Conserved currents and T T ¯ s irrelevant deformations of 2D integrable field theories
PPrepared for submission to JHEP
Conserved currents and T ¯ T s irrelevant deformationsof 2D integrable field theories Riccardo Conti, Stefano Negro, Roberto Tateo Dipartimento di Fisica and Arnold-Regge Center, Università di Torino, and INFN, Sezione diTorino, Via P. Giuria 1, I-10125 Torino, Italy. C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, NY11794-3840. U.S.A.
E-mail: [email protected] , [email protected] , [email protected] Abstract:
It has been recently discovered that the T ¯ T deformation is closely-related toJackiw-Teitelboim gravity. At classical level, the introduction of this perturbation inducesan interaction between the stress-energy tensor and space-time and the deformed EoMs canbe 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-perturbativelydescribed by an inhomogeneous Burgers equation. In this paper, we point out that there existinfinite families of models where the geometry couples instead to generic combinations of localconserved currents labelled by the Lorentz spin. In spirit, these generalisations are similarto the J ¯ T model as the resulting theories and the corresponding scattering phase factors arenot Lorentz invariant. The link with the J ¯ T model is discussed in detail. While the classicalsetup described here is very general, we shall use the sine-Gordon model and its CFT limit asexplanatory quantum examples. Most of the final equations and considerations are, however,of broader validity or easily generalisable to more complicated systems. a r X i v : . [ h e p - t h ] S e p ontents ¯ T-deformed higher spin conserved currents 4 ¯ T-deformed higher conserved currents 42.2 The massless free boson 6 s > s < s → and the J ¯ T-type models 17 ¯ T model 394.6 A simple example involving a pair of scattering phase factors 40
A.1 Explicit Lorentz breaking in a simple example 42
B Burgers equations 45
The deformation of 2D quantum field theories [1, 2] by the Zamolodchikov’s T ¯ T operator [3],has recently attracted the attention of theoretical physicists due to the many important linkswith effective string theory [4–7] and the AdS/CFT correspondence [8–17]. A remarkableproperty of this perturbation, discovered in [1, 2], concerns the evolution of the quantumspectrum at finite volume R , with periodic boundary conditions, in terms of the T ¯ T couplingconstant τ . The spectrum is governed by the hydrodynamic-type equation ∂ τ E ( R, τ ) = 12 ∂ R (cid:0) E ( R, τ ) − P ( R ) (cid:1) , (1.1)– 1 –here E ( R, τ ) and P ( R ) are the eigenvalues of the energy and momentum operator, respec-tively, on a generic eigenstate | n (cid:105) . Important for the current purposes is that, under theperturbation, the evolution of the spectrum is equivalently encoded in the following Lorentz-type transformation (cid:32) E ( R, τ ) P ( R ) (cid:33) = (cid:32) cosh ( θ ) sinh ( θ )sinh ( θ ) cosh ( θ ) (cid:33) (cid:32) E ( R ) P ( R ) (cid:33) , (1.2)with R and θ defined through sinh θ = τ P ( R ) R = τ P ( R ) R , cosh θ = R + τ E ( R, τ ) R = R − τ E ( R ) R . (1.3)From (1.2), it follows that the solution to (1.1) can be written, in implicit form, as E ( R, τ ) − P ( R ) = E ( R ) − P ( R ) , (1.4)with the additional constraint ∂ τ R = − E ( R, τ ) , (1.5)at fixed R , obtainable directly from (1.3) (cf. with the s = 1 case of (4.50)).As extensively discussed in [18] (see also [19]), the solutions to the classical EoMs associatedto the T ¯ T-deformed Lagrangians [2, 18, 20, 21] are obtained from the τ = 0 ones by a field-dependent coordinate transformation dx µ = (cid:16) δ µν + τ (cid:101) T µν ( y ) (cid:17) dy ν , y = ( y , y ) = ( x (cid:48) , t (cid:48) ) , (1.6)with (cid:101) T µν ( y ) = − g µδ (cid:15) δρ (cid:15) σν T ρσ ( y ) , (cid:15) µν = (cid:32) − (cid:33) µν , (1.7)where T µν ( y ) is the Hilbert stress-energy tensor associated to the undeformed theory, canon-ically defined as T µν ( y ) = − √ g ∂ L g ( y ) ∂g µν , √ g = (cid:113) det ( g µν ) , (1.8)and L g ( y ) is the undeformed Lagrangian in the set of cartesian coordinates y , minimallycoupled to gravity through the metric g µν . As shown in [18], equation (1.6) can be invertedas dy µ = (cid:16) δ µν − τ (cid:101) T µν ( x , τ ) (cid:17) dx ν , x = ( x , x ) = ( x, t ) , (1.9)with (cid:101) T µν ( x , τ ) = − g µδ (cid:15) δρ (cid:15) σν T ρσ ( x , τ ) , (1.10) In (1.9), we corrected a sign typo made in [18]. – 2 –here T µν ( x , τ ) is now the Hilbert stress-energy tensor associated to the T ¯ T-deformed theory T µν ( x , τ ) = − √ g ∂ L g ( x , τ ) ∂g µν , (1.11)and L g ( x , τ ) is the deformed Lagrangian in the set of cartesian coordinates x . Following theconvention of [18], we will switch from cartesian to complex coordinates according to (cid:40) z = x + i x ¯ z = x − i x , (cid:40) w = y + i y ¯ w = y − i y , (1.12)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 shownthat JT gravity coupled to matter leads to a scattering phase matching that associated to theT ¯ T perturbation [1, 2, 4–6]. Equations (1.6) and (1.9) were also obtained in [18] starting fromthe deformed EoMs following, therefore, a completely independent line of thoughts comparedto [19]. However, the final results turn out to be fully consistent with the proposal of [19]. Inthis paper, we shall argue that the approach of [18] admits natural generalisations correspond-ing 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 mixedderivatives implies ∂ x µ ∂y ρ ∂y σ = ∂ x µ ∂y σ ∂y ρ ⇐⇒ ∂ µ T µν = 0 . (1.13)Equation (1.13) suggests that a consistent and natural generalisation of (1.6) can be obtainedby 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 T µν with a matrix built using the higher-spin conserved currents,typically present in free or integrable theories. In complex coordinates z , the spin s conservedcurrents T s +1 , ¯T s +1 , Θ s − and ¯Θ s − are related through the continuity equations ¯ ∂ T s +1 = ∂ Θ s − , ∂ ¯T s +1 = ¯ ∂ ¯Θ s − , ( s ∈ N ) , (1.14)where ∂ = ∂ z and ¯ ∂ = ∂ ¯ z , the subscripts s + 1 and s − are the Lorentz spins of thecorresponding field and the s = 1 case of (1.14) corresponds to the energy and momentumconservation law. The replacement we shall perform in (1.6)–(1.7) is T −→ T s , (1.15)where the cartesian and complex components of T s are related through ( T s ) = − π (cid:0) ¯T s +1 + T s +1 − ¯Θ s − − Θ s − (cid:1) , ( T s ) = i π (cid:0) ¯T s +1 − T s +1 + ¯Θ s − − Θ s − (cid:1) , ( T s ) = i π (cid:0) ¯T s +1 − T s +1 − ¯Θ s − + Θ s − (cid:1) , ( T s ) = 12 π (cid:0) ¯T s +1 + T s +1 + ¯Θ s − + Θ s − (cid:1) . (1.16)– 3 –or simplicity, it will be useful to define the re-scaled quantities T s +1 = T s +1 π , Θ s − = Θ s − π , ¯T s +1 = ¯T s +1 π , ¯Θ s − = ¯Θ s − π . (1.17)One of the main consequences of the replacement (1.15) is that the resulting generaliseddeformations break explicitly the (eventual) Lorentz symmetry of the original models. Seeappendix A for an explicit example where the difference between the T ¯ T and the s (cid:54) = 1 cases clearly emerge. The corresponding gravity-like systems have therefore many features incommon with the J ¯ T (T ¯ J) models recently studied in [22–28]. As we shall see in section 3.1,in the reference frame w , the deformed Hamiltonian – formally integrated over the space-timein the z variable – can be written as the integrated bare Hamiltonian plus a perturbing field: (cid:90) H ( s ) ( z , τ ) dz ∧ d ¯ z = (cid:90) H ( w ) dw ∧ d ¯ w + τ (cid:90) Φ , s ( w ) dw ∧ d ¯ w , (1.18)where the perturbing operator Φ , s ( w ) reduces to Φ , s ( w ) = T( w ) ¯T s +1 ( ¯ w ) + ¯T( ¯ w ) T s +1 ( w ) , (1.19)when the original theory is a CFT. For this reason we shall denote this newly-introduced classof systems as geometrically T ¯ T s -deformed theories. A precise connection between the s = 0 case and the J ¯ T models will be established in section 4.5. As in [2, 18, 21], starting fromsection 4.2, we shall use the sine-Gordon model as a specific quantum example. However, wewould like to stress that the techniques adopted and some of the results, i.e. the generalisedBurgers equations (4.50)–(4.52), are at least formally, of much wider validity. ¯ T-deformed higher spin conserved currents
The aim of this section is to introduce an efficient method, based on the field-dependentcoordinate transformation derived in [18], to reconstruct the local Integrals of Motion (IMs)associated to the T ¯ T deformation of a generic integrable field theory. The application of theseideas to the family of classical geometrically T ¯ T s deformed models will be described in section3. ¯ T-deformed higher conserved currents
Let us consider the following pair of conjugated 1-forms I k = T k +1 ( z , τ ) dz + Θ k − ( z , τ ) d ¯ z , ¯ I k = ¯T k +1 ( z , τ ) d ¯ z + ¯Θ k − ( z , τ ) dz , ( k ∈ N ) , (2.1) Notice that this convention differs by a factor compared to [2, 18]. This choice allows for a moretransparent match between the classical and the quantum results of section 4. It turns out that these perturbations are, in general, different from the ones recently studied in [29].Currently, we do not know if it exists a link between the class of deformations considered in [29] and ours. Inparticular, it would be nice to understand whether or not the models of [29] possess a geometric interpretationin terms of specific space-time coordinate transformations on the plane. – 4 –here the components T k +1 ( z , τ ) , Θ k − ( z , τ ) and their complex conjugates are the higherconserved currents of the T ¯ T-deformed integrable theory which fulfil the continuity equations ¯ ∂ T k +1 ( z , τ ) = ∂ Θ k − ( z , τ ) , ∂ ¯T k +1 ( z , τ ) = ¯ ∂ ¯Θ k − ( z , τ ) , (2.2)and k denotes the Lorentz spin. Using (2.2), it is easy to check that (2.1) are closed forms d I k = (cid:0) ∂ Θ k − − ¯ ∂ T k +1 (cid:1) dz ∧ d ¯ z = 0 , d ¯ I k = (cid:0) ∂ ¯T k +1 − ¯ ∂ ¯Θ k − (cid:1) dz ∧ d ¯ z = 0 , (2.3)therefore, for any given integration contour C , the following integrals (cid:90) C I k , (cid:90) C ¯ I k , (2.4)do not depend on deformations of C , at fixed end-points. Expressions (2.4) can be used todefine local IMs. From their very definition, differential forms are the right objects to beintegrated over manifolds since they are independent of coordinates. In fact, since -formsremain closed also under field-dependent coordinate transformations, we can construct the T ¯ T-deformed local IMs from the undeformed ones, using the change of coordinates introduced in[18, 19]. The strategy is the following: • Start from the -forms (2.1) expressed in w coordinates I k = T k +1 ( w ) dw + Θ k − ( w ) d ¯ w , ¯ I k = ¯T k +1 ( w ) d ¯ w + ¯Θ k − ( w ) dw ; (2.5)where T k +1 ( w ) and Θ k − ( w ) and their complex conjugates are the higher conserved cur-rents of the undeformed theory which fulfil the continuity equations ∂ ¯ w T k +1 ( w ) = ∂ w Θ k − ( w ) , ∂ w ¯T k +1 ( w ) = ∂ ¯ w ¯Θ k − ( w ) . (2.6) • Consider the change of coordinates w = w ( z ) (see [18]), which at differential level acts asfollows (cid:32) dwd ¯ w (cid:33) = J T (cid:32) dzd ¯ z (cid:33) , (cid:32) ∂ w f∂ ¯ w f (cid:33) = J − (cid:32) ∂f ¯ ∂f (cid:33) , ( ∀ f : R → R ) , (2.7)where the Jacobian and its inverse are J = (cid:32) ∂w ∂ ¯ w ¯ ∂w ¯ ∂ ¯ w (cid:33) = 1∆( w ) (cid:32) τ Θ ( w ) − τ T ( w ) − τ ¯T ( w ) 1 + 2 τ ¯Θ ( w ) (cid:33) , (2.8) J − = (cid:32) ∂ w z ∂ w ¯ z∂ ¯ w z ∂ ¯ w ¯ z (cid:33) = (cid:32) τ ¯Θ ( w ) 2 τ T ( w )2 τ ¯T ( w ) 1 + 2 τ Θ ( w ) (cid:33) , (2.9)and ∆( w ) = (cid:0) τ Θ ( w ) (cid:1)(cid:0) τ ¯Θ ( w ) (cid:1) − τ T ( w ) ¯T ( w ) . Thus they are locally exact by the Poincaré lemma. – 5 –
Use the explicit expressions of T ( w ) , Θ ( w ) and their complex conjugates in terms of thefundamental fields of the theory to explicitly invert the map w ( z ) at differential level. Then,use the first expression of (2.7) in (2.5) I k = T k +1 ( w ( z )) + 2 τ (cid:0) T k +1 ( w ( z )) Θ ( w ( z )) − Θ k − ( w ( z )) T ( w ( z )) (cid:1) ∆( w ( z )) dz + Θ k − ( w ( z )) + 2 τ (cid:0) Θ k − ( w ( z )) ¯Θ ( w ( z )) − T k +1 ( w ( z )) ¯T ( w ( z )) (cid:1) ∆( w ( z )) d ¯ z , (2.10)where F ( w ( z )) indicates that the fundamental fields in w coordinates involved in an arbi-trary function F , have been replaced with fundamental fields in z coordinates according tothe map w ( z ) . • Read the T ¯ T-deformed higher conserved currents as components of (2.10) in z coordinates: T k +1 ( z , τ ) = T k +1 ( w ( z )) + 2 τ (cid:0) T k +1 ( w ( z )) Θ ( w ( z )) − Θ k − ( w ( z )) T ( w ( z )) (cid:1) ∆( w ( z )) , Θ k − ( z , τ ) = Θ k − ( w ( z )) + 2 τ (cid:0) Θ k − ( w ( z )) ¯Θ ( w ( z )) − T k +1 ( w ( z )) ¯T ( w ( z )) (cid:1) ∆( w ( z )) . (2.11)By definition, the integrals (2.4) are invariant under coordinate transformations, provided theintegration contour C is mapped into C (cid:48) accordingly (cid:90) C T k +1 ( z , τ ) dz + Θ k − ( z , τ ) d ¯ z = (cid:90) C (cid:48) T k +1 ( w ) dw + Θ k − ( w ) d ¯ w . (2.12)The conserved charges are obtained by integrating over the whole volume at constant time, inthe x reference frame. However, as clearly emerges from (1.6) and (1.9), the equal time struc-ture in the x − plane gets distorted in the y − plane and viceversa, which causes the emergingof a τ − dependent flow in the charges. In the current setup, the resulting integrals of motioncoincide with T ¯ T-deformed ones. The application of this scheme to the more general changeof variables defined through the replacement (1.15), will lead instead to different, but equallyinteresting, families of deformed classical Hamiltonians (see, section 3).In order to make the above strategy more concrete, in the following section, we will explicitlydiscuss the construction of the T ¯ T-deformed higher currents for the massless free boson theoryand comment on more general cases.
