Celestial sector in CFT: Conformally soft symmetries

Abstract

We show that time intervals of width \Delta \tau <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>τ</mml:mi> </mml:mrow> </mml:math> in 3-dimensional conformal field theories (CFT _3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mn>3</mml:mn> </mml:msub> </mml:math> ) on the Lorentzian cylinder admit an infinite dimensional symmetry enhancement in the limit \Delta \tau → 0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>τ</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . The associated vector fields are approximate solutions to the conformal Killing equations in the strip labelled by a function and a conformal Killing vector on the sphere. An Inonu-Wigner contraction yields a set of symmetry generators obeying the extended BMS _4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mn>4</mml:mn> </mml:msub> </mml:math> algebra. We analyze the shadow stress tensor Ward identities in CFT _d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mi>d</mml:mi> </mml:msub> </mml:math> on the Lorentzian cylinder with all operator insertions in infinitesimal time intervals separated by \pi <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>π</mml:mi> </mml:math> . We demonstrate that both the leading and subleading conformally soft graviton theorems in (d-1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:math> -dimensional celestial CFT (CCFT _{d-1} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> ) can be recovered from the transverse traceless components of these Ward identities in the limit \Delta \tau → 0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>τ</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . A similar construction allows for the leading conformally soft gluon theorem in CCFT _{d-1} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> to be recovered from shadow current Ward identities in CFT _d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mi>d</mml:mi> </mml:msub> </mml:math> .

Topics & Concepts