symmetries is an outstanding open problem. Recent progress on this problem was made in [4] for tree-level gravity and gauge theory. In this work symmetries were compactly represented by 2D (higher-spin) currents in the celestial conformal field theory (CCFT) living on the celestial sphere at null infinity. An infinite tower of such currents and their algebra were derived at tree level using positive-helicity soft theorems and the celestial operator product expansion (OPE).1 Related work appears in [5–8]. The somewhat lengthy results of [4] were displayed in a basis with manifest covariance under the global Lorentz-conformal group of the celestial sphere. The currents include an ‘SL(2, R)w’ Kac- Moody algebra arising from the subleading soft graviton theorem. In this paper we find dramatic simplifications by reorganizing the generators according to representations of this SL(2, R)w, which is accomplished with a version of the light ray transform [9]. The entire tower of currents is assembled into a chiral Kac-Moody symmetry of the wedge algebra of w1+∞! This same Kac- Moody algebra has appeared previously including in discrete states of the c = 1 string [10] and w1+∞-gravity [11]. Moreover we find, unlike for previous presentations, in the SL(2, R)w covariant presentation only positive half-integral conformal weights appear. In the next section we present explicity the algebraic transformation from the conventional to the SL(2, R)w covariant basis, and show that the entire algebra is the Kac-Moody symmetry of the wedge algebra of w1+∞. In section 3 the infinite tower of soft symmetries for gauge theory are written in SL(2, R)w covariant form and found to simplify. Again only positive SL(2, R)w weights appear. These results suggest that w1+∞ - or perhaps its quantization W1+∞ - provides an organizing principle for CCFT. We conclude with speculations on this role in section 4.

1The analysis of [4] is incomplete in several aspects. It omits a presumed second inﬁnite tower arising from negative- helicity soft theorems. These may lead to yet more symmetries, but their inclusion must confront ambiguities in the double soft limit. Deformations of the OPE from one-loop corrections, higher dimension operators or inclusion of Goldstone modes are further expected to deform the algebra or produce central terms.

1 2 Gravity

+ Let us recap the basic results and notation of [4] for gravity. Let G∆(z, z¯) denote the positive-helicity conformal-primary graviton operator with 2D conformal weight ∆ which crosses the celestial sphere at a point (z, z¯). Deﬁne a discrete family of conformally soft positive-helicity gravitons

Hk = lim εG+ , k = 2, 1, 0, −1,..., (2.1) ε→0 k+ε with weights k + 2 k − 2 (h, h¯) = , , (2.2) 2 2 and a consistently-truncated antiholomorphic mode expansion2

2−k X2 Hk(z) Hk(z, z¯) = n . (2.3) n+ k−2 k−2 z¯ 2 n= 2

k Each Hn(z) is a 2D symmetry-generating conserved current whose Ward identity is a soft theorem. The factor of ε in (2.1) cancels the soft poles, allowing a ﬁnite OPE. Throughout this paper z andz ¯ are treated as independent, which amounts to continuing (3,1) Minkowski space to (2,2) Klein space, the celestial sphere to the celestial torus [12] and Lorentz-SL(2, C) to SL(2, R)L⊗SL(2, R)R. The n- index in (2.3) then transforms in the 3−k dimensional representation of the SL(2, R)R. The simplest 1 P 1 −m−3/2 example of the k = 1 term generates supertranslations. Expanding H 1 (z) = m H 1 z , ± 2 ± 2 ,m 1 the four modes H 1 1 generate the four global translations. Deﬁning the commutator for holo- ± 2 ,± 2 morphic objects3 I dw [A, B](z) = A(w)B(z), (2.4) z 2πi the soft current algebra found in [4] for gravity is4

h i κ ( 2−k − m + 2−l − n − 1)! ( 2−k + m + 2−l + n − 1)! Hk ,Hl = − [n(2 − k) − m(2 − l)] 2 2 2 2 Hk+l , m n 2 2−k 2−l 2−k 2−l m+n ( 2 − m)!( 2 − n)! ( 2 + m)!( 2 + n)! (2.5) To write the algebra (2.5) in a simpler form we deﬁne

1 wp = (p − n − 1)!(p + n − 1)!H−2p+4. (2.6) n κ n

2 Outside the speciﬁed range of n, the SL(2, R)R-invariant norm vanishes. Such operators may still have contact interactions but in this paper operators are always at distinct points. 3Note that this is a 2D celestial commutator on a 1D circle, not to be mistaken for a 4D commutator on a 3D slice. 4 2 We note that the (2, 0) current H0 (z) commutes with all other generators and is a central term in the super- translation algebra. It has been taken to vanish in most applications but is natural to include here.

