Prepared for submission to JHEP

Conserved currents and TTs irrelevant deformations of

2D integrable field theories

Riccardo Conti,1 Stefano Negro,2 Roberto Tateo1

1Dipartimento di Fisica and Arnold-Regge Center, Universita di Torino, and INFN, Sezione di

Torino, Via P. Giuria 1, I-10125 Torino, Italy.2C.N. Yang Institute for Theoretical Physics, New York Stony Brook, NY 11794-3840. U.S.A.

E-mail: riccardo.conti@to.infn.it, stefano.negro@stonybrook.edu,

tateo@to.infn.it

Abstract: It has been recently discovered that the TT deformation is closely-related to

Jackiw-Teitelboim gravity. At classical level, the introduction of this perturbation induces

an interaction between the stress-energy tensor and space-time and the deformed EoMs can

be mapped, through a field-dependent change of coordinates, onto the corresponding unde-

formed ones. The effect of this perturbation on the quantum spectrum is non-perturbatively

described by an inhomogeneous Burgers equation. In this paper, we point out that there exist

infinite families of models where the geometry couples instead to generic combinations of local

conserved currents labelled by the Lorentz spin. In spirit, these generalisations are similar

to the JT model as the resulting theories and the corresponding scattering phase factors are

not Lorentz invariant. The link with the JT model is discussed in detail. While the classical

setup described here is very general, we shall use the sine-Gordon model and its CFT limit as

explanatory quantum examples. Most of the final equations and considerations are, however,

of broader validity or easily generalisable to more complicated systems.arX

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Contents

1 Introduction 1

2 TT-deformed higher spin conserved currents 4

2.1 A strategy to reconstruct the TT-deformed higher conserved currents 4

2.2 The massless free boson 5

3 Deformations induced by conserved currents with higher Lorentz spin 7

3.1 Deformations related to charges with positive spin s > 0 7

3.2 Deformations related to charges with negative spin s < 0 9

3.3 The TT deformation of the massless free boson 10

3.4 The s = −1 deformation of the massless free boson 11

3.5 Deformations related to charges with spin s→ 0 and the JT-type models 12

4 Deformed classical solutions 17

4.1 Explicit Lorentz breaking in a simple example 18

5 The quantum spectrum 20

5.1 Burgers-type equations for the spectrum 21

5.2 The CFT limit of the NLIE 25

5.3 Further deformations involving the topological charge 31

5.4 The quantum JT model 33

5.5 A simple example involving a pair of scattering phase factors 34

6 Conclusions 35

1 Introduction

The deformation of 2D quantum field theories [1, 2] by the Zamolodchikov’s TT operator [3],

has recently attracted the attention of theoretical physicists due to the many important links

with string theory [4–7] and AdS/CFT [8–17]. A remarkable property of this perturbation,

discovered in [1, 2], concerns the evolution of the quantum spectrum at finite volume R,

with periodic boundary conditions, in terms of the TT coupling constant τ . The spectrum is

governed by the hydrodynamic-type equation

∂τE(R, τ) =1

2∂R(E2(R, τ)− P 2(R)

), (1.1)

– 1 –

where E(R, τ) and P (R) are the total energy and momentum of a generic state |n〉, respec-

tively. Important for the current purposes is that, under the perturbation, the evolution of

the spectrum is equivalently encoded in the following Lorentz-type transformation(E(R, τ)

P (R)

)=

(cosh (θ0) sinh (θ0)

sinh (θ0) cosh (θ0)

)(E(R0, 0)

P (R0)

), (1.2)

with

sinh θ0 =τ P (R)

R0=τ P (R0)

R, cosh θ0 =

R+ τ E(R, τ)

R0=R0 − τ E(R0, 0)

R. (1.3)

From (1.2), it follows that the solution to (1.1) can be written in implicit form as

E2(R, τ)− P 2(R) = E2(R0, 0)− P 2(R0) , (1.4)

with the additional constraint

∂τR = −E(R, τ) , (1.5)

at fixed R0, obtainable directly from (1.3) (cf. with the s = 1 case of (5.30)).

As extensively discussed in [18] (see also [19]), the solutions to the classical EoMs associated

to the TT-deformed Lagrangians [2, 18, 20, 21] are obtained from the τ = 0 ones by a

field-dependent coordinate transformation

dxµ =(δµν + τ Tµ

ν(y, 0))dyν , y = (y1, y2) , (1.6)

with

Tµν(y, 0) = −εµρ εσνTρ

σ(y, 0) , εµν =

(0 1

−1 0

), (1.7)

where T(y, 0) is the stress-energy tensor of the unperturbed theory in the set of Euclidean

coordinates y. Equation (1.6) can be inverted as

dyµ =(δµν + τ Tµ

ν(x, τ))dxν , x = (x1, x2) , (1.8)

with

Tµν(x, τ) = −εµρ εσνTρ

σ(x, τ) , (1.9)

where T(x, τ) is now the stress-energy tensor of the TT-deformed theory in the set of Eu-

clidean coordinates x. Following the convention of [18], we will switch from Euclidean to

complex coordinates according to{z = x1 + i x2

z = x1 − i x2,

{w = y1 + i y2

w = y1 − i y2, (1.10)

and we shall denote with z = (z, z) and w = (w, w) the two set of complex coordinates.

An important link with topological gravity was noticed and studied in [19], where it was shown

– 2 –

that JT gravity coupled to matter leads to a scattering phase matching that associated to

the TT perturbation [1, 2, 4–6]. Equations (1.6) and (1.8) were also obtained in [18] starting

from the deformed EoMs following, therefore, a completely independent line of thoughts

compared to [19]. However, the final results turn out to be fully consistent with the proposal

of [19]. In this paper, we shall argue that the approach of [18] admits natural generalisations

corresponding to infinite families of geometric-type deformations of classical and quantum

field theories.

Our analysis starts from (1.6) and the observation that the equality between the second mixed

derivatives implies∂2xµ

∂yρ∂yσ=

∂2xµ

∂yσ∂yρ⇐⇒ ∂µT

µν = 0 . (1.11)

Equation (1.11) suggests that a consistent and natural generalisation of (1.6) can be obtained

by replacing the stress-energy tensor with an arbitrary (rank-two) conserved current. Thus,

the main objective of this work is to study the generalisations of the change of coordinates

(1.6) obtained by replacing Tµν with a matrix built using the higher-spin conserved currents,

typically present in free or integrable theories. In light-cone coordinates z, the continuity

equations are

∂Ts+1 = ∂Θs−1 , ∂Ts+1 = ∂Θs−1 , (s ∈ N) , (1.12)

where the subscripts s + 1 and s − 1 are the Lorentz spins of the corresponding field and

the s = 1 case of (1.12) corresponds to the conservation of energy and momentum. The

replacement we shall perform in (1.6)–(1.7) is

T −→ Ts , (1.13)

where the Euclidean and complex components of Ts are related as follows

(Ts)11 = − 1

2π

(Ts+1 + Ts+1 −Θs−1 − Θs−1

), (Ts)12 =

i

2π

(Ts+1 − Ts+1 + Θs−1 −Θs−1

),

(Ts)21 =i

2π

(Ts+1 − Ts+1 − Θs−1 + Θs−1

), (Ts)22 =

1

2π

(Ts+1 + Ts+1 + Θs−1 + Θs−1

).

(1.14)

For simplicity, it will be useful to define the re-scaled quantities1

τs+1 =Ts+1

2π, θs−1 =

Θs−1

2π, τs+1 =

Ts+1

2π, θs−1 =

Θs−1

2π. (1.15)

One of the main consequences of the replacement (1.13) is that the resulting generalised

deformations break explicitly the (eventual) Lorentz symmetry of the original models. The

corresponding gravity-like systems have therefore many features in common with the JT (TJ)

models recently studied in [22–28]. We shall denote this newly-introduced class of systems

1Notice that this convention differs by a factor 2 compared to [2, 18]. This choice allows for a more

transparent match between the classical and the quantum results of section 5.

– 3 –

as TTs-deformed theories. A precise connection between the s = 0 case and the JT models

will be established in section 5.4. As in [2, 18, 21], starting from section 5.1, we shall use

the sine-Gordon model as a specific quantum example. However, we would like to stress

that the techniques adopted and some of the results, i.e. the generalised Burgers equations

(5.30)–(5.32), are at least formally, of much wider validity.

2 TT-deformed higher spin conserved currents

The aim of this section is to introduce an efficient method, based on the field-dependent change

of coordinates derived in [18], to reconstruct the local Integrals of Motion (IMs) associated

to the TT deformation of a generic integrable field theory. The application of these ideas to

the family of classical TTs-deformed models will be described in section 3.

2.1 A strategy to reconstruct the TT-deformed higher conserved currents

Let us consider the following pair of conjugated 1-forms

Ik = τk+1(z, τ) dz + θk−1(z, τ) dz , Ik = τk+1(z, τ) dz + θk−1(z, τ) dz , (k ∈ N) , (2.1)

where the components τk+1, τk+1, θk−1 and θk−1 are the higher conserved currents of the

TT-deformed integrable theory which fulfil the continuity equations

∂τk+1(z, τ) = ∂θk−1(z, τ) , ∂τk+1(z, τ) = ∂θk−1(z, τ) , (2.2)

and k denotes the Lorentz spin. Using (2.2), it is easy to check that (2.1) are closed forms2,

in fact

dIk =(∂θk−1 − ∂τk+1

)dz ∧ dz = 0 , dIk =

(∂τk+1 − ∂θk−1

)dz ∧ dz = 0 , (2.3)

therefore the quantities

Ik =

∫CIk , Ik =

∫CIk , (2.4)

define local IMs for any given integration contour C in the complex plane.

From their very definition, differential forms are the right objects to be integrated over man-

ifolds since they are independent of coordinates. This property allows for the construction

of the TT-deformed local IMs from the undeformed ones, using the change of coordinates

introduced in [18, 19]. The strategy is the following:

• start from the higher conserved currents of the undeformed theory, expressed in the w

coordinates

Ik = τk+1(w, 0) dw + θk−1(w, 0) dw , Ik = τk+1(w, 0) dw + θk−1(w, 0) dw ; (2.5)

2Thus they are locally exact by the Poincare lemma.

– 4 –

• consider the change of coordinates w = w(z) (see [18]), which at differential level acts as

follows (dw

dw

)= J T

(dz

dz

),

(∂wf

∂wf

)= J −1

(∂f

∂f

), (∀f : R2 → R) , (2.6)

with

J =

(∂w ∂w

∂w ∂w

)=

1

∆(w)

(1 + 2τ θ0(w, 0) −2τ τ2(w, 0)

−2τ τ2(w, 0) 1 + 2τ θ0(w, 0)

), (2.7)

J −1 =

(∂wz ∂wz

∂wz ∂wz

)=

(1 + 2τ θ0(w, 0) 2τ τ2(w, 0)

2τ τ2(w, 0) 1 + 2τ θ0(w, 0)

), (2.8)

and

∆(w) =(1 + 2τ θ0(w, 0)

)(1 + 2τ θ0(w, 0)

)− 4τ2 τ2(w, 0) τ2(w, 0) ;

• starting from (2.6), express the derivatives ∂wf and ∂wf in terms of ∂f and ∂f , which

corresponds to the explicit inversion of the map w(z), at differential level. Plug the first

expression of (2.6) into (2.5), obtaining

Ik =τk+1(w(z), 0) + 2τ

(τk+1(w(z), 0) θ0(w(z), 0)− θk−1(w(z), 0) τ2(w(z), 0)

)∆(w(z))

dz

+θk−1(w(z), 0) + 2τ

(θk−1(w(z), 0) θ0(w(z), 0)− τk+1(w(z), 0) τ2(w(z), 0)

)∆(w(z))

dz ; (2.9)

• read the TT-deformed higher conserved currents as components of (2.9) in z coordinates:

τk+1(z, τ) =τk+1(w(z), 0) + 2τ

(τk+1(w(z), 0) θ0(w(z), 0)− θk−1(w(z), 0) τ2(w(z), 0)

)∆(w(z))

,

θk−1(z, τ) =θk−1(w(z), 0) + 2τ

(θk−1(w(z), 0) θ0(w(z), 0)− τk+1(w(z), 0) τ2(w(z), 0)

)∆(w(z))

.

