Metric approach to a $\mathrm{T}\bar{\mathrm{T}}-$like deformation in arbitrary dimensions

We consider a one-parameter family of composite fields -- bi-linear in the components of the stress-energy tensor -- which generalise the $\mathrm{T}\bar{\mathrm{T}}$ operator to arbitrary space-time dimension $d\geq 2$. We show that they induce a deformation of the classical action which is equivalent -- at the level of the dynamics -- to a field-dependent modification of the background metric tensor according to a specific flow equation. Even though the starting point is the flat space, the deformed metric is generally curved for any $d>2$, thus implying that the corresponding deformation can not be interpreted as a coordinate transformation. The central part of the paper is devoted to the development of a recursive algorithm to compute the coefficients of the power series expansion of the solution to the metric flow equation. We show that, under some quite restrictive assumptions on the stress-energy tensor, the power series yields an exact solution. Finally, we consider a class of theories in $d=4$ whose stress-energy tensor fulfils the assumptions above mentioned, namely the family of abelian gauge theories in $d=4$. For such theories, we obtain the exact expression of the deformed metric and the vierbein. In particular, the latter result implies that ModMax theory in a specific curved space is dynamically equivalent to its Born-Infeld-like extension in flat space. We also discuss a dimensional reduction of the latter theories from $d=4$ to $d=2$ in which an interesting marginal deformation of $d=2$ field theories emerges.


Introduction
The recent discovery that specific irrelevant perturbations [1] of field theories in dimension d = 2 can be addressed using exact flow equations [2,3] and other powerful mathematical tools [4][5][6], has triggered a fair amount of research activity. Particularly striking are the observed links with string theory [3] and topological gravity [7], together with the AdS/CFT interpretation of these perturbations [8]. The main motivations to study these novel class of models are the deepening of our general knowledge on non-renormalisable Quantum Field Theories and to clarify aspects of quantum gravity.
In this paper, we work within the framework of Lagrangian field theories in space-time dimension d ≥ 2 equipped with a metric tensor g µν = g µν (x) with Euclidean signature, where x = (x 0 , x 1 , . . . , x d−1 ) is a set of local coordinates. We denote as a generic covariant action whereL := √ g L is the Lagrangian density that depends on x through a generic collection of N fields {Φ I } I∈{1,...,N } and their higher-order derivatives {∂ µ 1 . . . ∂ µ i Φ I } (I,i)∈{1,...,N }×{1,...,n} for some n ≥ 1 and with ∂ µ = ∂ ∂x µ . The field content of the theory is arbitrary, unless otherwise stated. Indexes of tensors are lowered and raised using the metric g µν and its inverse g µν , respectively, and repeated indexes are summed according to the Einstein notation. We shall denote with η ab the flat metric with the same (Euclidean) signature of g µν . Following the standard convention, we use latin (Lorentz) and greek (Einstein) indexes to distinguish between flat and curved reference frames, respectively, and we adopt the tetrad formalism to move from one to the other as customary.
This paper focuses on a family of deformations defined by the flow equation where τ ∈ R is the flow parameter and τ 0 is a fixed value; A τ = d d xL τ denotes the deformed action andL τ the corresponding Lagrangian density; T τ = (T µ τ,ν ) µ,ν∈{0,...,d−1} is a d×d matrix and T µν τ are the components of the (symmetric) Hilbert stress-energy tensor associated to A τ according to the standard prescription We start by reviewing some facts about the most studied representative among this family of deformations, namely the TT deformation of field theories in d = 2, from which the present paper draws inspiration. The TT deformation [2,3] is described by the flow equation (1.2) with (r, d) = (1, 2), i.e. the TT operator is given by where in the last equality we used the Cayley-Hamilton Theorem. The TT−deformed action A τ can be obtained either directly by solving explicitly the flow equation (1.2) or indirectly using a field-dependent coordinate transformation [7,9] (see also [10]) which provides an efficient tool to derive also solutions of the TT−deformed equations of motion [9,11] and integrals of motion [12]. Let us also mention that an alternative method to compute TTdeformed actions is given by the light-cone gauge approach developed in [13,14].
As it was noted in [9], the coordinate transformation induces a specific field-dependent deformed metric that defines a modified background in which the solutions of the seed theory are equivalent to the corresponding TT−deformed ones in flat space. In other words, there exists a deformed metric that makes the seed theory dynamically equivalent 2 to the deformed theory in flat space. Strictly speaking, this deformed metric is a pseudo-metric since, for a generic field configuration there might exist a range of values of the deformation parameter for which it becomes degenerate (see [9]).
Throughout the paper we will handle with pseudo-metrics -see (3.11) and (3.12) -associated to the generalised operators (1.3) in arbitrary dimension d ≥ 2. However, we shall refer to them simply as metrics neglecting the issue related to the degeneracy, since it does not affect the general conclusions that emerge from our analysis.
In contrast to the TT operator, the geometric properties of the operators (1.3) for d > 2 are essentially unknown. The interest toward such deformations is partially due to the discovery first made in [15], that the operator O [ 1 2 ,4] τ surprisingly links the Maxwell theory with Maxwell Born-Infeld [16] and, in [17,18], it was proven that the same link exists between the ModMax theory [19] and its Born-Infeld-like extension [20], thus generalising the result of [15].
The aim of this paper is to study the geometric properties of the family of deformations (1.2) through a metric approach. In section 2.1, we start by showing that (1.2) can be interpreted as a modification of the background metric -at dynamical level -according to a specific flow equation. In section 2.2 we prove that, for a generic field configuration, such deformed metric is curved except for the specific case (r, d) = (1, 2) -corresponding to the TT deformation -in which it remains flat, in accordance with the existence of a coordinate transformation. In section 3.1, we develop a perturbative algorithm to solve the flow equation for the metric and, in section 3.2, we show that under some assumptions on the stress-energy tensor, the series yields an exact solution for the metric. In section 3.3 we consider the class of abelian gauge theories in d = 4, whose stress-energy tensors meet the conditions abovementioned, and we derive an exact expression for the deformed metric and the vierbein; appendix A contains the details of the derivation of the vierbein. Finally, in section 4 we construct a class of modified scalar theories in d = 2 and their corresponding TT deformation, as a dimensional reduction of the ModMax theory and its Born-Infeld-like extension.

