(Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes

Published for SISSA by Springer Received: July 11, 2022 Accepted: September 18, 2022 Published: September 28, 2022 Shamik Banerjee,a,b Sudip Ghoshc and Partha Pauld a Institute of Physics, Sachivalaya Marg, Bhubaneshwar 751005, India b Homi Bhabha National Institute, Anushakti Nagar, Mumbai 400085, India c Okinawa Institute of Science and Technology, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan d Chennai Mathematical Institute, SIPCOT IT Park, Siruseri 603103, India E-mail: , , Abstract: In this paper we continue our study of the tree level MHV graviton scattering amplitudes from the point of view of celestial holography. In arXiv:2008.04330 we showed that the celestial OPE of two gravitons in the MHV sector can be written as a linear combination of SL(2, C) current algebra and supertranslation descendants. In this note we show that the OPE is in fact manifestly invariant under the infinite dimensional Virasoro algebra as is expected for a 2-D CFT. This is consistent with the conjecture that the holographic dual in 4-D asymptotically flat space time is a 2-D CFT. Since we get only one copy of the Virasoro algebra we can conclude that the holographic dual theory which computes the MHV amplitudes is a chiral CFT with a host of other infinite dimensional global symmetries including SL(2, C) current algebra, supertranslations and subsubleading soft graviton symmetry. We also discuss some puzzles related to the appearance of the Virasoro symmetry. Keywords: Conformal and W Symmetry, Scale and Conformal Symmetries, Scattering Amplitudes ArXiv ePrint: 2108.04262 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP09(2022)236 JHEP09(2022)236 (Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes Contents 1 2 2 2 2 2 Virasoro invariance of the OPE in the MHV sector 2.1 L2 invariance 2.2 Reorganising the OPE in terms of Virasoro primaries and descendants 3 3 4 3 Discussion 6 A Virasoro invariance of positive helicity OPE: further checks A.1 O(z 2 ) terms in positive helicity OPE A.2 L2 invariance of O(z 2 ) terms in positive helicity OPE A.3 Reorganising OPE in terms of Virasoro primaries and descendants 7 7 10 10 B Virasoro invariance of mixed helicity OPE B.1 O(z z̄) term in mixed helicity OPE using symmetries B.2 L2 invariance of O(z z̄) terms in mixed helicity OPE B.3 Reorganising OPE in terms of Virasoro primaries and descendants 11 12 13 13 1 Introduction and results In four dimensional asymptotically flat space time the asymptotic symmetries are infinite dimensional [1–11, 13–21] and so in a holographic description the dual theory must have infinite dimensional global symmetry. It has been conjectured [6–10, 21] that the global symmetry group of the dual theory contains the local conformal or Virasoro symmetry. As a result the dual theory is a two dimensional CFT. In this note we study this conjecture in the context of celestial holography [22–63]. We revisit the global symmetries of the tree level MHV graviton scattering amplitudes calculated in General Relativity. In [13] we studied the celestial MHV amplitudes in great detail and showed that the celestial OPE between two graviton primaries in the MHV sector can be written as a linear combination of SL(2, C) current algebra, supetranslations –1– JHEP09(2022)236 1 Introduction and results 1.1 Virasoro algebra 1.2 SL(2, C) current algebra 1.3 Supertranslation 1.4 Mixed commutators