A Path Integral Realization of Joint $J\bar T$, $T\bar J$ and $T\bar T$ Flows

We recast the joint $J\bar{T}$, $T\bar{J}$ and $T\bar{T}$ deformations as coupling the original theory to a mixture of topological gravity and gauge theory. This geometrizes the general flow triggered by irrelevant deformations built out of conserved currents and the stress-energy tensor, by means of a path integral kernel. The partition function of the deformed theory satisfies a diffusion-like flow equation similar to that found in the pure $T\bar{T}$ case. Our proposal passes two stringent tests. Firstly, we recover the classical deformed actions from the kernel, reproducing the known expressions for the free boson and fermion. Secondly, we explicitly compute the torus path integral along the flow and show it localizes to a finite-dimensional, one-loop exact integral over base space torus moduli. The dressed energy levels so obtained match exactly onto those previously reported in the literature.


Introduction
The Wilsonian paradigm has taught us that characterising trajectories on the space of Quantum Field Theories (QFT) is difficult but imperative to understand. These trajectories are triggered by a wide variety of deformations, whose form depends on the particular details of the conformal fixed point at which they are turned on. A UV complete scheme posses a well defined UV fixed point described by a Conformal Field Theory (CFT). In principle, we can track the flow generated by a relevant or marginal deformation. However, this picture usually breaks down for irrelevant deformations, which give rise to non-renormalizable interactions. This obscures whether the UV physics is well defined or not.
In 2-dimensional quantum field theories, certain composite operators built out of conserved currents stand as an exception to the rule. Starting with the remarkable TT operator [1] and its Lorentz preserving higher spin relatives [2], to the manifestly Lorentz breaking JT and TJ deformations [4], these special operators all share two remarkable properties. Besides being composite operators, they are unambiguously defined ,i.e. free of short-distance singularities. Secondly, they give rise to exactly solvable trajectories in the space of theories.
Much recent work has succeeded in describing the dynamics TT flow and elucidating the underlying origin of its solubility. In a nutshell, general properties of the QFT stress tensor (subject to very mild assumptions) lead to the explicit factorization of its expectation value in terms of bilocal products of single trace operators. The flow of the deformed energy levels then satisfies the inviscid Burgers equation [2,3]. [5,6] provided a rather different perspective by studying the flow infinitesimally. They showed how the TT deformation could be accounted for by coupling the seed theory to random metrics, whose action turns out to be topological. [7,8] pushed this analysis beyond the infinitesimal regime, and recast the flow as coupling the undeformed theory to a variant of JT gravity.
This powerful geometric interpretation further allows one to obtain the classical deformed actions in terms of a field-dependent coordinate transformation [9,10,11]. For interested readers, we refer to the pedagogical review [12] which further explains many of the concepts listed above.
The main point of this present paper is to similarly geometrize the JT and TJ flows, in Euclidean space. The so-called JT and TJ operators provide an equally interesting set of solvable, Lorentz breaking deformations. These operators are defined as where J andJ are the Noether currents associated to some U (1) symmetries of the underlying seed theory. T µ a = T µ ν e ν a , where T µ ν is the stress-energy tensor and e a µ the vielbein satisfying e a µ e b ν η ab = g µν . Finally, n,ñ will be taken to be two light-like vectors, satisfying n · n =ñ ·ñ = 0 , n ·ñ = 1, δ a b = n añ b +ñ a n b (1.3) These operators, in requiring two light-like vectors of opposite chirality, explicitly break Lorentz invariance. This light-like nature in Euclidean space means the vectors cannot be real-valued. In Euclidean space the only vector with norm 0 is the null vector 0. Nonetheless, we make this choice to facilitate comparison with past papers working in Lorentzian space and choosing real light-like vectors in that context.
In terms of the above operators, the flow is described by the following expressions 1 with Z i denoting the partition function and i parameters of length dimension one parametrizing the curve throughout the space of theories. Note that we are making explicit the dependence on both sides of (1.4) and (1.5), emphasizing the recursive nature of the flow and its non-linear dependence on the deformation parameter. We will drop this excessive notation in the rest of the paper. This flow was first introduced in [4]. Within the last two years, a considerable amount of work has been devoted to its study. This includes the construction of a holographic realization [13] involving an AdS bulk with mixed boundary conditions for the metric and Chern-Simons gauge fields. Moreover, the deformed energy levels for a general combination of JT , TJ and TT flows were recently obtained in [17]. They ingeniously recast it as a marginal deformation of a suitable string worldsheet theory. The latter describes a target space geometry which interpolates between AdS 3 and a linear dilaton [14,15,16,17].
