The $\textrm{T}\bar{\textrm{T}}$ perturbation and its geometric interpretation

Starting from the recently-discovered $\textrm{T}\bar{\textrm{T}}$-perturbed Lagrangians, we prove that the deformed solutions to the classical EoMs for bosonic field theories are equivalent to the unperturbed ones but for a specific field-dependent local change of coordinates. This surprising geometric outcome is fully consistent with the identification of $\textrm{T}\bar{\textrm{T}}$-deformed 2D quantum field theories as topological JT gravity coupled to generic matter fields. Although our conclusion is valid for generic interacting potentials, it first emerged from a detailed study of the sine-Gordon model and in particular from the fact that solitonic pseudo-spherical surfaces embedded in $\mathbb R^3$ are left invariant by the deformation. Analytic and numerical results concerning the perturbation of specific sine-Gordon soliton solutions are presented.

A remarkable property of this perturbation, discovered in [1,2], concerns the evolution of the quantum spectrum at finite volume R, with periodic boundary conditions, in terms of the TT coupling constant τ . The spectrum is governed by the inhomogeneous Burgers equation where E n (R, τ ) and P n (R) are the total energy and momentum of a generic energy eigenstate |n , respectively. Equation (1.1) is valid also for non-integrable models.
Notice that (1.1) reveals an important feature of TT-deformed QFTs: the interaction between the perturbing operator and the geometry, through the coupling τ . The latter property is a basic requirement for any sensible theory of gravity but in the current case it naturally emerges, non perturbatively and at full quantum level, from a specific irrelevant perturbation of Lorentz-invariant Quantum Field Theories (QFTs). An important link with JT topological gravity was noticed and studied in [18], where it was shown that JT gravity coupled to matter leads to a scattering phase matching that associated to the TT perturbation [1, 2, 4-6, 19, 20].
Studies of partition functions [2,[21][22][23] have led to a proof of the uniqueness of this perturbation [24] under the assumption that the theory on the torus is invariant under modular transformations and that the energy of a given eigenstate is a function only of τ and of the energy and momentum of the corresponding state at τ = 0. Furthermore, starting from the JT-gravity setup, in [22] the hydrodynamic-type equation (1.1) was recovered. The latter result together with [18] confirms, beyond any reasonable doubt, the equivalence between the TT deformation and JT topological gravity coupled to generic matter field.
The aim of this paper is to address the problem concerning the classical interpretation of the TT perturbation following the more direct approach proposed in [2] and further developed in [25,26]. The current analysis is based on the observation [1,2] that (1.1) directly implies a self-consistent flow equation for the deformed Lagrangian L (τ ) where g = Det (g µν ) and TT = −π 2 Det (T µν ) is the classical counterpart of Zamolodchikov's operator.
There are many reasons to study these newly-discovered set of classical Lagrangians. First of all, according to [18,22], these systems should correspond to JT gravity coupled to non-topological matter, a fact that is by no mean evident from the Lagrangian point of view.
Secondly, when the starting model is integrable, there should be a general way to deform the whole integrable model machinery. For example, a generalisation of the ODE/IM correspondence [28][29][30] should lead to an alternative method to obtain the quantum spectrum at finite volume [31,32] and it may open the way to the inclusion of the TT inside the Wilson Loops/Scattering Amplitudes setup, in AdS 5 /CFT 4 [33,34] and perhaps also to consistently deform the Argyres-Douglas theory [35][36][37].
The main purpose of this article is to prove that, for bosonic theories with arbitrary interacting potentials, the TT perturbation has indeed the alternative interpretation as a space-time deformation. In Euclidean coordinates the change of variables is is the unperturbed stress-energy tensor in the set of coordinates y.
Then, any solution of the perturbed EoMs con be mapped onto the τ = 0 corresponding solution, i.e.
where the r.h.s. of (1.4) 1 is defined on a deformed space-time with metric In fact (1.3) corresponds to a natural generalization of the Virasoro conditions used in the GGRT treatment of the NG string [38], 2 and it matches precisely the generalisation corresponding to classical JT gravity [18,22].