A TT−like deformation in d dimensions
For the purposes of the current paper, it is convenient to rewrite (1.3) as follows where we introduced the tensor 2) and f µνρσ := r g µν g ρσ − g µσ g νρ .
It is immediate to check that f µνρσ fulfils the following properties Notice that r = 0 and r = 2 d seems to be special cases since the last formula in (2.4) simplifies.

Metric approach
In this section, we prove that (1.2) amounts to a modification of the background metric at the level of the dynamics and we identify the flow equation that describes the evolution of the metric. To this aim, we adopt the same logic followed by [6] in the TT context. Under an infinitesimal deformation δτ of the parameter τ , (1.2) can be written as where we explicitly reported the dependence of the action on the background metric for future convenience. Let δg µν = δτ h µν be an infinitesimal deformation of the metric where h µν is dynamical, and consider the action where c is a real constant and we defined the tensor e µνρσ := q g µν g ρσ − g µσ g νρ , q ∈ R . (2.7) Notice that e µνρσ has the same form as f µνρσ , thus it fulfils the same properties (2.4) with the substitution (f µνρσ , r) → (e µνρσ , q). Moreover, the two tensors are trivially related via e µνρσ = f µνρσ + (q − r) g µν g ρσ . (2.8) In the following we will fix the parameters (c, q) in (2.6) in terms of (r, d) by requiring that the actions (2.5) and (2.6) are dinamically equivalent, i.e. they have the same equations of motion where we introduced the symbol to denote the dynamical equivalence between the actions. Notice that in (2.9) we used the fact that 10) and the variation of A τ (g µν + δτ h µν ) w.r.t. Φ I is performed before evaluating h µν to its on shell value h * µν .
We first compute the variation ofÂ(h µν ) w.r.t. h µν and set it to zero to obtain the equation of motion for h µν . Using the analogous of the properties (2.4) in which (f µνρσ , r) → (e µνρσ , q) and formula (2.8) we have Multiplying both sides of the last equation in (2.11) by g µν we obtain .
The next step is to compute the variations of bothÂ(h µν ) and , (2.14) while for A τ +δτ (g µν ) we have (2.15) Notice that in the last equality of (2.15) we used the fact that where X is any element of the set {∂ µ 1 . . . ∂ µ i Φ I } (I,i)∈{1,...,N }×{1,...,n} . From (2.13), (2.14) and (2.15) it is immediate to see that the equivalence (2.9) holds, in general, only if the parameters (c, q) are chosen as follows In the following, we shall impose the constraint (2.17). Using the identity in (2.9), we obtain the following (constrained) dynamical equivalence which has the following physical interpretation: the deformed theory A τ +δτ with background metric g µν (τ ) is dynamically equivalent to the theory A τ with deformed background metric g µν (τ + δτ ), which evolves according to the second equation of (2.19).
Notice that (2.19) can be equivalently written as which has the following physical interpretation: the theory A τ with background metric g µν (τ ) is dynamically equivalent to the deformed theory A τ +δτ with deformed background metric g µν (τ + δτ ), which evolves according to the second equation of (2.20).