and global SL(2, C) descendants. Moreover, due to the existence of the null states [13],1 h i 1 Ψ∆ (z, z̄) = J−1 P−1,−1 − (∆ − 1)P−2,0 G+ ∆ (z, z̄) = 0 (1.1) and h i 0 Φ∆ (z, z̄) = L−1 P−1,−1 + 2J−1 P−1,−1 − (∆ + 1)P−2,−1 − L̄−1 P−2,0 G+ ∆ (z, z̄) = 0 (1.2) 1.1 Virasoro algebra c n(n2 − 1)δm+n,0 m, n ∈ Z . (1.3) 12 Although we have written down the central charge term in the Virasoro algebra, our demonstration of Virasoro invariance of the celestial OPE does not allow us to determine the value of the central charge c. So we leave the determination of c to future work. [Lm , Ln ] = (m − n)Lm+n + 1.2 SL(2, C ) current algebra h i a+b a Jm , Jnb = (a − b)Jm+n , J01 = L̄1 , 1.3 m, n ∈ Z (1.4) J00 = L̄0 , J0−1 = L̄−1 . (1.5)  n, n0 = 0, −1 . (1.6) Supertranslation  1.4 a, b = 0, ±1, Pm,n , Pm0 ,n0 = 0, Mixed commutators 1 0 [Jm , Pn,−1 ] = Pm+n,−1 , 2 h i h i 1 1 0 Jm , Pn,0 = 0, Jm , Pn,0 = − Pm+n,0 , 2 a a [Lm , Jn ] = −nJm+n , m ∈ Z, n ∈ Z 1 [Jm , Pn,−1 ] = Pm+n,0 ,  [Ln , Pa,b ] = n−1 − a Pa+n,b , 2 −1 [Jm , Pn,−1 ] = 0 −1 [Jm , Pn,0 ] = −Pm+n,−1 (1.7) (1.8) (1.9)  n ∈ Z, a ∈ Z, b = 0, −1 . (1.10) The generators of the SL(2, C) current algebra are denoted by Jna where a = 0, ±1 and n ∈ Z. Similarly we denote the generators of the supertranslations coming from the positive helicity soft graviton by Pn,b where n ∈ Z and b = 0, −1. For further details on the symmetry algebra and null states we refer the reader to [13]. 2 See [71] for a potential derivation of this symmetry algebra from the bulk gravity point of view. 1 –2– JHEP09(2022)236 one can write down differential equations which can be solved to determine the MHV graviton scattering amplitudes. Therefore the MHV amplitudes are completely determined by the SL(2, C) current algebra, supetranslations and the global SL(2, C) symmetries. In this paper we point out that the celestial OPE of two graviotns in the MHV sector is actually invariant under the infinite dimensional Viarasoro algebra whose global part is the SL(2, C) algebra. This shows that the symmetries of the MHV graviton scattering amplitudes is a semi-direct product of the Virasoro algebra, SL(2, C) current algebra and supertranslations.2 Since we get only one copy of the Virasoro algebra, the holographic dual theory which computes the MHV graviton scattering amplitudes is a chiral (celestial) CFT. For the sake of completeness let us now write down the symmetry algebra. The symmetry algebra that we have shown can be further extended by including the subsubleading soft graviton symmetry [14–16] and an infinite number of other soft symmetries [14, 16] which appear when the scaling dimension of a (positive helicity) graviton primary assumes negative integer values. 2 Virasoro invariance of the OPE in the MHV sector Consider the celestial OPE between two positive helicity outgoing gravitons in the MHV sector. This is given by [13], MHV h   z̄ 0 = B(∆1 − 1, ∆2 − 1) −P−1,−1 + z c1 J−1 P−1,−1 + c2 P−2,−1 z   0 +z 2 c3 J−2 P−1,−1 + c4 P−3,−1 + c5 2L−1 P−2,−1 − 2L̄−1 P−3,0 + 2L−1 L̄−1 P−2,0 − L2−1 P−1,−1  i +O(z 3 ) G+ ∆1 +∆2 −1 (0) + · · · (2.1) The descendant OPE coefficients denoted above by ci are given by c1 = 2(∆1 − 1) , ∆1 + ∆ 2 − 2 ∆1 ∆2 c4 = − , ∆1 + ∆ 2 − 1 c2 = −∆1 , c3 = 2(∆1 − 1)(∆2 − 1) , (∆1 + ∆2 − 2)(∆1 + ∆2 − 1) ∆1 (∆1 − 1) c5 = . 2(∆1 + ∆2 − 2)(∆1 + ∆2 − 1) (2.2) In (2.1) we have, for simplicity, kept only terms of the O(z̄z n ) where n ≥ −1 because under the actio (...truncated)