$T\bar{T}$ flow as characteristic flows

We show that method of characteristics provides a powerful new point of view on $T\bar{T}$-and related deformations. Previously, the method of characteristics has been applied to $T\bar{T}$-deformation mainly to solve Burgers' equation, which governs the deformation of the \emph{quantum} spectrum. In the current work, we study \emph{classical} deformed quantities using this method and show that $T\bar{T}$ flow can be seen as a characteristic flow. Exploiting this point of view, we re-derive a number of important known results and obtain interesting new ones. We prove the equivalence between dynamical change of coordinates and the generalized light-cone gauge approaches to $T\bar{T}$-deformation. We find the deformed Lagrangians for a class of $T\bar{T}$-like deformations in higher dimensions and the $(T\bar{T})^{\alpha}$-deformation in 2d with generic $\alpha$, generalizing recent results in arXiv:2206.03415 and arXiv:2206.10515.


Introduction
Recently, solvable irrelevant deformations have been under intensive study, for good reasons. The best studied example is TT -deformation [3,4], which is defined in the Lagrangian formulation by where L λ is the Lagrangian density and O TT is a composite operator constructed from the stress-energy tensor T µ ν as Such a deformation can be defined for any relativistic quantum field theories in 1+1 dimensions and exhibit interesting new features [5]. One of the reasons that TT -deformation raised broad interest is that it lies at the intersection of several research directions in theoretical physics.
The original motivation for studying the composite operator (1.2) [3,4] and the deformation triggered by it comes from integrability. For integrable models, TT -deformation belongs to a family of infinitely many irrelevant deformations which preserve integrability [3]. These deformations modify the S-matrix by CDD factors and does not change the IR properties also discussed in [24].
The fact that TT -deformation lies at the intersection of several research areas makes it possible to formulate it from various different point of views. Apart from the original definition (1.1), several alternative formulations of TT -deformation have been proposed.
In this work, we clarify the relation between various aforementioned methods by offering yet another point of view on TT -deformation. We study TT -deformation using method of characteristics and view TT flow as a characteristic flow. Method of characteristics is a general approach to solve first order partial differential equations. Previously, it has been applied in TT -deformation to solve inviscid Burgers' equation, which gives the quantum spectrum of deformed CFTs and the Lagrangian of some specific models [34,35]. In the current work, we consider classical quantities using this approach for general TT -like deformations.
The basic idea is simple. We rewrite the flow equation of the deformed quantity as a first order non-linear partial differential equation (PDE) and then investigate the equation by method of characteristics. In this approach, we can view the deformation parameter λ as time and the deformation as a 'time evolution' of the original theory. In order to find analytic solutions, the key point is finding certain quantities which are constant along the flow. As we shall see, the dynamical change of coordinate of TT -deformation can be obtained rather straightforwardly from these constants. The uniform lightcone approach can also be investigated from the point of view of characteristic flow. In this way, we prove the equivalence of the two approaches, at least classically.
More importantly, we show that the applicability of the method goes beyond TT -deformation. Recently, an interesting TT -like deformation in higher dimensions has been proposed in [1] where the authors pointed out the deformation is equivalent to a metric deformation. Using method of characteristics, we can rederive the results in [1] with a different approach. The authors of [1,2,36] considered the root TT -deformation and obtained the classical deformed Lagrangian 2 . We see that method of characteristic can be successfully applied to solve a wider class of deformations of the form (TT ) α with generic power α.
In section 2, we give a brief review on the method of characteristics. In section 3, we prove that the TT flow is just the characteristic flow and the method of characteristics is equivalent to the dynamical coordinate transformation. Using the result of the characteristics, we prove the trace flow equation of TT -deformation. In section 4, we prove that the light-cone gauge method is equivalent to the method of characteristics and get a dual description of the lightcone gauge method. In section 5, we study (TT ) α -deformation, where √ TT -deformation is a special case. In section 6, we use the method of characteristics to prove that TT -like deformation in arbitrary dimensions is equivalent to the dynamical metric transformation.
Then we generalize the equivalence to (TT ) α -deformation in arbitrary dimensions.