Deformation of the Riemann tensor
In this section we briefly discuss the infinitesimal deformation of the Riemann tensor δR ρ σµν induced by the infinitesimal deformation δg µν = − 4 d δτ T τ,µν of the metric. Assuming that the starting point is a d−dimensional flat space with metric η ab , i.e. the associated Riemann tensor is R i jab = 0, then δg µν := δη ab e a µ e b ν where we defined δη ab = − 4 d δτ T τ,ab and e a µ = δ a µ is the trivial vierbein.
A standard computation leads to for the Riemann tensor, for the Ricci tensor and for the scalar curvature. In (2.22) and (2.23) we used the additional constraint coming from the conservation of the stress-energy tensor in flat space, i.e. ∂ a T ab τ = 0. From (2.23) it follows that Let us consider separately the cases d = 2 and d > 2.
• case d > 2: from (2.21), it emerges that the deformation of the Riemann tensor depends on the field configuration through the stress-energy tensor and it is,in general, nonvanishing. Therefore, we conclude that the deformation induced by (1.3) modifies the geometry of the space in a non-trivial way for d > 2.
• case d = 2: in this case the Riemann tensor has only one independent component, i.e. the scalar curvature R. From (2.25) it follows that the operator O [r,2] τ modifies the geometry of the space for any r = 1. The case r = 1 is special and corresponds to the which does not affect the geometry, in agreement with the existence of a coordinate transformation.

Metric flow equation
In this section, we derive a system of differential equations that completely defines the flow of the metric. Moreover, we develop a perturbative algorithm to find a power series expansion for the solution to the metric flow equation.
The equivalence (2.20) leads to the following system of differential equations, where the second equation descends from (1.2) and (1.4). Using the properties and the second equation of (3.1) yields explicitly where we denoted T n,µν The key point of the computation is that T µν s depends on s both explicitly and implicitly through g µν . Using the property ∂T µν From the latter expression, the first equation of (3.1) and formula we can easily compute the total derivative of T s,µν as Upon explicit computation, we arrive to the system where we denoted T n s,µν = T s,µµ 1 g µ 1 µ 2 T s,µ 2 µ 3 . . . g µ n−1 µn T s,µnν and we defined (3.10) The system (3.9) completely defines the flow of the metric once an initial condition has been chosen. The idea is to solve it for g µν (s) := g µν (s; s 0 ) with initial condition g µν (s 0 ) = η ab e a µ e b ν for some s 0 , where e a µ = δ a µ is the trivial vierbein. Such solution provides the deformed background metrics that allow to boost or absorb the deformation of the action, depending on the choice of the parameters s 0 and s. In fact, 1) if s 0 = τ and s = τ 0 : the action A τ with metric η ab is dynamically equivalent to A τ 0 with metricĝ 2) if s 0 = τ 0 and s = τ : the action A τ 0 with metric η ab is dynamically equivalent to A τ with metricǧ µν (τ ) = g µν (τ ; τ 0 ) . (3.12)

Algorithm to solve the metric flow equation
In this section we compute the solution g µν (s) := g µν (s; s 0 ) of (3.9) by means of a perturbative approach which can be made algorithmic and implemented in a computer software. The idea is to Taylor expand g µν (s) := g µν (s; s 0 ) around s = s 0 as where e a µ = δ a µ is the trivial vierbein and (3.14) We impose g  ab (s 0 )} n≥1 . The first two coefficients g (1) ab (s 0 ) and g (2) ab (s 0 ) descend trivially from (3.1) and yields where α s and β s are defined as per (3.10). To get {g  µν can be written as ..,n} are polynomials in the variables α s and β s with real coefficients. Therefore, the computation of g (n) µν has been reduced to the computation of the coefficients {c (n) k } k∈{1,...,n} . Differentiating (3.22) w.r.t. s and using (3.19), we easily obtain the recurrence relations where dc For example, the coefficients of g µν are (3.26) The implementation of the recurrence relations (3.23) corresponds to a couple of lines in a Mathematica notebook. The first n = 100 terms of the sequence {c (n) k } k∈{1,...,n} can be obtained in less than a minute on a standard laptop. However, the task of finding a close expression for g µν valid for all n ∈ N is highly non-trivial. In the next section we shall consider special cases in which this task becomes feasible.

