Carrollian approach to 1 + 3D flat holography
Amartya Saha
Abstract
A bstract The isomorphism between the (extended) BMS 4 algebra and the 1 + 2D Carrollian conformal algebra hints towards a co-dimension one formalism of flat holography with the field theory residing on the null-boundary of the asymptotically flat space-time enjoying a 1 + 2D Carrollian conformal symmetry. Motivated by this fact, we study the general symmetry properties of a source-less 1 + 2D Carrollian CFT, adopting a purely field-theoretic approach. After deriving the position-space Ward identities, we show how the 1 + 3D bulk super-translation and the super-rotation memory effects emerge from them, manifested by the presence of a temporal step-function factor in the same. Temporal-Fourier transforming these memory effect equations, we directly reach the bulk null-momentum-space leading and sub-leading soft graviton theorems. Along the way, we construct six Carrollian fields $$ {S}_0^{\pm } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>0</mml:mn> <mml:mo>±</mml:mo> </mml:msubsup> </mml:math> , $$ {S}_1^{\pm } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> <mml:mo>±</mml:mo> </mml:msubsup> </mml:math> , T and $$ \overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> corresponding to these soft graviton fields and the Celestial stress-tensors, purely in terms of the Carrollian stress-tensor components. The 2D Celestial shadow-relations and the null-state conditions arise as two natural byproducts of these constructions. We then show that those six fields consist of the modes that implement the super-rotations and a subset of the super-translations on the quantum fields. The temporal step-function allows us to relate the operator product expansions (OPEs) with the operator commutation relations via a complex contour integral prescription. We deduce that not all of those six fields can be taken together to form consistent OPEs. So choosing $$ {S}_0^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> , $$ {S}_1^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> and T as the local fields, we form their mutual OPEs using only the OPE-commutativity property, under two general assumptions. The symmetry algebra manifest in these holomorphic-sector OPEs is then shown to be Vir $$ \overset{\wedge }{\ltimes \overline{\textrm{sl}\left(2,{\mathbb{R}}\right)}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mrow> <mml:mo>⋉</mml:mo> <mml:mover> <mml:mrow> <mml:mi>sl</mml:mi> <mml:mfenced> <mml:mn>2</mml:mn> <mml:mi>ℝ</mml:mi> </mml:mfenced> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mo>∧</mml:mo> </mml:mover> </mml:math> with an abelian ideal.