Method of characteristics
The method of characteristics is a general technique for solving first-order PDEs. In the method, a PDE is converted into a system of ordinary differential equations (ODEs). In the section, we give a brief review of the method. More details could be found in [48].

Linear equation: a simple example
Let us first consider a first-order linear PDE, whose equation is given by where Γ is a boundary. It is a Cauchy problem with the boundary curve Γ and the boundary condition u| Γ = φ. Suppose u(x, y) is a solution of the equation. Then at each point (x 0 , y 0 ), (2.1) can be written as 2) has a nice geometrical interpretation as shown in figure 1. For a solution plane S = (x 0 , y 0 , u(x 0 , y 0 )), whose normal vecter is given by n = (∂ x u(x 0 , y 0 ), ∂ y u(x 0 , y 0 ), −1), the vecter (a(x 0 , y 0 ), b(x 0 , y 0 ), c(x 0 , y 0 )) lies in the tangent plane of S. Now we want to construct the solution plane by the vector (a(x, y), b(x, y), c(x, y)). Let us look for a curve C parametrized by s, C = (x(s), y(s), z(s)), whose tangent vecter is given by (a(x(s), y(s)), b(x(s), y(s)), c(x(s), y(s))). Then C is the integral curve for the vecter field The characteristic curve C lies in the solution plane S. At each point (x, y, u(x, y)), the tangent vector of C is given by v = (a(x, y), b(x, y), c(x, y)) and the normal vector is given by n = (∂ x u(x, y), ∂ y u(x, y), −1).
(a(x, y), b(x, y), c(x, y)) and is called the characteristic curve, see figure 1. From the definition of the tangent vector, we can get dz(s) ds = c(x(s), y(s)), which are called characteristic equations. We can solve the ODEs with initial conditions, where x 0 , y 0 and z 0 satisfy the boundary condition z 0 | Γ(x 0 ,y 0 ) = φ. As the initial point (x 0 , y 0 ) moves, the curve C sweeps the solution plane S = (x, y, u(x, y)), where u(x, y) is given by u(u, y) = z(x 0 (x, y), y 0 (x, y), s(x, y)).

(2.5)
Let us explain how the method of characteristics works by an example. Example The characteristic equations are given by with initial conditions, The solution is given by To find the solution u(x, y), we need to eliminate s and x 0 in z from above equations, s = y, (2.10) Therefore, u(x, y) = z(x 0 (x, y), y 0 (x, y), s(x, y)), = −x + xe −y + 1. (2.11)

Fully nonlinear equations
For the general case, let us consider the first-order fully nonlinear equation, where x is a collection of n variables and Γ is a (n − 1)-dimensional manifold in R n .
If we get the solution of characteristic equations, ( x( r, s), z( r, s), p( r, s)), and we can find the inverse functions of the solution such that r = R( x) and s = S( x), the solution of the original PDE is given by u( x) ≡ z( r, s) = z( R( x), S( x)). (2.17)

TT flow as characteristic flows
In this section, we show that TT flow is the characteristic flow of the PDE of the defining equation. And we derive how the fields change on the flow. Then we prove that the dynamical coordinate transformation is equivalent to the method of characteristics. Finally, as an example, we use the result to re-derive the trace flow equation of the stress-energy tensor.

