Fix the dual geometries of $$T\bar{T}$$ deformed CFT$$_2$$ and highly excited states of CFT$$_2$$
Peng Wang, Houwen Wu, Haitang Yang
Abstract
Abstract In previous works, we have developed an approach to fix the leading behaviors of the pure AdS $$_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow/><mml:mn>3</mml:mn></mml:msub></mml:math> and BTZ black hole from the entanglement entropies of the free CFT $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow/><mml:mn>2</mml:mn></mml:msub></mml:math> and finite temperature CFT $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow/><mml:mn>2</mml:mn></mml:msub></mml:math> , respectively. We exclusively use holographic principle only and make no restriction about the bulk geometry, not only the kinematics but also the dynamics. In order to verify the universality and correctness of our method, in this paper, we apply it to the $$T\bar{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mover><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:math> deformed CFT $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow/><mml:mn>2</mml:mn></mml:msub></mml:math> , which breaks the conformal symmetry. In terms of the physical arguments of the $$T\bar{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mover><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:math> deformed CFT $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow/><mml:mn>2</mml:mn></mml:msub></mml:math> , the derived metric is a deformed BTZ black hole. The requirement that the CFT $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow/><mml:mn>2</mml:mn></mml:msub></mml:math> lives on a conformally flat boundary leads to $$r_{c}^{2}=\ 6R_{AdS}^{4}/(\pi c\mu )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>r</mml:mi><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mspace/><mml:mn>6</mml:mn><mml:msubsup><mml:mi>R</mml:mi><mml:mrow><mml:mi>AdS</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msubsup><mml:mo>/</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>π</mml:mi><mml:mi>c</mml:mi><mml:mi>μ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> naturally, in perfect agreement with previous conjectures in literature. The energy spectum and propagation speed calculated with this deformed BTZ metric are the same as these derived from $$T\bar{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>T</mml:mi><mml:mover><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:math> deformed CFT $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow/><mml:mn>2</mml:mn></mml:msub></mml:math> . We furthermore fix the dual geometry of highly excited states with our approach. The result contains the descriptions for the conical defect and BTZ black hole.