TT-deformed classical and quantum field theories Trieste, October - PowerPoint PPT Presentation

TT-deformed classical and quantum field theories Trieste, October 2018 Tateo Roberto Based on: M. Caselle, D. Fioravanti, F. Gliozzi, RT, JHEP 1307 (2013) 071 A. Cavaglià, S. Negro, I. Szécsényi, RT, JHEP 1610 (2016) 112 R. Conti, L. Iannella, S. Negro, RT, JHEP11(2018)007

Early works A.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz , Nucl.Phys. B358 (1991) 524; V. Fateev, S.L. Lukyanov, A. B. Zamolodchikov, Al. B. Zamolodchikov, Expectation values of local fields in Bullough-Dodd model and integrable perturbed conformal field theories , Nucl.Phys. B516 (1998) 652; G. Mussardo, P. Simon, Bosonic type S matrix, vacuum instability and CDD ambiguities , Nucl.Phys. B578 (2000) 527; A.B. Zamolodchikov, Expectation value of composite field TT in two- dimensional quantum field theory , [hep-th/0401146]; G. Delfino, G. Niccoli, The composite operator TT in sinh-Gordon and a series of massive minimal models , JHEP 0605 (2006) 035;

More recent references S. Dubovsky, R. Flauger and V. Gorbenko, Solving the simplest theory of quantum gravity , JHEP 1209 (2012) 133; F. A. Smirnov and A. B. Zamolodchikov, On space of integrable quantum field theories, Nucl.Phys. B915 (2017); L. McGough, M. Mezei, H. Verlinde, Moving the CFT into the bulk with TT, JHEP 1804 (2018) 010; O. Aharony, S. Datta, A. Giveon, Y. Jiang, D. Kutasov, Modular invariance and uniqueness of TT deformed CFT, [arXiv:1808.02492]; A. Dei, A. Sfondrini, Integrable spin chain for stringy Wess-Zumino-Witten models, JHEP 1807 (2018) 109; G. Bonelli, N. Doroud, M. Zhu, -d T eformations in closed form, JHEP 1807 (2018) 109; J. Cardy, The TT deformation of quantum field theory as random geometry, [arXiv:1801.06895]

Main motivations The effective string theory for the quark-antiquark potential; [Dubovsky-Flauger-Gorbenko (2012) , Caselle-Gliozzi-RT (2013)] Emergence of singularities in RG/TBA flows with irrelevant perturbations; CFT UV ? CFT IR CFT IR Relation between irrelevant perturbations and S-matrix CDD (scalar phase factor) ambiguity; [Zamolodchikov (1991), Mussardo-Simon (2000), Smirnov-Zamolodchikov (2016),...]

Exact S-matrix and CDD ambiguity Consider a relativistic integrable field theory with factorised scattering: C astillejo- D alitz- D yson ambiguity: The simplest possibility, consistent with the crossing and unitarity relations is:

The sine-Gordon NLIE [1991: Klủmper-Batchelor-Pearce; Destri-DeVega] The finite-size properties of the sine-Gordon model are encoded in the single counting function f(θ), solution to the following nonlinear integral equation: For the ground state and , but more more complicated contours appear for excited states. and

replacing we get with Therefore:

and Then which allows to compute the exact form of the τ-deformed energy level once its R-dependence is known at τ = 0. The result is: therefore with

We now have an implicit form of the solution of the inviscid Burgers equation with a source term: E(R) (Typical τ=0 finite-volume spectrum) R where, c eff = c – 24Δ is the “effective central charge” of the UV CFT state.

Real part of E(R,τ) for τ = 0 (dashed line) and τ = 0.025 (solid line), for c eff = 1 Real part of E(R,τ) for τ = 0 (dashed line) and τ = 0.025 (solid line), for c eff = −1

The CFT case An extra CDD factor couples left (-) with right (+) movers scattering, any NLIE or TBA equation leads to a pair of coupled algebraic equations: c eff = c – 24 Δ(primary), obtained by an energy-dependent shift: The total energy: which matches the form of the (D=26, c eff =24) Nambu Goto spectrum, for generic CFT, with τ=1/(2 ) s , where s is the string tension.

Identification of the perturbing operator Start from the equation: and use the standard relations since then with

Zamolodchikov's composite operator fulfils the following factorization property: Putting all this information together: Therefore, up to total derivatives:

Classical Lagrangians Starting from: the deformed Lagrangian coincides with the bosonic Born-Infeld model or, equivalently, the Nambu-Goto Lagrangian in the static gauge:

Boson field theories with generic potential [Bonelli-Doroud-Zhu, Conti- Negro-Iannella-RT (2018)] Also:

The sine-Gordon model with and EoM (Lax consistency equation) Deformed Conserved charges [2018: Conti-Negro-Iannella-RT] (expansion in the spectral parameter λ )

Pseudo-spherical surfaces in R 3 Gauss curvature: Mean curvature: Where are the principal curvatures. We found: Beltrami-pseudosphere (λ : spectral parameter)

A local change of coordinates Where:  Jackiw–Teitelboim gravity! [2017: Dubovsky-Gorbenko-Mirbabayi]

The kink Deformed breather Start with the breather solution with envelope speed v = 0

Deformed 2-kink solution

Generic TT-deformed models In Minkowski or Euclidean coordinates:

Further gravity-type models Consider a model with further conserved currents : The corresponding conserved charges are: with where

? Similar to M. Guica model, [arXiv:1710.08415]!

Conclusions  Adding CDD factors generates interesting, often UV incomplete, QFTs.  The research on this topic may clarify important aspects concerning the appearance of singularities in effective QFT (Landau pole and tachyon singularity).  At classical level it corresponds to a sort of topological gravity theory.  Studies of the torus partition function: uniqueness and relation with Jackiw–Teitelboim gravity! [Dubovsky, Gorbenko, Mirbabayi, H-Chifflet, Cardy, Aharony, Datta, Giveon, Jiang, Kutasov,...]  Connection with AdS(3)/CFT(2) [McGough, Mezei, H. Verlinde,...].  Connection to random-geometry [Cardy].  SUSY models! [Baggio,Sfondrini, Tartaglino-Mazzucchelli, Walsh]  Generalization to D>2 dimensions? [Cardy,M.Taylor,Bonelli,Doroud, Zhu, us]