Characteristic flows
Consider a Lagrangian which only depends on N fields ϕ and its first order derivative. The TT flow equation is given by where the stress-energy tensor is defined as a · b means the inner product of two vectors, a · b = G ij a i b j , where G ij is the target space metric. In two dimensions, the determinant can be expended as Here we take the Lorentz index µ = 1, 2. Let u = L, Then the flow equation becomes Notice that here we view the fields like ∂ µ φ as variables in the PDE. The "coordinate" are x = ( x 1 , x 2 , x 3 ) ∈ R 2N +1 . The initial conditions are given by where r = ( r 1 , r 2 ) ∈ R 2N and L 0 is a function of r 1 , r 2 . Here r 1 , r 2 are the undeformed fields ∂ 1 φ, ∂ 2 φ respectively. The characteristic equations are given by We show how to solve the characteristic equations in Appendix A. The solution is summarized as follows, If the inverse functions are exist, i.e. ∃ r 1 ( x 1 , x 2 , s), r 2 ( x 1 , x 2 , s), then we can plug the inverse functions into the expression of z, (3.7g) and express TT -deformed Lagrangian z by x 1 , x 2 , x 3 . At λ = 0, the Lagrangian z is undeformed and expressed by variables r 1 , r 2 , which satisfy However, we can also express z by r 1 , r 2 , i.e. (3.7g). It is worth emphasizing that (3.7g) is not the undeformed Lagrangian, but the TT -deformed Lagrangian expressed by the undeformed coordinate r 1 , r 2 , see figure 2.
Example: N scalars with a potential As an example, let us consider N scalars with a potential, whose undefomed Lagrangian is given by Then, the undeformed canonical momenta can be got by definition, The solution of the characteristic equations (3.7) becomes The next step is to get the inverse functions in this example, we don't need to get r µ = r µ ( x µ , λ) but just need to express r µ · r ν by (3.13)

Dynamical coordinate transformations
In [29,30], the authors proposed that TT -deformation is equivalent to a coordinate transformation, (w,w) − → (z,z). It is an unconventional change of coordinates because it is field dependent. We prove that the dynamical coordinate transformation is equivalent to the method of characteristics.
In [29,30], the coordinate transformation is defined as (3.14) And the Lagrangian is given by Let us translate them into our notations, From the definition of the stress-energy tensor, Under the dynamical coordinate transformation [29,30], the fields satisfy It is the same as (3.7a)(3.7b) in the method of characteristics. The Lagrangian (3.15) can be written as It matches perfectly with (3.7g) in the method of characteristics if we take λ = s.
Therefore, the two methods are equivalent. In fact, our method can be seen as a first principal derivation of the dynamical coordinate transformation, at least classically. And we find that in the view of the method of characteristics, TT -deformation is just the flow along the characteristic curve from r 1 , r 2 to x 1 , x 2 and the TT flow parameter is just the intrinsic parameter of the characteristic curve.

Trace flow equation
Now, we derive the trace flow equation of the stress-energy tensor by the method of characteristics, which matches the results in [49]. The deformed stress-energy tensor is given by By (A.4) and (3.7), we can get 3 where and we have used the relation that The expression is different from one in [49].

Light-cone gauge method as characteristics
In the section, we show that the uniform light-cone gauge method in [21] is equivalent to the method of characteristics, and as a result, equivalent to the dynamical coordinate transformation. As a byproduct, we present a dual description of the uniform light-cone gauge method.