2 This is essentially the light-transform or right-shadow [9] in one dimension adapted to ﬁnite SL(2, R) representations. (2.5) then becomes

p q p+q−2 [wm, wn] = [m(q − 1) − n(p − 1)] wm+n . (2.7)

k Since the index k on Hm runs over k = 2, 1, 0, ...., p runs over the positive half integral values

3 5 p = 1, , 2, , .... (2.8) 2 2

k−2 2−k The restriction 2 ≤ m ≤ 2 becomes

1 − p ≤ m ≤ p − 1. (2.9)

This is one of our main results. The commutators (2.7) were ﬁrst written down by Ioannis Bakas in 1989 [13]. In this work m is an arbitrary integer and the resulting algebra is now referred to as w1+∞. The closed p = 2 subalgebra of w1+∞ is the c = 0 Virasoro algebra. This algebra in (2.7) is a subgroup of w1+∞ with m restricted according to (2.9). The resulting restricted algebra is known as the wedge subalgebra of w1+∞, and can also be rewritten as GL(∞, R) [11]. p Moreover, each wm(z) acts as a current on the celestial sphere. Hence we have a GL(∞, R) Kac-Moody algebra. The modes of the Kac-Moody currents act on the states of the 2D CCFT associated by the state-operator correspondence to operator insertions in the celestial sphere [14]. In fact precisely this Kac-Moody algebra has been studied previously both as the symmetry group 5 of c = 1 string theory [10] and 2D w1+∞-gravity [15]. A review of this and other fascinating aspects of W algebras can be found in [16]. We return to this connection in the last section. The closed subgroup generated by

2 Lm ≡ wm, m = −1, 0, 1 (2.10) is an ‘SL(2, R)w’ current algebra implied by the subleading soft graviton theorem. It is related to the original generators in (2.5) by

1 2 L = H0,L = H0 . (2.11) 0 κ 0 ±1 κ ±1

q wn transforms under this SL(2, R)w as

q q [Lm, wn] = [m(q − 1) − n] wm+n. (2.12) like the modes of an SL(2, R)w primary operator of SL(2, R)w weight q. Since q ≥ 1 here, these k are all positive. However, unlike the Hn, they do not transform canonically under the original

5 Here we take the classical limit W1+∞ →w1+∞ of the quantum symmetry W1+∞.

3 SL(2, R)R because of the mode-dependent relation between the two. Indeed the normalization of k the Hn modes was chosen so that they transform canonically under the original SL(2, R)R conformal ¯ k−2 generators with weights h = 2 . p Evidently SL(2, R)R and SL(2, R)w are not quite the the same thing. wn on the LHS of (2.6) −2p+4 lies in a positive weight hw = p representation of SL(2, R)w, while Hn on the RHS lies in a ¯ 2p + 1 dimensional, negative weight h = 1 − p representation of SL(2, R)R. (2.6) is the relation between them. Both representations are 2p + 1 dimensional, because ﬁnite weight h SL(2, R) representations are 2h − 1 dimensional for positive half integral h and −2h − 1 dimensional for negative half integral h.

Signiﬁcantly, the SL(2, R)w representations which appear here are all positive weight.

3 Gauge theory

a,+ Now we turn to nonabelian gauge theory. Let Ok (z, z¯) denote a positive helicity, conformal weight k gluon operator with adjoint group index a at the point (z, z¯) on the celestial sphere. Mode-expanding inz ¯ on the right

a,+ X Ok,n (z) Oa,+(z, z¯) = , (3.1) k n+ k−1 n z¯ 2 conformally soft currents are deﬁned by

k,a a,+ k − 1 1 − k Rn (z) := lim εO (z), k = 1, 0, −1, −2,..., ≤ n ≤ . (3.2) ε→0 k+ε,n 2 2

This has SL(2, R)L⊗SL(2, R)R weights

k + 1 k − 1 (h, h¯) = , . (3.3) 2 2

These values of conformal weights ∆ = k include all the conformally soft poles encountered in the OPE [17,18]. The factor of ε incorporated in (3.2) is needed to cancel these poles, leading to ﬁnite OPEs for the rescaled Rk,a. The soft current algebra for gauge theory is [4]

h i ( 1−k − n + 1−l − n0)! ( 1−k + n + 1−l + n0)! Rk,a,Rl,b = −if ab 2 2 2 2 Rk+l−1,c. (3.4) n n0 c 1−k 1−l 0 1−k 1−l 0 n+n0 ( 2 − n)!( 2 − n )! ( 2 + n)!( 2 + n )!