Classical integrable equations and embedded surfaces
It is an established fact that integrable equations in two dimensions admit an interpretation in terms of surfaces embedded inside an N -dimensional space. The two oldest examples of this connection, dating back to the works of 19th century geometers [39,40], are the sine-Gordon and Liouville equations. They appear as the Gauss-Mainardi-Codazzi (GMC) system of equations (A.14) for, respectively, pseudo-spherical and minimal surfaces embedded in the Euclidean space R 3 . As proved by Bonnet [41], any surface embedded in R 3 is uniquely determined (up to its position in the ambient space) by two rank 2 symmetric tensors: the metric g µν (A.4) and the second fundamental tensor d µν (A.6). Their intuitive role is to measure, respectively, the length of an infinitesimal curve and the displacement of its endpoint from the tangent plane at the starting point. One can then use g µν and d µν to study the motion of a frame anchored to the surface. The result is a system of linear differential equations, known as Gauss-Weingarten equations (A.9, A.10).
The GMC system appears then as the consistency condition for this linear system, effectively constraining the "moduli space" consisting of the two tensors g µν and d µν .
The search for a general correspondence originated in the works of Lund, Regge, Pohlmeyer and Getmanov [42][43][44] and was subsequently formalised by Sym [45][46][47][48][49] who showed that any integrable system whose associated linear problem is based on a semi-simple Lie algebra g can be put in the form of a GMC system for a surface embedded in a dim(g)-dimensional surface. 3 1 Notice that from (1.4) it follows that φ (τ ) (x) fulfills the Burgers-type equation which may justify the wave-breaking phenomena observed in section 5. In our results x µ is always linear in τ , however we could not find an explicit expression for ∂τ x µ valid in general. In this section we will shortly review Sym's results for the general setting and concentrate on the case of sine-Gordon model. We will use the following conventions

Construction of the solitonic surfaces
Let us consider a generic 2-dimensional system of non-linear partial differential equations for a set of real fields {φ i (z)} admitting a Zero Curvature Representation (ZCR) for a pair of functions L 1 and L 2 taking values in a d-dimensional representation of a semi-simple Lie algebra 4 g (dim(g) = N ): The functions L µ depend on z through the fields φ i (z) and their derivatives and on a real spectral parameter λ: The Zero Curvature Representation can be interpreted as the compatibility condition for a system of first-order linear partial differential equations involving an auxiliary commonly known as associated linear problem. Assuming Φ (z 0 |λ) ∈ G as initial condition, with G being the Lie group associated to g, equation (2.3) allows, in principle, to recover a single-valued function Φ ∈ G in the whole R 2 . This function can then be used to construct the following object which is interpreted as the coordinate description of a λ-family of surfaces embedded into the N -dimensional affine space g. Moreover, equipping the affine space g with a non-degenerate scalar product (i.e. the Killing form of the semi-simple Lie algebra), we can convert g into an N -dimensional flat space. In other words, we can find an orthonormal basis e i of g with respect to the Killing form and then extract the quantities r i from the identity The vector r = r 1 , r 2 , . . . , r N T is then the position vector of a family of surfaces embedded in N -dimensional flat space, 5 parametrised by λ. These are called solitonic surfaces and satisfy the following properties: 1. their GMC system reduces to the ZCR (2.1), meaning that any integrable system can be associated to a particular class of surfaces; 2. they are invariant with respect to λ-independent gauge transformation of the pair L µ . This fact provides a way to prove the equivalence of distinct soliton systems up to gauge transformations and independent coordinate redefinitions, see [48]; 3. their metric tensor (induced by the flat space g) is explicitly computed from the pair L µ as where Ad denotes the adjoint representation of the algebra g. Consequently, any intrinsic property of the soliton surface is determined uniquely by the ZCR.

The case of sine-Gordon
Let us now consider the specific case of the sine-Gordon equation where we set z = (z 1 , z 2 ) = (z,z). The ZCR for this model is well known where S j are the generators of su (2) Since dim (su (2)) = 3, we know that we are dealing with a surface embedded in the Euclidean plane R 3 (su (2) is compact). As mentioned in section 2, Bonnet theorem [41] tells us that any surface in R 3 is completely specified (modulo its position) by its first and second fundamental quadratic forms, which can be computed easily: 6 These can be recovered by plugging (2.5) in the classical geometry formulae where n is the normal unit vector to the plane spanned by ∂1r and ∂2r: From (2.11) and 2.12) one can then extract the Gaussian and the mean curvatures using (A.8): with g sG µν g sG νρ = δ ρ µ . The fact that K sG is constant negative tells us that we are dealing with a pseudo-spherical surface, which we were expecting from the old results of Bour [39]. Thus, for this specific case, the solitonic surfaces correspond to pseudo-spherical ones, with the spectral parameter λ playing the role of Gaussian curvature.