Uniform light-cone gauge method
In [21], the authors consider a type of theories, whose Lagrangian can be written as the form V is a function of P µ i and is independent of ∂ γ Ψ a . The TT -deformed Lagrangian with TT parameter s is given by We can rewrite the TT -deformed Lagrangian as Here, index 1, 2 can be Euclidean coordinate t, x or other coordinates such as We find that P µ i are the conjugate momentum in undeformed theories, i.e. P µ = ψ µ and ∂ ν Ψ a are fields in TT -deformed theories, i.e. ∂ ν Ψ = x ν . By our notation used in the method of characteristics, (4.6) And we introduce another variable Firstly, we want to get the relation between K µν and k µν . Multiply two sides of (3.7a), (3.7a) by ψ 1 , ψ 2 and get [(1 + s (k 11 − L 0 )) k 11 + sk 21 k 12 ] , [(1 + s (k 22 − L 0 )) k 12 + sk 12 k 11 ] , where det(J −1 ) is defined by (3.8). We can also expressed k µν by K µν by solving equations (4.8). The solution is given by , , Then, we prove the TT -deformed Lagrangian in the method of characteristics and the light-cone gauge method are equivalent. The TT -deformed light-cone gauge Lagrangian (4.5) is given by Plugging (4.8) into the above expression, we can get which is precisely the deformed Lagrangian from the method of characteristics.
For the light-cone gauge method, one needs to eliminate P µ i by the equation of motion of the light-cone gauge Lagrangian (4.5). Finally, we prove the solution of characteristics, (3.7a) and (3.7e) , is just the solution of the equation of motion of the light-cone gauge Lagrangian (4.5). According to (4.4), we get This is a multiple Legendre transformation. It is easy to get Hamilton's equations, According to the definition of K µν = ψ µ · x ν ,(4.6), one get (4.14) The equations of motion in the light-cone gauge method are given by where L is given by (4.5). Using (4.13) and (4.14), the solution of (4.15) is It is worth noting that the undeformed variable p µ (0) = ψ µ and the deformed variable x ν (s) are mixed in the light-cone gauge method. The TT -deformed Lagrangian can be expressed by four groups of variables, i.e. L λ ( ψ µ , r ν ), L λ ( ψ µ , x ν ), L λ ( p µ , r ν ) and L λ ( p µ , x ν ).
What we ultimately want is L λ ( p µ , x ν ), which is the Lagrangian expressed by deformed variables ( p µ , x ν ). The light-cone gauge method is to derive L λ ( p µ , x ν ) from L λ ( ψ µ , x ν ). The dynamical coordinate transformation is to derive L λ ( p µ , x ν ) from L λ ( ψ µ , r ν ). And there is a new method to derive L λ ( p µ , x ν ) from L λ ( p µ , r ν ), which is called the dual description of the light-cone gauge method in the next subsection.

Dual description of the light-cone gauge method
By the multiple Legendre transformation, we can introduce another new method, which is dual to the light-cone gauge method, to get TT -deformation.
The dual description of (4.5) is given by Using the definition ofK µν (4.18) and ∂L 0 ∂ rµ = ψ µ , the solution of the equations of motion (4.19) about p 1 , p 2 is (4.20) Plugging the solution back to (4.17), we can get The dual description is equivalent to the method of characteristics. The solution (4.20) is the same as the solution of the method of characteristics (3.7d) and (3.7e). Considering (3.7), then F (4.21) becomes which is just the multiple Legendre transformation of the TT -deformed Lagrangian L. When

Example: one free scalar
Let us show how the dual description works by an example. Consider the simplest example, the free scalar theory, whose Lagrangian is given by, Then (4.25) The (4.17) becomes F = p 1 · r 1 + p 2 · r 2 − r 1 · r 2 1 − s r 1 · r 2 . (4.26) Vary (4.26) by r 1 , r 2 to get the equations about r 1 , r 2 and solve them. With the initial conditions (4.25), the solution is (4.27) Plug the solution into (4.26) and we get (4.28) (4.30)

(TT ) α -deformation in two dimensions
The method of characteristics is a powerful tool to solve all kind of the first order differential deformations. In the section, we apply this method to (TT ) α -deformation.

(TT ) α -deformation
The flow equation of the generalized deformation is given by The flow equation can be rewritten as where a = 1/α. And the characteristic equations are (3.6a)(3.6b)(3.6e)(3.6f)(3.6g) and where λ = x 3 . Plugging the solution back into (3.6g), we can get (5.8) The above system of equations can be rewritten as a matrix form, where X = ( x 1 , x 2 ) T and A is the coefficient matrix. The general solution of (5.9) is where P means the time-ordering integral. For this case, [A(λ 1 ), A(λ 2 )] = 0, the timeordering integral degenerates into the ordinary integral, which can be calculated explicitly.
It is noticed that it is very special when a = 2, which is just √ TT -deformation. In the case, above expressions are valid in the sense of limit a → 2. As for a = 1, the deformation is just TT -deformation. Actually, we can take a inC. It can be understood in the sense of x y = e y ln x .