(2.10)

In order to make the above strategy more concrete, in the following section we will explicitly

discuss the TT-deformed conserved currents for the massless free boson and comment on

more general cases.

2.2 The massless free boson

In the massless free boson theory the components of (2.5) are somehow trivial. However,

without further external constraints, there exists an infinite number of options for the choice

of the basis of conserved currents. For example, both

τk+1(w, 0) = −1

2(∂wφ)k+1 , θk−1(w, 0) = 0 , (k ∈ N) , (2.11)

– 5 –

and3

τk+1(w, 0) = −1

2

(∂

1+k2

w φ)2

, θk−1(w, 0) = 0 , (k ∈ 2N + 1) , (2.12)

are possible sets of higher conserved currents since they fulfil the continuity equations

∂τk+1(w, 0) = ∂θk−1(w, 0) , ∂τk+1(w, 0) = ∂θk−1(w, 0) . (2.13)

In general, any functional F (∂wφ, ∂2wφ, . . . ) automatically defines a conserved current. More-

over, since the change of variables (1.6) is non-linear, different choices of the current in (1.13)

should, at least in principle, give rise to totally different deformations of the original theory.

For simplicity, we will consider the two sets separately. From the second equation of (2.6),

the differential map w(z) is, in this case:

∂wf = ∂f − 1

4τ

(−1 + S∂φ

)2

∂f , ∂wf = − 1

4τ

(−1 + S∂φ

)2

∂f + ∂f , (∀f : R2 → R) ,

(2.14)

with S =√

1 + 4τ ∂φ ∂φ .

Considering the set of currents (2.11) and performing the change of variables (2.14) in the

general expressions (2.10), one finds

τk+1(z, τ) = −(∂φ)k+1

2S

(2

1 + S

)k−1

, θk−1(z, τ) = −τ (∂φ)k+1(∂φ)2

2S

(2

1 + S

)k+1

.

(2.15)

A result which was first obtained in [2] by resummation of the perturbative series.

Considering instead the set (2.12), we can again derive exactly all the TT-deformed higher

conserved currents, however their analytic expressions are more and more involved as k in-

creases and we were unable to find a compact formula valid for arbitrary spin k ∈ N. We

report here, as an example, the level k = 3 deformed current of the hierarchy (2.12):

τ4(z, τ) = −(∂φ)2

2S

((S − 1)4∂2φ− 16τ2(∂φ)4 ∂2φ

4τ(S − 1) (S2 + 1) (∂φ)3

)2

,

θ2(z, τ) = −(S − 1)2

8τ S

((S − 1)4∂2φ− 16τ2(∂φ)4 ∂2φ

4τ(S − 1) (S2 + 1) (∂φ)3

)2

. (2.16)

Finally, it is important to stress that the method presented in this section is completely

general and can be applied to a generic integrable model, provided the change of variables is

known. We have explicitly computed the TT-deformed conserved currents with k = 3, 5 for

the sine-Gordon model using the generalised change of variables described in [18]. Again the

resulting expressions are extremely complicated.

3The set of currents (2.12) can be obtained as the massless limit of the Klein-Gordon hierarchy:

τk+1(w, 0) = −1

2

(∂

1+k2

w φ)2, θk−1(w, 0) = −m

2

2

(∂

k−12

w φ)2, (k ∈ 2N + 1) ,

with Lagrangian LKG = ∂wφ∂wφ+m2φ2.

– 6 –

3 Deformations induced by conserved currents with higher Lorentz spin

In the following sections we shall generalise the change of variables (2.6) to deformations built

from conserved currents with generic positive and negative integer spins. Conventionally, we

shall denote with s = |s| ≥ 0 (k = |k| ≥ 0) the absolute value of the spin, and set s′ = −s ≤ 0

(k′ = −k ≤ 0). We will discuss separately perturbations induced by conserved currents with

spin s = s > 0, s′ = s < 0 and s→ 0, providing explicit examples.

3.1 Deformations related to charges with positive spin s > 0

The natural generalisation of (2.6) which ensures the commutativity of partial derivatives is(dw

dw

)=(J (s)

)T ( dzdz

),

(∂wf

∂wf

)=(J (s)

)−1(∂f

∂f

), (∀f : R2 → R) , (3.1)

where

J (s) =

(∂w ∂w

∂w ∂w

)=

1

∆(s)(w)

(1 + 2τ θs−1(w, 0) −2τ τs+1(w, 0)

−2τ τs+1(w, 0) 1 + 2τ θs−1(w, 0)

), (3.2)

(J (s)

)−1=

(∂wz ∂wz

∂wz ∂wz

)=

(1 + 2τ θs−1(w, 0) 2τ τs+1(w, 0)

2τ τs+1(w, 0) 1 + 2τ θs−1(w, 0)

), (3.3)

with s > 0,

∆(s)(w) =(1 + 2τ θs−1(w, 0)

)(1 + 2τ θs−1(w, 0)

)− 4τ2 τs+1(w, 0) τs+1(w, 0) , (3.4)

and s = 1 corresponds to the TT deformation, J (1) = J . In fact, using the continuity

equations (2.13) in (3.3), one finds that the second mixed partial derivatives are identical

∂w(∂wz) = ∂w(∂wz) , ∂w(∂wz) = ∂w(∂wz) . (3.5)

Consider now the 1-forms

Ik = τ(s)k+1(z, τ) dz + θ

(s)k−1(z, τ) dz , Ik = τ

(s)k+1(z, τ) dz + θ

(s)k−1(z, τ) dz , (3.6)

where the components τ(s)k+1, τ

(s)k+1, θ

(s)k−1 and θ

(s)k−1 are the higher spin conserved currents of

the integrable theory, deformed according to the generalised change of variables (3.1). They

fulfil the continuity equations

∂τ(s)k+1(z, τ) = ∂θ

(s)k−1(z, τ) , ∂τ

(s)k+1(z, τ) = ∂θ

(s)k−1(z, τ) . (3.7)

Using the strategy described in section 2, we perform the change of variables (3.1) in (2.5)

and obtain

Ik =τk+1(w(z), 0) + 2τ

(τk+1(w(z), 0) θs−1(w(z), 0)− θk−1(w(z), 0) τs+1(w(z), 0)

)∆(s)(w(z))

dz

+θk−1(w(z), 0) + 2τ

(θk−1(w(z), 0) θs−1(w(z), 0)− τk+1(w(z), 0) τs+1(w(z), 0)

)∆(s)(w(z))

dz ,

(3.8)

– 7 –

from which the components of the deformed currents can be extracted

τ(s)k+1(z, τ) =

τk+1(w(z), 0) + 2τ(τk+1(w(z), 0) θs−1(w(z), 0)− θk−1(w(z), 0) τs+1(w(z), 0)

)∆(s)(w(z))

,

θ(s)k−1(z, τ) =

θk−1(w(z), 0) + 2τ(θk−1(w(z), 0) θs−1(w(z), 0)− τk+1(w(z), 0) τs+1(w(z), 0)

)∆(s)(w(z))

.

(3.9)

Furthermore, let us consider the following combinations of currents4

H(s)k (z, τ) = −

(τ

(s)k+1(z, τ) + θ

(s)k−1(z, τ) + c.c.

)=

1

∆(s)(w(z))

[Hk(w(z), 0) + 2τ

(T+,k(w(z), 0) T−,s(w(z), 0) + c.c.

)], (3.10)

P(s)k (z, τ) = −

(τ

(s)k+1(z, τ) + θ

(s)k−1(z, τ)− c.c.

)=

1

∆(s)(w(z))

[Pk(w(z), 0) + 2τ

(T−,k(w(z), 0) T−,s(w(z), 0)− c.c.

)], (3.11)

where

T±,n(w, 0) = τn+1(w, 0)± θn−1(w, 0) , T±,n(w, 0) = τn+1(w, 0)± θn−1(w, 0) , (3.12)

and

Hk(w, 0) = −(τk+1(w, 0)+θk−1(w, 0)+c.c.

), Pk(w, 0) = −

(τk+1(w, 0)+θk−1(w, 0)−c.c.

).

(3.13)

H(s)k and P(s)

k can be considered to be the Hamiltonian and the momentum, at level k, in the

integrable hierarchy. Integrating (3.10) and (3.11) we find∫H(s)k (z, τ) dz ∧ dz =

∫ [Hk(w, 0) + 2τ

(T+,k(w, 0) T−,s(w, 0) + c.c.

)]dw ∧ dw

=

∫Hk(w, 0) dw ∧ dw − 2τ

∫(Ik + Ik) ∧ (Is − Is) , (3.14)

∫P(s)k (z, τ) dz ∧ dz =

∫ [Pk(w, 0) + 2τ

(T−,k(w, 0) T−,s(w, 0)− c.c.

)]dw ∧ dw

=

∫Pk(w, 0) dw ∧ dw − 2τ

∫(Ik − Ik) ∧ (Is − Is) . (3.15)

It is now very temping to interpret the result (3.14) as follows: the k-th Hamiltonian of the

integrable hierarchy perturbed with the s-th charge formally evolves following the flux of the

operator

T+,k(w, 0) T−,s(w, 0) + T+,k(w, 0) T−,s(w, 0) , (3.16)

4In the following, “c.c.” denotes the replacement {τ(z, τ), θ(z, τ)} → {τ(z, τ), θ(z, τ)}.

– 8 –

dressed by the change of variables (3.1). The latter interpretation is consistent with the

well studied TT example. However, the coordinate transformation (3.1) also introduces O(τ)

corrections to the bare Hamiltonians and these contributions can, in principle, completely

spoil this naive picture. Examples of this phenomena will neatly emerge from the study of the

s ≤ 0 deformations, but we conjecture that this interpretation remains valid for deformations

induced by conserved currents with strictly positive spin s > 0.