Exact solutions for the metric
In this section we show that for some values of r and under some assumptions on the stressenergy tensor, it is possible to obtain a close expression for the coefficient g ab (s 0 ) valid for all n ≥ 1 and we are able to formally sum the series (3.13).
It is convenient to work with the matrix notation. Let us introduce the d × d matrices ρν µ,ν∈{0,...,d−1} , n ≥ 1 . Assume that the matrix T s 0 is diagonalisable, i.e. there exist an invertible matrix P and a diagonal matrix D such that T s 0 = PDP −1 . Moreover, assume that T s 0 has 2 (resp. 1) independent eigenvalues of multiplicity d 2 (resp. d) if d is even (resp. odd), namely where (x) n = Γ(x+n) Γ(x) is the Pochhammer symbol. Whence Let us make a few remarks: • in d = 2, the condition (3.28) does not constraint the stress-energy tensor which has, in general, 2 distinct eigenvalues. Therefore, (3.31) is the deformed metric associated to the TT deformation of a generic theory in d = 2. Moreover, using the identifications where H s 0 and P s 0 are the energy and momentum densities and setting dx 0 = 0, the line element d 2 = g µν (s; s 0 ) dx µ dx ν becomes Formula (3.34) resembles the modification of the "effective size" of the system at quantum level (see, for example, equation (2.8) in [9]), which is ultimately a consequence of the Zamolodchikov's factorisation Theorem [1].
• in d = 4, (3.31) takes a particularly simple expression, being linear in s. It is natural to ask whether also in this case the information of the quantum theory is hidden in the line element d 2 , in analogy to the d = 2 case.
• all formulas can be analytically continued to d = 1, in which the field theory reduces to a mechanical system. In this case, the tensors g µν and T µ s,ν reduce to the scalars g 00 = g and T 0 s,0 := −E s respectively, where E s is the energy, while the perturbing operator is O . Moreover, from (3.32) using T s,00 = −gE s we get which matches the result of [21]. Notice that this is also the expression of the deformed energy density of a Yang-Mills theory in d = 2 [15,22,23].
• it would be interesting to look for a match between the series expansion of the metric proposed here and the perturbative results obtained in [24] for the Lagrangians associated to abelian gauge theories in d ∈ 2N deformed by the operator O  might have a close expression as well. In the following, we will address the computation of the deformed Lagrangian density focusing on the simple case of a noninteracting scalar field in d = 2. We leave the analysis of more complicated theories in d = 2 as well as the extension to theories in d > 2 to a future publication.
It is possible to show that the solution to the flow equation is given by where W (x) is the Lambert function. It is surprising that corresponds to the generating function associated to Cayley's formula in graph theory. 5

Abelian gauge theories in d = 4: the exact vierbein
The assumption (3.28) restricts the range of applicability of the results obtained in section 3.2 for d > 2. However, in d = 4 there exists a whole class of field theories whose stress-energy tensors fulfil the constraint (3.28): the abelian gauge theories, describing the dynamics of one gauge field, i.e. the electromagnetic four-potential A a (in flat space with metric η ab ). We briefly review the proof of this fact, which can be found also in [18,Appendix A]. Let F ab = ∂ a A b − ∂ b A a be the field-strength associated to the gauge field and F ab = 1 2 abij F ij the dual field-strength, where abij is the Levi-Civita symbol with the choice 0123 = 1. Following [18], the stress-energy tensor of a generic abelian gauge theory can be decomposed as where a (0) , a (1) and a (2) are functions of tr[F 2 ] and tr[F 4 ] with F = F a b a,b∈{0,...,3} . A straightforward computation shows that the eigenvalues {λ i } i∈{1,...,4} of the matrix F are such that λ 2 = −λ 1 and λ 4 = −λ 3 (independently of the signature of the metric η ab ) whence it follows that the eigenvalues {λ i } i∈{1,...,4} of T = {T a b } a,b∈{0,...,3} arē where the tensors T τ,ab and T τ 0 ,ab are both evaluated in flat space with metric η ab . Notice that the expression of (3.45) matches that of the (pseudo) metric found many years ago (see the lecture notes [16]) for the Maxwell Born-Infeld theory, using a completely different approach. From the discussion of section 2.2 we know that, in general, (3.45) are curved. In appendix A, we show that it is possible to give an exact expression for the pair of vierbeinê a µ :=ê a µ (τ ) andě a µ :=ě a µ (τ ) that fulfilĝ Decomposing T τ 0 ,ab and T τ,ab as per (3.43) with coefficients a (0) τ , a (2) τ , respectively, we obtain e a µ (τ ) = Σ (τ 0 ; τ ) e a µ + where we defined Relevant examples of models such that a s = 0 are ModMax and its Born-Infeld-like extension, that we shall briefly discuss in the next section.