Let us deﬁne q,a 3−2q,a Sm = (q − m − 1)!(q + m − 1)!Rm , (3.5)

3 where q = 1, 2 , 2, .... One then ﬁnds the simple algebra

h q,a p,bi ab q+p−1,c Sn ,Sn0 = −if cSn+n0 . (3.6)

4 Moreover using [4] h i κ ( 2−k − m + 1−l − n − 1)! ( 2−k + m + 1−l + n − 1)! Hk ,Rl,a = − [n(2 − k) − m(1 − l)] 2 2 2 2 Rk+l,a, m n 2 2−k 1−l 2−k 1−l m+n ( 2 − m)!( 2 − n)! ( 2 + m)!( 2 + n)! (3.7) one ﬁnds the irreducible representation

p q,a p+q−2,a [wm,Sn ] = [m(q − 1) − n(p − 1)] Sm+n . (3.8)

q The SL(2, R)w transformation of Sm

q,a q,a [Lm,Sn ] = [m(q − 1) − n] Sm+n. (3.9) is that of a primary of weight q. Again we ﬁnd only positive weights.

4 Speculations

The appearance of W-algebras in the 2D celestial symmetry group connects celestial holography to several other research areas. We ﬁnd it irresistible to speculate on what might lie ahead as these connections unfold.

w1+∞ has a natural deformation to W1+∞, a signiﬁcantly more complicated algebra. This deformation can be understood as arising from quantization. In the context of speciﬁc classical 2D realizations of w1+∞, anomalies encountered in quantization deform the classical w1+∞ algebra to the quantum W1+∞ [19]. While currently not well-understood, the tree-level soft algebra of 4D quantum gravity also gets deformed on quantization, as implied among other things by one-loop corrections to the soft theorems. So it is natural to speculate that the action of w1+∞ on this soft algebra is deformed to W1+∞ in the 4D quantum theory of gravity. One might have expected a Virasoro algebra rather than an SL(2, R)w current algebra among the symmetries. However the two are closely related in many similar contexts. For example the Chern-

Simon formulation of AdS3 gravity, at first sight gives an SL(2, R)R⊗SL(2, R)L current algebra on the boundary. However, the AdS3 boundary conditions implement a Hamiltonian reduction to 6 VirR⊗VirL [20, 21]. The Virasoro generators are field-dependent SL(2, R) transformations which lie in the enveloping algebra of the SL(2, R)R⊗SL(2, R)L current algebra. In another example, 2D gravity in light cone gauge exhibits an SL(2, R)R current algebra [22], but the same theory in conformal gauge exhibits a Virasoro symmetry [23], again indicating a relation between the two symmetry actions. This type of relation has an infinite-dimensional uplift to the present context. SL(2, R) is the wedge algebra of Virasoro, just as GL(∞, R) is the wedge algebra of w1+∞ [11]. Just as Virasoro lies in the enveloping algebra of SL(2, R) Kac-Moody, w1+∞ (or W1+∞ at the quantum level) lies in the enveloping algebra of GL(∞, R) Kac-Moody [15]. So it is natural to speculate that W1+∞ 6A similar reduction could perhaps be operative here from constraints related to IR divergences.

5 is a celestial symmetry of 4D quantum gravity.

w1+∞ also appears as the symmetry group of classical self-dual gravity in (2,2) signature Klein space [24], whose sole degree of freedom is given by the Kahler potential. The self-duality suggests that the dual CCFT should be chiral, and acted on (at bulk tree level) by w1+∞. This is a natural context in which to study celestial holography, because the complicated interactions of opposite chirality sectors and double soft-limit ambiguities are absent. The N = 2 string describes the quantization of this theory [25], where w1+∞ is potentially deformed to W1+∞. Perhaps the holographic CCFT dual of N = 2 string theory in 4D Klein space is given by a 2D quantum

W1+∞-gravity theory on the celestial torus. We leave these thoughts to future explorations.

Acknowledgements

I am grateful to Ofer Aharony, Adam Ball, Alfredo Guevara, Mina Himwich, Sruthi Narayanan and Monica Pate for useful conversations. This work was supported by DOE grant de-sc/0007870.

References

[1] A. Strominger, “Asymptotic Symmetries of Yang-Mills Theory,” JHEP 07 (2014) 151, arXiv:1308.0589 [hep-th].

[2] A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 07 (2014) 152, arXiv:1312.2229 [hep-th].

[3] T. He, V. Lysov, P. Mitra, and A. Strominger, “BMS Supertranslations and Weinberg’s Soft Graviton Theorem,” JHEP 05 (2015) 151, arXiv:1401.7026 [hep-th].