3.2 Deformations related to charges with negative spin s < 0

In order to define perturbations induced by higher conserved currents with negative spin s,

we flip s→ s′ in the definition of the generalised Jacobian (3.2) and (3.3), obtaining

J (s′) =1

∆(s′)(w)

(1 + 2τ τs+1(w, 0) −2τ θs−1(w, 0)

−2τ θs−1(w, 0) 1 + 2τ τs+1(w, 0)

), (3.17)

(J (s′)

)−1=

(1 + 2τ τs+1(w, 0) 2τ θs−1(w, 0)

2τ θs−1(w, 0) 1 + 2τ τs+1(w, 0)

), (3.18)

with s′ = s < 0 , s = |s| and

∆(s′)(w) =(1 + 2τ τs+1(w, 0)

)(1 + 2τ τs+1(w, 0)

)− 4τ2 θs−1(w, 0) θs−1(w, 0) . (3.19)

We arrived to (3.17)–(3.18) by implementing the spin-flip symmetry (see, for example, [1])

θs′−1 = τs+1 , τs′+1 = θs−1 , (3.20)

which corresponds to the following reflection property at the level of the 1-forms

Is′ = Is , Is′ = Is . (3.21)

Using the continuity equations (2.13), it is easy to verify that (3.18) fulfils again the conditions

(3.5), therefore it defines a consistent field-dependent change of variables. Repeating the

computations (3.8)–(3.15) using (3.17)–(3.18) one finds∫H(s′)k (z, τ) dz ∧ dz =

∫ [Hk(w, 0)− 2τ

(T+,k(w, 0) T−,s(w, 0) + c.c.

)]dw ∧ dw

=

∫Hk(w, 0) dw ∧ dw + 2τ

∫(Ik + Ik) ∧ (Is − Is) , (3.22)

∫P(s′)k (z, τ) dz ∧ dz =

∫ [Pk(w, 0)− 2τ

(T−,k(w, 0) T−,s(w, 0)− c.c.

)]dw ∧ dw

=

∫Pk(w, 0) dw ∧ dw + 2τ

∫(Ik − Ik) ∧ (Is − Is) , (3.23)

– 9 –

which are formally equal to (3.14) and (3.15), except for the sign of τ . Notice that, using

(3.12) and (3.20), expressions (3.22) and (3.23) can be rewritten as∫H(s′)k (z, τ) dz ∧ dz =

∫ [Hk(w, 0) + 2τ

(T+,k(w, 0) T−,s′(w, 0) + c.c.

)]dw ∧ dw , (3.24)∫

P(s′)k (z, τ) dz ∧ dz =

∫ [Pk(w, 0) + 2τ

(T−,k(w, 0) T−,s′(w, 0)− c.c.

)]dw ∧ dw , (3.25)

where the conventional sign of the perturbing parameter τ is restored.

However, the positive and negative spin sectors are deeply different, especially in what con-

cerns the non-zero momentum states. They are not simply related by a change of sign in

the coupling constant τ . In fact, equations (3.22), (3.23) and (3.14), (3.15) are only formally

identical: the change of variables and the corresponding perturbing operators are different.

By studying in detail the s = −1 perturbation of the massless free boson model (see sec-

tion 3.4 below), the difference with respect to the s = 1 perturbation, i.e. the TT, clearly

emerges. Although the identification of the perturbing operator is not always straightforward,

we checked on particular solutions that the generic k-th higher charges of the hierarchy evolve

– already at classical level – according to the generalised Burgers equations derived in section

5.1.

3.3 The TT deformation of the massless free boson

From the results obtained in section 2, the TT-deformed Hamiltonian of the massless free

boson is

H(z, τ) = −(τ2(z, τ) + θ0(z, τ) + c.c

)=−1− τ

(∂φ+ ∂φ

)2+ S

2τ S, (3.26)

and the corresponding Lagrangian is [2]

L(z, τ) =−1 + S

2τ. (3.27)

From the small τ expansion of (3.26), we see that the O(τ) contribution apparently does not

coincide with the TT operator of the undeformed theory. As we shall show, the proper iden-

tification of the perturbing operator in Hamiltonian formalism must be done upon Legendre

transforming (3.26). Repeating the computation of [21], the inversion of the Legendre map

gives

Π = i∂L(x, τ)

∂(∂2φ)−→ ∂2φ = 2iΠ

√1 + τ (∂1φ)2

1 + 4τ Π2, (3.28)

where ∂µ = ∂xµ , (µ = 1, 2) , and L(x, τ) is the Lagrangian (3.27) evaluated in Euclidean

coordinates x. Plugging (3.28) in (3.26) we obtain

H(x, τ) = iΠ ∂2φ+ L(x, τ) =−1 +

√1 + 4τ H(x) + 4τ2P2(x)

2τ, (3.29)

– 10 –

whereH(x) = 4Π2+(∂1φ)2 is the undeformed Hamiltonian and P(x) = Π ∂1φ is the conserved

momentum density. We notice that (3.29) has the same formal expression of the quantum

spectrum (see [2, 21]), where the integrated quantities are replaced here by the classical

densities. Using (3.29), it clearly emerges that the perturbing operator is indeed TT.

3.4 The s = −1 deformation of the massless free boson

Consider the perturbation of the massless free boson theory with spin s′ = s = −1. Expres-

sions (3.17)–(3.18) become explicitly

J (−1) =1(

1− τ (∂wφ)2)(

1− τ (∂wφ)2) ( 1− τ (∂wφ)2 0

0 1− τ (∂wφ)2

),

(J (−1)

)−1=

(1− τ (∂wφ)2 0

0 1− τ (∂wφ)2

), (3.30)

and the differential map w(z) is

∂wf =2

1 + S ′∂f , ∂wf =

2

1 + S ′∂f , (3.31)

where we have defined

S ′ =√

1 + 4τ (∂φ)2 , S ′ =√

1 + 4τ (∂φ)2 . (3.32)

Using equations (3.31), one can prove that the undeformed EoMs are mapped into themselves,

namely

∂w∂wφ = 0 −→ ∂∂φ = 0 . (3.33)

This reflects the fact, common to all the negative spin perturbations of CFT’s, that the

deformation does not mix the holomorphic and anti-holomorphic sectors, as clearly seen also

from (3.31). The components of the stress-energy tensor of the deformed theory are

τ(−1)2 (z, τ) = −−1 + S ′

4τ, θ

(−1)0 (z, τ) = 0 ,

τ(−1)2 (z, τ) = −−1 + S ′

4τ, θ

(−1)0 (z, τ) = 0 .

and, more generally, the deformation of the set of higher conserved currents (2.11) computed

using the method presented in section 3 is

τ(−1)k+1 (z, τ) = − 1

2(∂φ)k−1

(−1 + S ′

2τ

)k, θ

(−1)k−1 (z, τ) = 0 . (3.34)

As already discussed in the TT case, in order to identify the perturbing operator we should

express the deformed Hamiltonian in terms of the conjugated momentum, which implies the

knowledge of the Lagrangian. However, starting from

H(−1)(z, τ) = −(τ

(−1)2 (z, τ) + θ

(−1)0 (z, τ) + c.c

)= −2− S ′ − S ′

4τ, (3.35)

– 11 –

we were not able to reconstruct the corresponding Lagrangian with the constraint that the

associated EoMs are (3.33). Finally, notice that

H(−1)(z, τ) = −2− S ′ − S ′

4τ=

1

2(∂φ)2 +

1

2(∂φ)2 − 1

2

(∂φ)4 + (∂φ)4

)τ +O(τ2)

= −(τ2(z, 0) + τ2(z, 0)

)− 2

(τ2

2 (z, 0) + τ22 (z, 0)

)τ +O(τ2) . (3.36)

Therefore, the leading correction to the free boson theory corresponds to the Lorentz breaking

operator typically appearing in effective field theories for discrete lattice models (see, for

example [29]). The study of the quantum spectrum performed in section 5.2 will lead to the

same conclusion.

3.5 Deformations related to charges with spin s→ 0 and the JT-type models

In this section we shall consider perturbations of the massless free boson model induced by

the U(1)L × U(1)R currents, related to the case k = 0 in (2.11)

J+(w, 0) = −2i τ1(w, 0) = i ∂wφ , J−(w, 0) = −2i θ−1(w, 0) = 0 ,

J+(w, 0) = 2i τ1(w, 0) = −i ∂wφ , J−(w, 0) = 2i θ−1(w, 0) = 0 , (3.37)

which are components of the closed 1-forms

I0 = J+(w, 0) dw + J−(w, 0) dw , I0 = J+(w, 0) dw + J−(w, 0) dw . (3.38)

Notice that the additional factor i in (3.37) allows us to write the holomorphic and anti-

holomorphic components of the stress-energy tensor in Sugawara form (see [26])

τ2(w, 0) =1

2J2

+(w, 0) , τ2(w, 0) =1

2J2

+(w, 0) . (3.39)

Following the same spirit of the s > 0 and s < 0 deformations, the most general change of

coordinates built out of the U(1) currents (3.37) is of the form

J (0) =1

∆(0)(w)

(1 + τ (4) ∂wφ −τ (2) ∂wφ

−τ (3) ∂wφ 1 + τ (1) ∂wφ

), (3.40)

(J (0)

)−1=

(1 + τ (1) ∂wφ τ (2) ∂wφ

τ (3) ∂wφ 1 + τ (4) ∂wφ

), (3.41)

where τ (i) , (i = 1, . . . 4), are four different deformation parameters and

∆(0)(w) = 1 + τ (1) ∂wφ+ τ (4) ∂wφ+(τ (1)τ (4) − τ (2)τ (3)

)∂wφ∂wφ . (3.42)

Since the general case is quite cumbersome, we will restrict our analysis to some particular

limits.

– 12 –

Case τ (1) = τ (3) = δ | τ (2) = τ (4) = τ

In this case (3.40)–(3.41) reduces to

J (0) =

(∂w ∂w

∂w ∂w

)=

(1− δ ∂φ −τ ∂φ−δ ∂φ 1− τ ∂φ

),

(J (0)

)−1=

(∂wz ∂wz

∂wz ∂wz

)=

(1 + δ ∂wφ τ ∂wφ

δ ∂wφ 1 + τ ∂wφ

), (3.43)

which corresponds to a change of variables w(z) of the form

w = z − δφ , w = z − τφ . (3.44)

In [18], it was observed that the TT-deformed solutions fulfil a non-linear evolution equation.

Its extension to generic spin s in complex coordinates is

∂τφ(s)(z, τ) + (∂τz) ∂φ

(s)(z, τ) + (∂τ z) ∂φ(s)(z, τ) = 0 , (3.45)

with5

∂τz = 2

∫ z

Is , ∂τ z = 2

∫ z

Is . (3.46)

The multi-parameter variant of (3.45) is

∂τ (i)φ(z, ~τ) +(∂τ (i)z

)∂φ(z, ~τ) +

(∂τ (i) z

)∂φ(z, ~τ) = 0 , ~τ = {τ (i)} . (3.47)

In general, equations (3.45)–(3.47) cannot be explicitly integrated, however, considering the

change of coordinates (3.44), equations (3.47) become a set of inviscid Burgers equations

∂τφ(z, τ, δ) +1

2∂(φ2(z, τ, δ)

)= 0 , ∂δφ(z, τ, δ) +

1

2∂(φ2(z, τ, δ)

)= 0 , (3.48)

whose solution can be expressed in implicit form as

φ(z, z, τ, δ) = φ(z − δφ, z − τφ, 0, 0) . (3.49)

Using the method discussed in sections 2 and 3, we can derive the tower of deformed higher

conserved currents from the undeformed ones. In particular the components of the U(1)

5Starting from (4.2), i.e. φ(s)(z, τ) = φ(w(z(s)), 0), we let z depending on τ such that ddτ

w(z(s)) = 0:

d

dτφ(s)(z, τ) = 0 = ∂τφ

(s)(z, τ) + (∂τz)∂zφ(s)(z, τ) + (∂τ z)∂zφ

(s)(z, τ) .

