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Infinite pseudo-conformal symmetries of classical $T \bar T$, $J \bar T $ and $J T_a$ - deformed CFTs

by Monica Guica, Ruben Monten

This is not the current version.

Submission summary

As Contributors: Ruben Monten
Arxiv Link: (pdf)
Date submitted: 2020-12-10 03:17
Submitted by: Monten, Ruben
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We show that $T \bar T, J \bar T$ and $J T_a$ - deformed classical CFTs possess 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 $U(1)$ 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 $U(1)$ 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 $J \bar T$ and $J T_a$ deformations, we find two copies of a functional Witt - Kač-Moody algebra. In the case of the $T \bar T$ deformation, we show that it is also possible to obtain two commuting copies of the Witt algebra.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2021-1-18 (Invited Report)


This paper studies an infinite set of underlying symmetries in $T\bar{T}$, $J\bar{T}$, and $JT_a$ deformed conformal field theories (CFTs) in two-dimensions. In particular, the authors showed that the symmetries at the classical level are two commuting copies of a (functional) Wit algebra for the $T\bar{T}$-deformed CFTs and a (functional) Wit and a $U(1)$ Kac-Moody algebras for the $J\bar{T}$ and $JT_a$-deformed CFTs. In the case of deformed CFTs with large central charges, there exist gravity duals and these results corroborate the infinite symmetries previously found by the authors and a collaborator in the holographic setting.

As the authors emphasize, to some extent, the results are expected, without holography, from an alternative interpretation of the deformed theories; namely, a deformed theory can be thought of as the undeformed theory living on a field-dependent (or operator or state-dependent) deformed space. Their results match in detail with this expectation.

Technically, in order to discuss the precise algebras of these symmetries, the authors used and developed the Hamiltonian formalism of these deformed theories and computed the Poisson brakets of the symmetry generators (and the field-dependent coordinates). Not only do these analyses show useful detailed structures of the algebras, but they also reveal a subtle issue in the case of the $T\bar{T}$-deformed CFTs on the cylinder due to the field-dependent radius. The issue remains in the paper, but it gives useful data for the future and further studies.

I am slightly confused about the appearance of the Kac-Moody level in their classical analyses. If I am not mistaken, the Kac-Moody level is quantum in nature and furthermore, the central charge of the Virasoro algebra is related to the Kac-Moody level via Sugawara construction. I would then expect that especially in the free boson example, one would be able to find the Virasoro, rather than Wit, algebra by considering the Sugawara form of the Virasoro generators. This might well be simply my confusion and ignorance, but it would be better if the authors can comment on this point.

The paper is thorough and clearly written, and the authors made it easy to read. It clearly deserves a publication in SciPost and I thus recommend the paper for publication (hopefully with the above point on the KM level and the central charge being addressed).

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Author:  Ruben Monten  on 2021-08-13  [id 1667]

(in reply to Report 2 on 2021-01-18)
answer to question

We would like to thank the referee for their comments.

Concerning the level of the Kac–Moody algebra and its relation to the Virasoro central charge through the Sugawara construction, the following argument readily shows that there is a nonzero level already for classical bosonic fields, using the (undeformed) classical free boson as an example. Here, we can use the Poisson bracket algebra to find the classical analog of the Kac–Moody level. Using the classical analogue of the Sugawara construction, one can use this to recover the Witt terms in the stress tensor algebra.

More explicitly, we can use the free boson's chiral current $J^+ = \partial_U \phi \, d U$ together with the undeformed Poisson bracket $\{ \phi(x_1), \dot{\phi}(x_2) \} = \delta(x_1 - x_2)$ to find the Kac–Moody Poisson bracket algebra
\{ J^+(x_1), J^+(x_2) \} &= \frac12 \delta'(x_1 - x_2)
\ .
Expanding the current in modes $J^+(U) = \sum_n e^{i n U} J_n^+$, we find
\{ J_m^+, J_n^+ \}
= \frac1{4\pi^2} \int dx_1 dx_2 e^{-i(m U_1 + n U_2)} \frac12 \delta'(x_1 - x_2)
= i \pi m \delta_{m+n}
\ ,
which shows that the Kac–Moody algebra has a nonzero (although classically unquantized) level.

The left-moving stress tensor component $T \equiv T_{UU} = (\partial_U \phi)^2 = (J^+)^2$ is indeed “pure Sugawara”. Its Poisson bracket is
\{ T(x_1), T(x_2) \} = 2 J^+(x_1) J^+(x_2) \delta'(x_1 - x_2)
\ ,
or in terms of the modes $T(U) = \sum_n e^{i n U} T_n$ we find
\{ T_m, T_n \}
&= \frac1{2\pi^2} \int d x \, e^{-i (m + n) U} \left( i m [J^+(x)]^2 - J^+(x) J^+{}'(x) \right)
&= \frac{i}{2\pi} (m - n) T_{m+n}
\ .
We thus find the Witt part of the Virasoro algebra. Note however that one cannot recover the central term in the Virasoro algebra, as this is related to the normal-ordering inherent in the quantum theory.

Anonymous Report 1 on 2020-12-30 (Invited Report)


This paper continues the investigations concerning properties
of families of solvable, non-local, irrelevant deformations of two-dimensional CFTs -- $T\bar T$, $J\bar T$ and $JT_a$ --
which have, in particular, applications in various fields,
such as string theory, holography, black holes and QCD
-- concretely, their infinite symmetries, which are in one to one correspondence with the initial conformal and the affine $U(1)$ symmetries of the undeformed CFT.
It is an interesting work, which not only adds new ingredients to known facts, but also raises questions for future research,
and it is thus suitable for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Ruben Monten  on 2021-08-13  [id 1668]

(in reply to Report 1 on 2020-12-30)

We would like to thank the referee for their comments.

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