The TT-deformed sine-Gordon model and its associated surfaces
Let us now apply the Sym formalism sketched above to the TT-deformed sine-Gordon model [26] and derive the geometric properties of the associated surfaces. We start with the ZCR, which was found in [26] L TT where with Again we have a ZCR based on the algebra su (2) and thus a surface embedded in R 3 . We need then to recover the fundamental forms I and II, whose computation, although straightforward as in the case of sine-Gordon, is lengthy and cumbersome. Sparing the uninteresting details, we present directly the results where the matricesĝ µν andd µν arê (3.11) One easily verifies that in the τ → 0 limit, which implies S → 1, one recovers the fundamental forms of sine-Gordon What is striking about the matrices (3.11) is that, although their dependence on τ is complicated, they recombine in such a way that the Gaussian and mean curvature do not depend explicitly on it! In fact these two geometric invariants are exactly the same as the unperturbed sine-Gordon model: This suggests that the solitonic surface corresponding to a particular solution of the TTdeformed sine-Gordon equation is the same as the one associated to the undeformed model, what changes should be the coordinate system used to describe it. For the sake of completeness, we have reported in Figure 1 examples of embedded pseudo-spherical surfaces related to one-kink solutions, a stationary breather and a two-kink solution. The plots were obtained implementing the method described in [50]. The embedded surfaces in R 3 , as we have just argued and will be explicitly shown in the next section, are independent of the deformation parameter τ , being it reabsorbable through a local change of coordinates. The corresponding soliton solutions, described in section 5, are instead affected by the TT in a highly non-trivial way. For instance, they generally possess critical values in τ corresponding to shock-wave phenomena, i.e. branching of the solutions. The latter property can be interpreted as the classical microscopic version of the Hagedorn-type phase transition characterising the finite temperature/volume quantum spectrum [1,2,[4][5][6]. Figure 1. Pseudo-spherical solitonic surfaces associated to kink and breather solutions. Figure 1a represents the Dini surface, corresponding to a moving kink, while in Figure 1b the famous Beltrami pseudo-sphere is represented. The latter surface is obtained from Dini's surface by taking the stationary limit of the kink solution. Figures 1c and 1d correspond to the pseudo-spherical surfaces associated to a stationary breather and to a two-kink solution, respectively.

From the deformed to the undeformed model through a local change of coordinates
Thus we have inferred that there must exist a coordinate system w = (w 1 (z), w 2 (z)) = (w(z),w(z)) in which the matrices g TT µν and d TT µν assume the same form as g sG µν and d sG µν , respectively. In formulae It is now a matter of simple algebraic manipulations to obtain the following equations for the new coordinates Let us now use the latter relations to find the partial derivatives of the field φ in the coordinates w: The result is where we have defined the following function With the help of (3.20), we can now find the expression for S in the coordinates w We can then write the Jacobian matrix J and its inverse J −1 in terms of w as This results allows us to express the partial derivatives of any function f (z) as partial derivatives with respect to the new coordinates and we can then apply all the above formulae to the equation (3.1), obtaining The equality of (3.25) and (3.26) yields then (3.27)

A geometric map for N-boson fields and arbitrary potential
We have seen, in the preceding section, how the TT deformation of the sine-Gordon model can be interpreted as a field-dependent coordinate transformation. We arrived at this interesting conclusion by exploiting the relation existing amongst ZCR of soliton equations and the classical geometry of surfaces embedded in flat space. Although this connection was pivotal in guiding us to the map (3.17, 3.18), from that point on we did not make explicit mention to the form of the potential. In other words, we can consider all formulae from (3.17) to (3.27) to be valid for any 2-dimensional single scalar system. More generally, the results (3.23, 3.27) admit a straightforward generalisation to the case of N bosonic fields φ i , (i = 1, . . . , N ) interacting with a generic derivative-independent potential V (φ i ) The generalization of (3.23) to the N -boson case is In fact we have verified that the deformed EoMs resulting from (4.1) are mapped by (4.4) into the undeformed EoMs associated to L (0) N . It is instructive to translate (4.4) in Euclidean coordinates. Considering and moving to Euclidean coordinates both in the z and in the w frames we find where T µ ν (y) is the stress energy tensor of the undeformed theory, T = T (0) . Expressions (4.7) can be more compactly rewritten as