From the definition of the change of coordinates (3.1)

z = w(z) + 2τ

w(z)∫ (θs−1 (w, 0) dw + τs+1 (w, 0) dw

), z = w(z) + 2τ

w(z)∫(τs+1 (w, 0) dw + θs−1 (w, 0) dw) ,

using the definition (2.5) and the coordinate independence property of differential forms, we arrive to (3.46).

– 13 –

currents (3.37) become

J(0)+ (z, τ, δ) = i ∂φ

1− δ ∂φ1− δ ∂φ− τ ∂φ

, J(0)− (z, τ, δ) = −i δ ∂φ ∂φ

1− δ ∂φ− τ ∂φ,

J(0)+ (z, τ, δ) = −i ∂φ 1− τ ∂φ

1− δ ∂φ− τ ∂φ, J

(0)− (z, τ, δ) = i τ

∂φ ∂φ

1− δ ∂φ− τ ∂φ, (3.50)

while the components of the stress-energy tensor are

τ(0)2 (z, τ, δ) = −1

2(∂φ)2 1− δ ∂φ

(1− δ ∂φ− τ ∂φ)2, θ

(0)0 (z, τ, δ) =

1

2δ

(∂φ)2 ∂φ

(1− δ ∂φ− τ ∂φ)2,

τ(0)2 (z, τ, δ) = −1

2(∂φ)2 1− τ ∂φ

(1− δ ∂φ− τ ∂φ)2, θ

(0)0 (z, τ, δ) =

1

2τ

(∂φ)2 ∂φ

(1− δ ∂φ− τ ∂φ)2.

(3.51)

Therefore, the deformed Hamiltonian and conserved momentum density are

H(0)(z, τ, δ) = −(τ

(0)2 (z, τ, δ) + θ

(0)0 (z, τ, δ) + c.c.

)= −

∂φ ∂φ(δ ∂φ+ τ ∂φ

)− (∂φ)2(1− δ ∂φ)− (∂φ)2(1− τ ∂φ)

2(1− δ ∂φ− τ ∂φ

)2 , (3.52)

P(0)(z, τ, δ) = −(τ

(0)2 (z, τ, δ) + θ

(0)0 (z, τ, δ)− c.c.

)=

(∂φ+ ∂φ)(∂φ (1− δ ∂φ)− ∂φ (1− τ ∂φ)

)2(1− δ ∂φ− τ ∂φ

)2 , (3.53)

and it is easy to check that the corresponding deformed Lagrangian is

L(0)(z, τ, δ) =∂φ ∂φ

1− δ ∂φ− τ ∂φ. (3.54)

Notice that, while the deformed action is mapped exactly into the undeformed one under

(3.44), the integral of (3.52) transforms with an additional term∫H(0)(z, τ, δ) dz ∧ dz =

∫H(0)(z(w), τ, δ) Det

((J (0)

)−1)dw ∧ dw

=

∫ [H(w, 0) + i (δ − τ)

(J+(w, 0) τ2(w, 0)− c.c.

)]dw ∧ dw .

(3.55)

As already discussed in the previous section, in order to unambiguously identify the perturbingoperator we must perform the Legendre transform on (3.52). First of all, we move from light-cone z coordinates to Euclidean coordinates x = (x1, x2) according to the convention (1.10).Then, inverting the Legendre map one finds

Π = i∂L(0)(x, τ, δ)

∂ (∂2φ),

∂2φ = i

(1 + 2 Π (τ − δ)

)(−2 + (τ + δ) ∂1φ

)+ 2√(

1 + 2 Π (τ − δ))(

1− τ ∂1φ)(

1− δ ∂1φ)

(τ − δ)(1 + 2 Π (τ − δ)

) . (3.56)

– 14 –

Plugging (3.56) in (3.52) and (3.53) one gets

H(0)(x, τ, δ) = −(τ(0)2 (x, τ, δ) + θ

(0)0 (x, τ, δ) + c.c.

)= iΠ ∂2φ+ L(0)(x, τ, δ)

= −

(1 + Π (τ − δ)

)(−2 + (τ + δ) ∂1φ

)+ 2√(

1 + 2 Π (τ − δ))(

1− τ ∂1φ)(

1− δ ∂1φ)

(τ − δ)2,

(3.57)

P(0)(x, τ, δ) = −(τ

(0)2 (x, τ, δ) + θ

(0)0 (x, τ, δ)− c.c.

)= −Π ∂1φ . (3.58)

From (3.58), we find that the momentum density is unaffected by the perturbation, since

P(0)(x, τ, δ) = P(x) = −Π ∂1φ , (3.59)

where P(x) is the unperturbed momentum density. This, in turn, implies that the deformation

preserves space translations. Finally, expanding (3.57) at the first order in τ and δ, we can

identify the perturbing operator

H(0)(x, τ, δ) ∼τ→0δ→0

H(x) + 2(τ J2(x) τ2(x) + δ J2(x) τ2(x)

)+O(τδ) , (3.60)

where H(x) = −(τ2(x) + τ2(x)

)= Π2 + 1

4 (∂1φ)2 is the unperturbed Hamiltonian and

τ2(x) = −1

8(2 Π− ∂1φ)2 , τ2(x) = −1

8(2 Π + ∂1φ)2 . (3.61)

are the (anti-)holomorphic components of the undeformed stress energy tensor, respectively.

In (3.60), we denoted as Jµ(x) and Jµ(x) , (µ = 1, 2), the Euclidean components of the

undeformed U(1) currents (3.37)

J1(x) = J+(x, 0)− J−(x, 0) =i

2(−2Π + ∂1φ) ,

J2(x) = i(J+(x, 0) + J−(x, 0)

)=

1

2(2Π− ∂1φ) ,

J1(x) = J+(x, 0)− J−(x, 0) =i

2(−2Π− ∂1φ) ,

J2(x) = −i(J+(x, 0) + J−(x, 0)

)=

1

2(−2Π− ∂1φ) , (3.62)

which fulfil the continuity equations

∂µJµ(x) = 0 , ∂µJµ(x) = 0 . (3.63)

In the same way, we define the Euclidean components of the deformed U(1) currents (3.50)

as

J(0)1 (x, τ, δ) = J

(0)+ (x, τ, δ)− J (0)

− (x, τ, δ) , J(0)2 (x, τ, δ) = i

(J

(0)+ (x, τ, δ) + J

(0)− (x, τ, δ)

),

J(0)1 (x, τ, δ) = J

(0)+ (x, τ, δ)− J (0)

− (x, τ, δ) , J(0)2 (x, τ, δ) = −i

(J

(0)+ (x, τ, δ) + J

(0)− (x, τ, δ)

),

(3.64)

– 15 –

which again fulfil the continuity equations

∂µJ(0)µ (x, τ, δ) = 0 , ∂µJ

(0)µ (x, τ, δ) = 0 . (3.65)

Therefore, from (3.65), one finds that the quantities

Q(R, τ, δ) =

∫ R

0J

(0)2 (x, τ, δ) dx , Q(R, τ, δ) =

∫ R

0J

(0)2 (x, τ, δ) dx . (3.66)

are the conserved charges associated to the U(1)L × U(1)R symmetry.

Case τ (1) = τ (3) = 0 | τ (2) = τ (4) = τ

This case can be easily retrieved from the previous one by sending δ → 0. It corresponds

to the change of coordinates associated to the JT deformation (see [22, 26]). First we notice

that, setting δ = 0 in (3.50) and (3.51), the deformation preserves the Sugawara construction

for the holomorphic sector

τ2(z, τ) =1

2

−(∂φ)2

(1− τ ∂φ)2=

1

2J2

+(z, τ) , τ2(z, τ) = τ(0)2 (z, τ, 0) , J+(z, τ) = J

(0)+ (z, τ, 0) ,

(3.67)

but this is not true for the anti-holomorphic sector. Then, we observe that the Lagrangian

(3.54) is

LJT(z, τ) = L(0)(z, τ, 0) =∂φ ∂φ

1− τ ∂φ, (3.68)

and the corresponding Legendre transformed Hamiltonian (3.57) becomes

HJT(x, τ) = H(0)(x, τ, 0) = P(x) +2

τ2

((1 + τ J2(x)

)+ SJT

), (3.69)

where

SJT =

√(1 + τ J2(x)

)2+ 2τ2 τ2(x) . (3.70)

Writing (3.69) as HJT(x, τ) = IJT(x, τ) + IJT(x, τ) with

IJT(x, τ) = −(τ2(x, τ) + θ0(x, τ)

)= P(x) +

1

τ2

((1 + τ J2(x)

)+ SJT

),

IJT(x, τ) = −(τ2(x, τ) + θ0(x, τ)

)=

1

τ2

((1 + τ J2(x)

)+ SJT

), (3.71)

we find that the Euclidean component J2 of the deformed U(1) currents fulfils

J2(x, τ) = −1

τ(1 + SJT) = J2(x)− τ IJT(x, τ) . (3.72)

As already discussed in [21] for the TT deformation, we argue that also in the JT case

the energy density of the right and left movers (3.71) has the same formal expression of

the spectrum obtained in [23] (cf. (5.92)), where the classical densities are replaced by the

corresponding integrated quantities. In addition, we notice that also the deformation of the

U(1) current density (3.72) admits a straightforward generalisation at the quantum level (cf.

(5.94)). Analogous considerations apply to the case τ (1) = τ (3) = τ , τ (2) = τ (4) = 0 , which

corresponds to the TJ deformation.

– 16 –

Case τ (1) = τ (2) = τ (3) = τ (4) = τ

Again all the equations (3.43)–(3.57) can be obtained setting δ = τ . In this case, the square

root in (3.57) disappears, and the Hamiltonian takes the simple form

H(0)(x, τ) = H(0)(x, τ, τ) =H(x) + τ P(x)

(J2(x) + J2(x)

)+ τ2 P2(x)

1 + τ(J2(x)− J2(x)

) . (3.73)

Again we split the Hamiltonian as H(0)(x, τ) = I(0)(x, τ) + I(0)(x, τ) with

I(0)(x, τ) = −(τ

(0)2 (x, τ) + θ

(0)0 (x, τ)

)=I(0)(x, 0) + τ J2(x)P(x) + τ2

2 P2(x)

1 + τ(J2(x)− J2(x)

) ,

I(0)(x, τ) = −(τ

(0)2 (x, τ) + θ

(0)0 (x, τ)

)=I(0)(x, 0) + τ J2(x)P(x) + τ2

2 P2(x)

1 + τ(J2(x)− J2(x)

) . (3.74)

Finally, we notice that the deformed U(1) currents fulfil

J(0)2 (x, τ) = J2(x) + τ P(x) , J

(0)2 (x, τ) = J2(x) + τ P(x) . (3.75)

We will see in section 5.4 that the quantum version of this perturbation (cf. (5.98)–(5.100))

can be obtained by introducing two different scattering factors in the NLIEs (5.33).

4 Deformed classical solutions

Following [18], we shall now describe how to derive explicitly a deformed solution φ(s)(z, τ)

associated to the change of variables (3.1) starting from the undeformed solution φ(w, 0).