Consider the Lagrangian of a single massless boson field φ in complex coordinates w L ( w ) = ∂ w φ ∂ ¯ w φ . (2.13)The EoMs are ∂ w ∂ ¯ w φ = 0 , (2.14)– 6 –herefore, without further external constraints, there exists an infinite number of options forthe choice of the basis of conserved currents. For example, both T ( POW ) k +1 ( w ) = −
12 ( ∂ w φ ) k +1 , Θ ( POW ) k − ( w ) = 0 , ( k ∈ N ) , (2.15)and T ( KG ) k +1 ( w ) = − (cid:16) ∂ k w φ (cid:17) , Θ ( KG ) k − ( w ) = 0 , ( k ∈ N + 1) , (2.16)are possible sets of higher conserved currents since they fulfil (2.6) on-shell. In general, anylinear combination of the form T ( GEN ) k +1 ( w ) = k +1 (cid:88) j =0 c ( k ) j ( ∂ w φ ) k +1 − j ∂ jw φ , Θ ( GEN ) k − ( w ) = k − (cid:88) j =0 ¯ c ( k ) j ( ∂ ¯ w φ ) k − − j ∂ j ¯ w φ , (2.17)automatically defines a conserved current with spin k . Moreover, since the change of variables(1.6) is non-linear, different choices of the current in (1.15) should, at least in principle, giverise to different classical deformations of the original theory.For simplicity, we will consider the sets (2.15)–(2.16) separately. Following the strategy de-scribed in the previous section, we shall first derive the differential map, which means toexpress ( ∂ w f, ∂ ¯ w f ) T in terms of ( ∂f, ¯ ∂f ) T , ∀ f : R → R . Setting f = φ in the second ex-pression of (2.7), we first write ( ∂ w φ, ∂ ¯ w φ ) T as a function of ( ∂φ, ¯ ∂φ ) T by solving the set ofalgebraic equations (cid:32) ∂φ ¯ ∂φ (cid:33) = J (cid:32) ∂ w φ∂ ¯ w φ (cid:33) ←→ ∂φ = ∂ w φ − τ ( ∂ w φ ) ¯ ∂φ = ∂ ¯ w φ − τ ( ∂ ¯ w φ ) , (2.18)where we used T ( w ) = −
12 ( ∂ w φ ) , Θ ( w ) = 0 . (2.19)The solution to (2.18) is ∂ w φ = ∂φ − τ (cid:18) − S ¯ ∂φ (cid:19) ¯ ∂φ , ∂ ¯ w φ = − τ (cid:18) − S ∂φ (cid:19) ∂φ + ¯ ∂φ , (2.20)with S = (cid:113) τ ∂φ ¯ ∂φ . (2.21) The set of currents (2.16) can be obtained as the massless limit of the Klein-Gordon hierarchy: T ( KG ) k +1 ( w ) = − (cid:0) ∂ k w φ (cid:1) , Θ ( KG ) k − ( w ) = − m (cid:0) ∂ k − w φ (cid:1) , ( k ∈ N + 1) , with Lagrangian L ( KG ) ( w ) = ∂ w φ ∂ ¯ w φ + m φ . – 7 –ow, plugging (2.20) into the second expression of (2.7), we find the differential map ∂ w f = ∂f − τ (cid:18) − S ¯ ∂φ (cid:19) ¯ ∂f , ∂ ¯ w f = − τ (cid:18) − S ∂φ (cid:19) ∂f + ¯ ∂f , ( ∀ f : R → R ) . (2.22)Using (2.22), we can easily derive the T ¯ T-deformed version of the original EoMs as follows ∂ w ( ∂ ¯ w φ ) = 0 −→ (cid:32) ∂ − τ (cid:18) − S ¯ ∂φ (cid:19) ¯ ∂ (cid:33) (cid:32) − τ (cid:18) − S ∂φ (cid:19) ∂φ + ¯ ∂φ (cid:33) = 0 , (2.23)which, after some manipulations, can be recast into ∂ ¯ ∂φ = τ ∂ φ ( ¯ ∂φ ) + ¯ ∂ φ ( ∂φ ) τ ∂φ ¯ ∂φ . (2.24)The T ¯ T-deformed currents are obtained using (2.22) in the general expressions (2.11). Con-sidering the set of currents (2.15), namely setting T k +1 = T ( POW ) k +1 , Θ k − = Θ ( POW ) k − and thesame for their complex conjugates in (2.11), one finds T ( POW ) k +1 ( z , τ ) = − ( ∂φ ) k +1 S (cid:18)
21 + S (cid:19) k − , Θ ( POW ) k − ( z , τ ) = − τ ( ∂φ ) k +1 ( ¯ ∂φ ) S (cid:18)
21 + S (cid:19) k +1 , (2.25)which coincides with the result first obtained in [2] through perturbative computations. Ob-serve that, using (2.22) in (2.8), the Jacobian can be rewritten as J = (cid:32) − τ ¯Θ ( z , τ ) − τ T ( z , τ ) − τ ¯T ( z , τ ) 1 − τ Θ ( z , τ ) (cid:33) , (2.26)where T ( z , τ ) , Θ ( z , τ ) and their complex conjugates are the components of the T ¯ T-deformedstress-energy tensor, which can be read from (2.25), setting k = 1 . Switching from complex z to cartesian x coordinates, it is easy to realize that expression (2.26) leads to the inverse map(1.9). The latter result can be generalised to the case of N -boson fields with generic potential(see [18]).For the set of currents (2.16), we can again derive exactly the associated T ¯ T-deformed currents,however their analytic expressions are more and more involved as k increases and we wereunable to find a compact formula valid for arbitrary spin k ∈ N . We report here, as anexample, the level k = 3 deformed current of the hierarchy (2.16): T ( z , τ ) = − ( ∂φ ) S (cid:18) ( S − ¯ ∂ φ − τ ( ¯ ∂φ ) ∂ φ τ ( S −
1) ( S + 1) ( ¯ ∂φ ) (cid:19) , Θ ( z , τ ) = − ( S − τ S (cid:18) ( S − ¯ ∂ φ − τ ( ¯ ∂φ ) ∂ φ τ ( S −
1) ( S + 1) ( ¯ ∂φ ) (cid:19) . (2.27)Finally, it is important to stress that the method presented in this section is completelygeneral and can be applied to a generic integrable model, provided the stress-energy tensor– 8 –nd the conserved currents are known in terms of fundamental fields. We have explicitlycomputed the T ¯ T-deformed conserved currents with k = 3 , for the sine-Gordon model usingthe generalised change of variables described in [18]. Again the resulting expressions areextremely complicated and we will not present them here. In the following sections we shall generalise the change of variables (2.7) to deformations builtfrom conserved currents with generic positive and negative integer spins. Conventionally, wewill denote with s = | s | ≥ ( k = | k | ≥ ) the absolute value of the spin, and set s (cid:48) = − s ≤ ( k (cid:48) = − k ≤ ). We will discuss separately perturbations induced by conserved currents withspin s = s > , s (cid:48) = s < and s → , providing explicit examples. s > The natural generalisation of (2.7), which ensures the equality of mixed partial derivatives is (cid:32) dwd ¯ w (cid:33) = (cid:16) J ( s ) (cid:17) T (cid:32) dzd ¯ z (cid:33) , (cid:32) ∂ w f∂ ¯ w f (cid:33) = (cid:16) J ( s ) (cid:17) − (cid:32) ∂f ¯ ∂f (cid:33) , ( ∀ f : R → R ) , (3.1)where J ( s ) = (cid:32) ∂w ∂ ¯ w ¯ ∂w ¯ ∂ ¯ w (cid:33) = 1∆ ( s ) ( w ) (cid:32) τ Θ s − ( w ) − τ T s +1 ( w ) − τ ¯T s +1 ( w ) 1 + 2 τ ¯Θ s − ( w ) (cid:33) , (3.2) (cid:16) J ( s ) (cid:17) − = (cid:32) ∂ w z ∂ w ¯ z∂ ¯ w z ∂ ¯ w ¯ z (cid:33) = (cid:32) τ ¯Θ s − ( w ) 2 τ T s +1 ( w )2 τ ¯T s +1 ( w ) 1 + 2 τ Θ s − ( w ) (cid:33) , (3.3)with s > , ∆ ( s ) ( w ) = (cid:0) τ Θ s − ( w ) (cid:1)(cid:0) τ ¯Θ s − ( w ) (cid:1) − τ T s +1 ( w ) ¯T s +1 ( w ) , (3.4)and s = 1 corresponds to the T ¯ T deformation, J (1) = J . In fact, using the continuityequations (2.6) in (3.3), one finds that the second mixed partial derivatives are identical ∂ ¯ w ( ∂ w z ) = ∂ w ( ∂ ¯ w z ) , ∂ ¯ w ( ∂ w ¯ z ) = ∂ w ( ∂ ¯ w ¯ z ) . (3.5)Consider now the 1-forms I k = T ( s ) k +1 ( z , τ ) dz + Θ ( s ) k − ( z , τ ) d ¯ z , ¯ I k = ¯T ( s ) k +1 ( z , τ ) d ¯ z + ¯Θ ( s ) k − ( z , τ ) dz , (3.6)where the components T ( s ) k +1 ( z , τ ) , Θ ( s ) k − ( z , τ ) and their complex conjugates are the level- k conserved currents of the integrable theory, deformed according to the generalised change ofvariables (3.1). They fulfil the continuity equations ¯ ∂ T ( s ) k +1 ( z , τ ) = ∂ Θ ( s ) k − ( z , τ ) , ∂ ¯T ( s ) k +1 ( z , τ ) = ¯ ∂ ¯Θ ( s ) k − ( z , τ ) . (3.7)– 9 –sing the strategy described in section 2, we perform the change of variables (3.1) in (2.5)and obtain I k = T k +1 ( w ( z )) + 2 τ (cid:0) T k +1 ( w ( z )) Θ s − ( w ( z )) − Θ k − ( w ( z )) T s +1 ( w ( z )) (cid:1) ∆ ( s ) ( w ( z )) dz + Θ k − ( w ( z )) + 2 τ (cid:0) Θ k − ( w ( z )) ¯Θ s − ( w ( z )) − T k +1 ( w ( z )) ¯T s +1 ( w ( z )) (cid:1) ∆ ( s ) ( w ( z )) d ¯ z , (3.8)from which the components of the deformed currents can be extracted T ( s ) k +1 ( z , τ ) = T k +1 ( w ( z )) + 2 τ (cid:0) T k +1 ( w ( z )) Θ s − ( w ( z )) − Θ k − ( w ( z )) T s +1 ( w ( z )) (cid:1) ∆ ( s ) ( w ( z )) , Θ ( s ) k − ( z , τ ) = Θ k − ( w ( z )) + 2 τ (cid:0) Θ k − ( w ( z )) ¯Θ s − ( w ( z )) − T k +1 ( w ( z )) ¯T s +1 ( w ( z )) (cid:1) ∆ ( s ) ( w ( z )) . (3.9)Let us consider the following combinations of the components of level- k conserved currents I k ( w ) = − (cid:0) T k +1 ( w ) + Θ k − ( w ) (cid:1) , ¯ I k ( w ) = − (cid:0) ¯T k +1 ( w ) + ¯Θ k − ( w ) (cid:1) , (3.10) I ( s ) k ( z , τ ) = − (cid:0) T ( s ) k +1 ( z , τ ) + Θ ( s ) k − ( z , τ ) (cid:1) , ¯ I ( s ) k ( z , τ ) = − (cid:0) ¯T ( s ) k +1 ( z , τ ) + ¯Θ ( s ) k − ( z , τ ) (cid:1) . (3.11)Then, following the standard convention, the level- k Hamiltonian and momentum density are H ( s ) k ( z , τ ) = I ( s ) k ( z , τ ) + ¯ I ( s ) k ( z , τ )= 1∆ ( s ) ( w ( z )) (cid:20) H k ( w ( z )) + 2 τ (cid:0) T + ,k ( w ( z )) ¯ T − ,s ( w ( z )) + c.c. (cid:1)(cid:21) , (3.12) P ( s ) k ( z , τ ) = I ( s ) k ( z , τ ) − ¯ I ( s ) k ( z , τ )= 1∆ ( s ) ( w ( z )) (cid:20) P k ( w ( z )) + 2 τ (cid:0) T − ,k ( w ( z )) ¯ T − ,s ( w ( z )) − c.c. (cid:1)(cid:21) , (3.13)where H k ( w ) = I k ( w ) + ¯ I k ( w ) , P k ( w ) = I k ( w ) − ¯ I k ( w ) , (3.14)are the undeformed Hamiltonian and momentum density, and the quantities T ± ,n and ¯ T ± ,n correspond to the combinations T ± ,n ( w ) = T n +1 ( w ) ± ¯Θ n − ( w ) , ¯ T ± ,n ( w ) = ¯T n +1 ( w ) ± Θ n − ( w ) , (3.15) In the following, “c.c.” denotes the replacement { T( z , τ ) , Θ( z , τ ) } → { ¯T( z , τ ) , ¯Θ( z , τ ) } . – 10 –ntegrating (3.12) and (3.13) we find (cid:90) H ( s ) k ( z , τ ) dz ∧ d ¯ z = (cid:90) (cid:20) H k ( w ) + 2 τ (cid:0) T + ,k ( w ) ¯ T − ,s ( w ) + c.c. (cid:1)(cid:21) dw ∧ d ¯ w = (cid:90) H k ( w ) dw ∧ d ¯ w − τ (cid:90) ( I k + ¯ I k ) ∧ ( I s − ¯ I s ) , (3.16) (cid:90) P ( s ) k ( z , τ ) dz ∧ d ¯ z = (cid:90) (cid:20) P k ( w ) + 2 τ (cid:0) T − ,k ( w ) ¯ T − ,s ( w ) − c.c. (cid:1)(cid:21) dw ∧ d ¯ w = (cid:90) P k ( w ) dw ∧ d ¯ w − τ (cid:90) ( I k − ¯ I k ) ∧ ( I s − ¯ I s ) . (3.17)We now interpret the result (3.16) as follows: H ( s ) k ( z , τ ) dz ∧ d ¯ z coincides with the correspond-ing bare quantity H k ( w ) dw ∧ d ¯ w deformed by the operator Φ k,s ( w ) dw ∧ d ¯ w = − I k + ¯ I k ) ∧ ( I s − ¯ I s ) , (3.18) Φ k,s ( w ) = 2 (cid:0) T + ,k ( w ) ¯ T − ,s ( w ) + ¯ T + ,k ( w ) T − ,s ( w ) (cid:1) , (3.19)together with a non-trivial dressing given by the change of variables (3.1). In the s = 1 case, i.e. the T ¯ T example, the operator (3.19) associated to the k = 1 Hamiltonian becomes Φ , ( w ) = T ( w ) ¯T ( w ) − Θ ( w ) ¯Θ ( w ) , (3.20)which, but for the change of coordinates, coincides with the bare T ¯ T operator. In analogywith the T ¯ T result [18, 19], one may be tempted to interpret (3.19) as the perturbing operatorof the level- k Hamiltonian. However, the coordinate transformation (3.1) also introduces O ( τ ) corrections which can, in principle, completely spoil this naive picture. In addition, even whenthe initial theory is a CFT and the bare operator (3.19) is completely symmetric in s and k Φ k,s ( w ) = T k +1 ( w ) ¯T s +1 ( ¯ w ) + ¯T k +1 ( ¯ w ) T s +1 ( w ) , (3.21)the change of variables spoils the s ↔ k symmetry, since it involves only the level- s currents.Examples of this phenomena will neatly emerge from the study of the s ≤ deformations ofthe free massless boson theory.From the explicit examples discussed in section 3.3, it will clearly emerge that the τ defor-mations of the original momentum and Hamiltonian, obtained using the change of variables,do not always coincide with the generators of space and time translations, in the cartesian x = ( x , x ) = ( x, t ) directions. Although all the deformed charges I ( s ) k ( R, τ ) = (cid:90) R I ( s ) k ( x , τ ) dx , ¯ I ( s ) k ( R, τ ) = (cid:90) R ¯ I ( s ) k ( x , τ ) dx , (3.22)are conserved in t , they evolve the system along “generalised space-time” directions, whichdiffer from the original undeformed ones for k (cid:54) = s . Due to this fact, the Lagrangians associated– 11 –o the deformed EoMs do not, in general, correspond to the deformed Hamiltonians. Therefore,these theories are more complicated compared to the T ¯ T and J ¯ T examples studied in theprevious literature.One of the main objectives of the following sections will be the identification of the additionalscattering phase factors needed for the characterisation of the finite-size quantum spectrumof the deformed charges (3.22). s < To gain precise information about the spectrum, it turns out to be particularly convenient tofirst extend the current setup to the s < cases. In order to define perturbations inducedby higher conserved currents with negative spin s , we replace s → s (cid:48) in the definition of thegeneralised Jacobian (3.2) and (3.3), obtaining J ( s (cid:48) ) = 1∆ ( s (cid:48) ) ( w ) (cid:32) τ ¯T s +1 ( w ) − τ ¯Θ s − ( w ) − τ Θ s − ( w ) 1 + 2 τ T s +1 ( w ) (cid:33) , (3.23) (cid:0) J ( s (cid:48) ) (cid:1) − = (cid:32) τ T s +1 ( w ) 2 τ ¯Θ s − ( w )2 τ Θ s − ( w ) 1 + 2 τ ¯T s +1 ( w ) (cid:33) , (3.24)with s (cid:48) = s < , s = | s | and ∆ ( s (cid:48) ) ( w ) = (cid:0) τ T s +1 ( w ) (cid:1)(cid:0) τ ¯T s +1 ( w ) (cid:1) − τ Θ s − ( w ) ¯Θ s − ( w ) . (3.25)We arrived to (3.23)–(3.24) by implementing the spin-flip symmetry (see, for example, [1]) Θ s (cid:48) − = ¯T s +1 , T s (cid:48) +1 = ¯Θ s − , (3.26)which corresponds to the following reflection property at the level of the -forms I s (cid:48) = ¯ I s , ¯ I s (cid:48) = I s , (3.27)or, equivalently I s (cid:48) = ¯ I s , ¯ I s (cid:48) = I s . (3.28)Using the continuity equations (2.6), it is easy to verify that (3.24) fulfils again the conditions(3.5), therefore it defines a consistent field-dependent change of variables. Repeating thecomputations (3.8)–(3.17) using (3.23)–(3.24) one finds that (3.9) become T ( s (cid:48) ) k +1 ( z , τ ) = T k +1 ( w ( z )) + 2 τ (cid:0) T k +1 ( w ( z )) ¯T s +1 ( w ( z )) − Θ k − ( w ( z )) ¯Θ s − ( w ( z )) (cid:1) ∆ ( s (cid:48) ) ( w ( z )) , Θ ( s (cid:48) ) k − ( z , τ ) = Θ k − ( w ( z )) + 2 τ (Θ k − ( w ( z )) T s +1 ( w ( z )) − T k +1 ( w ( z )) Θ s − ( w ( z )))∆ ( s (cid:48) ) ( w ( z )) , (3.29)– 12 –hile (3.16)–(3.17) become (cid:90) H ( s (cid:48) ) k ( z , τ ) dz ∧ d ¯ z = (cid:90) (cid:20) H k ( w ) − τ (cid:0) T + ,k ( w ) ¯ T − ,s ( w ) + c.c. (cid:1)(cid:21) dw ∧ d ¯ w = (cid:90) H k ( w ) dw ∧ d ¯ w + 2 τ (cid:90) ( I k + ¯ I k ) ∧ ( I s − ¯ I s ) , (3.30) (cid:90) P ( s (cid:48) ) k ( z , τ ) dz ∧ d ¯ z = (cid:90) (cid:20) P k ( w ) − τ (cid:0) T − ,k ( w ) ¯ T − ,s ( w ) − c.c. (cid:1)(cid:21) dw ∧ d ¯ w = (cid:90) P k ( w ) dw ∧ d ¯ w + 2 τ (cid:90) ( I k − ¯ I k ) ∧ ( I s − ¯ I s ) , (3.31)which are formally equal to (3.16) and (3.17), except for the sign of τ .However, the positive and negative spin sectors are deeply different, especially in what concernsthe non-zero momentum states. They are not simply related by a change of sign in the couplingconstant τ as the comparison between (3.16)–(3.17) and (3.30)–(3.31) would naively suggests.In fact, the change of variables and the corresponding perturbing operators are different. Bystudying in detail the s < perturbations of the massless free boson model (see section 3.3below), the difference with respect to the s = 1 perturbation, i.e. the T ¯ T, clearly emerges.