From (4.8) the inverse Jacobian in Euclidean coordinates reads
and thus the metric, in the set of coordinates y, is where we used the fact that g ρσ = δ ρσ . Finally let us conclude this section with a couple of remarks: • Consider the transformation of the Lagrangian 7 (4.1) under the on-shell map (4.4) (4.12) Using the latter expression together with we find that the action transforms as  . (4.11) where TT (0) (w) = (K N ) 2 − V 2 . Thus, we conclude that the action is not invariant under the change of variables. This is somehow expected, since the map (4.4) is on-shell, but we find very surprising that such a simple expression involving explicitly the bare TT (0) field is emerging.
• Notice that the EoMs associated to (4.1) for a generic potential V are invariant under the with c constant and γ = 1/(1 − τ c), which corresponds to the following change of variables at the level of the solutions where the notation φ is solution to the deformed EoMs with potential V .

TT-deformed soliton solutions in the sine-Gordon model
In this section we show how to compute TT-deformed solutions of the sG model by explicitly evaluating the change of variables on specific solutions φ(w) of the undeformed theory. The idea is to solve the following sets of differential equations derived from the inverse Jacobian for z(w) andz(w). Then from the inverse map, i.e. w(z), we evaluate the expression of the deformed solution as φ (τ ) (z) = φ (0) (w(z)) .

(5.2)
In the following we will deal only with some of the simplest solutions of the sG model. In principle our approach applies for all the solutions, although we could not find an explicit result for the integrated map in the cases involving more than two solitons. For sake of clarity, the computations shown in the following sections will be carried on in light cone coordinates, i.e. (z,z) and (w,w), however the plots will be displayed using space and time coordinates (x, t) = x 1 , x 2 . which are easily inverted as

The one-kink solution
Finally, plugging (5.7) into (5.3) we find which is exactly the deformed one-kink solution found in [26]. In Figure 2 (Figures 2c and 2d) it bends and becomes multi-valued. In particular τ = 1/8 (Figure 2c) is the delimiting value corresponding to a shock wave singularity.

The two-kink solution
Consider now the solution which describes the scattering between two kinks with velocities where again a i = 1−v i 1+v i , i = 1, 2 , and k i , i = 1, 2, are constant phases. Compared to the one-kink case, this time the sets of differential equations (5.1) are more complicated to integrate. It is useful to parametrize the solutions z(w) andz(w) of (5.1) in terms of the combinations  2-kink (w), we obtain two sets of differential equations which can be solved for z(u), giving .

(5.12)
As in the previous section, the constants of integration in (5.12) are fixed by imposing the consistency with the τ = 0 case. In order to find the deformed two-kink solution φ 2-kink (u(z)), we should solve (5.12) for u(z). Since this is analytically very complicated, we resort to numerical inversion. In Figure 3 the deformed solution φ  (Figures 3c and 3d) it bends and again it becomes multi-valued. Unlike the one-kink case, here it is not possible to find analytically the delimiting value of τ corresponding to the shock singularity.

The breather
Another interesting solution is the breather with envelope speed v = 0 where ψ is a parameter related to the period T of one full oscillation via T = 2π cos ψ and k,k are constant phases. In analogy with the two-kink case, it is useful to use the same strategy and parametrize the solutions z(w) of (5.1) in terms of Performing the change of variables u(w) = (u(w),ū(w)), one finds  (Figure 4b). However, notice that in this case the shock phenomenon occurs in both positive and negative directions of τ , and consequently the solution becomes multi-valued (Figures 4a and 4d) for |τ | sufficiently large.