First of all we compute the relation between the sets of coordinates w and z by integrating,

for s ≥ 0, (3.3) for z{∂z(s)(w)∂w = 1 + 2τ θs−1(w, 0) ,

∂z(s)(w)∂w = 2τ τs+1(w, 0) ,

{∂z(s)(w)∂w = 2τ τs+1(w, 0) ,

∂z(s)(w)∂w = 1 + 2τ θs−1(w, 0) ,

(4.1)

where we have denoted z = z(s) to distinguish between the different perturbations. The com-

ponents of the higher charges τs+1(w, 0), θs−1(w, 0) along with their conjugates are implicitly

evaluated on the specific field configuration φ(w, 0). Inverting the relation z(s)(w), we find

the deformed solution as

φ(s)(z, τ)

= φ(w(z(s)), 0). (4.2)

Similarly, we can apply the same strategy to derive a deformed solution associated to the

change of variables (3.2), for s ≤ 0. In this case we should, instead, integrate the system∂z(s′)(w)

∂w = 1 + 2τ τs+1(w, 0) ,

∂z(s′)(w)∂w = 2τ θs−1(w, 0) ,

∂z(s′)(w)

∂w = 2τ θs−1(w, 0) ,

∂z(s′)(w)∂w = 1 + 2τ τs+1(w, 0) ,

(4.3)

and repeat the same procedure described above.

– 17 –

4.1 Explicit Lorentz breaking in a simple example

The purpose of this section is to highlight the difference between the TT perturbation and

those corresponding to spins s 6= 1, studying a specific solution of the massless free boson

model. Since perturbations with higher spins should break explicitly Lorentz symmetry, it

is convenient to consider a rotational invariant solution in Euclidean space-time. The initial

solution is [30]:

φ(w, 0) = d ln

(w + ξ

w + ξ

), (ξ, ξ ∈ C , d ∈ R) , (4.4)

where w and w are complex conjugated variables related to the Euclidean coordinates y =

(y1, y2) through (1.10). Using

∂w1

w + ξ= −π δ(w + ξ) , ∂w

1

w + ξ= −π δ(w + ξ) , (4.5)

one can show that (4.4) is indeed solution to the undeformed EoMs: ∂w∂wφ = 0.

For sake of brevity, we will only consider perturbations induced by the set of charges (2.12),

the deformations associated to (2.11) can be obtained from the results described here through

a simple redefinition of the coupling τ .

Integrating (4.1) using the set of charges (2.12) evaluated on the solution (4.4) we obtain

z(s)(w) = w + τΓ(s+1

2

)2s

d2

(w + ξ)s, z(s)(w) = w + τ

Γ(s+1

2

)2s

d2

(w + ξ)s. (4.6)

The latter equations cannot explicitly be inverted, for generic spin s, as w(z(s)). However,

for s = 1, i.e. the TT perturbation, (4.6) can be written as

z + ξ = (w + ξ)

[1 + τ

d2

(w + ξ)(w + ξ)

], z + ξ = (w + ξ)

[1 + τ

d2

(w + ξ)(w + ξ)

],(4.7)

with z = z(1)(w), from which

φ(w, 0) = d ln

(w + ξ

w + ξ

)= d ln

(z + ξ

z + ξ

)= φ(z, τ) . (4.8)

Therefore, the solution (4.4) is a fixed point of the TT flow. To clearly see how the s 6= 1

perturbations affect the solution (4.4), we shall restrict to the s < 0 perturbations, where the

deformed solutions can be found explicitly. Integrating (4.3) using the set of charges (2.12),

again evaluated on (4.4), we find

z(s′)(w) = w + τΓ(s+1

2

)2s

d2

(w + ξ)s, z(s′)(w) = w + τ

Γ(s+1

2

)2s

d2

(w + ξ)s. (4.9)

Comparing (4.6) with (4.9) we see that the difference between the s > 0 and s < 0 perturba-

tions lies in the substitution w + ξ ←→ w + ξ in the term proportional to τ , which implies

– 18 –

that, for s < 0, there is no mixing between holomorphic and anti-holomorphic components.

Since holomorphic and anti-holomorphic parts are completely decoupled, we can integrate

explicitly (4.9) for w and w. The result is

w(z(s′)

)= z(s′) +

s

1 + s

[−1 +Gs

((1 + s

s

)1+s

Γ

(1 + s

2

)2 τ d2(z(s′) + ξ

)1+s

)](z(s′) + ξ

),

w(z(s′)

)= z(s′) +

s

1 + s

[−1 +Gs

((1 + s

s

)1+s

Γ

(1 + s

2

)2 τ d2(z(s′) + ξ

)1+s

)](z(s′) + ξ

),

(4.10)

with

Gs(x) = sFs−1

(− 1

1 + s,

1

1 + s, . . . ,

s− 1

1 + s;1

s, . . . ,

s− 1

s;x

). (4.11)

Finally, the deformed solutions are recovered by plugging (4.10) into (4.4)

φ(s′)(z, τ) = d ln

(z + ξ

z + ξ

)+ d ln

1 + sGs

((1+ss

)1+sΓ(

1+s2

)2 τ d2

(z+ξ)1+s

)1 + sGs

((1+ss

)1+sΓ(

1+s2

)2 τ d2

(z+ξ)1+s

) . (4.12)

Let us discuss in detail the case s = −1 of (4.12),

φ(−1)(z, τ) = d ln

(z + ξ

z + ξ

)+ d ln

1 +√

1− 4τ d2

(z+ξ)2

1 +√

1− 4τ d2

(z+ξ)2

. (4.13)

We observe that, as soon as the perturbation is switched on, a pair of square root branch

points appears at z = ±2d√τ − ξ. Considering for simplicity ξ, ξ ∈ R, they form a branch

cut on the real axis of the complex plane of z

C =(−2d√τ − ξ; +2d

√τ − ξ

), (4.14)

i.e. the black line in Figure 1b. Instead, the logarithmic singularity of the undeformed

solution (4.4) at w = −ξ cancels out with the singularities coming from the additional term

in (4.13). Therefore, the logarithmic cut of (4.4) which runs, in our convention, along the real

axis from w = −∞ to w = −ξ, now connects z = −∞ on the first sheet to z = −∞ on the

secondary branches reached by passing through C (see the red line in Figure 1b). This implies

that the behaviour of (4.13) at z =∞ is different according to the choice of the branch. On

the first sheet one has

φ(−1)(z, τ) ∼z→∞

(1-th sheet)

d ln

(z + ξ

z + ξ

)+O(z0) , (4.15)

while, on the second sheet, flipping the + sign in front of the square roots in (4.13) into a −sign one finds

φ(−1)(z, τ) ∼z→∞

(2-nd sheet)

−d ln

(z + ξ

z + ξ

)+O(z0) . (4.16)

– 19 –

(a) s = 1 (b) s = −1 (c) s = −3

Figure 1. Analytic structure of the deformed solution (4.12) in the complex plane of z for different

values of s, with ξ = ξ = 0. The black lines correspond to the square root branch cuts connecting the

singularities (4.19), while the red lines correspond to the logarithmic cuts.

In Figure 2, is represented the Riemann surface of the solution (4.13) (Figure 2b) together

with that of the bare solution (Figure 2a), which coincides with the TT deformed solution.

Notice that the analytic structure of (4.13) can be read out from the implicit map (4.9). In

fact, for s = −1, equation (4.9) reduces to the Zhukovsky transformation

z + ξ = (w + ξ) + τd

(w + ξ), z + ξ = (w + ξ) + τ

d

(w + ξ), (4.17)

from which we see that z =∞ on the first sheet is mapped into w =∞, while z =∞ on the

second sheet is mapped into w = −ξ. Moreover (4.17) captures the large z behaviour of the

solution. In fact,

d ln

(w + ξ

w + ξ

)∼

w→∞d ln

(z + ξ

z + ξ

), d ln

(w + ξ

w + ξ

)∼

w→−ξ−d ln

(z + ξ

z + ξ

). (4.18)

Let us now consider the generic solution (4.12). The hypergeometric functions appearing in

(4.12) are of the form p+1Fp(a1, . . . , ap+1; b1, . . . , bp;x), with coefficients {ai}p+1i=1 , {bj}

pj=1 ∈ Z.

Generally, these hypergeometric functions have branch points at x =∞ and x = 1, which in

our case are mapped into x = 0 and

xn =1 + s

s

(dΓ

(1 + s

2

)) 21+s

τ1

1+s e2πi1+s

n − ξ , (n = 0, . . . , s) , (4.19)

respectively. The branch points (4.19) are all of square root type. In conclusion, roughly

speaking, starting from a rotational-symmetric solution, the perturbations with s < 0 have

explicitly broken the original U(1) symmetry down to a discrete Z2s.

5 The quantum spectrum

This second part of the paper is devoted to the study of the quantum version of the per-

turbations of classical field theories described in the preceding sections. The NLIEs will be

the starting point for the derivation of the spectral flow equations for the conserved charges

– 20 –

(a) (b)

Figure 2. Riemann surface describing the deformed solution φ(s)(z, τ) , (ξ = ξ = 0), in the complex

plane of z for s = 1 (a), and s = −1 (b).

subjected to infinitesimal variations of the deforming parameters. Although most of our final

results are expected to hold in general, we will consider for simplicity the sine-Gordon model

as explicit example.

5.1 Burgers-type equations for the spectrum

The current purpose is to try to build the quantum version of the classical models described

in section 3, by including specific scattering phase factors into the NLIE for the sine-Gordon

model confined on a infinite cylinder of circumference R. The sine-Gordon NLIE is

fν(θ) = ν(R,α0 | θ)−∫C1dθ′K(θ − θ′) ln

(1 + e−fν(θ′)

)+

∫C2dθ′K(θ − θ′) ln

(1 + efν(θ′)

),

(5.1)

where we have set

ν(R,α0 | θ) = i2πα0 − imR sinh(θ) , (5.2)

to denote the driving term. In (5.1)–(5.2), m is the sine-Gordon soliton mass, α0 is the

quasi-momentum and K(θ) is the kernel defined as

K(θ) =1

2πi∂θ lnSsG(θ) , (5.3)

where

lnSsG(θ) = −i∫R+

dp

psin(pθ)

sinh (πp(ζ − 1)/2)

cosh (πp/2) sinh (πpζ/2), ζ =

β2

1− β2. (5.4)

The quasi-momentum α0 emerges [31, 32] by imposing periodic boundary conditions on the

sine-Gordon field: ϕ(x + R, t) = ϕ(x, t). Due to the periodicity of the potential in the sine-

– 21 –

Gordon model

LsG =1

8π

((∂tϕ)2 − (∂xϕ)2

)+ 2µ cos(

√2βϕ) ,

(µ ∝ (m)2−2β2

), (5.5)

the Hilbert space splits into orthogonal subspaces Hα0 , characterised by the quasi-momentum

α0,

ϕ→ ϕ+2π√2β

←→ |Ψα0〉 → ei2πα0 |Ψα0〉 , ( |Ψα0〉 ∈ Hα0 ) . (5.6)

with α0 ∈ [−1/2, 1/2]. Twisted boundary conditions of the form:

ϕ(x+R, t) = ϕ(x, t) +2π√2βn , n ∈ Z (5.7)

are also natural in the sine-Gordon model since they correspond, in the infinite volume limit,

to field configurations with non-trivial topological charge:

Qx =

√2β

2π

∫ R

0∂xϕdx . (5.8)

Energy levels in the twisted sectors are also described by the same NLIE at specific valuesof the quasi-momentum. Furthermore, α0 can also be related to a background charge (cf.equation (5.38)). Therefore, (5.1) also describes minimal models of the Virasoro algebra,Mp,q perturbed by the operator φ13 [33]. In the following, we shall equivalently refer to α0

as quasi-momentum or vacuum parameter.The integration contours C1, C2 are state dependent. For the ground-state in an arbitrarysubspace Hα0 , one may take them to be straight lines slightly displaced from the real axis:C1 = R+i0+, C2 = R−i0+. Equations describing excited states have the same form [32, 34–36]but with integration contours encircling a number of singularities {θi} with

(1 + ef(θi)

)= 0.