In this section, we consider deformations of the massless free boson theory induced by thecoordinate transformations (3.23)–(3.24), and we derive the higher conserved currents of thedeformed models. As already discussed in section 2, the most general level- k current of thehierarchy can be expressed in the form (2.17). While the structure of the deformed currentsdoes not emerge clearly by working with the general combination (2.17), we observed that thesubset (2.15) is analytically much easier to treat since it does not mix with the others. Thisproperty allows to obtain compact expressions for the deformed currents which are formallyidentical to the exact quantum results of section 4.2.Using (2.15) in (3.23)–(3.24), the coordinate transformations read explicitly J ( s (cid:48) ) = (cid:32) − τ ( ∂ w φ ) s +1 − τ ( ∂ ¯ w φ ) s +1 (cid:33) , (3.32) (cid:0) J ( s (cid:48) ) (cid:1) − = (cid:32) − τ ( ∂ w φ ) s +1
00 1 − τ ( ∂ ¯ w φ ) s +1 (cid:33) . (3.33)Repeating the same computation performed in section 2.2, we first express ( ∂ w φ, ∂ ¯ w φ ) T interms of ( ∂φ, ¯ ∂φ ) T , by solving the set of equations (cid:32) ∂φ ¯ ∂φ (cid:33) = J ( s (cid:48) ) (cid:32) ∂ w φ∂ ¯ w φ (cid:33) ←→ ∂φ = ∂ w φ − τ ( ∂ w φ ) s +1 ¯ ∂φ = ∂ ¯ w φ − τ ( ∂ ¯ w φ ) s +1 . (3.34)– 13 –he solutions to (3.34) can be written in terms of generalized hypergeometric functions forany value of s (cid:48) = − s as ∂ w φ = ˜ F s (cid:18) − τ ( s + 1) s +1 s s ( ∂φ ) s +1 (cid:19) ∂φ , ∂ ¯ w φ = ˜ F s (cid:18) − τ ( s + 1) s +1 s s ( ¯ ∂φ ) s +1 (cid:19) ¯ ∂φ , (3.35)with ˜ F n ( x ) = n F n − (cid:18) n + 1 , . . . , nn + 1 ; 2 n , . . . , n − n , n + 1 n ; x (cid:19) , ( n ∈ N − { } ) . (3.36)Plugging (3.35) into the second expression of (3.1) (with the replacement s → s (cid:48) = − s ), onefinds the differential map ∂ w f = ˜ F s (cid:18) − τ ( s + 1) s +1 s s ( ∂φ ) s +1 (cid:19) ∂f , ∂ ¯ w f = ˜ F s (cid:18) − τ ( s + 1) s +1 s s ( ¯ ∂φ ) s +1 (cid:19) ¯ ∂f , (3.37) ∀ f : R → R . From (3.37), it follows immediately that the deformed EoMs are ∂ ¯ ∂φ = 0 , (3.38)which reflects the fact that the s < perturbations of CFT’s do not mix the holomorphic andanti-holomorphic derivatives, as already emerged from (3.34). Using the technique describedin section 2, we can now derive the deformed currents. Plugging the differential map (3.37)into (3.29), we obtain T ( s (cid:48) ) s +1 ( z , τ ) = − s τ ( s + 1) (cid:20) − F s (cid:18) − τ ( s + 1) s +1 s s ( ∂φ ) s +1 (cid:19)(cid:21) , Θ ( s (cid:48) ) s − ( z , τ ) = 0 , ¯T ( s (cid:48) ) s +1 ( z , τ ) = − s τ ( s + 1) (cid:20) − F s (cid:18) − τ ( s + 1) s +1 s s ( ¯ ∂φ ) s +1 (cid:19)(cid:21) , ¯Θ ( s (cid:48) ) s − ( z , τ ) = 0 , (3.39)with F n ( x ) = n F n − (cid:18) − n + 1 , n + 1 , . . . , n − n + 1 ; 1 n , n , . . . , n − n ; x (cid:19) , ( n ∈ N − { } ) , (3.40)where we used the following relation between generalised hypergeometric functions F n ( x ) = 1 n − n + 1) n +2 ( n + 1) n +1 + n n x (cid:16) ˜ F n ( x ) (cid:17) n +1 , (3.41)to trade ˜ F n ( x ) with F n ( x ) . From (3.38), it follows that φ ( z , τ ) = ϕ ( z, τ )+ ¯ ϕ (¯ z, τ ) and therefore T ( s (cid:48) ) s +1 ( z , τ ) and ¯T ( s (cid:48) ) s +1 ( z , τ ) depend only on z and ¯ z , respectively. Again we observe that, using(3.37) in (3.32), the Jacobian can be rewritten in terms of the deformed components (3.39) as J ( s (cid:48) ) = (cid:32) − τ T ( s (cid:48) ) s +1 ( z , τ ) − τ ¯Θ ( s (cid:48) ) s − ( z , τ ) − τ Θ ( s (cid:48) ) s − ( z , τ ) 1 − τ ¯T ( s (cid:48) ) s +1 ( z , τ ) (cid:33) , (3.42) Relation (3.41) can be easily checked expanding at every perturbative order around x = 0 . – 14 –hich confirms that the coordinate transformation is invertible.In terms of the quantities (3.10) and (3.11), (3.39) can be more transparently written as I ( s (cid:48) ) s ( z , τ ) = s τ ( s + 1) (cid:20) − F s (cid:18) − τ ( s + 1) s +1 s s I s ( z ) (cid:19)(cid:21) , ¯ I ( s (cid:48) ) s ( z , τ ) = s τ ( s + 1) (cid:20) − F s (cid:18) − τ ( s + 1) s +1 s s ¯ I s ( z ) (cid:19)(cid:21) . (3.43)Quite remarkably, the latter expressions are solutions to simple algebraic equations of theform I ( s (cid:48) ) s ( z , τ ) = I s ( z ) (cid:16) τ I ( s (cid:48) ) s ( z , τ ) (cid:17) s , ¯ I ( s (cid:48) ) s ( z , τ ) = ¯ I s ( z ) (cid:16) τ ¯ I ( s (cid:48) ) s ( z , τ ) (cid:17) s . (3.44)Moreover, one can show that the combinations I ( s (cid:48) ) k ( z , τ ) and ¯ I ( s (cid:48) ) k ( z , τ ) of generic level- k deformed currents are I ( s (cid:48) ) k ( z , τ ) = I k ( z ) (cid:26) ss + 1 (cid:20) − F s (cid:18) − τ ( s + 1) s +1 s s I s ( z ) (cid:19)(cid:21)(cid:27) − k , ¯ I ( s (cid:48) ) k ( z , τ ) = ¯ I k ( z ) (cid:26) ss + 1 (cid:20) − F s (cid:18) − τ ( s + 1) s +1 s s ¯ I s ( z ) (cid:19)(cid:21)(cid:27) − k , (3.45)and fulfil the following equations I ( s (cid:48) ) k ( z , τ ) = I k ( z ) (cid:16) τ I ( s (cid:48) ) s ( z , τ ) (cid:17) k , ¯ I ( s (cid:48) ) k ( z , τ ) = ¯ I k ( z ) (cid:16) τ ¯ I ( s (cid:48) ) s ( z , τ ) (cid:17) k , (3.46)which generalise (3.44).Before moving to the next section, let us make a few important remarks:1. The T ¯ T and the J ¯ T examples discussed in [14, 22], taught us that, at least formally,the evolution equations for the quantized spectra already emerge at classical level afterreplacing the classical densities with their average value over the volume R : I k ( z ) −→ I k ( R ) R = I (+) k ( R ) R , ¯ I k ( z ) −→ ¯ I k ( R ) R = I ( − ) k ( R ) R , I ( s ) k ( z , τ ) −→ I ( s ) k ( R, τ ) R = I ( s , +) k ( R, τ ) R , ¯ I ( s ) k ( z , τ ) −→ ¯ I ( s ) k ( R, τ ) R = I ( s , − ) k ( R, τ ) R . (3.47)In (3.47), the labels (+) and ( − ) stand for the right and left light-cone component ofthe conserved charges, respectively (cf. section 4).Implementing (3.47) in (3.46) gives I ( s (cid:48) , ± ) k ( R, τ ) = R k I ( ± ) k ( R ) (cid:16) R + 2 τ I ( s (cid:48) , ± ) s ( R, τ ) (cid:17) k , (3.48)which coincides with the CFT quantum result (4.79) of section 4.3.– 15 –. Although (3.48) were derived for the s < case, it is natural to conjecture that theycan be extended also to s ≥ . From the reflection property (3.28), we find: I ( s, ± ) k ( R, τ ) = R k I ( ± ) k ( R ) (cid:16) R + 2 τ ¯ I ( s, ∓ ) s ( R, τ ) (cid:17) k , (3.49)which again match the s > CFT quantum result quoted in (4.85).3. It is straightforward to check that (3.48) is a solution to the generalised Burgers-typeequations (4.47)–(4.50) for the deformed quantum spectrum, which hold also for massivemodels. In fact, as shown in Appendix B, in the CFT case equations (4.47)–(4.50) canbe recast into the simpler form ∂ τ I ( s, ± ) k ( R, τ ) = 2 I ( s, ∓ ) s ( R, τ ) ∂ R I ( s, ± ) k ( R, τ ) , ( s > ,∂ τ I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) = 2 I ( s (cid:48) , ± ) s (cid:48) ( R, τ ) ∂ R I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) , ( s (cid:48) = − s < . (3.50)4. From the point of view of the infinite tower of conservation laws, the map (3.32)-(3.33)corresponds to a non trivial τ -dependent mixing of the charges. Every conserved quan-tity, including the deformations of the Hamiltonian and momentum densities H ( s (cid:48) ) ( z , τ ) = I ( s (cid:48) )1 ( z , τ ) + ¯ I ( s (cid:48) )1 ( z , τ ) , (3.51) P ( s (cid:48) ) ( z , τ ) = I ( s (cid:48) )1 ( z , τ ) − ¯ I ( s (cid:48) )1 ( z , τ ) , (3.52)are smoothly deformed, leading to a trajectory in the space of models which sharethe same set of conserved charges. As already mentioned at the end of section 3.1, adirect consequence of the mixing among the conserved quantities, and of the associated generalised time variables, is that the interpretation of E ( s (cid:48) ) ( R, τ ) = (cid:90) R H ( s (cid:48) ) ( x , τ ) dx , P ( s (cid:48) ) ( R, τ ) = (cid:90) R P ( s (cid:48) ) ( x , τ ) dx , (3.53)as the generators of translations in t and x is no longer valid at τ (cid:54) = 0 . Formally, thegenerators of translations in x correspond to the unperturbed ( τ = 0 ) Hamiltonian andmomentum.5. The result (3.38) shows that the coordinate transformation (3.32)-(3.33) is, on the plane,an automorphism of the space of classical solutions of the free boson theory. In addition,the undeformed action is mapped into itself by the change of variables (3.32): (cid:90) L ( w ) dw ∧ d ¯ w = (cid:90) L ( z ) dz ∧ d ¯ z , L ( z ) = ∂φ ¯ ∂φ . (3.54)– 16 – .4 Deformations related to charges with spin s → and the J ¯ T-type models
In this section we shall consider perturbations of the massless free boson model induced bythe U (1) L × U (1) R currents defined through J + ( w ) = − i T ( w ) = i ∂ w φ , J − ( w ) = − i Θ − ( w ) = 0 , ¯ J + ( w ) = 2 i ¯T ( w ) = − i ∂ ¯ w φ , ¯ J − ( w ) = 2 i ¯Θ − ( w ) = 0 , (3.55)where T ( w ) , Θ − ( w ) and their complex conjugates corresponds to the case k = 0 in (2.15),while the additional factor i in (3.55) is used to write the holomorphic and anti-holomorphiccomponents of the stress-energy tensor in Sugawara form (see [26]) T ( w ) = 12 ( J + ( w )) , ¯T ( w ) = 12 (cid:0) ¯ J + ( w ) (cid:1) . (3.56)Since also the currents (3.55) are components of closed -forms I = J + ( w ) dw + J − ( w ) d ¯ w , ¯ I = ¯ J + ( w ) d ¯ w + ¯ J − ( w ) dw , (3.57)we shall use the strategy described in section 2 to derive the corresponding deformations.Following the same spirit of the s > and s < deformations, the most general change ofcoordinates built out of the U (1) currents (3.55) is of the form J (0) = 1∆ (0) ( w ) (cid:32) τ (4) ∂ ¯ w φ − τ (2) ∂ w φ − τ (3) ∂ ¯ w φ τ (1) ∂ w φ (cid:33) , (3.58) (cid:0) J (0) (cid:1) − = (cid:32) τ (1) ∂ w φ τ (2) ∂ w φτ (3) ∂ ¯ w φ τ (4) ∂ ¯ w φ (cid:33) , (3.59)where τ ( i ) , ( i = 1 , . . . , are four different deformation parameters and ∆ (0) ( w ) = 1 + τ (1) ∂ w φ + τ (4) ∂ ¯ w φ + (cid:0) τ (1) τ (4) − τ (2) τ (3) (cid:1) ∂ w φ ∂ ¯ w φ . (3.60)Since the general case is quite cumbersome, we will restrict our analysis to some particularlimits. Case τ (1) = τ (4) = 0 | τ (2) = τ (3) = − τ With this choice, (3.58)–(3.59) are reduced to J (0) = 11 − τ ∂ w φ ∂ ¯ w φ (cid:32) τ ∂ w φτ ∂ ¯ w φ (cid:33) , (3.61) (cid:16) J (0) (cid:17) − = (cid:32) − τ ∂ w φ − τ ∂ ¯ w φ (cid:33) , (3.62)– 17 –hich correspond to the case s = 0 in (3.2)–(3.3). Following the procedure described in detailin section 2.2, one can easily derive the differential map ∂ w f = ∂f − τ ∂φ τ ¯ ∂φ ¯ ∂f , ∂ ¯ w f = ¯ ∂f − τ ¯ ∂φ τ ∂φ ∂f , (3.63)from which the deformed EoMs are ∂ ¯ ∂φ = τ ∂φ ¯ ∂ φ (1 + τ ∂φ ) + ¯ ∂φ ∂ φ (cid:0) τ ¯ ∂φ (cid:1) τ (cid:0) ∂φ + ¯ ∂φ (cid:1) + 2 τ ∂φ ¯ ∂φ . (3.64)Then, setting s = 0 in (3.9) and using (3.63), we get both the deformed level- currents T (0)1 ( z , τ ) = − ∂φ (1 + τ ∂φ )1 + τ (cid:0) ∂φ + ¯ ∂φ (cid:1) , Θ (0) − ( z , τ ) = − τ ∂φ ¯ ∂φ τ (cid:0) ∂φ + ¯ ∂φ (cid:1) , (3.65)and the components of the deformed stress-energy tensor T (0)2 ( z , τ ) = −