Conclusions
Starting from the TT-deformed Lagrangians proposed in [2,25,26], the main result of this article is the direct derivation of the exact one-to-one map between solutions of the unperturbed and deformed equations of motion, which takes the general form (1.3,1.4). The result matches the topological gravity predictions of [18,22] but it should be possible to obtain the fundamental equations (1.3,1.4,1.6) also by working within the framework introduced by Cardy in [21].
We initially arrived to this conclusion by studying the well known classical relation between sine-Gordon, the associated Lax operators and pseudo-spherical surfaces embedded in R 3 . We think that this alternative and more explicit approach to the problem may provide a complementary point of view compared to [18,22] and open the way to the implementation of further integrable model tools, such as the Inverse Scattering Method and the ODE/IM correspondence within the TT/JT framework.
There are many theoretical aspects that deserve to be further explored. First of all, it would be conceptually very important to study fermionic theories and supersymmetric sigma models. In [25], it was argued that for the TT-perturbed Thirring model the Lagrangian truncates at second order in τ , such a truncation is not totally surprising, however the sine-Gordon Lagrangian is instead deformed in an highly non trivial way and it would be nice to identify the mechanism which allows to preserve the quantum equivalence between the two systems. Secondly, it would important to continue the investigation of deformed 2D Yang-Mills [26], along the lines started in the interesting recent work [51]. These studies might also serve as a guide for the inclusion of the TT inside the Wilson Loop/Scattering Amplitude setup [33,34] (see also the remarks in the outlook section of [51]).
Finally, it would also be interesting to study the generalisation of our results to the JT case described in [52][53][54][55][56] and to check whether for any of the higher-dimensional models discussed in [21,25,26,57] there could exist a map, between deformed and undeformed solutions, similar to equations (1.3,1.4).

Note:
We have recently been informed that the coordinate map between deformed and undeformed classical Lagrangian systems was also independently introduced by Chih-Kai Chang and studied in an on-going research project involving also Christian Ferko and Savdeep Sethi.
The surface Σ inherits a metric structure from the ambient space R 3 and its line element, also known as first fundamental quadratic form, is The tensor is called first fundamental tensor or metric tensor of the surface Σ. According to the classical theorem by Bonnet [41] any surface embedded in flat 3-space is uniquely determined, up to isometries, by the first and the second fundamental quadratic form, defined as The tensor describes the projection of the vectors ∂ ∂z µ r ν (P ) on the normal direction and tells us how much the surface curves away from the tangent space in an infinitesimal interval around the point P . These two tensors can be combined into the object known as shape or Weingarten operator, whose eigenvalues κ 1 , κ 2 are the principal curvatures of the surface Σ. The latter quantities are geometric invariants, meaning that they do not change under coordinate transformations. Usually they are combined into the Gauss and mean curvatures K = κ 1 κ 2 = Det s ν µ , H = The tensors g µν and d µν determine the structural equations for embedded surfaces, comprising the Gauss equations ∂ ∂z µ r ν = Γ ρ µν r ρ + d µν n , (A.9) and the Weingarten equations ∂ ∂z µ n = s ν µ r ν , (A.10) where we introduced the Christoffel symbols for the metric These equations describe how the frame σ moves on the surface and can be collected into the following linear system ∂ ∂z µ σ = U µ σ , (A.12) These structural equations are subject to a set of compatibility conditions called Gauss-Mainardi-Codazzi (GMC) system, which takes the form of a zero curvature condition on the matrices U µ In the case in which we are dealing with a pseudo-spherical surface, that is when K = − 1 ρ 2 < 0, with ρ a constant, one can choose as parametric curves the asymptotic lines and express the fundamental forms as follows I = dz 1 2 − 2 cos ω dz 1 dz 2 + dz 2 2 , (A.15) II = − 2 ρ sin ω dz 1 dz 2 , (A. 16) with ω representing the angle between the parametric curves. It is then a simple matter of computation to show that the GMC system (A.14) becomes the sine-Gordon equation which is known as modified sinh-Gordon equation. Its Gauss curvature is Rescaling the field as ω → ω + 2 ln H, the functions A i as A i → HA i and sending H → 0 yields a minimal surface and reduces the GMC system to Liouville equation 10 Note that Γ ρ µν = Γ ρ νµ and dµν = dνµ.

B Computation of the fundamental quadratic forms from sine-Gordon ZCR
Here we wish to explicitely show how to obtain (2.11, 2.12) starting from sine-Gordon ZCR (2.8, 2.9). First of all we need to find a basis of su (2) with respect to the Killing form (a, b) K = Tr (Ad(a) Ad(b)) , a, b ∈ su (2) . (B.1) In the adjoint representation one has T i = Ad S i , with Now we need the partial derivatives of r (2.4) where we have used the linear system ∂ µ Φ = L µ Φ. We have then We can immediately compute the metric tensor g µν