The similarity to the TT deformation motivates us to seek a similar geometrical interpretation of the JT and TJ flows. We require the following of our proposal. At the classical level, it should lead to a well-defined procedure to derive the deformed classical action by means of coordinate and gauge transformations. We expect to write down a kernel as a topological action for random spacetime metrics and gauge fields. Finally, it should naturally lead to a well defined prescription for constructing the quantum partition function, encoding physically relevant properties of the deformed theory such as the energy spectrum. Our kernel does all of the above.

Summary of Results
Here we list the main results of this paper.
• By closely following the arguments developed for TT deformations [7,8], we capture the effects of the JT and TJ deformations by coupling the seed theory to topological gauge and gravitational degrees of freedom. In fact, we can easily incorporate a joint flow with TT , and via a suitable limit of the parameters, even reproduce the JJ deformation.
To do so, we first write the partition function of the undeformed theory as Z 0 [e a , A,Ã]. This is a functional of the background spacetime metric (written in terms of the first order formalism vielbeins e a µ ) and background gauge fields A µ andÃ µ which couple to the U (1) currents of the seed theory.
We obtain the partition function of the deformed theory living on a spacetime with metric vielbeins f a µ and background gauge fields B µ andB µ via the following path integral where the kernel's action is given by Note that n,ñ have switched places; this is because of the off-diagonal decomposition of identity in (1.3). Here, γ is related to the deformation parameter λ of the TT deformation as First off, readers might object to this being called a kernel, as it is not quite of the schematic form f (x) = dyK(x, y)f 0 (y). Instead, there are extra integrations over the fields Y a , α and α. The purpose of these additional integrals is to impose important constraints on the range of integration for our kernel. From our path integral perspective, these constraints lie at the heart of the solubility of these flows.
It is helpful to introduce the distinction between the "target" space, where the deformed theory lives, and the "base" space on which the undeformed theory is defined. The base space is parametrized by the coordinates σ µ . We thus say the f a µ denote target space vielbeins, which will be taken as flat throughout all this work. Similarly, B µ andB µ denote background gauge fluxes defined in target space. The fundamental purpose of the kernel is to establish a map between base and target space variables. 2 The additional fields carry an interesting physical interpretation. Physically, the Y a , α andα denote fluctuations around the above described target space backgrounds. They parametrize coordinate and gauge transformations respectively. Finally, they make the diffeomorphism and gauge invariance of the kernel manifest.
• Section 3 first tests our proposal at the classical level, i.e. the deformation given by replacing log Z in (1.4), (1.5) by the action S. We introduce X µ 's which play the role of a "dynamical" coordinate parametrizing the target space. At this stage, B andB do not play any role , and are set to 0.
To probe the classical structure of the deformed theory, we can evaluate the path integral in (1.7) via saddle point. All fluctuating fields are set to their saddle point configurations, in particular the X µ . In nutshell, we rewrite the original action on a space with coordinates X µ . In addition, we also perform a gauge transformation determined by the classical configuration of the fields α andα. This gives rise to a well-defined geometrical procedure to obtain the exact classical deformed action by solving an algebraic system of equations, in complete analogy with [11]. We successfully recover the known deformed actions for the free boson and free fermion. Further, we propose a particular modification to account for the joint JT , TJ and TT flow and revisit the two examples discussed above.
• We compute explicitly the path integral on the torus. The TT analog of this computation was done in [8]. We attempt to make this rather technical calculation as accessible as possible. We confirm the validity of our result on multiple fronts. Our deformed partition function satisfies the torus version of the flow equation: where L a µ and b µ ,b µ denote respectively the lengths of the torus and the holonomies of the background gauge fields B,B that couple to J,J. We have also defined Z = Z/A with A the corresponding area of the torus.
• The explicit evaluation of (1.6) for the torus provides a further stringent test because it allows us to access the spectrum of the deformed theory. Indeed, by setting the target space gauge fields to zero, we know the torus partition function can be written as the sum with R denoting the length of the spatial circle of the target space torus. τ 1 , τ 2 are its modular parameters with P n = n/R the quantized spatial momentum (i.e. n ∈ Z). From (1.10), we can extract E n = 2E (R) n + P n with right-moving deformed energy levels given by where the (R) n are the right-moving energies of the seed theory. 3 Up to notational conventions, this agrees with the spectrum in the existing literature, obtained via very different approaches. Moreover, we can similarly compute the spectrum along the joint flow of JT and TT , again finding precise agreement with previous results.
