Infinite pseudo-conformal symmetries of classical $T \bar T$, $J \bar T $ and $J T_a$ - deformed CFTs
Monica Guica, Ruben Monten
Abstract
We show that T\bar{T}, J\bar{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>J</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> </mml:mrow> </mml:math> and JT_a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>J</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>a</mml:mi> </mml:msub> </mml:mrow> </mml:math> - deformed classical CFTs posses an infinite set of symmetries that take the form of a field-dependent generalization of two-dimensional conformal transformations. If, in addition, the seed CFTs possess an affine U(1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> symmetry, we show that it also survives in the deformed theories, again in a field-dependent form. These symmetries can be understood as the infinitely-extended conformal and U(1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> symmetries of the underlying two-dimensional CFT, seen through the prism of the ``dynamical coordinates’’ that characterise each of these deformations. We also compute the Poisson bracket algebra of the associated conserved charges, using the Hamiltonian formalism. In the case of the J\bar{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>J</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> </mml:mrow> </mml:math> and JT_{a} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>J</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>a</mml:mi> </mml:msub> </mml:mrow> </mml:math> deformations, we find two copies of a functional Witt - Kac-Moody algebra. In the case of the T\bar{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> </mml:mrow> </mml:math> deformation, we show that it is also possible to obtain two commuting copies of the Witt algebra.