12 ( ∂φ ) (1 + τ ∂φ ) (cid:0) τ ¯ ∂φ (cid:1) (cid:0) τ (cid:0) ∂φ + ¯ ∂φ (cid:1)(cid:1) , Θ (0)0 ( z , τ ) = − τ ¯ ∂φ ( ∂φ ) (cid:0) τ ¯ ∂φ (cid:1) (cid:0) τ (cid:0) ∂φ + ¯ ∂φ (cid:1)(cid:1) . (3.66)Plugging (3.63) into (3.61), we notice that the Jacobian can be rewritten in terms of thedeformed level- currents (3.65) as J (0) = (cid:32) − τ ¯Θ (0) − ( z , τ ) − τ T (0)1 ( z , τ ) − τ ¯T (0)1 ( z , τ ) 1 − τ Θ (0) − ( z , τ ) (cid:33) , (3.67)which again confirms that the coordinate transformation is invertible. In terms of the quan-tities (3.10) and (3.11), one finds the following relations I (0)0 ( z , τ ) = 12 ∂φ = I ( z ) , ¯ I (0)0 ( z , τ ) = 12 ¯ ∂φ = ¯ I ( z ) , I (0)1 ( z , τ ) = 12 ( ∂φ ) τ ¯ ∂φ = I ( z )1 + 2 τ ¯ I (0)0 ( z , τ ) , ¯ I (0)1 ( z , τ ) = 12 ( ¯ ∂φ ) τ ∂φ = ¯ I ( z )1 + 2 τ I (0)0 ( z , τ ) . (3.68) Case τ (1) = τ (4) = − τ | τ (2) = τ (3) = 0 With this choice (3.58)–(3.59) reduced to J (0) = 1(1 − τ ∂ w φ ) (1 − τ ∂ ¯ w φ ) (cid:32) − τ ∂ ¯ w φ
00 1 − τ ∂ w φ (cid:33) , (3.69) (cid:16) J (0) (cid:17) − = (cid:32) − τ ∂ w φ
00 1 − τ ∂ ¯ w φ (cid:33) , (3.70)– 18 –hich corresponds to the case s (cid:48) = − s = 0 in (3.23)–(3.24). The deformed level- currents andthe components of the deformed stress energy tensor can be immediately read from (3.39),setting s (cid:48) = − s = 0 : T (0)1 ( z , τ ) = − ∂φ , Θ (0) − ( z , τ ) = 0 , (3.71) T (0)2 ( z , τ ) = −
12 ( ∂φ ) τ ∂φ , Θ (0)0 ( z , τ ) = 0 , (3.72)and, in terms of the quantities (3.11) and (3.10), from (3.45) one has I (0)0 ( z , τ ) = 12 ∂φ = I ( z ) , ¯ I (0)0 ( z , τ ) = 12 ¯ ∂φ = ¯ I ( z ) , I (0)1 ( z , τ ) = 12 ( ∂φ ) τ ∂φ = I ( z )1 + 2 τ I (0)0 ( z , τ ) , ¯ I (0)1 ( z , τ ) = 12 ( ¯ ∂φ ) τ ¯ ∂φ = ¯ I ( z )1 + 2 τ ¯ I (0)0 ( z , τ ) . (3.73) Case τ (1) = τ (3) = δ | τ (2) = τ (4) = τ In this particular case (3.58)–(3.59) reduce to J (0) = (cid:32) − δ ∂φ − τ ∂φ − δ ¯ ∂φ − τ ¯ ∂φ (cid:33) , (cid:16) J (0) (cid:17) − = (cid:32) δ ∂ w φ τ ∂ w φδ ∂ ¯ w φ τ ∂ ¯ w φ (cid:33) , (3.74)which corresponds to an explicit change of variables w ( z ) of the form w = z − δφ , ¯ w = ¯ z − τ φ . (3.75)In [18], it was observed that the T ¯ T-deformed solutions fulfil a non-linear evolution equation.Its extension to generic spin s is, in complex coordinates z ∂ τ φ ( s ) ( z , τ ) + ( ∂ τ z ) ∂φ ( s ) ( z , τ ) + ( ∂ τ ¯ z ) ¯ ∂φ ( s ) ( z , τ ) = 0 , (3.76)with ∂ τ z = 2 (cid:90) z ¯ I s , ∂ τ ¯ z = 2 (cid:90) z I s . (3.77) Starting from (A.2), i.e. φ ( s ) ( z , τ ) = φ ( w ( z ( s ) ) , , we let z depend on τ such that ddτ w ( z ( s ) ) = 0 : ddτ φ ( s ) ( z , τ ) = 0 = ∂ τ φ ( s ) ( z , τ ) + ( ∂ τ z ) ∂ z φ ( s ) ( z , τ ) + ( ∂ τ ¯ z ) ∂ ¯ z φ ( s ) ( z , τ ) . From the definition of the change of coordinates (3.1) z = w ( z ) + 2 τ w ( z ) (cid:90) (cid:0) ¯Θ s − ( w ) dw + ¯T s +1 ( w ) d ¯ w (cid:1) , ¯ z = ¯ w ( z ) + 2 τ w ( z ) (cid:90) (T s +1 ( w ) dw + Θ s − ( w ) d ¯ w ) , using the definition (2.5) and the coordinate independence property of differential forms, we arrive to (3.77). – 19 –he multi-parameter variant of (3.76) associated to the coordinate transformation (3.58) is,instead ∂ τ ( i ) φ ( z , (cid:126)τ ) + (cid:0) ∂ τ ( i ) z (cid:1) ∂φ ( z , (cid:126)τ ) + (cid:0) ∂ τ ( i ) ¯ z (cid:1) ¯ ∂φ ( z , (cid:126)τ ) = 0 , (cid:126)τ = { τ ( i ) } . (3.78)In general, equations (3.76)–(3.78) cannot be explicitly integrated, however, in the case of thechange of coordinates (3.75), equations (3.78) become a set of inviscid Burgers equations forthe function φ ( z, ¯ z, τ, δ ) in the variables z , ¯ z , τ and δ∂ τ φ ( z, ¯ z, τ, δ ) + 12 ¯ ∂ (cid:0) φ ( z, ¯ z, τ, δ ) (cid:1) = 0 , ∂ δ φ ( z, ¯ z, τ, δ ) + 12 ∂ (cid:0) φ ( z, ¯ z, τ, δ ) (cid:1) = 0 , (3.79)whose solution can be expressed, in implicit form, as φ ( z, ¯ z, τ, δ ) = φ ( z − δφ, ¯ z − τ φ, , . (3.80)Using the method discussed in sections 2 and 3, we can write down the deformed EoMs, i.e. ∂ ¯ ∂φ = − τ ∂φ ¯ ∂ φ (1 − δ ∂φ ) + δ ¯ ∂φ ∂ φ (cid:0) − τ ¯ ∂φ (cid:1) − δ ∂φ − τ ¯ ∂φ + 2 τ δ ∂φ ¯ ∂φ , (3.81)and the tower of deformed higher conserved currents from the undeformed ones. In particular,the components of the deformed U (1) currents (3.55) are J (0)+ ( z , τ, δ ) = i ∂φ − δ ∂φ − δ ∂φ − τ ¯ ∂φ , J (0) − ( z , τ, δ ) = − i δ ∂φ ¯ ∂φ − δ ∂φ − τ ¯ ∂φ , ¯ J (0)+ ( z , τ, δ ) = − i ¯ ∂φ − τ ∂φ − δ ∂φ − τ ¯ ∂φ , ¯ J (0) − ( z , τ, δ ) = i τ ∂φ ¯ ∂φ − δ ∂φ − τ ¯ ∂φ , (3.82)while the components of the deformed stress-energy tensor are T (0)2 ( z , τ, δ ) = −
12 ( ∂φ ) − δ ∂φ (1 − δ ∂φ − τ ¯ ∂φ ) , Θ (0)0 ( z , τ, δ ) = 12 δ ( ∂φ ) ¯ ∂φ (1 − δ ∂φ − τ ¯ ∂φ ) , ¯T (0)2 ( z , τ, δ ) = −
12 ( ¯ ∂φ ) − τ ¯ ∂φ (1 − δ ∂φ − τ ¯ ∂φ ) , ¯Θ (0)0 ( z , τ, δ ) = 12 τ ( ¯ ∂φ ) ∂φ (1 − δ ∂φ − τ ¯ ∂φ ) . (3.83)Therefore, the deformed Hamiltonian and momentum density are H (0) ( z , τ, δ ) = I (0)1 ( z , τ, δ ) + ¯ I (0)1 ( z , τ, δ )= − ∂φ ¯ ∂φ (cid:0) δ ∂φ + τ ¯ ∂φ (cid:1) − ( ∂φ ) (1 − δ ∂φ ) − ( ¯ ∂φ ) (1 − τ ¯ ∂φ )2 (cid:0) − δ ∂φ − τ ¯ ∂φ (cid:1) , (3.84) P (0) ( z , τ, δ ) = I (0)1 ( z , τ, δ ) − ¯ I (0)1 ( z , τ, δ )= ( ∂φ + ¯ ∂φ ) (cid:0) ∂φ (1 − δ ∂φ ) − ¯ ∂φ (1 − τ ¯ ∂φ ) (cid:1) (cid:0) − δ ∂φ − τ ¯ ∂φ (cid:1) , (3.85)– 20 –nd the corresponding deformed Lagrangian L (0) ( z , τ, δ ) = ∂φ ¯ ∂φ − δ ∂φ − τ ¯ ∂φ , (3.86)whose associated EoMs coincide with (3.81). Notice that, while the deformed action is mappedexactly into the undeformed one under (3.75), the integral of (3.84) transforms with an addi-tional term (cid:90) H (0) ( z , τ, δ ) dz ∧ d ¯ z = (cid:90) (cid:2) H ( w ) + i ( δ − τ ) (cid:0) J + ( w ) ¯T ( w ) − c.c. (cid:1)(cid:3) dw ∧ d ¯ w . In order to unambiguously identify the perturbing operator we must Legendre transform(3.84). First of all, we move from complex z coordinates to cartesian coordinates x =( x , x ) = ( x, t ) according to the convention (1.12). Then, inverting the Legendre map onefinds Π = i ∂ L (0) ( x , τ, δ ) ∂φ t ,φ t = i (cid:0) τ − δ ) (cid:1)(cid:0) − τ + δ ) φ x (cid:1) + 2 (cid:113)(cid:0) τ − δ ) (cid:1)(cid:0) − τ φ x (cid:1)(cid:0) − δ φ x (cid:1) ( τ − δ ) (cid:0) τ − δ ) (cid:1) . (3.87) Plugging (3.87) in (3.84)–(3.85), or equivalently, performing the canonical Legendre transfor-mation, one gets H (0) ( x , τ, δ ) = I (0)1 ( x , τ, δ ) + ¯ I (0)1 ( x , τ, δ ) = i Π φ t + L (0) ( x , τ, δ )= − (cid:0) τ − δ ) (cid:1)(cid:0) − τ + δ ) φ x (cid:1) + 2 (cid:113)(cid:0) τ − δ ) (cid:1)(cid:0) − τ φ x (cid:1)(cid:0) − δ φ x (cid:1) ( τ − δ ) , (3.88) and P (0) ( x , τ, δ ) = I (0)1 ( x , τ, δ ) − ¯ I (0)1 ( x , τ, δ ) = − Π φ x . (3.89)From (3.89), we find that the momentum density is unaffected by the perturbation, since P (0) ( x , τ, δ ) = − Π φ x = P ( x ) , (3.90)where P ( x ) is the unperturbed momentum density. Finally, expanding (3.88) at the first orderin τ and δ , we can identify the perturbing operator at first order H (0) ( x , τ, δ ) ∼ τ → δ → H ( x ) + 2 (cid:0) τ J ( x ) ¯T ( x ) + δ ¯ J ( x ) T ( x ) (cid:1) + O ( τ δ ) , (3.91)where H ( x ) = I ( x ) + ¯ I ( x ) = Π + φ x is the undeformed Hamiltonian and I ( x ) = − T ( x ) = 18 (2 Π − φ x ) , ¯ I ( x ) = − ¯T ( x ) = 18 (2 Π + φ x ) . (3.92) A straightforward way to obtain the Lagrangian (3.86) from the Hamiltonian (3.84), is to start from aformal series expansion of L (0) ( z , τ, δ ) around τ = δ = 0 and fix the unknown coefficients by matching theLegendre transformation of L (0) ( z , τ, δ ) with (3.84). – 21 –re the holomorphic and anti-holomorphic components of the undeformed stress energy tensor,respectively. In (3.91), we denoted as J µ ( x ) and ¯ J µ ( x ) , ( µ = 1 , , the cartesian componentsof the undeformed U (1) currents (3.55) J ( x ) = J + ( x ) − J − ( x ) = i −
2Π + φ x ) ,J ( x ) = i (cid:0) J + ( x ) + J − ( x ) (cid:1) = 12 (2Π − φ x ) , ¯ J ( x ) = ¯ J + ( x ) − ¯ J − ( x ) = i − − φ x ) , ¯ J ( x ) = − i (cid:0) ¯ J + ( x ) + ¯ J − ( x ) (cid:1) = 12 ( − − φ x ) , (3.93)which fulfil the continuity equations ∂ µ J µ ( x ) = 0 , ∂ µ ¯ J µ ( x ) = 0 . (3.94)In a similar way, we define the cartesian components of the deformed U (1) currents (3.82) as J (0)1 ( x , τ, δ ) = J (0)+ ( x , τ, δ ) − J (0) − ( x , τ, δ ) , J (0)2 ( x , τ, δ ) = i (cid:0) J (0)+ ( x , τ, δ ) + J (0) − ( x , τ, δ ) (cid:1) , ¯ J (0)1 ( x , τ, δ ) = ¯ J (0)+ ( x , τ, δ ) − ¯ J (0) − ( x , τ, δ ) , ¯ J (0)2 ( x , τ, δ ) = − i (cid:0) ¯ J (0)+ ( x , τ, δ ) + ¯ J (0) − ( x , τ, δ ) (cid:1) , (3.95)which again fulfil the continuity equations ∂ µ J (0) µ ( x , τ, δ ) = 0 , ∂ µ ¯ J (0) µ ( x , τ, δ ) = 0 . (3.96)Therefore, from (3.96), one finds that the quantities Q ( R, τ, δ ) = (cid:90) R J (0)2 ( x , τ, δ ) dx , ¯ Q ( R, τ, δ ) = (cid:90) R ¯ J (0)2 ( x , τ, δ ) dx . (3.97)are the conserved charges associated to the U (1) L × U (1) R symmetry in the deformed theory. Case τ (1) = τ (3) = 0 | τ (2) = τ (4) = τ This case can be easily retrieved from the previous one by sending δ → . It correspondsto the change of coordinates associated to the J ¯ T deformation (see [22, 26]). First we noticethat, setting δ = 0 in (3.82) and (3.83), the deformation preserves the Sugawara constructionfor the holomorphic sector T ( J ¯ T )2 ( z , τ ) = T (0)2 ( z , τ,
0) = −
12 ( ∂φ ) (1 − τ ¯ ∂φ ) = 12 (cid:16) J ( J ¯ T )+ ( z , τ ) (cid:17) , J ( J ¯ T )+ ( z , τ ) = J (0)+ ( z , τ, , (3.98)– 22 –ut this is not true for the anti-holomorphic sector. Then, we observe that the Lagrangian(3.86) reduces to L ( J ¯ T ) ( z , τ ) = L (0) ( z , τ,
0) = ∂φ ¯ ∂φ − τ ¯ ∂φ , (3.99)and the corresponding Legendre transformed Hamiltonian (3.88) becomes H ( J ¯ T ) ( x , τ ) = H (0) ( x , τ,
0) = P ( x ) + 2 τ (cid:16)(cid:0) τ J ( x ) (cid:1) − S ( J ¯ T ) (cid:17) , (3.100)where S ( J ¯ T ) = (cid:113)(cid:0) τ J ( x ) (cid:1) − τ ¯ I ( x ) . (3.101)Writing (3.100) as H ( J ¯ T ) ( x , τ ) = I ( J ¯ T )1 ( x , τ ) + ¯ I ( J ¯ T )1 ( x , τ ) with I ( J ¯ T )1 ( x , τ ) = − (cid:16) T ( J ¯ T )2 ( x , τ ) + Θ ( J ¯ T )0 ( x , τ ) (cid:17) = P ( x ) + 1 τ (cid:16)(cid:0) τ J ( x ) (cid:1) − S ( J ¯ T ) (cid:17) , ¯ I ( J ¯ T )1 ( x , τ ) = − (cid:16) ¯T ( J ¯ T )2 ( x , τ ) + ¯Θ ( J ¯ T )0 ( x , τ ) (cid:17) = 1 τ (cid:16)(cid:0) τ J ( x ) (cid:1) − S ( J ¯ T ) (cid:17) , (3.102)we find that J ( J ¯ T )2 ( x , τ ) = − τ (cid:16) S ( J ¯ T ) (cid:17) = J ( x ) − τ ¯ I ( J ¯ T )1 ( x , τ ) . (3.103)As already discussed in [21] for the T ¯ T deformation, we argue that also in the J ¯ T case theenergy density of the right and left movers (3.102) has the same formal expression of thespectrum obtained in [23] (cf. (4.118)), where the classical densities are replaced by thecorresponding integrated quantities. In addition, also the deformation of the U (1) currentdensity (3.103) admits a straightforward generalisation at the quantum level (cf. (4.120)).Analogous considerations apply to the case τ (1) = τ (3) = τ , τ (2) = τ (4) = 0 , which correspondsto the T ¯ J deformation.