Example: one free scalar
We solve the equations(5.8) about the simplest case, where the seed theory is a free scalar, In the case, and we can get the solution of (5.8) by (5.11). .

(5.13)
Here, we just need to express L 0 by X ≡ x 1 x 2 . From the above solution, we can get L 0 and X satisfy the equation, (5.14) The equation cannot always be solved explicitly. However, for some special a, such as a = 1, 2, −1, −2, ..., L 0 can be expressed by X explicitly and the deformed Lagrangian z can be expressed by X.

TT -deformation
The flow equation of √ TT -deformation is given by It is a special case of (TT ) 1/a when a = 2. Taking the limit, a → 2, in (5.7) and (5.8), we can get and It is worth emphasizing that the coefficients on the right hand side in (5.19) is independent of λ, so (5.19) can be always solved easily. In the following, we solve (5.19) in two examples.

Example: N free scalars
Consider the N free scalars as the seed theory, whose Lagrangian is given by The initial conditions become The characteristic equations (5.19) become The result is the same as one in [2] if we change λ → −iγ. The difference, −i, comes from the sign of the definition of the stress-energy tensor.

Example: one scalar with a potential
Consider the one scalar with a potential as the seed theory, whose Lagrangian is given by The initial conditions become The characteristic equations (5.19) become The solution of the above equations is Plugging the solution into (5.18), we get where R = r 1 r 2 is the solution of the equation For the model, there is no explicit function form of the deformed Lagrangian. By the iterative method, we can get the first few orders of the deformed Lagrangian, (5.30)

TT -like deformation in arbitrary dimensions
In the section, we get the TT -like deformation in [1] by the method of characteristics. Then we generalize the definition to (TT ) α -deformation in arbitrary dimensions.

Characteristics for TT -like deformation
In d dimensions with Euclidean signature, the action is given by where L ≡ √ gL is the Lagrangian density. The flow equation of the TT -like deformation is given by We take the symmetric Hilbert stress-energy tensor When r = 1, d = 2, the deformation becomes TT -deformation and We exploit the method of characteristics to obtain the deformed Lagrangian. The coordinates on the characteristic flow curve are (g µν , λ) and the conjugate momenta are (p µν = ∂L ∂gµν , p λ = ∂L ∂λ ), respectively, and z = L. The flow equation (6.2) becomes where µ, ν are Lorentz index and λ is the parameter of the deformation. The initial conditions satisfy the constraint Notice that η µν is the initial value of g µν but is not necessarily the metric of the flat spacetime.
The characteristic equations (2.14) are given by We have used the flow equation (6.6) to get (6.9c) and (6.9e). (6.9a), (6.9c) and (6.9d) can be solved readily, yielding We assume that for the seed theory, the metric and the stress-energy tensor are symmetric, i.e. η µν = η νµ and ψ µν = ψ νµ . Then we can derive that g µν = g νµ and p µν = p νµ are always correct along the flow by (6.9b) and (6.9e). Using (6.9b), (6.9e) and the above property, we get dp α Therefore, By (6.9b), we can get where we use the formula δg δg αβ = gg αβ . The solution of the equation is given by Plug the all above results into (6.9b), we can get the matrix form of (6.9b), dG ds = GA, (6.15) where G is the matrix form of the metric, G = (g µν ). A is the matrix, where g(s) is given by (6.14). Because [A(s 1 ), A(s 2 )] = 0, the general solution of (6.15) is which can be calculated explicitly. Now the characteristic equations have been solved.
By (6.9e) and (6.13), we can also calculate the flow equation of T µν , And (6.9b) can be rewrite as The (6.18) and (6.20) are the same as (3.9) in [1].
It should be noted that although the matrix integral in (6.17) could be calculated explicitly, it is too complicated. For some models, we don't need to calculate the integral, such as a free scalar model.