However, in this paper we can ignore these subtleties, since the final evolution equations donot depend explicitly on the specific details of the integration contours. Setting

b(±)k =

∫C1

dθ

2πi

(±r

2e±θ)k

ln(

1 + e−fν(θ))−∫C2

dθ

2πi

(±r

2e±θ)k

ln(

1 + efν(θ)), (k ∈ 2N + 1) ,

(5.9)

with r = mR, and

I(±)k =

(2π

R

)k b(±)k

Ck, (5.10)

where

Ck =1

2k

(4π

β2

)k+12 Γ

(k2 (ζ + 1)

)Γ(k+3

2

)Γ(k2ζ) ( Γ(1 + ζ/2)

Γ(3/2 + ζ/2)

)k, (5.11)

then,{I

(+)k , I

(−)k

}are the quantum versions of the local integrals of motion {Ik, Ik} introduced

in (2.4), i.e.

I(+)k ←→ Ik = −

∫ R

0

dx

2π(Tk+1 + Θk−1) , I

(−)k ←→ Ik = −

∫ R

0

dx

2π

(Tk+1 + Θk−1

). (5.12)

– 22 –

It is also convenient to define the alternative set of conserved charges

Ek(R) = I(+)k (R) + I

(−)k (R) , Pk(R) = I

(+)k (R)− I(−)

k (R) , (5.13)

where E(R) = E1(R) and P (R) = P1(R) are the total energy and momentum of the state,

respectively. In addition, the following reflection property holds

I(±)−k (R) = I

(∓)k (R) , (k > 0) , (5.14)

which is the quantum analog of (3.21). We conjecture that the classical geometric-type

deformations described in the previous sections – associated to generic combinations of the

higher spin conserved currents – correspond, at the quantum level, to the inclusion of extra

scattering phases of the form

δ(θ, θ′) =∑s

τ (s) γ1γs sinh(θ − s θ′) , (5.15)

where the τ (s) are independent coupling parameters and

γs =(2πm)s

Cs, γ1 = m, (5.16)

with s = |s|. In (5.15) the sum runs, in principle, over positive and negative odd integers.

However, since the whole analysis can be straightforwardly analytically extended to arbitrary

values of s, in the following we shall relax this constraint, at least to include the case s→ 0 and

the set of nonlocal conserved charges [37]. The scattering phase factor (5.15) leads to multi-

parameter perturbations of the spectrum of the original QFT which, due to the intrinsic non-

linearity of the problem, can effectively be studied only on the case-by-case basis. A detailed

analytic and numerical study of specific multi-parameter deformations of a massive QFT,

such as the sine-Gordon model, appears to be a very challenging long-term objective. Most

of our checks have been performed considering conformal field theories deformed (explicitly

at leading order) by a single irrelevant composite field. Therefore, in this paper, we shall

restrict the analysis to scattering phase factors with only a single non vanishing irrelevant

coupling:

δ(s)(θ, θ′) = τ γ1γs sinh(θ − s θ′) , (5.17)

which modifies the kernel appearing in the NLIE (5.1) as

K(θ − θ′)→ K(θ − θ′)− 1

2π∂θδ

(s)(θ − s θ′) = K(θ − θ′)− τ m γs2π

cosh(θ − s θ′) . (5.18)

Inserting (5.18) in (5.1), after simple manipulations, we find that the deformed version of

fν(θ) fulfils (5.1) with

ν = ν(R0, α0 | θ − θ0) , (5.19)

where R0 and θ0 are defined through

R0 cosh (θ0) = R+ τ E(s)s (R, τ) , R0 sinh (θ0) = τ P(s)

s (R, τ) . (5.20)

– 23 –

Equations (5.20) imply

(R0)2 =(R+ τ E(s)

s (R, τ))2−(τ P(s)

s (R, τ))2

. (5.21)

The quantities E(s)k (R, τ) and P(s)

k (R, τ) denote the k-th higher conserved charges of the

theory deformed with the s-th perturbation

E(s)k (R, τ) = I

(s,+)k (R, τ) + I

(s,−)k (R, τ) , P(s)

k (R, τ) = I(s,+)k (R, τ)− I(s,−)

k (R, τ) , (5.22)

and I(s,±)k are again defined through (5.9)–(5.11) but with the deformed driving term (5.19).

Formula (5.19) shows that the solutions of the deformed NLIE are modified simply by a

redefinition of the length R and by a rapidity shift. One can easily show that the deformed

charges are related to the undeformed ones through

I(s,±)k (R, τ) = e±kθ0I

(±)k (R0) , (5.23)

which, together with (5.22), leads to a generalisation of the Lorentz-type transformation (1.2)

derived in [21] (E(s)k (R, τ)

P(s)k (R, τ)

)=

(cosh (k θ0) sinh (k θ0)

sinh (k θ0) cosh (k θ0)

)(Ek(R0)

Pk(R0)

). (5.24)

From (5.24) one has(E(s)k (R, τ)

)2−(P(s)k (R, τ)

)2= (Ek(R0))2 − (Pk(R0))2 , (5.25)

and, in particular, for k = 1

E2(R, τ)− P 2(R, τ) = E2(R0)− P 2(R0) , (5.26)

where E(R, τ) = E(s)1 (R, τ) and P (R, τ) = P(s)

1 (R, τ). To find the generalisations of the

Burgers equation (1.1) describing the evolution of the spectrum under the deformations with

generic Lorentz spin s, we differentiate both sides of (5.23) w.r.t. τ at fixed R0, getting

∂τI(s,±)k (R, τ) +R′∂RI

(s,±)k (R, τ) = k θ′0I

(s,±)k (R, τ) . (5.27)

All that is left to do is compute ∂τR = R′ and ∂τθ0 = θ′0. We start by rewriting (5.20) as{R0 = Re−θ0 + 2τ e(s−1)θ0I

(+)s (R0) ,

R0 = Reθ0 + 2τ e−(s−1)θ0I(−)s (R0) ,

(5.28)

then, differentiating both equations in (5.28) w.r.t. τ we obtain{0 = R′ e−θ0 −Re−θ0θ′0 + 2e(s−1)θ0I

(+)s (R0) + 2τe(s−1)θ0(s− 1) θ′0 I

(+)s (R0) ,

0 = R′ eθ0 +Reθ0θ′0 + 2e−(s−1)θ0I(−)s (R0)− 2τe−(s−1)θ0(s− 1) θ′0 I

(−)s (R0) ,

(5.29)

– 24 –

whose solution is

∂τR = R′ = −E(s)s (R, τ) +

(s− 1)τ(P(s)s (R, τ)

)2

(s− 1)τ E(s)s (R, τ)−R

,

∂τθ0 = θ′0 = − P(s)s (R, τ)

(s− 1)τ E(s)s (R, τ)−R

. (5.30)

The equations for the total energy and momentum are then{∂τE(R, τ) +R′∂RE(R, τ) = θ′0P (R, τ) ,

∂τP (R, τ) +R′∂RP (R, τ) = θ′0E(R, τ) ,(5.31)

which are driven, through R′ and θ′0 by the evolution equations for E(s)s and P(s)

s{∂τE(s)

s (R, τ) +R′∂RE(s)s (R, τ) = s θ′0 P

(s)s (R, τ) ,

∂τP(s)s (R, τ) +R′∂RP(s)

s (R, τ) = s θ′0 E(s)s (R, τ) .

(5.32)

We conclude this section by stressing again that, within the current framework, the most

general geometric-type theory corresponds to a deformation with an infinite number of irrel-

evant couplings {τi}, described by the scattering phase factor (5.15). The method described

here can be straightforwardly generalised to encompass cases with more than one parameter.

However, the study of these more complicated examples goes far beyond the scope of this

paper.

5.2 The CFT limit of the NLIE

The CFT limit of the sine-Gordon model is described by a pair of decoupled NLIEs cor-

responding to the right- (+) and the left- (−) mover sectors. The latter equations can be

obtained by sending m to zero as m = mε with ε → 0+ and simultaneously θ to ±∞ as

θ → θ± ln(ε) such that me±θ remain finite. The resulting equations are identical to (5.1) but

with the term m sinh(θ) replaced by m2 e

θ and − m2 e−θ, for the right- and left-mover sectors,

respectively

fν(±)(θ) = ν(±)(R,α(±)0 | θ)

−∫C1dθ′K(θ − θ′) ln

(1 + e−fν(±) (θ′)

)+

∫C2dθ′K(θ − θ′) ln

(1 + efν(±) (θ′)

),

(5.33)

where

ν(±)(R,α(±)0 | θ) = i2πα

(±)0 ∓ imR

2e±θ , (5.34)

and m sets the energy scale. In (5.34), we have allowed for the possibility of two independent

vacuum parameters α(±)0 in the two chiral sectors. The resulting NLIEs are, in principle,

– 25 –

suitable for the description of twisted boundary conditions and more general states in thec = 1 CFT, compared to the set strictly emerging from the m → 0 sine-Gordon model.For example, they can accommodate the states with odd fermionic numbers of the masslessThirring model, which require anti-periodic boundary conditions. The integration contoursC1, C2 are still state dependent. In particular, they should be deformed away from the initial

ground-state configuration C1 = R + i0+, C2 = R − i0+ when the parameters ±α(±)0 are

analytically extended to large negative values. Setting

b(±)k =

∫C1

dθ

2πi

(± r

2e±θ)k

ln(

1 + e−fν(±) (θ))−∫C2

dθ

2πi

(± r

2e±θ)k

ln(

1 + efν(±) (θ)), k ∈ 2N + 1 ,

(5.35)

with r = mR, then

I(±)k (R) =

(2π

R

)k b(±)k

Ck=

(1

R

)k2πa

(±)k , (5.36)

where a(±)k do not depend on R. Using again the spin-flip symmetry I

(±)−k (R) = I

(∓)k (R), we

can extend the discussion to k ∈ Z, with k = |k|. Some of the state-dependent coefficients

I(±)k (2π) can be found in [31]. In particular, the energy and momentum of a generic state are

E(R) = I(+)1 (R) + I

(−)1 (R) =

2π

R

(n(+) − c

(+)0

24

)+

2π

R

(n(−) − c

(−)0

24

),

P (R) = I(+)1 (R)− I(−)

1 (R) =2π

R

(h(+) − h(−)

), (5.37)

with effective central charges

c(±)0 = 1− 24β2

(α

(±)0

)2= 1− 24h

(±)0 , (5.38)

where h(±)0 are the holomorphic and antiholomorphic highest weights and h(±) = h

(±)0 +n(±),

n(±) ∈ N.

