On the classification of UV completions of integrable $$ T\overline{T} $$ deformations of CFT
Changrim Ahn, André LeClair
Abstract
A bstract It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant $$ T\overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> perturbation of dimension 4 has universal properties. In particular, for the most interesting cases, the theory develops a singularity in the ultra-violet (UV), signifying a shortest possible distance, with a Hagedorn transition in applications to string theory. We show that by adding an infinite number of higher [ $$ T\overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> ] s> 1 irrelevant operators of positive integer scaling dimension 2( s +1) with tuned couplings, this singularity can be resolved and the theory becomes UV complete with a Virasoro central charge c UV > c IR consistent with the c-theorem. We propose an approach to classifying the possible UV completions of a given CFT perturbed by [ $$ T\overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> ] s that are integrable. The main tool utilized is the thermodynamic Bethe ansatz. We study this classification for theories with scalar (diagonal) factorizable S-matrices. For the Ising model with c IR = $$ \frac{1}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> we find 3 UV completions based on a single massless Majorana fermion description with c UV = $$ \frac{7}{10} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>7</mml:mn> <mml:mn>10</mml:mn> </mml:mfrac> </mml:math> and $$ \frac{3}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> , which both have $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1 SUSY and were previously known, and we argue that these are the only solutions to our classification problem based on this spectrum of particles. We find 3 additional ones with a spectrum of 8 massless particles related to the Lie group E 8 appropriate to a magnetic perturbation with c UV = $$ \frac{21}{22},\frac{15}{12} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>21</mml:mn> <mml:mn>22</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mn>15</mml:mn> <mml:mn>12</mml:mn> </mml:mfrac> </mml:math> , and $$ \frac{31}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>31</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> . We argue that it is likely there are more cases for this E 8 spectrum. We also study simpler cases based on su(3) and su(4) where we can propose complete classifications. For su(3) the infrared (IR) theory is the 3-state Potts model with c IR = $$ \frac{4}{5} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>5</mml:mn> </mml:mfrac> </mml:math> and we find 3 completions with $$ \frac{4}{5} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>5</mml:mn> </mml:mfrac> </mml:math> < c UV ≤ $$ \frac{16}{5} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>16</mml:mn> <mml:mn>5</mml:mn> </mml:mfrac> </mml:math> . For the su(4) case, which has 3 particles and c IR = 1, and we find 11 UV completions with 1 < c UV ≤ 5, most of which were previously unknown.