Abstract
We study the random geometry approach to the Toverline{T} deformation of 2d conformal field theory developed by Cardy and discuss its realization in a gravity dual. In this representation, the gravity dual of the Toverline{T} deformation becomes a straightforward translation of the field theory language. Namely, the dual geometry is an ensemble of AdS3 spaces or BTZ black holes, without a finite cutoff, but instead with randomly fluctuating boundary diffeomorphisms. This reflects an increase in degrees of freedom in the renormalization group flow to the UV by the irrelevant Toverline{T} operator. We streamline the method of computation and calculate the energy spectrum and the thermal free energy in a manner that can be directly translated into the gravity dual language. We further generalize this approach to correlation functions and reproduce the all-order result with universal logarithmic corrections computed by Cardy in a different method. In contrast to earlier proposals, this version of the gravity dual of the Toverline{T} deformation works not only for the energy spectrum and the thermal free energy but also for correlation functions.
Highlights
Deformation by an oft-uncontrollable irrelevant operator, the T T-deformed theories are remarkably tractable and marginally well-behaved.1 To study further and better understand the T T-deformed theories, it would be desirable to demystify these somewhat surprising features of the T Tdeformation and make the underlying simplicity manifest
We study the random geometry approach to the T Tdeformation of 2d conformal field theory developed by Cardy and discuss its realization in a gravity dual
The dual geometry is an ensemble of AdS3 spaces or BTZ black holes, without a finite cutoff, but instead with randomly fluctuating boundary diffeomorphisms
Summary
An illuminating view of the T Tdeformation was suggested and developed by Cardy [9], refining his own earlier idea [25].3 This is the key perspective as well as the main technical tool we adopt in this work. An illuminating view of the T Tdeformation was suggested and developed by Cardy [9], refining his own earlier idea [25].3 This is the key perspective as well as the main technical tool we adopt in this work. An obvious but important fact is that the stress tensor is conserved in the absence of local operator singularities:. An obvious but important fact is that the stress tensor is conserved in the absence of local operator singularities:4 These can be solved by hij = ∂iαj + ∂j αi with ij ∂iαj = 0. In the absence of local operator singularities, there can be effects of the T Tdeformation only if the 2d manifold has boundaries or nontrivial cycles. As commented in footnote 4, when local operators are inserted, there are effects of the T Tdeformation from singularities even if the manifold is a topologically trivial infinite plane R2. (Or, alternatively, we can say that, upon removing points of insertions, R2 has gained nontrivial topology.)
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