For the current purposes, it is convenient to think about the massless limit of the sine-

Gordon model as a perturbation by a relevant field of the compactified free boson with

lagrangian given by (5.5) with µ = 0 (see [36] for more details):

ϕ(x+R, t) = ϕ(x, t) +2π n√

2β, n ∈ Z . (5.39)

In (5.39), r = (√

2β)−1 is standard compactification radius of the bosonic field. Then, the

highest weights are now labelled by a pair of integers (n, n), where nβ2/√

4π is the quantized

charge associated to the total field momentum

Qt =

√2

2πβ

∫ R

0∂tϕdx =

2√

2

β

∫ R

0Π dx , (5.40)

and n is the winding number corresponding to the topological charge

Qx =

√2β

2π

∫ R

0∂xϕdx . (5.41)

– 26 –

Then the combinations

Q(±)0 = π (Qx ±Qt) = π

(n± nβ2

), (5.42)

are the two different charges, associated to the U(1)R × U(1)L symmetry of the c = 1 com-

pactified boson. Notice that Q(±)0 differ from the the standard Kac-Moody U(1)R × U(1)L

charges by a multiplicative factor β which spoils explicitly the β → 1/β symmetry. We

adopted this unconventional definition for the topological charges since, as a reminiscence of

the sine-Gordon model, it emerges more naturally from the current setup. The anagolous of

the Bloch wave states in (5.6) are now created by the action on the CFT vacuum state of the

vertex operators

V(n,n)(z) = exp

( √2

2πβQ

(+)0 φ(z) +

√2

2πβQ

(−)0 φ(z)

), ϕ(z) = φ(z) + φ(z) , (5.43)

with left and right conformal dimensions given by

h(±)0 =

1

4π2β2

(Q

(±)0

)2=

1

4

(n

β± nβ

)2

. (5.44)

Considering (5.38), in section 5.3 we will make the following identification:

Q(±)0 = 2πα

(±)0 β2 . (5.45)

However, relation (5.45) is valid only at formal level since α(±)0 are continuous parameters

which can also account, for example, for twisted boundary conditions while, at fixed β, the

charges Q(±)0 can only assume the discrete set of values given in (5.42).

Using (5.45) in (5.43) we find

V(n,n)(z) = exp(√

2βα(+)0 φ(z) +

√2βα

(−)0 φ(z)

), (5.46)

which, for n = 0 and under the field-shift (5.39), display the same quasi-periodicity properties

of the finite volume sine-Gordon Bloch states (5.6). In this limit, the τ dependent phase factor

(5.17) splits, for s > 0, into

δ(s)(±,∓)(θ, θ

′) = ±τ m2 γs e

±(θ−s θ′) , δ(s)(±,±)(θ, θ

′) = 0 , (5.47)

with

γs =(2πm)s

Cs, γ1 = m , (5.48)

breaking conformal invariance by explicitly introducing a coupling between the right and the

left mover sectors. Setting

R(s,±)0 = R+ 2τ I(s,∓)

s (R, τ) , (5.49)

the resulting NLIE is identical to (5.33) with driving term ν(±)(R(s,±)

0 , α0 | θ). For s < 0, the

two chiral sectors remain decoupled

δ(s′)(±,∓)(θ, θ

′) = 0 , δ(s′)(±,±)(θ, θ

′) = ±τ m2γs′ e

±(θ−s′ θ′) , (s′ = s < 0) , (5.50)

and, due to the reflection property I(s′,±)s = I

(s′,∓)s′ , the driving terms become ν(±)

(R(s′,∓)

0 , α0 | θ).

– 27 –

Perturbations with s < 0 :

For simplicity, let us discuss first the cases with s < 0. Since the generic k-th charges of a

CFT scale as6

I(±)k (R) =

1

Rk2π a

(±)k , (k > 0) , (5.51)

with a(±)k constants, the corresponding deformed charges I

(s′,±)k (R, τ) fulfil the set of equations

I(s′,±)k (R, τ) = I

(s′,∓)k′ (R, τ) =

2π a(∓)k′(

R(s′,±)0

)k =2π a

(∓)k′(

R+ 2τ I(s′,∓)s′ (R, τ)

)k , (5.52)

where a(±)k = a

(∓)k′ , and k′ = −k, s′ = s. In order to find the solution to (5.52) for generic k,

one must first solve (5.52) for k = s (k′ = s′). In this case, the solution can be reconstructed

perturbatively as

I(s′,±)s (R, τ) = I

(s′,∓)s′ (R, τ) =

∞∑j=0

(−1)j

j + 1

(j(1 + s) + (s− 1)

j

)(2τ)j

(2π a

(±)s

)j+1

Rj(1+s)+s. (5.53)

This expression can be resummed as

• s = −1 :

I(−1,±)1 (R, τ) =

R

4τ

−1 +

√1 + 8τ

2π a(±)1

R2

. (5.54)

The latter result is in full agreement with the classical analysis performed in section 3.4. It

confirms the interesting fact that the leading perturbing operator is that typically associated

to a lattice regularisation [29].

• s = −2 :

I(−2,±)2 (R, τ) =

4R

6τsinh

1

3arcsinh

3√

3

2

(2τ

2π a(±)2

R3

)1/2 . (5.55)

For generic spin s < 0 the result can be written in terms of a single generalised hypergeometric

function:

I(s′,±)s (R, τ) = I

(s′,∓)s′ (R, τ) =

sR

2τ (1 + s)

(−1 + Fs

(−2τ

(1 + s)1+s

ss2π a

(±)s

R1+s

)), (5.56)

with

Fk(x) = kFk−1

(− 1

k + 1,

1

k + 1,

2

k + 1, . . . ,

k − 1

k + 1;

1

k,

2

k, . . . ,

k − 1

k;x

). (5.57)

6Separately, the NLIEs in (5.33), with generic parameters β and α(±)0 can be also associated to the quantum

KdV theory, as extensively discussed in [31, 38]. The coefficients ak for k = 3, 5 can be recovered from [31].

– 28 –

The total momentum and energy are

E(R) = I(s′,+)1 (R, τ) + I

(s′,−)1 (R, τ) , P (R) = I

(s′,+)1 (R, τ)− I(s′,−)

1 (R, τ) , (5.58)

where I(s′,±)1 (R, τ) are obtained by solving (5.52) with k′ = −k = −1 using (5.56).

Finally, notice that even spin charges do not, in general, correspond to local conserved currents

in the sine-Gordon model. They can occasionally emerge from the set of non-local charges,

at specific rational values of β2. Our results concerning the exact quantum spectrum can

be smoothly deformed in s, therefore they should formally also describe deformations of the

sine-Gordon model by non-local currents [39]. Moreover, there are many integrable systems

with extended symmetries where even spin charges appear. The sign of the corresponding

eigenvalues depends on the internal flavor of the specific soliton configuration considered.

Since the flow equations (5.30)–(5.32), should properly describe the evolution of the spectrum

driven by analogous deformations in a very wide class of systems, perturbations by currents

with s even, should lead to interesting quantum gravity toy models where the effective sign

of the perturbing parameter τ depends on the specific state under consideration.

Perturbations with s > 0 :

In the case s > 0, the left- and right-mover sectors are now coupled and the solution of the

generalised Burgers equations become equivalent to the set of equations

I(s,±)k (R, τ) =

2π a(±)k(

R+ 2τ I(s,∓)s (R, τ)

)k . (5.59)

While, for models with P(s)k = I

(s,+)k (R, τ)− I(s,−)

k (R, τ) = 0 the left- and right-mover sectors

are completely symmetric and the relations (5.59) formally reduce to the equations (5.52),

for P(s)k 6= 0 the solution to (5.59) for k = s are

I(s,±)s (R, τ) =

2π a(±)s

Rs

[1 + 2π a(∓)

s

∞∑k=1

k−1∑l=0

1

l + 1

((k − l)s+ l − 1

l

) ((l + 1)s+ k − l − 1

k − l

)

×(

2π a(±)s

)l (2π a(∓)

s

)k−l−1 (−1)k(2τ)k

R(s+1)k

]. (5.60)

We were not able to find a general compact expression of (5.60), except for the already known

(s = 1) TT-related result

E(1)1 (R, τ) =

R

2τ

−1 +

√√√√1 +4τ

R

(2π(a

(+)1 + a

(−)1

)R

)+

4τ2

R2

(2π(a

(+)1 − a(−)

1

)R

)2 ,

P(1)1 (R) =

2π(a

(+)1 − a(−)

1

)R

. (5.61)

– 29 –

Perturbations with s→ 0 :

Let us first perform the s → 0 limit in the massive case. In this limit γs → 0 but, rescaling

τ , we can recast the driving term in the form

ν(R,α0 | θ) = i2πα0 − imR sinh(θ) + iτG cosh(θ) , (5.62)

where

G = −1

2

(∫C1

dθ

2πiln(

1 + e−fν(θ))−∫C2

dθ

2πiln(

1 + efν(θ)))

. (5.63)

Next, in the limit m → 0, the leading contribution to G in (5.35) is coming from a large

plateau of the integrand (g(θ) in figure 3) of height

− ifν(θ) ∼ 4πα0β2 , (5.64)

and width growing as ∼ 2 ln(mR/2):

G ∼ ln

(mR

2

)2α0β

2 . (5.65)

Therefore, at fixed finite R, after a further rescaling τ → τπ/ ln(mR/2), as m tends to zero,

we obtain for the driving terms in the NLIEs:

ν(±) = ν(±)(R(±)0 , α0 | θ) = i2πα0 ∓ i

m

2e±θR(±)

0 , (5.66)

with

R(±)0 = R∓ τQ0 , Q0 = 2πα0β

2 . (5.67)

Thus, perturbing the theory with the phase factor (5.17) with s = 0 is equivalent, in an

appropriate scaling limit, to a constant shift of the volume R. At fixed normalisation of

the deforming parameter τ , the shift turns out to be directly proportional to the topological

charge Q0. We see, from (5.66) that the left- and right-mover sectors remain decoupled also

at τ 6= 0. However, starting directly from the CFT limit, one can argue that there exist

less trivial ways to couple the two sectors. The most general variant involves four different

coupling constants ~τ = {τ (a|b)} with a, b ∈ {+,−}:

R(±)0 (~τ) = R∓

(τ (±|+)Q

(+)0 + τ (±|−)Q

(−)0

), Q

(±)0 = 2πα

(±)0 β2 . (5.68)

Notice that in (5.68) we allowed the possibility for two different topological charges Q(±)0 ,

associated to the U(1)R × U(1)L symmetry of the c = 1 free (compactified) boson model,

corresponding to the CFT limit of the sine-Gordon theory.

– 30 –

Figure 3. The formation of the plateau in g(θ) = − 12π=m

(ln(

1 + e−fν(θ+i0+)))

as m→ 0.

5.3 Further deformations involving the topological charge

A further, natural extension of the quantum models studied in section 5.1, corresponds to

scattering phase factors of the form

δ(s,s)(θ, θ′) = τ γsγs sinh(s θ − s θ′) . (5.69)

Deforming the kernel according to (5.18) and using (5.69) instead of (5.17), the driving term

of (5.1) becomes

ν = ν(R,α0 | θ)− iτ s γs(es θI

(s,s,−)s − e−s θI(s,s,+)

s

), (5.70)

where I(s,s,±)k are defined through (5.35) and (5.36) with the deformed driving term (5.70).

The cases with s = ±1, have been extensively discussed in the previous sections, they all

correspond to gravity-like theories, where the effect of the perturbation can be re-absorbed

into a redefinition of the volume R plus a shift in the rapidity θ, according to (5.19). From

equation (5.70), we see that for generic values of s 6= ±1 it is not no longer possible to

re-absorb the perturbation in the same fashion. However, another interesting example is

recovered by considering the scaling limit {s, γs,m} → {0, 0, 0}, such that s γs remains finite.