[4] A. Guevara, E. Himwich, M. Pate, and A. Strominger, “Holographic Symmetry Algebras for Gauge Theory and Gravity,” arXiv:2103.03961 [hep-th].

[5] S. Banerjee, S. Ghosh, and P. Paul, “Mhv graviton scattering amplitudes and current algebra on the celestial sphere,” Journal of High Energy Physics 2021 no. 2, (Feb, 2021) . http://dx.doi.org/10.1007/JHEP02(2021)176.

[6] Z.-Z. Li, H.-H. Lin, and S.-Q. Zhang, “Inﬁnite soft theorems from gauge symmetry,” Physical Review D 98 no. 4, (Aug, 2018) . http://dx.doi.org/10.1103/PhysRevD.98.045004.

[7] Y. Hamada and G. Shiu, “Inﬁnite set of soft theorems in gauge-gravity theories as ward-takahashi identities,” Physical Review Letters 120 no. 20, (May, 2018) . http://dx.doi.org/10.1103/PhysRevLett.120.201601.

[8] S. Pasterski, A. Puhm, and E. Trevisani, “Celestial Diamonds: Conformal Multiplets in Celestial CFT,” arXiv:2105.03516 [hep-th].

6 [9] P. Kravchuk and D. Simmons-Duﬃn, “Light-ray operators in conformal ﬁeld theory,” JHEP 11 (2018) 102, arXiv:1805.00098 [hep-th].

[10] I. R. Klebanov and A. M. Polyakov, “Interaction of discrete states in two-dimensional string theory,” Mod. Phys. Lett. A 6 (1991) 3273–3281, arXiv:hep-th/9109032.

[11] C. N. Pope, L. J. Romans, and X. Shen, “W (inﬁnity) and the Racah-wigner Algebra,” Nucl. Phys. B 339 (1990) 191–221.

[12] A. Atanasov, A. Ball, W. Melton, A.-M. Raclariu, and A. Strominger, “(2, 2) Scattering and the Celestial Torus,” arXiv:2101.09591 [hep-th].

[13] I. Bakas, “The Large n Limit of Extended Conformal Symmetries,” Phys. Lett. B 228 (1989) 57.

[14] E. Crawley, N. Miller, S. A. Narayanan, and A. Strominger, “State-Operator Correspondence in Celestial Conformal Field Theory,” arXiv:2105.00331 [hep-th].

[15] C. N. Pope, X. Shen, K. W. Xu, and K.-j. Yuan, “SL(inﬁnity,R) symmetry of quantum W(inﬁnity) gravity,” Nucl. Phys. B 376 (1992) 52–74.

[16] C. N. Pope, “Lectures on W algebras and W gravity,” in Summer School in High-energy Physics and Cosmology. 12, 1991. arXiv:hep-th/9112076.

[17] W. Fan, A. Fotopoulos, and T. R. Taylor, “Soft Limits of Yang-Mills Amplitudes and Conformal Correlators,” JHEP 05 (2019) 121, arXiv:1903.01676 [hep-th].

[18] M. Pate, A.-M. Raclariu, A. Strominger, and E. Y. Yuan, “Celestial Operator Products of Gluons and Gravitons,” arXiv:1910.07424 [hep-th].

[19] E. Bergshoeff, P. S. Howe, C. N. Pope, E. Sezgin, X. Shen, and K. S. Stelle, “Quantization deforms w(infinity) to W(infinity) gravity,” Nucl. Phys. B 363 (1991) 163–184.

[20] M. Banados, “Global charges in Chern-Simons ﬁeld theory and the (2+1) black hole,” Phys. Rev. D 52 (1996) 5816–5825, arXiv:hep-th/9405171.

[21] O. Coussaert, M. Henneaux, and P. van Driel, “The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant,” Class. Quant. Grav. 12 (1995) 2961–2966, arXiv:gr-qc/9506019.

[22] A. M. Polyakov, “Quantum Gravity in Two-Dimensions,” Mod. Phys. Lett. A 2 (1987) 893.

[23] F. David, “Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge,” Mod. Phys. Lett. A 3 (1988) 1651.

[24] C. P. Boyer and J. F. Plebanski, “An inﬁnite hierarchy of conservation laws and nonlinear superposition principles for selfdual Einstein spaces,” J. Math. Phys. 26 (1985) 229–234.

7 [25] H. Ooguri and C. Vafa, “Geometry of N=2 strings,” Nucl. Phys. B 361 (1991) 469–518.