Case τ (1) = τ (2) = τ (3) = τ (4) = τ All the equations (3.74)–(3.88) can be obtained setting δ = τ . In this case, the square root in(3.88) disappears, and the Hamiltonian takes the simple form H (0) ( x , τ ) = H (0) ( x , τ, τ ) = H ( x ) + τ P ( x ) (cid:0) J ( x ) + ¯ J ( x ) (cid:1) + τ P ( x )1 + τ (cid:0) J ( x ) − ¯ J ( x ) (cid:1) . (3.104)Again, we split the Hamiltonian as H (0) ( x , τ ) = I (0)1 ( x , τ ) + ¯ I (0)1 ( x , τ ) with I (0)1 ( x , τ ) = − (cid:0) T (0)2 ( x , τ ) + Θ (0)0 ( x , τ ) (cid:1) = I ( x ) + τ J ( x ) P ( x ) + τ P ( x )1 + τ (cid:0) J ( x ) − ¯ J ( x ) (cid:1) , ¯ I (0)1 ( x , τ ) = − (cid:0) ¯T (0)2 ( x , τ ) + ¯Θ (0)0 ( x , τ ) (cid:1) = ¯ I ( x ) + τ ¯ J ( x ) P ( x ) + τ P ( x )1 + τ (cid:0) J ( x ) − ¯ J ( x ) (cid:1) . (3.105)– 23 –inally, the deformed U (1) currents fulfil J (0)2 ( x , τ ) = J ( x ) + τ P ( x ) , ¯ J (0)2 ( x , τ ) = ¯ J ( x ) + τ P ( x ) . (3.106)We will see in section 4.5 that the quantum version of this perturbation (cf. (4.124)–(4.126))can be obtained by introducing two different scattering factors in the NLIEs (4.56). This second part of the paper is devoted to the study of the quantum version of the pertur-bations of classical field theories described in the preceding sections. As in [2], a NLIE willbe the starting point for the derivation of the Burgers-type equations for the spectrum of thedeformed conserved charges I ( s , ± ) k ( R, τ ) . Although we arrived to (3.48) by considering the freeboson model and only a very specific set of conserved currents, in the following we will assumethe general validity of (3.48) and of (3.49), obtained from (3.48) using the reflection property(3.28). In order to put our classical/quantum identification on a more solid foundation, itwould be very important to extend the proof of (3.48) to the set of conserved currents (2.17)and also to massive theories. We will now argue that the results of section 3.3 unambiguously suggest that the quantisationof these deformations is associated to a specific family of non-Lorentz invariant scatteringphase factors. Consider first the quantum version of (3.48), where the holomorphic and anti-holomorphic sectors are not coupled together by the interaction. Then, the level- k Hamiltonianand momentum operators factorise as ˆ E ( s (cid:48) ) k ( R, τ ) = ˆ I ( s (cid:48) , +) k ( R, τ ) ⊗ I + I ⊗ ˆ I ( s (cid:48) , − ) k ( R, τ ) , (4.1) ˆ P ( s (cid:48) ) k ( R, τ ) = ˆ I ( s (cid:48) , +) k ( R, τ ) ⊗ I − I ⊗ ˆ I ( s (cid:48) , − ) k ( R, τ ) , (4.2)and their action on a generic multi-particle state | N (+) , N ( − ) (cid:105) τ = | N (+) (cid:105) τ ⊗ | N ( − ) (cid:105) τ = | θ (+)1 , θ (+)2 , . . . , θ N (+) (cid:105) τ ⊗ | θ ( − )1 , θ ( − )2 , . . . , θ ( − ) N ( − ) (cid:105) τ , (4.3)is, in the asymptotically in the R → ∞ limit, determined by ˆ I ( s (cid:48) , ± ) k ( R, τ ) | N ( ± ) (cid:105) τ = ˆ γ k N ( ± ) (cid:88) i =1 e ± kθ ( ± ) i | N ( ± ) (cid:105) τ . (4.4)Notice that, in the deformed massless boson theory under consideration, there is only onespecies of elementary excitations and the set of rapidities { θ ( ± ) i } completely characterises an We have adopted here the convention of [30], where the single particle energy and momentum for right (+) and left ( − ) movers are parametrised as (cid:0) ˆ m e ± θ , ± ˆ m e ± θ (cid:1) . – 24 –symptotic quantum state. In addition ˆ I ( ± ) k ( R ) | N ( ± ) (cid:105) = (cid:18) ˆ γ k (cid:19) (cid:18) m (cid:19) k N ( ± ) (cid:88) i =1 (cid:32) πn ( ± ) i R (cid:33) k | N ( ± ) (cid:105) , (4.5)where n ( ± ) i ∈ Z + , ˆ m = ˆ γ and, we used the fact that the undeformed theory is free. Consid-ering now equations (3.48), assuming that, at least asymptotically in the R → ∞ limit: [ ˆ I ( s (cid:48) , ± ) k ( R, τ ) , ˆ I ( s (cid:48) , ± ) k (cid:48) ( R, τ )] = 0 , (4.6)we have (cid:104) N ( ± ) | ˆ I ( s (cid:48) , ± ) k ( R, τ ) (cid:16) R + 2 τ ˆ I ( s (cid:48) , ± ) s ( R, τ ) (cid:17) k | N ( ± ) (cid:105) τ = (cid:104) N ( ± ) | ˆ I ( ± ) k ( R ) | N ( ± ) (cid:105) τ R k , (4.7)and using (4.4) and (4.5), we find N ( ± ) (cid:88) i =1 e ± kθ ( ± ) i R + 2 τ ˆ γ s N ( ± ) (cid:88) j =1 e ∓ s (cid:48) θ ( ± ) j k = (cid:18) m (cid:19) k N ( ± ) (cid:88) i =1 (cid:16) πn ( ± ) i (cid:17) k , ( ∀ k ∈ Z ) , (4.8)with s (cid:48) = s < . The only consistent solutions to (4.8) are ± R ˆ m e ± θ ( ± ) i ± τ ˆ m γ s (cid:48) N ( ± ) (cid:88) j =1 e ± ( θ ( ± ) i − s (cid:48) θ ( ± ) j ) = ± πn ( ± ) i , ( i = 1 , , . . . , N ( ± ) ) , (4.9) i.e. the asymptotic Bethe Ansatz (BA) equations for our models. The two body scatteringamplitudes involving right and left movers are: δ ( s (cid:48) )( ± , ∓ ) ( θ, θ (cid:48) ) = 0 , δ ( s (cid:48) )( ± , ± ) ( θ, θ (cid:48) ) = ± τ ˆ m γ s (cid:48) e ± ( θ − s (cid:48) θ (cid:48) ) , ( s (cid:48) = s < . (4.10)Similarly, starting from (3.49) we find: δ ( s )( ± , ∓ ) ( θ, θ (cid:48) ) = ± τ ˆ m γ s e ± ( θ − s θ (cid:48) ) , δ ( s )( ± , ± ) ( θ, θ (cid:48) ) = 0 , ( s = s > . (4.11)The results presented in this section, strongly support the idea that the classical theoriesintroduced in this paper through a field dependent change of coordinates, can be consistentlyquantised within the exact S-matrix approach through the introduction of specific Lorentzbreaking phase factors. The natural generalisation of (4.10) and (4.11) for a massive fieldtheory is: δ ( s ) ( θ, θ (cid:48) ) = τ mγ s sinh( θ − s θ (cid:48) ) , (4.12)with asymptotic BA equations R m sinh( θ i ) + τ mγ s N (cid:88) j =1 sinh( θ i − s θ j ) = 2 πn i , ( n i ∈ Z , i = 1 , , . . . , N ) . (4.13)– 25 –umming over all the rapidities we see that, apart for s = 1 , the kinetic total momentum isnot quantised, i.e. P ( R, τ ) = P ( s )1 ( R, τ ) = N (cid:88) i =1 m sinh( θ i ) (cid:54) = 2 πR n , n = N (cid:88) i =1 n i , ( s (cid:54) = 1) , (4.14)since, at classical level, there is no trace of translational invariance breaking, this result showsthat the generator of space translations ˇ P and the deformed momentum P do not, in general,coincide. This fact is in agreement with the discussion of section 3.3. From (4.13) it is easyto show that a natural definition of quantised momentum is, in the large R limit: ˇ P ( R ) = P ( R, τ ) + τR (cid:16) P ( R, τ ) E ( s ) s ( R, τ ) − E ( R, τ ) P ( s ) s ( R, τ ) (cid:17) = 2 πR n , ( n ∈ Z ) , (4.15)with E ( s ) k = N (cid:88) i =1 γ s cosh( k θ i ) , P ( s ) k = N (cid:88) i =1 γ s sinh( k θ i ) , E = E ( s )1 . (4.16)Relation (4.15) can also be written as: ˇ P = ˇ P (+) − ˇ P ( − ) , ˇ P ( ± ) = I ( s , ± )1 + 2 τR I ( s , ± )1 I ( s , ∓ ) s . (4.17)Finally, notice that the quantisation of P on a circle would be preserved by taking symmetrisedversions of the scattering phases (4.10), (4.11) or (4.12). In the massive case: δ (cid:48) (˜ s , s ) ( θ, θ (cid:48) ) = τ mγ s (cid:0) sinh(˜ s θ − s θ (cid:48) ) + sinh( s θ − ˜ s θ (cid:48) ) (cid:1) . (4.18)with ˜ s = 1 . However, this is not the phase factor that our classical analysis suggests for thespectrum of the deformed charges (3.22). We have some concrete evidences [31] that the phasefactors (4.18), with arbitrary integers ˜ s and s , correspond instead to the T s +1 ¯ T ˜ s +1 + ¯ T s +1 T ˜ s +1 − Θ s − ¯Θ ˜ s − − ¯Θ s − Θ ˜ s − , (4.19)perturbations, recently discussed in [32]. Furthermore, the scattering phase ˜ δ ( s ) ( θ, θ (cid:48) ) = τ mγ s sinh( s θ − θ (cid:48) ) , (4.20)appears to be related, instead, to the spectrum of the corresponding mirror deformed Hamil-tonians (See, for example, [33] for a rigorous definition of mirror theory in a similar, nonrelativistic invariant, Bethe Ansatz context.) of the models under consideration [31]. The next goal is to try to build the quantum version of the classical models described insection 3, by including the scattering phase factors (4.12) into the NLIE for the sine-Gordonmodel confined on a infinite cylinder of circumference R . The sine-Gordon NLIE is [34–38] f ν ( θ ) = ν ( R, α | θ ) − (cid:90) C dθ (cid:48) K ( θ − θ (cid:48) ) ln (cid:16) e − f ν ( θ (cid:48) ) (cid:17) + (cid:90) C dθ (cid:48) K ( θ − θ (cid:48) ) ln (cid:16) e f ν ( θ (cid:48) ) (cid:17) , (4.21)– 26 –here we have set ν ( R, α | θ ) = i πα − i mR sinh( θ ) , (4.22)to denote the driving term . In (4.21)–(4.22), m is the sine-Gordon soliton mass, α is thequasi-momentum and K ( θ ) is the kernel defined as K ( θ ) = 12 πi ∂ θ ln S sG ( θ ) , (4.23)where ln S sG ( θ ) = − i (cid:90) R + dpp sin( pθ ) sinh ( πp ( ζ − / πp/
2) sinh ( πpζ/ , ζ = β − β . (4.24)The quasi-momentum or vacuum parameter α emerges [39, 40] by imposing periodic boundaryconditions on the sine-Gordon field: ϕ ( x + R, t ) = ϕ ( x, t ) . Due to the periodicity of thepotential in the sine-Gordon model L sG = 18 π (cid:16) ( ∂ t ϕ ) − ( ∂ x ϕ ) (cid:17) + 2 µ cos( √ βϕ ) , (cid:16) µ ∝ ( m ) − β (cid:17) , (4.25)the Hilbert space splits into orthogonal subspaces H α , characterised by the quasi-momentum α , ϕ → ϕ + 2 π √ β ←→ | Ψ α (cid:105) → e i πα | Ψ α (cid:105) , ( | Ψ α (cid:105) ∈ H α ) , (4.26)with α ∈ [ − / , / . Twisted boundary conditions of the form: ϕ ( x + R, t ) = ϕ ( x, t ) + 2 π √ β n , ( n ∈ Z ) , (4.27)are also natural in the sine-Gordon model since they correspond, in the infinite volume limit,to field configurations with non-trivial topological charge: Q x = √ β π (cid:90) R ∂ x ϕ dx . (4.28)Energy levels in the twisted sectors are also described by the same NLIE at specific valuesof the quasi-momentum. Furthermore, α can also be related to a background charge (cf.equation (4.61)). Therefore, (4.21) also describes minimal models of the Virasoro algebra, M p,q perturbed by the operator φ [41].The integration contours C , C in (4.21) are state dependent. For the ground-state in anarbitrary subspace H α , one may take them to be straight lines slightly displaced from thereal axis: C = R + i + , C = R − i + . Equations describing excited states have the sameform [37, 38, 40, 42] but with integration contours encircling a number of singularities { θ i } with (cid:0) e f ( θ i ) (cid:1) = 0 . However, in this paper we can ignore these subtleties, since the finalevolution equations do not depend explicitly on the specific details of the integration contours.Setting b ( ± ) k ( r ) = (cid:90) C dθ πi (cid:16) ± r e ± θ (cid:17) k ln (cid:16) e − f ν ( θ ) (cid:17) − (cid:90) C dθ πi (cid:16) ± r e ± θ (cid:17) k ln (cid:16) e f ν ( θ ) (cid:17) , ( k ∈ N + 1) , (4.29) – 27 –ith r = mR , and I ( ± ) k ( R, m ) = (cid:18) πR (cid:19) k b ( ± ) k ( r ) C k , (4.30)where C k = 12 k (cid:18) πβ (cid:19) k +12 Γ (cid:0) k ( ζ + 1) (cid:1) Γ (cid:0) k +32 (cid:1) Γ (cid:0) k ζ (cid:1) (cid:18) Γ(1 + ζ/ / ζ/ (cid:19) k , (4.31)then, (cid:110) I (+) k , I ( − ) k (cid:111) are the eigenvalues of the quantum operators associated to the classicalconserved charges defined in (3.47), i.e I (+) k ←→ (cid:90) R I k ( x ) dx , I ( − ) k ←→ (cid:90) R ¯ I k ( x ) dx . (4.32)It is also convenient to define the alternative set of conserved charges E k ( R ) = I (+) k ( R ) + I ( − ) k ( R ) , P k ( R ) = I (+) k ( R ) − I ( − ) k ( R ) , (4.33)where E ( R ) = E ( R ) and P ( R ) = P ( 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 > , (4.34)which is the quantum analog of (3.28).Motivated by the results of section 4.1, we conjecture that the geometric-type deformationsdescribed in section 3, associated to generic combinations of the higher spin conserved currents,correspond to the inclusion of extra scattering phases of the form δ ( θ, θ (cid:48) ) = (cid:88) s τ ( s ) γ γ s sinh( θ − s θ (cid:48) ) , (4.35)where the τ ( s ) are independent coupling parameters and γ s = (2 πm ) s C s , γ = m , (4.36)with s = | s | . In (4.35) the sum runs, in principle, over positive and negative odd integers.However, since the whole analysis can be straightforwardly analytically extended to arbitraryvalues of s , in the following we shall relax this constraint, at least to include the case s → andthe set of nonlocal conserved charges [43]. The scattering phase factor (4.35) 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 detailedanalytic 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 The explicit dependence on some parameters, e.g. the mass m , will be sometime omitted not to weighdown the notation. – 28 –ur checks have been performed considering conformal field theories deformed (explicitly atleading order) by a single irrelevant composite field. Therefore, in this paper, we shall restrictthe analysis to scattering phase factors with only a single non vanishing irrelevant coupling: δ ( s ) ( θ, θ (cid:48) ) = τ γ γ s sinh( θ − s θ (cid:48) ) , (4.37)which modifies the kernel appearing in the NLIE (4.21) as K ( θ − θ (cid:48) ) → K ( θ − θ (cid:48) ) − π ∂ θ δ ( s ) ( θ − s θ (cid:48) ) = K ( θ − θ (cid:48) ) − τ m γ s π cosh( θ − s θ (cid:48) ) . (4.38)Inserting (4.38) in (4.21), after simple manipulations, we find that the deformed version of f ν ( θ ) fulfils (4.21) with ν = ν ( R , α | θ − θ ) , (4.39)where R and θ are defined through R cosh ( θ ) = R + τ E ( s ) s ( R, τ ) , R sinh ( θ ) = τ P ( s ) s ( R, τ ) . (4.40)Equations (4.40) imply ( R ) = (cid:16) R + τ E ( s ) s ( R, τ ) (cid:17) − (cid:16) τ P ( s ) s ( R, τ ) (cid:17) . (4.41)The quantities E ( s ) k ( R, τ ) and P ( s ) k ( R, τ ) denote the k -th higher conserved charges of the theorydeformed 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 ) , (4.42)and I ( s , ± ) k are again defined through (4.29)–(4.31) but with the deformed driving term (4.39).Formula (4.39) shows that the solutions of the deformed NLIE are modified simply by aredefinition of the length R and by a rapidity shift. One can