ModMax and its Born-Infeld-like extension
Let us recall that the ModMax (MM) theory [19] represents a marginal deformation of the Maxwell theory described by the Euclidean action A MM where γ is a real parameter and we defined the invariants 6 One can associate to (4.1) a Born-Infeld-like extension (MMBI) [20] that is described by the action A MMBI Using (4.4) and (4.5) it is possible to show (see [18]) that the action A MMBI with initial condition A MBI τ at γ = 0 for any value of τ . This fact has been shown in [17] by means of a perturbative expansion around γ = 0. Let us briefly report here the exact computation at finite values of γ. Using (4.4) and (4.5) one has On the other hand, It is well known [26] that there is a deep connection between the theories of Nambu-Goto in d = 2 and Maxwell Born-Infeld in d = 4. In fact, particular solutions of Maxwell Born-Infeld are also solutions of Nambu-Goto in static gauge with two transversal scalar fields. In this section, we show that this link can be lifted to the Born-Infeld-like extension of ModMax. In analogy with [26], we consider a specific field configuration consisting in the scattering of plane waves along the direction x 1 , which corresponds to the requirements wherex = (x 0 , x 1 ) denotes the restricted set of local coordinates on the plane. Let us identify φ a (x) := A a+1 (x) with a ∈ {1, 2}. Then, the constraint (4.10) implies the following reduction F ab (x) →F ab (x), whereF ab has only four non-vanishing (independent) components that depends on the derivative of the scalar fields {φ i } i∈{1,2} w.r.t.x: Consequently, the invariants (S, P ) as per (4.2) reduce to (S 2 ,P 2 ), where we defined 12) with H N,ab the following symmetric tensor and is the Nambu-Goto Lagrangian density in a d = N + 2 target space and imposing the static gauge condition. A simple computation shows that the components of the stress-energy tensor associated to (4.17) are with coefficients with L NG,N τ as initial condition at γ = 0, for any value of τ .
Finally, notice that a perturbation of a CFT in d = 2 with the square root of the TT operator was introduced in [27] in the study of the relation between relativistic and ultra/nonrelativistic conformal algebra (see also [28] for further interesting results on this subject).

Conclusions
This paper discusses important geometric features of specific TT-type deformations in arbitrary dimensions. Various aspects and open problems deserve further investigation.
The first natural question is whether there exist physically acceptable theories, in d = 1, 2, 4, whose stress-energy tensors fulfil the constraint (3.28). More generally, it would be important to find an exact formula for the deformed metric without imposing strong constraints on the stress-energy tensor eigenvalues. For example, a milder assumption would be the tracelessness of the stress-energy tensor in flat space, which drastically simplifies the perturbative series. Nevertheless, we were unable to obtain a closed expression for the deformed metric. In d > 2 and besides Born-Infeld nonlinear electrodynamics, the general properties of this TT-induced geometric deformation and its phenomenological features as a perturbation of classical field theory problems are unknown and certainly deserve some investigation. Moreover, it remains an important open question whether the simple expression for the truncated metric (3.45) could lead to some exact quantum result in d = 4, for example, the Casimir energy in specific geometries.
A further possible line of research concerns the extension of the current setup to encompass the TT-like deformation in arbitrary dimensions introduced in [29] and the deformation of supersymmetric theories [30][31][32]. Concerning the results of section 4, it would be nice to understand the properties of the Modified Scalar theories, viewed as marginal deformations of free boson CFTs.
Finally, let us mention that the article [33] about T 2 deformations of large N holographic CFTs appeared a day after the first version of the current paper was available on ArXiv. The perturbing operator considered in [33] is of the form (1.3) with r = 1 d−1 . The setup and methodologies adopted in the two papers are similar in spirit but slightly different, and it may be very instructive to explore the eventual connection between them. Moreover, [33] motivated the search for a one-parameter extension of our initial results, which were restricted to the case r = 2 d .
PTDC/MAT-PUR/30234/2017 "Irregular connections on algebraic curves and Quantum Field Theory". R.C. is also supported by the FCT Investigator grant IF/00069/2015 "A mathematical framework for the ODE/IM correspondence".
Note added -On June 22 nd 2022, the day the second version of this work was made public on arXiv, the manuscript [34] appeared. In [34], it is independently shown that the operator (4.20) generates the γ-flow of the Lagrangian (4.17), the setup is slightly more general, but the main equations and conclusions are in total agreement with ours.
A Deformed vierbein for abelian gauge theories in d = 4 Let us start from the general solution