w1+∞ and Carrollian holography
Amartya Saha
Abstract
A bstract In a 1 + 2D Carrollian conformal field theory, the Ward identities of the two local fields $$ {S}_0^{+} $$ <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> and $$ {S}_1^{+} $$ <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> , entirely built out of the Carrollian conformal stress-tensor, contain respectively up to the leading and the subleading positive helicity soft graviton theorems in the 1 + 3D asymptotically flat space-time. This work investigates how the subsubleading soft graviton theorem can be encoded into the Ward identity of a Carrollian conformal field $$ {S}_2^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> . The operator product expansion (OPE) $$ {S}_2^{+}{S}_2^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> is constructed using general Carrollian conformal symmetry principles and the OPE commutativity property, under the assumption that any time-independent, non-Identity field that is mutually local with $$ {S}_0^{+} $$ <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> , $$ {S}_2^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> has positive Carrollian scaling dimension. It is found that, for this OPE to be consistent, another local field $$ {S}_3^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> must automatically exist in the theory. The presence of an infinite tower of local fields $$ {S}_{k\ge 3}^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> is then revealed iteratively as a consistency condition for the $$ {S}_2^{+}{S}_{k-1}^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> </mml:msubsup> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> OPE. The general $$ {S}_k^{+}{S}_l^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>l</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> OPE is similarly obtained and the symmetry algebra manifest in this OPE is found to be the Kac-Moody algebra of the wedge sub-algebra of w 1+ ∞ . The Carrollian time-coordinate plays the central role in this purely holographic construction. The 2D Celestial conformally soft graviton primary $$ {H}^k\left(z,\overline{z}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mfenced> <mml:mi>z</mml:mi> <mml:mover> <mml:mi>z</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mfenced> </mml:math> is realized to be contained in the Carrollian conformal primary $$ {S}_{1-k}^{+}\left(t,z,\overline{z}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mfenced> <mml:mi>t</mml:mi> <mml:mi>z</mml:mi> <mml:mover> <mml:mi>z</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mfenced> </mml:math> . Finally, the existence of the infinite tower of fields $$ {S}_k^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> is shown to be directly related to an infinity of positive helicity soft graviton theorems.