easily show that the deformedcharges are related to the undeformed ones through I ( s , ± ) k ( R, τ ) = e ± k θ I ( ± ) k ( R ) , (4.43)which, together with (4.42), leads to a generalisation of the Lorentz-type transformation (1.2)derived in [21] (cid:32) E ( s ) k ( R, τ ) P ( s ) k ( R, τ ) (cid:33) = (cid:32) cosh ( k θ ) sinh ( k θ )sinh ( k θ ) cosh ( k θ ) (cid:33) (cid:32) E k ( R ) P k ( R ) (cid:33) . (4.44)From (4.44) one has (cid:16) E ( s ) k ( R, τ ) (cid:17) − (cid:16) P ( s ) k ( R, τ ) (cid:17) = ( E k ( R )) − ( P k ( R )) , (4.45)and, in particular, for k = 1 E ( R, τ ) − P ( R, τ ) = E ( R ) − P ( R ) , (4.46)– 29 –here E ( R, τ ) = E ( s )1 ( R, τ ) and P ( R, τ ) = P ( s )1 ( R, τ ) . To find the generalisations of theBurgers equation (1.1) describing the evolution of the spectrum under the deformations withgeneric Lorentz spin s , we differentiate both sides of (4.43) w.r.t. τ at fixed R , getting ∂ τ I ( s , ± ) k ( R, τ ) + R (cid:48) ∂ R I ( s , ± ) k ( R, τ ) = ± k θ (cid:48) I ( s , ± ) k ( R, τ ) . (4.47)All that is left to do is compute ∂ τ R = R (cid:48) and ∂ τ θ = θ (cid:48) . We start by rewriting (4.40) as (cid:40) R = R e − θ + 2 τ e ( s − θ I (+) s ( R ) R = R e θ + 2 τ e − ( s − θ I ( − ) s ( R ) , (4.48)then, differentiating both equations in (4.48) w.r.t. τ we obtain (cid:40) R (cid:48) e − θ − R e − θ θ (cid:48) + 2 e ( s − θ I (+) s ( R ) + 2 τ e ( s − θ ( s − θ (cid:48) I (+) s ( R )0 = R (cid:48) e θ + R e θ θ (cid:48) + 2 e − ( s − θ I ( − ) s ( R ) − τ e − ( s − θ ( s − θ (cid:48) I ( − ) s ( R ) , (4.49)whose solution is ∂ τ R = R (cid:48) = −E ( s ) s ( R, τ ) + ( s − τ (cid:16) P ( s ) s ( R, τ ) (cid:17) ( s − τ E ( s ) s ( R, τ ) − R ,∂ τ θ = θ (cid:48) = − P ( s ) s ( R, τ )( s − τ E ( s ) s ( R, τ ) − R . (4.50)The equations for the total energy and momentum are then (cid:40) ∂ τ E ( R, τ ) + R (cid:48) ∂ R E ( R, τ ) = θ (cid:48) P ( R, τ ) ∂ τ P ( R, τ ) + R (cid:48) ∂ R P ( R, τ ) = θ (cid:48) E ( R, τ ) , (4.51)which are driven, through R (cid:48) and θ (cid:48) by the evolution equations for E ( s ) s and P ( s ) s (cid:40) ∂ τ E ( s ) s ( R, τ ) + R (cid:48) ∂ R E ( s ) s ( R, τ ) = s θ (cid:48) P ( s ) s ( R, τ ) ∂ τ P ( s ) s ( R, τ ) + R (cid:48) ∂ R P ( s ) s ( R, τ ) = s θ (cid:48) E ( s ) s ( R, τ ) . (4.52)Again it is worth to stress that the deformed momentum P , defined through (4.51) is notalways quantized for τ (cid:54) = 0 , as it flows according to the complicated non-linear equations(4.51)-(4.52). One can argue (see for example [44]) that a quantised object is ˇ P ( R ) = 12 πR (cid:18)(cid:90) C dθ p ( θ ) ln (cid:16) e − f ν ( θ ) (cid:17) − (cid:90) C dθ p ( θ ) ln (cid:16) e f ν ( θ ) (cid:17)(cid:19) , (4.53)with p ( θ ) = ∂ θ ν ( R , α | θ − θ ) = − im R cosh( θ − θ ) . (4.54)Using (4.40) and (4.54) in (4.53), we find ˇ P = P ( R ) = P ( R, τ ) + τR (cid:16) P ( R, τ ) E ( s ) s ( R, τ ) − E ( R, τ ) P ( s ) s ( R, τ ) (cid:17) , (4.55)which is the same formula obtained in section 4.15 from BA considerations.– 30 – .3 The CFT limit of the NLIE The CFT limit of the sine-Gordon model is described by a pair of decoupled NLIEs correspond-ing to the right- (+) and the left- ( − ) mover sectors. The latter equations can be obtained bysending m to zero as m = ˆ m(cid:15) with (cid:15) → + and simultaneously θ to ±∞ as θ → θ ± ln( (cid:15) ) suchthat ˆ me ± θ remain finite. The resulting equations are identical to (4.21) but with the term m sinh( θ ) replaced by ˆ m e θ and − ˆ m e − θ , for the right- and left-mover sectors, respectively f ν ( ± ) ( θ ) = ν ( ± ) ( R, α ( ± )0 | θ ) − (cid:90) C dθ (cid:48) K ( θ − θ (cid:48) ) ln (cid:16) e − f ν ( ± ) ( θ (cid:48) ) (cid:17) + (cid:90) C dθ (cid:48) K ( θ − θ (cid:48) ) ln (cid:16) e f ν ( ± ) ( θ (cid:48) ) (cid:17) , (4.56)where ν ( ± ) ( R, α ( ± )0 | θ ) = i πα ( ± )0 ∓ i ˆ mR e ± θ , (4.57)and ˆ m sets the energy scale. In (4.57), we have allowed for the possibility of two independentvacuum parameters α ( ± )0 in the two chiral sectors. The resulting NLIEs are, in principle,suitable for the description of twisted boundary conditions and more general states in the c = 1 CFT, compared to the set strictly emerging from the m → 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 contours C , C are still state dependent. In particular, they should be deformed away from the initialground-state configuration C = R + i + , C = R − i + when the parameters ± α ( ± )0 areanalytically extended to large negative values. Setting ˆ b ( ± ) k = (cid:90) C dθ πi (cid:18) ± ˆ r e ± θ (cid:19) k ln (cid:16) e − f ν ( ± ) ( θ ) (cid:17) − (cid:90) C dθ πi (cid:18) ± ˆ r e ± θ (cid:19) k ln (cid:16) e f ν ( ± ) ( θ ) (cid:17) , ( k ∈ N + 1) , (4.58) with ˆ r = ˆ mR , then I ( ± ) k ( R ) = (cid:18) πR (cid:19) k ˆ b ( ± ) k C k = (cid:18) R (cid:19) k πa ( ± ) k , (4.59)where the a ( ± ) k are state-dependent constants, i.e. R and ˆ m independent. Using again thespin-flip symmetry I ( ± ) − k ( R ) = I ( ∓ ) k ( R ) , we can extend the discussion to k ∈ Z + 1 , with k = | k | . Some of the state-dependent coefficients I ( ± ) k (2 π ) can be found in [39]. In particular,the energy and momentum of a generic state are E ( R ) = I (+)1 ( R ) + I ( − )1 ( R ) = 2 πR (cid:32) n (+) − c (+)0 (cid:33) + 2 πR (cid:32) n ( − ) − c ( − )0 (cid:33) ,P ( R ) = I (+)1 ( R ) − I ( − )1 ( R ) = 2 πR (cid:16) h (+) − h ( − ) (cid:17) , (4.60)with effective central charges c ( ± )0 = 1 − β (cid:16) α ( ± )0 (cid:17) = 1 − h ( ± )0 , (4.61)– 31 –here 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 La-grangian given by (4.25) with µ = 0 (see [38] for more details): ϕ ( x + R, t ) = ϕ ( x, t ) + 2 π n √ β , ( n ∈ Z ) . (4.62)In (4.62), r = ( √ β ) − is standard compactification radius of the bosonic field. Then, thehighest weights are now labelled by a pair of integers ( n, ˜ n ) , where ˜ nβ / √ π is the quantizedcharge associated to the total field momentum Q t = √ πβ (cid:90) R ∂ t ϕ dx = 2 √ β (cid:90) R Π dx , (4.63)and n is the winding number corresponding to the topological charge Q x = √ β π (cid:90) R ∂ x ϕ dx . (4.64)Then the combinations Q ( ± )0 = π ( Q x ± Q t ) = π (cid:0) n ± ˜ nβ (cid:1) , (4.65)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 → /β symmetry. Weadopted this unconventional definition for the topological charges since, as a reminiscence ofthe sine-Gordon model, it emerges more naturally from the current setup. The anagolous ofthe Bloch wave states in (4.26) are now created by the action on the CFT vacuum state ofthe vertex operators V ( n, ˜ n ) ( z ) = exp (cid:32) √ πβ Q (+)0 φ ( z ) + √ πβ Q ( − )0 ¯ φ (¯ z ) (cid:33) , ϕ ( z ) = φ ( z ) + ¯ φ (¯ z ) , (4.66)with left and right conformal dimensions given by h ( ± )0 = 14 π β (cid:16) Q ( ± )0 (cid:17) = 14 (cid:18) nβ ± ˜ nβ (cid:19) . (4.67)Considering (4.61), in section 4.4 we will make the following identification: Q ( ± )0 = 2 πα ( ± )0 β . (4.68)However, relation (4.68) is valid only at formal level since α ( ± )0 are continuous parameterswhich can also account, for example, for twisted boundary conditions while, at fixed β , the– 32 –harges Q ( ± )0 can only assume the discrete set of values given in (4.65). Using (4.68) in (4.66)we find V ( n, ˜ n ) ( z ) = exp (cid:16) √ βα (+)0 φ ( z ) + √ βα ( − )0 ¯ φ (¯ z ) (cid:17) , (4.69)which, for ˜ n = 0 and under the field-shift (4.62), display the same quasi-periodicity propertiesof the finite volume sine-Gordon Bloch states (4.26). In this limit, the τ dependent phasefactor (4.37) splits, for s > , into δ ( s )( ± , ∓ ) ( θ, θ (cid:48) ) = ± τ ˆ m ˆ γ s e ± ( θ − s θ (cid:48) ) , δ ( s )( ± , ± ) ( θ, θ (cid:48) ) = 0 , (4.70)with ˆ γ s = (2 π ˆ m ) s C s , ˆ γ = ˆ m , (4.71)breaking conformal invariance by explicitly introducing a coupling between the right and theleft mover sectors. Setting R ( s, ± )0 = R + 2 τ I ( s, ∓ ) s ( R, τ ) , (4.72)the resulting NLIE is identical to (4.56) with driving term ν ( ± ) (cid:0) R ( s, ± )0 , α | θ (cid:1) . For s < , thetwo chiral sectors remain decoupled δ ( s (cid:48) )( ± , ∓ ) ( θ, θ (cid:48) ) = 0 , δ ( s (cid:48) )( ± , ± ) ( θ, θ (cid:48) ) = ± τ ˆ m γ s (cid:48) e ± ( θ − s (cid:48) θ (cid:48) ) , ( s (cid:48) = s < , (4.73)and, due to the reflection property I ( s (cid:48) , ± ) s = I ( s (cid:48) , ∓ ) s (cid:48) , the driving terms become ν ( ± ) (cid:0) R ( s (cid:48) , ∓ )0 , α | θ (cid:1) .In turn, the length redefinition (4.72) implies I ( s, ± ) k ( R, τ ) = I ( ± ) k (cid:16) R ( s, ± )0 (cid:17) , ( s > , (4.74) I ( s (cid:48) , ± ) k ( R, τ ) = I ( ± ) k (cid:16) R ( s (cid:48) , ∓ )0 (cid:17) , ( s (cid:48) = − s < , (4.75)which are equivalent to the evolution equations (3.50), deduced at classical level.Using the scaling property of the CFT charges , (3.50) can be further simplified to ∂ τ I ( s, ± ) k = − k (cid:16) R − τ ( s − I ( s, ± ) s (cid:17) I ( s, ± ) s I ( s, ∓ ) k τ (cid:16) R − τ ( s − I ( s, +) s (cid:17) I ( s, − ) s + R (cid:16) R + 2 τ I ( s, +) s (cid:17) , ( s > (4.76) ∂ τ I ( s (cid:48) , ± ) k = − k I ( s (cid:48) , ± ) k I ( s (cid:48) , ± ) s R + 2 τ ( s + 1) I ( s (cid:48) , ± ) s , ( s (cid:48) = − s < . (4.77)Notice that, setting k = s = 1 in (4.76) we get ∂ τ I ( ± )1 = − I (+)1 I ( − )1 R + 2 τ (cid:16) I (+)1 + I ( − )1 (cid:17) , (4.78)which matches with the T ¯ T result quoted in [45].– 33 – erturbations with s < : For simplicity, let us discuss first the cases with s < . Since the generic k -th charges of aCFT scale as R − k according to (4.59) and the corresponding deformed charges I ( s (cid:48) , ± ) k ( R, τ ) fulfil the same equation with R → R ( s (cid:48) , ± )0 , then I ( s (cid:48) , ± ) k ( R, τ ) = I ( s (cid:48) , ∓ ) k (cid:48) ( R, τ ) = R k I ( ∓ ) k (cid:48) ( R ) (cid:16) R ( s (cid:48) , ± )0 (cid:17) k = 2 π a ( ∓ ) k (cid:48) (cid:16) R ( s (cid:48) , ± )0 (cid:17) k , (4.79)where a ( ± ) k = a ( ∓ ) k (cid:48) , k (cid:48) = − k, s (cid:48) = s , and R ( s (cid:48) , ± )0 is defined in (4.72).In order to find the solution to (4.79) for generic k , one must first solve (4.79) for k = s ( k (cid:48) = s (cid:48) ) . In this case, the solution can be reconstructed perturbatively as I ( s (cid:48) , ± ) s ( R, τ ) = I ( s (cid:48) , ∓ ) s (cid:48) ( R, τ ) = ∞ (cid:88) j =0 ( − j j + 1 (cid:18) j (1 + s ) + ( s − j (cid:19) (2 τ ) j (cid:16) π a ( ± ) s (cid:17) j +1 R j (1+ s )+ s . (4.80)This expression can be resummed as • s = − : I ( − , ± )1 ( R, τ ) = R τ − (cid:115) τ π a ( ± )1 R . (4.81)Both the classical (see the s = − case in (3.43)) and quantum results suggest that the lead-ing perturbing operator corresponds to the Lorentz breaking operator typically appearingin effective field theories for discrete lattice models [47]. • s = − : I ( − , ± )2 ( R, τ ) = 4 R τ sinh arcsinh √ (cid:32) τ π a ( ± )2 R (cid:33) / . (4.82)For generic spin s < the result can be written in terms of a single generalised hypergeometricfunction: I ( s (cid:48) , ± ) s ( R, τ ) = I ( s (cid:48) , ∓ ) s (cid:48) ( R, τ ) = s R τ (1 + s ) (cid:32) − F s (cid:32) − τ (1 + s ) s s s π a ( ± ) s R s (cid:33)(cid:33) , (4.83)where F s ( x ) is defined in (3.40). The total momentum and energy are E ( R ) = I ( s (cid:48) , +)1 ( R, τ ) + I ( s (cid:48) , − )1 ( R, τ ) , P ( R ) = I ( s (cid:48) , +)1 ( R, τ ) − I ( s (cid:48) , − )1 ( R, τ ) , (4.84) Separately, the NLIEs in (4.56), with generic parameters β and α ( ± )0 can be also associated to the quantumKdV theory, as extensively discussed in [39, 46]. The coefficients a k for k = 3 , can be recovered from [39]. – 34 –here I ( s (cid:48) , ± )1 ( R, τ ) are obtained by solving (4.79) with k (cid:48) = − k = − using (4.83).Finally, notice that even spin charges do not, in general, correspond to local conserved currentsin the sine-Gordon model. They can occasionally emerge from the set of non-local charges,at specific rational values of β . Our results concerning the exact quantum spectrum can besmoothly deformed in s , therefore they formally also describe deformations of the sine-Gordonmodel by non-local currents [48]. Moreover, there are many integrable systems with extendedsymmetries where even spin charges appear. The sign of the corresponding eigenvalues dependson the internal flavor of the specific soliton configuration considered. Since the flow equations(4.50)–(4.52), should properly describe the evolution of the spectrum driven by analogousdeformations in a very wide class of systems, perturbations by currents with s even, may leadto interesting quantum gravity toy models where the effective sign of the perturbing parameter τ depends on the specific state under consideration. Perturbations with s > : In the case s > , the left- and right-mover sectors are now coupled and the solution of thegeneralised Burgers equations become equivalent to the set of equations I ( s, ± ) k ( R, τ ) = 2 π a ( ± ) k (cid:16) R + 2 τ I ( s, ∓ ) s ( R, τ ) (cid:17) k . (4.85)While, for models with P ( s ) k = I ( s, +) k ( R, τ ) − I ( s, − ) k ( R, τ ) = 0 the left- and right-mover sectorsare completely symmetric and the relations (4.85) formally reduce to the equations (4.79), for P ( s ) k (cid:54) = 0 the solution to (4.85) for k = s are I ( s, ± ) s ( R, τ ) = 2 π a ( ± ) s R s (cid:34) π a ( ∓ ) s ∞ (cid:88) k =1 k − (cid:88) l =0 l + 1 (cid:32) ( k − l ) s + l − l (cid:33) (cid:32) ( l + 1) s + k − l − k − l (cid:33) × (cid:16) π a ( ± ) s (cid:17) l (cid:16) π a ( ∓ ) s (cid:17) k − l − ( − k (2 τ ) k R ( s +1) k (cid:35) . (4.86)We were not able to find a general compact expression of (4.86), except for the already known( s = 1 ) T ¯ T-related result E (1)1 ( R, τ ) = R τ − (cid:118)(cid:117)(cid:117)(cid:116) τR (cid:32) π (cid:0) a (+)1 + a ( − )1 (cid:1) R (cid:33) + 4 τ R (cid:32) π (cid:0) a (+)1 − a ( − )1 (cid:1) R (cid:33) , P (1)1 ( R ) = 2 π (cid:0) a (+)1 − a ( − )1 (cid:1) R . (4.87)
