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

TT-deformed classical and quantum field theories

Tateo Roberto Trieste, October 2018

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

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;

Early works

• 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]

More recent references

Main motivations

The effective string theory for the quark-antiquark potential; Emergence of singularities in RG/TBA flows with irrelevant perturbations; Relation between irrelevant perturbations and S-matrix CDD (scalar phase factor) ambiguity;

CFTUV CFTIR CFTIR

?

[Dubovsky-Flauger-Gorbenko (2012) , Caselle-Gliozzi-RT (2013)] [Zamolodchikov (1991), Mussardo-Simon (2000), Smirnov-Zamolodchikov (2016),...]

Exact S-matrix and CDD ambiguity

Consider a relativistic integrable field theory with factorised scattering: The simplest possibility, consistent with the crossing and unitarity relations is: Castillejo-Dalitz-Dyson ambiguity:

The sine-Gordon NLIE

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 [1991: Klủmper-Batchelor-Pearce; Destri-DeVega]

replacing we get with Therefore:

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: and therefore with

R E(R)

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

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

An extra CDD factor couples left (-) with right (+) movers scattering, any NLIE or TBA equation leads to a pair of coupled algebraic equations: ceff= c – 24 Δ(primary), obtained by an energy-dependent shift:

which matches the form of the (D=26, ceff=24) Nambu Goto spectrum, for generic CFT, with τ=1/(2 ) s , where s is the string tension.

The CFT case

The total energy:

Identification of the perturbing operator

Start from the equation: and use the standard relations then

with since

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

Also:

[Bonelli-Doroud-Zhu, Conti- Negro-Iannella-RT (2018)]

The sine-Gordon model

with and EoM (Lax consistency equation) Deformed Conserved charges (expansion in the spectral parameter λ)

[2018: Conti-Negro-Iannella-RT]

Pseudo-spherical surfaces in R3

Beltrami-pseudosphere Gauss curvature: Mean curvature: Where are the principal curvatures. We found: (λ : spectral parameter)

A local change of coordinates

 Jackiw–Teitelboim gravity! [2017: Dubovsky-Gorbenko-Mirbabayi]

Where:

Deformed breather

Start with the breather solution with envelope speed v = 0

The kink

Deformed 2-kink solution

Generic TT-deformed models

In Minkowski or Euclidean coordinates:

Further gravity-type models

with Consider a model with further conserved currents : The corresponding conserved charges are: 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]