Example: one free scalar
Consider one free scalar model as the seed model, whose Lagrangian density is given by The conjugate momenta are where we use ∂g µν ∂g αβ = −g µα g νβ . We consider TT -deformation in 2 dimensions, where d = 2, r = 1. For the model, (6.14) becomes Introduce a new variable X ≡ g µν ∂ µ φ∂ ν φ. By (6.9b) and (6.13), we get Solve the system of equations (6.23), (6.24) and (6.25), we get And the Lagrangian density z is . (6.27) Here, z = L s (g µν (s)). We want the deformed Lagrangian density in the flatness space-time, i.e. g µν (s) = δ µν = diag(1, 1). Therefore, the TT -deformed Lagrangian density is given by There are two perspectives about TT -deformation. In this section, the fields ∂ µ φ is a constant along the characteristic flow but the metric g µν depends on the flow parameter s. In section 3, on the contrary, along the characteristic flow, ∂ µ φ depends on the flow parameter s but the metric g µν is a constant.

(TT ) α -deformation in arbitrary dimensions
On the analogy of the definition of TT -like deformation, we can define (TT ) α -deformation are defined as (6.1) and (6.3) respectively. In our notation, the flow equation becomes where a = 1/α ∈C. The initial conditions are the same as (6.7) and satisfy the constraint The characteristic equations (2.14) are given by 32c) dp λ ds = 0, (6.32d) The solution is given by and where G = (g µν ) and When d = 2, r = 1, the deformation degenerates into the case in section 5.

Conclusions
In this work, we use the method of characteristics to study TT -deformed theories. We find that the TT flow is just the characteristic flow in essential. In this viewpoint, we prove that the dynamical coordinate transformation and the light-cone gauge method are both equivalent to the method of characteristics. The method of characteristics can be seen as a first principal derivation of these methods. We also propose a new dual description of the light-cone gauge method and re-derive the trace flow equation. Exploiting our method to generalized TT -deformations, we find the deformed Lagrangians for TT -like deformation and (TT ) α -deformation with generic α in arbitrary dimensions.
It is interesting that in 2 dimensions, there are two equivalent perspectives about TTdeformation along the characteristic flow. In one perspective, the field ∂ µ φ evolves and the metric g µν is a constant along the flow. In another perspective, the metric evolves and the field is a constant.
There is a very interesting question how other physical quantities evolve along the characteristic flow. For integrable QFTs, some works show that Lax connections satisfy the rules of the dynamical coordinate transformation on the flow [29,52]. However, the result is not proven strictly, yet. Maybe the method of characteristics is a good point to prove the result.
Furthermore, there are other quantities for integrable theories, such as the R-matrix. We don't know how they evolve. Correlation functions of some TT -deformed models have been evaluated by perturbation in first several orders [27,51,[53][54][55][56]. Maybe we can explore how they evolve along the characteristic flow.
We are also interested in another question whether bosonization holds under TT -deformation.
We have tried the traditional method, which maps fermion fields to boson fields, to bosonized the TT -deformed fermion theories. However we can't get the correct dual boson theories. Maybe one can take the view in section 6 to study bosonization of TT -deformation.
The method can be also used to more generalized TT -like deformations, such as the multiple TT deformation [57].
Acknowledgements I am extremely grateful to Yunfeng Jiang for very helpful suggestions and two pictures in this paper. I also thank Roberto Tateo for valuable comments and Bin Chen and Yi-jun He for discussions.

A Solve characteristic equations about TT deformation
In the appendix, we solve the characteristic equations about TT -deformation(3.6) with initial conditions(3.5).