Perturbations with s → : – 35 –et us first perform the s → limit in the massive case. In this limit γ s → but, rescaling τ ,we can recast the driving term in the form ν ( R, α | θ ) = i πα − i mR sinh( θ ) + iτ G cosh( θ ) , (4.88)where G = − (cid:18)(cid:90) C dθ πi ln (cid:16) e − f ν ( θ ) (cid:17) − (cid:90) C dθ πi ln (cid:16) e f ν ( θ ) (cid:17)(cid:19) . (4.89)Next, in the limit m → , the leading contribution to G in (4.58) is coming from a largeplateau of the integrand ( g ( θ ) in figure 1) of height − if ν ( θ ) ∼ πα β , (4.90)and width growing as ∼ mR/ : G ∼ ln (cid:18) mR (cid:19) α β . (4.91)The emergence of the plateau can be deduced analytically from the NLIE (see for example[49] for a related discussion in the TBA context). Therefore, at fixed finite R , after a furtherrescaling τ → τ π/ ln( mR/ , as m tends to zero, we obtain for the driving terms in the NLIEs: ν ( ± ) = ν ( ± ) ( R ( ± )0 , α | θ ) = i πα ∓ i ˆ m e ± θ R ( ± )0 , (4.92)with R ( ± )0 = R ∓ τ Q , Q = 2 πα β . (4.93)Thus, perturbing the theory with the phase factor (4.37) with s = 0 is equivalent, in anappropriate scaling limit, to a constant shift of the volume R . At fixed normalisation of thedeforming parameter τ , the shift turns out to be directly proportional to the topological charge Q . We see, from (4.92) that the left- and right-mover sectors remain decoupled also at τ (cid:54) = 0 .However, starting directly from the CFT limit, one can argue that there exist less trivial waysto couple the two sectors. The most general variant involves four different coupling constants (cid:126)τ = { τ ( a | b ) } with a, b ∈ { + , −} : R ( ± )0 ( (cid:126)τ ) = R ∓ (cid:16) τ ( ±| +) Q (+)0 + τ ( ±|− ) Q ( − )0 (cid:17) , Q ( ± )0 = 2 πα ( ± )0 β . (4.94)Notice that in (4.94) 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.– 36 – igure 1 . The formation of the plateau in g ( θ ) = − π I m (cid:16) ln (cid:16) e − f ν ( θ + i + ) (cid:17)(cid:17) as m → at β = 1 / , α = 1 / . For this choice of the parameters, the height of the plateau of g ( θ ) is / . A further, natural extension of the quantum models studied in section 4.2, corresponds toscattering phase factors of the form δ (˜ s , s ) ( θ, θ (cid:48) ) = τ γ ˜ s γ s sinh(˜ s θ − s θ (cid:48) ) . (4.95)Deforming the kernel according to (4.38) and using (4.95) instead of (4.37), the driving termof (4.21) becomes ν = ν ( R, α | θ ) − iτ ˜ s γ ˜ s (cid:16) e ˜ s θ I (˜ s , s , − ) s − e − ˜ s θ I (˜ s , s , +) s (cid:17) , (4.96)where I (˜ s , s , ± ) k are defined through (4.58) and (4.59) with the deformed driving term (4.96).The cases with ˜ s = ± , have been extensively discussed in the previous sections, they allcorrespond to gravity-like theories, where the effect of the perturbation can be re-absorbedinto a redefinition of the volume R plus a shift in the rapidity θ , according to (4.39). Fromequation (4.96), we see that for generic values of ˜ s (cid:54) = ± it is not no longer possible tore-absorb the perturbation in the same fashion. However, another interesting example isrecovered by considering the scaling limit { ˜ s , γ ˜ s , m } → { , , } , such that ˜ s γ ˜ s remains finite.In the following we shall set ˜ s γ ˜ s = 2 π . This situation corresponds to the standard masslesslimit where, in the undeformed ( τ = 0) NLIE, the right- (+) and the left-mover ( − ) sectorsare completely decoupled, while a residual interaction between the two sectors is still presentat τ (cid:54) = 0 . In fact, the deformed versions of f ν ( ± ) ( θ ) fulfil (4.56) with ν ( ± ) = ν ( ± ) (cid:0) R, α ( ± )0 − α ( s , ± ) | θ (cid:1) , α ( s , ± ) = (cid:16) τ ( ± ) I ( s , − ) s ( R, (cid:126)τ ) − τ ( ∓ ) I ( s , +) s ( R, (cid:126)τ ) (cid:17) , (4.97)– 37 –here the charges I ( s , ± ) k are defined through (4.58) with driving term (4.97), and { τ ( − ) , τ (+) } = (cid:126)τ are two coupling parameters defined as τ ( ± ) = τ e ± σ , ± σ = lim θ →±∞ ˜ s → + ˜ s θ . (4.98)Therefore, the contributions of the nontrivial interaction can be formally re-absorbed in aredefinition of the vacuum parameters α ( ± )0 . In turn, this affects the value of the effectivecentral charges as c ( s , ± ) = c ( ± )0 + 24 β ( α ( ± )0 ) − β (cid:16) ( α ( ± )0 ) − α ( s , ± ) (cid:17) , c ( ± )0 = c − β ( α ( ± )0 ) . (4.99)Considering the formal identification Q ( ± )0 = 2 πα ( ± )0 β made in (4.93), the redefinition (4.97)corresponds to the following dressing of the topological charges Q ( s , ± ) ( R, (cid:126)τ ) = Q ( ± )0 − κ π (cid:16) τ ( − ) I ( s , − ) s ( R, (cid:126)τ ) − τ (+) I ( s , +) s ( R, (cid:126)τ ) (cid:17) , (4.100)with κ = 8 π β . (4.101)Let us focus on the s = 1 case. The equations for the spectrum are I ( ± )1 ( R, (cid:126)τ ) = 2 πR (cid:16) n ( ± ) − c ( ± ) (cid:17) , ( I ( ± )1 = I (1 , ± )1 , c ( ± ) = c (1 , ± ) ) , (4.102)which can be solved exactly for I ( ± )1 ( R, (cid:126)τ ) at any values of the parameters τ ( ± ) . However, sincethe general analytic expressions for I ( ± )1 are very cumbersome, we will restrict the discussionto specific scaling limits: • τ (+) = τ ( − ) = τ : I (+)1 ( R, τ ) − I ( − )1 ( R, τ ) = P ( R ) = 2 πR (cid:16) h (+) − h ( − ) (cid:17) , (4.103) I (+)1 ( R, τ ) = I (+)1 ( R ) + τ Q ( ± )0 R P ( R ) + τ κ πR P ( R ) , (4.104)with h ( ± ) = h ( ± )0 + n ( ± ) and (4.100) becomes Q ( ± ) ( R, τ ) = Q ( ± )0 + τ κ π P ( R ) . (4.105)Surprisingly, with the replacement I ( ± )1 ( R, τ ) = 2 πR (cid:16) h ( ± ) ( τ ) − c (cid:17) , (4.106)equations (4.103) and (4.104) lead to exact expressions for the conformal dimensions h ( ± ) ( τ ) ,which precisely match the form of the (left) conformal dimension in the J ¯ T model, as recentlyshown in [50]: h ( ± ) ( τ ) = h ( ± ) + τ Q ( ± )0 π P + τ κ π P . (4.107)– 38 –s we will shortly see, this is the first instance among many that link the phase factor (4.95)with s = 1 , in the scaling limit (4.98) to the quantum J ¯ T model. • τ (+) = 0 , τ ( − ) = τ : the two sectors ( ± ) are completely decoupled I ( ± )1 ( R, τ ) = I ( ± )1 ( R ) ± τ Q ( ± )0 R I ( ± )1 ( R, τ ) + τ κ πR (cid:16) I ( ± )1 ( R, τ ) (cid:17) , (4.108) Q ( ± ) ( R, τ ) = Q ( ± )0 ± τ κ π I ( ± )1 ( R, τ ) , (4.109)and the solutions to (4.108) are I ( ± )1 ( R, τ ) = 4 πκτ (cid:32) R ∓ τ Q ( ± )0 − (cid:114)(cid:16) R ∓ τ Q ( ± )0 (cid:17) − τ κ (cid:16) h ( ± ) − c (cid:17)(cid:33) . (4.110)We notice that (4.108) can be rewritten as I ( ± )1 ( R, τ ) = 2 π (cid:0) h ( ± ) − c (cid:1) R − τ (cid:16) ± Q ( ± )0 + τ κ π I ( ± )1 ( R, τ ) (cid:17) , (4.111)which suggests that, in this limit, the perturbation has a dual geometric description, sinceit can be interpreted as a redefinition of the length R . The dual deformation of the NLIEcorresponds to a deformed version of f ν ( ± ) ( θ ) which fulfils (4.56) with ν ( ± ) = ν ( ± ) (cid:16) R ( ± )0 , α θ (cid:17) , R ( ± )0 = R − τ (cid:16) ± Q ( ± )0 + τ κ π I ( ± )1 ( R, τ ) (cid:17) . (4.112)Expressions (4.110) are trivially solutions of two decoupled Burgers-like equations ∂ τ I ( ± )1 ( R, τ ) ± Q ( ± ) ( R, τ ) ∂ R I ( ± )1 ( R, τ ) = 0 . (4.113)Therefore, I ( ± )1 ( R, τ ) fulfils a Burgers-type equation analogous to (1.1), where Q ( ± ) ( R, τ ) play the role of velocities. ¯ T model
Formula (4.110) strongly resembles the expression for the energy of the right-movers of theJ ¯ T-deformed CFT derived in [23, 26]. To obtain the J ¯ T model, one must treat the ( ± ) sectorsin a non-symmetric way. For s = 1 , a possible asymmetric generalisation of (4.97) with fourfree parameters (cid:126)τ = { τ ( a | b ) } with a, b ∈ { + , −} is ν ( ± ) = ν ( ± ) (cid:0) R, α ( ± )0 − α ( ± ) ( (cid:126)τ ) | θ (cid:1) ,α ( ± ) ( (cid:126)τ ) = (cid:16) τ ( ±|− ) I ( − ) s ( R, (cid:126)τ ) − τ ( ∓| +) I (+) s ( R, (cid:126)τ ) (cid:17) . (4.114)In (4.114) we also included the possibility to have two different initial values of the twistparameters α ( ± )0 in the two sectors. From (4.114), the J ¯ T model is recovered with the choice τ ( ±|− ) = 0 , τ ( ∓| +) = τ , α (+)0 = α ( − )0 = α . (4.115)– 39 –orrespondingly, relations (4.114) become ν ( ± ) = ν ( ± ) J ¯ T (cid:16) R, α − α ( ± ) J ¯ T θ (cid:17) , α ( ± ) J ¯ T = − τ I (+) J ¯ T ( R, τ ) . (4.116)Alternatively, the (+) sector can be equivalently described with a redefinition of the length R as ν (+) = ν (+) ( R J ¯ T , α θ ) , R J ¯ T = R − τ (cid:16) Q + τ κ π I (+) J ¯ T ( R, τ ) (cid:17) . (4.117)The right-moving solution in (4.110) and the topological charge become I (+)1 ( R, τ ) = I (+) J ¯ T ( R, τ ) = 4 πκτ (cid:18) R − τ Q − (cid:114) ( R − τ Q ) − τ κ (cid:16) h (+) − c (cid:17)(cid:19) , (4.118) I (+) J ¯ T ( R, τ ) − I ( − ) J ¯ T ( R, τ ) = P ( R ) = 2 πR (cid:16) h (+) − h ( − ) (cid:17) , (4.119) Q (+) ( R, τ ) = Q (+) J ¯ T ( R, τ ) = Q + τ κ π I (+) J ¯ T ( R, τ ) . (4.120)Therefore, Q = 2 πα β and κ = 8 π β have been again consistently identified with thetopological charge and the chiral anomaly, respectively. The results described here are in fullagreement with [23] and the classical results presented in section 3. As already discussed, in principle one may introduce several scattering phase factors to deformthe NLIEs. In this section we will consider a particular combination of s → and ˜ s → scattering factors which allows us to match with the classical results (3.105)–(3.106). Weconsider a double perturbation made of a length redefinition (4.94) with τ ( ±| +) = ∓ τ and τ ( ±|− ) = ± τ , together with a shift of the twist parameter (4.114) with τ ( ∓| +) = + τ and τ ( ±|− ) = + τ . The corresponding deformed driving term is then ν ( ± ) = ν ( ± ) (cid:16) R ( ± )0 , α ( ± )0 − α ( ± ) ( R, τ ) θ (cid:17) , (4.121)with α ( ± ) ( R, τ ) = − τ P ( R ) , R ( ± )0 = R + τ (cid:16) Q (+) ( R, τ ) − Q ( − ) ( R, τ ) (cid:17) = R + τ (cid:16) Q (+)0 − Q ( − )0 (cid:17) , (4.122)where in the last equality we used the fact that Q ( ± ) ( R, τ ) = 2 π β (cid:16) α ( ± )0 − α ( ± ) ( R, τ ) (cid:17) . Sincethe central charges are affected by the deformation as c ( ± ) ( R, τ ) = c + 24 β (cid:0) α ( ± )0 (cid:1) − β (cid:16) α ( ± )0 − α ( ± ) ( R, τ ) (cid:17) , (4.123)one finds I ( ± )1 ( R, τ ) = 2 π R ( ± )0 (cid:32) n ( ± ) − c ( ± ) ( R, τ )24 (cid:33) = R I ( ± )1 ( R ) + τ Q ( ± )0 P ( R ) + κ π τ P ( R ) R + τ (cid:16) Q (+)0 − Q ( − )0 (cid:17) , (4.124) I (+)1 ( R, τ ) − I ( − )1 ( R, τ ) = P ( R ) , (4.125) Q ( ± ) ( R, τ ) = Q ( ± )0 + τ κ π P ( R ) , (4.126)– 40 –hich match exactly with the results (3.105)–(3.106) obtained at the classical level for thecorresponding densities. In this paper, we have introduced and studied novel types of systems obtained by a localspace-time deformation induced by higher-spin conserved charges. We have argued that thesegeometrical maps correspond to classical and quantum field theories perturbed by irrelevantLorentz-breaking composite fields. The possibility of non-Lorentz invariant generalisations ofthe T ¯ T operator was first briefly discussed in [1] and the J ¯ T and the T ¯ J models studied in [22–28] are concrete realisations of these ideas, though with some important difference in the actualdetails. Another concise discussion on non-Lorentz invariant perturbations has appeared morerecently in [51] were it was underlined the possibility to replace the T ¯ T operator with compositefields built out of a pair of conserved currents, a somehow obvious generalisation of the earliercomments [1] and results [22–28] which however also encompasses the systems effectivelystudied in the current paper. Here we would like to stress again that the spin s in thescattering phase factor (4.38) can also assume non-integer values, therefore the extension tonon-local conserved charges [46] appears to be straightforward, as also underlined in [1].Many open problems deserve further attention. First of all, the identification betweenthe classical and the quantum results is, in our opinion, very convincing, but certainly morework is needed to fully confirm the match. In particular, a Lagrangian description for thehigher-spin deformed models would lead the way to more stringent tests. Furthermore, it isnatural to try to identify within the AdS /CFT setup the corresponding (bulk) JJ and Yang-Baxter type deformations, as it was successfully done for their close relatives studied [22–28].It would be important to extend the gravity setup adopted in [19, 52] to accommodate alsothese novel geometric deformations.While the T ¯ T perturbation is compatible with supersymmetry [16, 27, 53–55], higher spinHamiltonians should, at least partially, lift the degeneracy of the states. Therefore, in general,supersymmetry should be explicitly broken by the more general perturbations discussed here.However, it would be interesting to find specific examples were a residual supersymmetrysurvives. Finally, it would be also nice to study the effect of these perturbations on 2D Yang-Mills [21, 56] and, following the suggestion of [57], to study the expectation values on curvedspacetimes, of the composite operators discussed here. Finally, while this paper was in its finalwriting stage the works [32, 58] appeared. The latter articles contain many interesting resultson the J ¯ T model and the combined J ¯ T/T ¯ T perturbations. There are only minor overlaps withthe current paper. As already mentioned, our methods are easily generalisable to encompasssuch a two-coupling extension of the models described here, both at classical and quantumlevels.