In the following we shall set s γs = 2π. This situation corresponds to the standard massless

limit where, in the undeformed (τ = 0) NLIE, the right- (+) and the left-mover (−) sectors

are completely decoupled, while a residual interaction between the two sectors is still present

at τ 6= 0. In fact, the deformed versions of fν(±)(θ) fulfil (5.33) with

ν(±) = ν(±)(R,α

(±)0 − α(s,±) | θ

), α(s,±) =

(τ (±)I

(s,−)s (R,~τ)− τ (∓)I

(s,+)s (R,~τ)

), (5.71)

– 31 –

where the charges I(s,±)k are defined through (5.35) with driving term (5.71), and {τ (−), τ (+)} =

~τ are two coupling parameters defined as

τ (±) = τe±σ , ±σ = limθ→±∞s→0+

s θ . (5.72)

Therefore, the contributions of the nontrivial interaction can be formally re-absorbed in a

redefinition of the vacuum parameters α(±)0 . In turn, this affects the value of the effective

central charges as

c(s,±) = c(±)0 + 24β2(α

(±)0 )2 − 24β2

((α

(±)0 )2 − α(s,±)

)2, c

(±)0 = c− 24β2(α

(±)0 )2 . (5.73)

Considering the formal identification Q(±)0 = 2πα

(±)0 β2 made in (5.67), the redefinition (5.71)

corresponds to the following dressing of the topological charge

Q(s,±)(R,~τ) = Q0 −κ

4π

(τ (−)I

(s,−)s (R,~τ)− τ (+)I

(s,+)s (R,~τ)

), (5.74)

with

κ = 8π2β2 . (5.75)

Let us focus on the s = 1 case. The equations for the spectrum are

I(±)1 (R,~τ) =

2π

R

(n(±) − c(±)

24

), (I

(±)1 = I

(1,±)1 , c(±) = c(1,±)) , (5.76)

which can be solved exactly for I(±)1 (R,~τ) at any values of the parameters τ (±). However, since

the general analytic expressions for I(±)1 are very cumbersome, we will restrict the discussion

to specific scaling limits:

• τ (+) = τ (−) = τ :

I(+)1 (R, τ)− I(−)

1 (R, τ) = P (R) =2π

R

(h(+) − h(−)

), (5.77)

I(+)1 (R, τ) = I

(+)1 (R) + τ

Q(±)0

RP (R) + τ2 κ

8πRP 2(R) , (5.78)

with h(±) = h(±)0 + n(±) and (5.74) becomes

Q(±)(R, τ) = Q(±)0 + τ

κ

4πP (R) . (5.79)

Surprisingly, with the replacement

I(±)1 (R, τ) =

2π

R

(h(±)(τ)− c

24

), (5.80)

equations (5.77) and (5.78) lead to exact expressions for the conformal dimensions h(±)(τ),

which precisely match the form of the (left) conformal dimension in the JT model, as

recently shown in [40]:

h(±)(τ) = h(±) + τQ

(±)0

2πP + τ2 κ

16π2P 2 . (5.81)

– 32 –

As we will shortly see, this is the first instance among many that link the phase factor

(5.69) with s = 1, in the scaling limit (5.72) to the quantum JT model.

• τ (+) = 0 , τ (−) = τ : the two sectors (±) are completely decoupled

I(±)1 (R, τ) = I

(±)1 (R)± τ Q

(±)0

RI

(±)1 (R, τ) + τ2 κ

8πR

(I

(±)1 (R, τ)

)2, (5.82)

Q(±)(R, τ) = Q(±)0 ± τ κ

4πI

(±)1 (R, τ) , (5.83)

and the solutions to (5.82) are

I(±)1 (R, τ) =

4π

κτ2

(R∓ τQ(±)

0 −√(

R∓ τQ(±)0

)2− τ2κ

(h(±) − c

24

)). (5.84)

We notice that (5.82) can be rewritten as

I(±)1 (R, τ) =

2π(h(±) − c

24

)R− τ

(±Q0 + τ κ

8π I(±)1 (R, τ)

) , (5.85)

which suggests that, in this limit, the perturbation has a dual gravitational description,

since it can be interpreted as a redefinition of the length R. The dual deformation of the

NLIE corresponds to a deformed version of fν(±)(θ) which fulfils (5.33) with

ν(±) = ν(±)(R(±)

0 , α0 θ), R(±)

0 = R− τ(±Q0 + τ

κ

8πI

(±)1 (R, τ)

). (5.86)

Expressions (5.84) are trivially solutions of two decoupled Burgers-like equations

∂τI(±)1 (R, τ)±Q(±)(R, τ) ∂RI

(±)1 (R, τ) = 0 . (5.87)

Therefore, I(±)1 (R, τ) fulfils a Burgers-type equation analogous to (1.1), where Q(±)(R, τ)

play the role of velocities.

5.4 The quantum JT model

Formula (5.84) strongly resembles the expression for the energy of the right-movers of the

JT-deformed CFT derived in [23, 26]. To obtain the JT model, one must treat the (±) sectors

in a non-symmetric way. For s = 1, a possible asymmetric generalisation of (5.71) with four

free parameters ~τ = {τ (a|b)} with a, b ∈ {+,−} is

ν(±) = ν(±)(R,α

(±)0 − α(±)(~τ) | θ

),

α(±)(~τ) =(τ (±|−)I(−)

s (R,~τ)− τ (∓|+)I(+)s (R,~τ)

). (5.88)

In (5.88) we also included the possibility to have two different initial values of the twist

parameters α(±)0 in the two sectors. From (5.88), the JT model is recovered with the choice

τ (±|−) = 0 , τ (∓|+) = τ , α(+)0 = α

(−)0 = α0 . (5.89)

– 33 –

Correspondingly, relations (5.88) become

ν(±) = ν(±)

JT

(R,α0 − α(±)

JTθ), α

(±)

JT= −τ I(+)

JT(R, τ) . (5.90)

Alternatively, the (+) sector can be equivalently described with a redefinition of the length

R as

ν(+) = ν(+) (RJT, α0 θ) , RJT = R− τ(Q0 + τ

κ

8πI

(+)

JT(R, τ)

). (5.91)

The right-moving solution in (5.84) and the topological charge become

I(+)1 (R, τ) = I

(+)

JT(R, τ) =

4π

κτ2

(R− τQ0 −

√(R− τQ0)2 − τ2κ

(h(+) − c

24

)), (5.92)

I(+)

JT(R, τ)− I(−)

JT(R, τ) = P (R) =

2π

R

(h(+) − h(−)

), (5.93)

Q(+)(R, τ) = Q(+)

JT(R, τ) = Q0 + τ

κ

4πI

(+)

JT(R, τ) . (5.94)

Therefore, Q0 = 2πα0β2 and κ = 8π2β2 have been again consistently identified with the

topological charge and the chiral anomaly, respectively. The results described here are in full

agreement with [23] and the classical results presented in section 3.

5.5 A simple example involving a pair of scattering phase factors

As already discussed, in principle one may introduce several scattering phase factors to deform

the NLIEs. In this section we will consider a particular combination of s → 0 and s →0 scattering factors which allows us to match with the classical results (3.74)–(3.75). We

consider a double perturbation made of a length redefinition (5.68) with τ (±|+) = ∓τ and

τ (±|−) = ±τ , together with a shift of the twist parameter (5.88) with τ (∓|+) = +τ and

τ (±|−) = +τ . The corresponding deformed driving term is then

ν(±) = ν(±)(R(±)

0 , α(±)0 − α(±)(R, τ) θ

), (5.95)

with

α(±)(R, τ) = −τ P (R) ,

R(±)0 = R+ τ

(Q(+)(R, τ)−Q(−)(R, τ)

)= R+ τ

(Q

(+)0 −Q(−)

0

). (5.96)

where in the last equality we used the fact that Q(±)(R, τ) = 2π β2(α

(±)0 − α(±)(R, τ)

). Since

the central charges are affected by the deformation as

c(±)(R, τ) = c+ 24β2(α

(±)0

)2 − 24β2(α

(±)0 − α(±)(R, τ)

)2, (5.97)

one finds

I(±)1 (R, τ) =

2π

R(±)0

(n(±) − c(±)(R, τ)

24

)=RI

(±)1 (R) + τ Q

(±)0 P (R) + κ

8π τ2 P 2(R)

R+ τ(Q

(+)0 −Q(−)

0

) , (5.98)

I(+)1 (R, τ)− I(−)

1 (R, τ) = P (R) , (5.99)

Q(±)(R, τ) = Q(±)0 + τ

κ

4πP (R) , (5.100)

– 34 –

which match exactly with the results (3.74)–(3.75) obtained at the classical level for the

corresponding densities.

6 Conclusions

In this paper, we have introduced and studied novel types of systems obtained by a local

space-time deformation induced by higher-spin conserved charges. We have argued that these

geometrical maps correspond to classical and quantum field theories perturbed by irrelevant

Lorentz-breaking composite fields. The possibility of non-Lorentz invariant generalisations

of the TT operator was first briefly discussed in [1] and the JT and the TJ models studied

in [22–28] are concrete realisations of these ideas, though with some important difference

in the actual details. Another concise discussion on non-Lorentz invariant perturbations

has appeared more recently in [41] were it was underlined the possibility to replace the TT

operator with composite fields built out of a pair of conserved currents, a somehow obvious

generalisation of the earlier comments [1] and results [22–28] which however also encompasses

the systems effectively studied in the current paper. Here we would like to stress again that the

spin s in the scattering phase factor (5.18) can also assume non-integer values, therefore the

extension to non-local conserved charges [38] appears to be straightforward, as also underlined

in [1].

Many open problems deserve further attention. First of all, it is natural to try to identify

within the AdS3/CFT2 setup the corresponding (bulk) JJ and Yang-Baxter type deforma-

tions, as it was successfully done for their close relatives studied [22–28, 28]. It would be

important to extend the gravity setup adopted in [19, 42] to accommodate also these novel

geometric deformations.

Furthermore, while the TT perturbation is compatible with supersymmetry [16, 27, 43–

45], higher spin Hamiltonians should, at least partially, lift the degeneracy of the states.

Therefore, in general, supersymmetry should be explicitly broken by the more general per-

turbations discussed here. However, it would be interesting to find specific examples were a

residual supersymmetry survives. Finally, it would be also nice to study the effect of these

perturbations on 2D Yang-Mills [21, 46] and, following the suggestion of [47], to study the

expectation values on curved spacetimes, of the composite operators discussed here. Finally,

while this paper was in its final writing stage the works [48, 49] appeared. The latter articles

contain many interesting results on the JT model and the combined JT/TT perturbations.

There are only minor overlaps with the current paper. As already mentioned, our methods

are easily generalisable to encompass such a two-coupling extension of the models described

here, both at classical and quantum levels.

Acknowledgments – We are especially grateful to Riccardo Borsato, Ferdinando Gliozzi,

Yunfeng Jiang, Zohar Komargodski, Bruno Le Floch, Sergei Lukyanov, Marc Mezei,

Domenico Orlando, Alessandro Sfondrini, Kentaroh Yoshida and Sasha Zamolodchikov for

useful discussions. This project was partially supported by the INFN project SFT, the EU

network GATIS+, NSF Award PHY-1620628, and by the FCT Project

– 35 –

PTDC/MAT-PUR/30234/2017 “Irregular connections on algebraic curves and Quantum

Field Theory”. RT gratefully acknowledges support from the Simons Center for Geometry

and Physics, Stony Brook University at which some of the research for this paper was

performed.

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