Acknowledgments –
We are especially grateful to Riccardo Borsato, Patrick Dorey, Ferdi-nando Gliozzi, Yunfeng Jiang, Zohar Komargodski, Bruno Le Floch, Sergei Lukyanov, MarcMezei, Domenico Orlando, Alessandro Sfondrini, Kentaroh Yoshida and Sasha Zamolodchikov– 41 –or useful discussions. This project was partially supported by the INFN project SFT, theEU network GATIS+, NSF Award PHY-1620628, and by the FCT Project 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, StonyBrook University at which some of the research for this paper was performed.
A 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 ) . Firstof all we compute the relation between the sets of coordinates w and z by integrating, for s ≥ , (3.3) for z (cid:40) ∂z ( s ) ( w ) ∂w = 1 + 2 τ ¯Θ s − ( w ) , ∂z ( s ) ( w ) ∂ ¯ w = 2 τ ¯T s +1 ( w ) , (cid:40) ∂ ¯ z ( s ) ( w ) ∂w = 2 τ T s +1 ( w ) , ∂ ¯ z ( s ) ( w ) ∂ ¯ w = 1 + 2 τ Θ s − ( w ) , (A.1)where we have denoted z = z ( s ) to distinguish between the different perturbations. Thecomponents of the higher charges T s +1 ( w ) , Θ s − ( w ) along with their complex conjugates areimplicitly evaluated on the specific field configuration φ ( w ) . Inverting the relation z ( s ) ( w ) ,we find the deformed solution as φ ( s ) (cid:0) z , τ (cid:1) = φ (cid:16) w (cid:0) z ( s ) (cid:1)(cid:17) . (A.2)Similarly, we can apply the same strategy to derive a deformed solution associated to thechange of variables (3.2), for s ≤ . In this case we should, instead, integrate the system ∂z ( s (cid:48) ) ( w ) ∂w = 1 + 2 τ T s +1 ( w ) , ∂z ( s (cid:48) ) ( w ) ∂ ¯ w = 2 τ Θ s − ( w ) , ∂ ¯ z ( s (cid:48) ) ( w ) ∂w = 2 τ ¯Θ s − ( w ) , ∂ ¯ z ( s (cid:48) ) ( w ) ∂ ¯ w = 1 + 2 τ ¯T s +1 ( w ) , (A.3)and repeat the same procedure described above. A.1 Explicit Lorentz breaking in a simple example
The purpose of this section is to highlight the difference between the T ¯ T perturbation andthose corresponding to spins s (cid:54) = 1 , through the study of a simple example. Since perturbationswith higher spins should break explicitly Lorentz symmetry, it is convenient to start from asolution of the Laplace equation which is particularly symmetric under rotations i.e. the“spiral staircase" solution of [59]: φ ( w ) = d ln (cid:18) w + ξ ¯ w + ¯ ξ (cid:19) , ( ξ, ¯ ξ ∈ C , d ∈ R ) , (A.4)where w and ¯ w are complex conjugated variables related to the cartesian coordinates y =( y , y ) through (1.12). Using ∂ ¯ w w + ξ = − π δ ( w + ξ ) , ∂ w w + ¯ ξ = − π δ ( ¯ w + ¯ ξ ) , (A.5)– 42 –ne can show that (A.4) is indeed solution to the undeformed EoMs: ∂ w ∂ ¯ w φ = 0 .For sake of brevity, we will only consider perturbations induced by the set of charges (2.16),the deformations associated to (2.15) can be obtained from the results described here througha simple redefinition of the coupling τ .Integrating (A.1) using the set of charges (2.16) evaluated on the solution (A.4) we obtain z ( s ) ( w ) = w + τ Γ (cid:0) s +12 (cid:1) s d ( ¯ w + ¯ ξ ) s , ¯ z ( s ) ( w ) = ¯ w + τ Γ (cid:0) s +12 (cid:1) s d ( w + ξ ) s . (A.6)The latter equations cannot explicitly be inverted, for generic spin s , as w (cid:0) z ( s ) (cid:1) . However,for s = 1 , i.e. the T ¯ T perturbation, (A.6) can be written as z + ξ = ( w + ξ ) (cid:34) τ d ( w + ξ )( ¯ w + ¯ ξ ) (cid:35) , ¯ z + ¯ ξ = ( ¯ w + ¯ ξ ) (cid:34) τ d ( w + ξ )( ¯ w + ¯ ξ ) (cid:35) , (A.7)with z = z (1) ( w ) , from which φ ( w ) = d ln (cid:18) w + ξ ¯ w + ¯ ξ (cid:19) = d ln (cid:18) z + ξ ¯ z + ¯ ξ (cid:19) = φ ( z , τ ) . (A.8)Therefore, the solution (A.4) is a fixed point of the T ¯ T flow. To clearly see how the s (cid:54) = 1 perturbations affect the solution (A.4), we shall restrict to the s < perturbations, where thedeformed solutions can be found explicitly. Integrating (A.3) using the set of charges (2.16),again evaluated on (A.4), we find z ( s (cid:48) ) ( w ) = w + τ Γ (cid:0) s +12 (cid:1) s d ( w + ξ ) s , ¯ z ( s (cid:48) ) ( w ) = ¯ w + τ Γ (cid:0) s +12 (cid:1) s d ( ¯ w + ¯ ξ ) s . (A.9)Comparing (A.6) with (A.9) we see that the difference between the s > and s < pertur-bations lies in the substitution w + ξ ←→ ¯ w + ¯ ξ in the term proportional to τ , which impliesthat, for s < , there is no mixing between holomorphic and anti-holomorphic components.Since holomorphic and anti-holomorphic parts are completely decoupled, we can integrateexplicitly (A.9) for w and ¯ w . The result is w (cid:0) z ( s (cid:48) ) (cid:1) = z ( s (cid:48) ) + s s (cid:34) − F s (cid:32)(cid:18) ss (cid:19) s Γ (cid:18) s (cid:19) τ d (cid:0) z ( s (cid:48) ) + ξ (cid:1) s (cid:33)(cid:35) (cid:0) z ( s (cid:48) ) + ξ (cid:1) , ¯ w (cid:0) z ( s (cid:48) ) (cid:1) = ¯ z ( s (cid:48) ) + s s (cid:34) − F s (cid:32)(cid:18) ss (cid:19) s Γ (cid:18) s (cid:19) τ d (cid:0) ¯ z ( s (cid:48) ) + ¯ ξ (cid:1) s (cid:33)(cid:35) (cid:0) ¯ z ( s (cid:48) ) + ¯ ξ (cid:1) , (A.10)Finally, the deformed solutions are recovered by plugging (A.10) into (A.4) φ ( s (cid:48) ) ( z , τ ) = d ln (cid:18) z + ξ ¯ z + ¯ ξ (cid:19) + d ln s F s (cid:16)(cid:0) ss (cid:1) s Γ (cid:0) s (cid:1) τ d ( z + ξ ) s (cid:17) s F s (cid:16)(cid:0) ss (cid:1) s Γ (cid:0) s (cid:1) τ d (¯ z +¯ ξ ) s (cid:17) . (A.11)– 43 – a) s = 1 (b) s = − (c) s = − Figure 2 . Analytic structure of the deformed solution (A.11) in the complex plane of z for differentvalues of s , with ξ = ¯ ξ = 0 . The black lines correspond to the square root branch cuts connecting thesingularities (A.18), while the red lines correspond to the logarithmic cuts. Let us discuss in detail the case s = − of (A.11), φ ( − ( z , τ ) = d ln (cid:18) z + ξ ¯ z + ¯ ξ (cid:19) + d ln (cid:113) − τ d ( z + ξ ) (cid:113) − τ d (¯ z +¯ ξ ) . (A.12)We observe that, as soon as the perturbation is switched on, a pair of square root branchpoints appears at z = ± d √ τ − ξ . Considering for simplicity ξ, ¯ ξ ∈ R , they form a branch cuton the real axis of the complex plane of z C = (cid:0) − d √ τ − ξ ; +2 d √ τ − ξ (cid:1) , (A.13) i.e. the black line in Figure 2b. Instead, the logarithmic singularity of the undeformed solution(A.4) at w = − ξ cancels out with the singularities coming from the additional term in (A.12).Therefore, the logarithmic cut of (A.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 secondarybranches reached by passing through C (see the red line in Figure 2b). This implies that thebehaviour of (A.12) at z = ∞ is different according to the choice of the branch. On the firstsheet one has φ ( − ( z , τ ) ∼ z →∞ (1 -th sheet ) d ln (cid:18) z + ξ ¯ z + ¯ ξ (cid:19) + O ( z ) , (A.14)while, on the second sheet, flipping the + sign in front of the square roots in (A.12) into a − sign one finds φ ( − ( z , τ ) ∼ z →∞ (2 -nd sheet ) − d ln (cid:18) z + ξ ¯ z + ¯ ξ (cid:19) + O ( z ) . (A.15)In Figure 3, is represented the Riemann surface of the solution (A.12) (Figure 3b) togetherwith that of the bare solution (Figure 3a), which coincides with the T ¯ T deformed solution.Notice that the analytic structure of (A.12) can be read out from the implicit map (A.9). Infact, for s = − , equation (A.9) reduces to the Zhukovsky transformation z + ξ = ( w + ξ ) + τ d ( w + ξ ) , ¯ z + ¯ ξ = ( ¯ w + ¯ ξ ) + τ d ( ¯ w + ¯ ξ ) , (A.16)– 44 – a) (b) Figure 3 . Riemann surface of the deformed solution φ ( s ) ( z , τ ) , ( ξ = ¯ ξ = 0) , in the complex plane of z for s = 1 (a), and s = − (b). from which we see that z = ∞ on the first sheet is mapped into w = ∞ , while z = ∞ on thesecond sheet is mapped into w = − ξ . Moreover (A.16) captures the large z behaviour of thesolution. In fact, d ln (cid:18) w + ξ ¯ w + ¯ ξ (cid:19) ∼ w →∞ d ln (cid:18) z + ξ ¯ z + ¯ ξ (cid:19) , d ln (cid:18) w + ξ ¯ w + ¯ ξ (cid:19) ∼ w →− ξ − d ln (cid:18) z + ξ ¯ z + ¯ ξ (cid:19) . (A.17)Let us now consider the generic solution (A.11). The hypergeometric functions appearing in(A.11) are of the form p +1 F p ( a , . . . , a p +1 ; b , . . . , b p ; x ) , with coefficients { a i } p +1 i =1 , { b j } pj =1 ∈ Q . Generally, these hypergeometric functions have branch points at x = ∞ and x = 1 , whichin our case are mapped into z = 0 and x n = 1 + ss (cid:18) d Γ (cid:18) s (cid:19)(cid:19) s τ s e πi s n − ξ , ( n = 0 , . . . , s ) , (A.18)respectively. The branch points (A.18) are all of square root type. In conclusion, roughlyspeaking, starting from a rotational-symmetric solution, the perturbations with s < haveexplicitly broken the original U (1) symmetry down to a discrete Z s . B Burgers equations
We want to prove that the general Burgers equations (4.47), which we report here for conve-nience ∂ τ I ( s , ± ) k ( R, τ ) + R (cid:48) ∂ R I ( s , ± ) k ( R, τ ) = ± k θ (cid:48) I ( s , ± ) k ( R, τ ) , (B.1)with R (cid:48) and θ (cid:48) defined through (4.50), reduce to ∂ τ I ( s, ± ) k ( R, τ ) = 2 I ( s, ∓ ) s ( R, τ ) ∂ R I ( s, ± ) k ( R, τ ) , ( s = s > , (B.2) ∂ τ I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) = 2 I ( s (cid:48) , ± ) s (cid:48) ( R, τ ) ∂ R I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) , ( s = s (cid:48) = − s < , (B.3)– 45 –n the CFT limit.Let us begin with the s < case. From the implicit relation (4.75) I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) = I ( s (cid:48) , ± ) k (cid:48) (cid:16) R ( s (cid:48) , ∓ )0 (cid:17) , (B.4)using (4.72), we have I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) = 2 πa k (cid:48) (cid:16) R + 2 τ I ( s (cid:48) , ± ) s (cid:48) ( R, τ ) (cid:17) k . (B.5)Differentiating (B.5) w.r.t. R for k (cid:48) = s (cid:48) ( k = s ) , we first express I ( s (cid:48) , ± ) s (cid:48) ( R, τ ) as a function of ∂ R I ( s (cid:48) , ± ) s (cid:48) ( R, τ ) as I ( s (cid:48) , ± ) s (cid:48) ( R, τ ) = − R ∂ R I ( s (cid:48) , ± ) s (cid:48) ( R, τ )2 τ s (1 + s ) ∂ R I ( s (cid:48) , ± ) s (cid:48) ( R, τ ) . (B.6)Then, differentiating again (B.5) w.r.t. R for generic k (cid:48) ( k ) and using (B.6), we express I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) as a function of ∂ R I ( s (cid:48) , ± ) s (cid:48) ( R, τ ) and ∂ R I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) as follows I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) = − sk R ∂ R I ( s (cid:48) , ± ) k (cid:48) ( R, τ )2 τ s (1 + s ) ∂ R I ( s (cid:48) , ± ) s (cid:48) ( R, τ ) . (B.7)Finally, using (B.7), it is a matter of simple algebraic manipulation to show that − R (cid:48) ∂ R I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) + k (cid:48) θ (cid:48) I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) = 2 I ( s (cid:48) , ± ) s (cid:48) ( R, τ ) ∂ R I ( s (cid:48) , ± ) k (cid:48) ( R, τ ) , (B.8)which proves that (4.47) for k = k (cid:48) and s = s (cid:48) reduce to (B.3).Considering the s > case, from the implicit relation (4.74) I ( s, ± ) k ( R, τ ) = I ( s, ± ) k (cid:16) R ( s, ± )0 (cid:17) , (B.9)using (4.72), we have I ( s, ± ) k ( R, τ ) = 2 πa k (cid:16) R + 2 τ I ( s, ∓ ) s ( R, τ ) (cid:17) k . (B.10)Repeating the same procedure as in the s < case, we express I ( s, ± ) k ( R, τ ) as a function of ∂ R I ( s, ± ) k and ∂ R I ( s, ± ) s as follows I ( s, ± ) k = − sk R ∂ R I ( s, ± ) k (cid:16) τ s ∂ R I ( s, ± ) s − τ ∂ R I ( s, ∓ ) k + s (cid:17) s (cid:16) τ ∂ R I ( s, +) s + 2 τ ∂ R I ( s, − ) s (cid:17) + 4 ( s − τ ∂ R I ( s, +) s ∂ R I ( s, − ) s , (B.11)Again, using (B.11), one can show that the following relation holds − R (cid:48) ∂ R I ( s, ± ) k + kθ (cid:48) I ( s, ± ) k = 2 I ( s, ∓ ) s ∂ R I ( s, ± ) k , (B.12)which proves that (4.47) for k = k and s = s reduce to (B.2).– 46 – eferences [1] F. A. Smirnov and A. B. Zamolodchikov, On space of integrable quantum field theories , Nucl.Phys.
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