• Not only does the torus path integral localize to a finite-dimensional one, it is furthermore one-loop exact. This means the exact integral can be computed from its saddle point approximation, even though it is not Gaussian. From the path integral point of view, this provides a further understand for the solubility of the JT and TJ flow. Our kernel shares this remarkable property with its TT predecessor in [8] . More importantly, we extract from the saddle-point equation important physical intuition and make contact with previous definitions of "chiral" charge appearing in [17] 2 An Introduction to the Kernel The main proposal of this paper is that a joint flow under TT , TJ and JT is implemented by the path integral and The purpose of this section is to introduce some important properties of this kernel. Along the way, we will carefully define the various quantities appearing in it. Before delving into the full story, we need to list a few choices that complete the definition of this theory.
1. All fields that appear in (2.1) are 'single-valued.' On the torus, this actually means singlevalued. On non-compact spaces, this means that it vanishes at the boundary. Either way, integration by parts in terms of these objects never generates boundary terms.
2. We impose the interpretation Y a = f a µ ξ µ , where ξ µ is a single-valued vector field; the motivation for this is elucidated later in this section, and this interpretation will be crucial in doing the path integral.
3. On a related note, we require α,α to be valued in the respective gauge groups. The relation between these requirements on α,α and Y should become clear as this section progresses.
4. On the torus, the gauge group volumes are only volumes of the groups of gauge transformations continuously connected to the identity. This is intimately linked with the single-valuedness of the fields, in a way that will be explained in section 4.
We can now move on to discussing more interesting qualitative questions. The first important fact to notice about this theory is key to its topological nature and underlies the factorisation of the bi-linear operators that define the flows. The fields Y a , α,α act as Lagrange multipliers setting the curvatures of the fixed and dynamical gauge fields to coincide: Together with the conditions that have been imposed on the external fields f, B,B, this implies the path integral runs only over flat vielbeins and gauge fields. The full path integral (2.1) therefore reduces to integral only over global modes of the vielbeins and gauge fields. We can parametrize the global modes of the vielbeins as overall scale, an uniform tangent-space rotation, and the torus moduli [8]. The holonomies around the non-trivial cylces of the torus play the analogous role for the gauge fields. There are two useful pictures to keep in mind when parsing the meaning of this path integral. In the first, we simply view the kernel as arising from some complicated action with sources f, B,B for the currents T, J,J. Because these are conserved currents, these background sources have special names: the vielbeins of the manifold and the background gauge fields, respectively. This is an unreasonably hard-nosed view. Of course, whatever words we associate with the kernel, they must be compatible with this picture.
A more intuitive picture arises by noticing that one of the dynamical variables is yet another set of vielbeins e a . These can be thought of as defining the metric on a secondary manifold on which the seed theory lives. We call the original, non-dynamical, manifold on which the deformed theory lives the target space and this new "dynamical" manifold on which the seed theory lives the base space. Nice as this picture is, it raises an important question. If there are two manifolds, there should exist two sets of diffeomorphisms, and similarly two sets of U (1) transformations.
These two sets of transformations have different interpretations. Invariance under U (1) transformations and diffeomorphisms of the non-dynamical target space fields encodes the conservation of the deformed currents. This is nothing new: invariance under such background field transformations always has this interpretation in QFT. As for the base space, invariance under U (1) transformations of the fields living here is a genuine gauge-invariance. The conservation of the seed currents is a necessary condition for this U (1) symmetry to be gauged. The same holds for base space diffeomorphism-invariance.
Let us see how the kernel encodes these transformations. We begin first with the U (1)'s. The action in (1.7) is readily seen to be invariant under a base space U (1) gauge transformation of the form and a target space transformation of the form The exact same transformation rules, with tilded quantities instead, describes the other U (1). α,α play the role of compensator fields. They transform linearly under these symmetries. They parametrize the difference in U (1) frames between the two manifolds. More plainly, these transformations suggest that moving any charged field from the base to the target space requires a gauge transformation with the parameter α,α. This underpins the construction of classical actions in section 3. We now turn to diffeomorphisms. These are more subtle. The action is manifestly invariant under base space diffeomorphisms, if all the fields transform under the usual rules compatible with a reparametrization of the coordinates σ µ . This includes a transformation of the background sources f a → f a , B → B andB →B . On the face of it, this would naively give Z[f a , B ,B ]. What we need to show is that Z[f a , B B ] = Z[f a , B,B]. This is tantamount to target space diffeomorphism invariance, whose infinitesimal version is just the conservation of the deformed stress tensor.
Under these target space diffeomorphisms, our fields transform as However, invariance still does not follow straightforwardly. To show this, we focus only on the term proportional to 1 -the other terms are similar and follow the same argument. Resorting to form language so as to avoid a proliferation of indices, we find that this part of the action transforms as Here, we have used the fact that Y a imposes ∂ [µ (f − e) a ν] = 0. 4 What we have found is that (2.5) is a symmetry if and only if In other words, the target space manifold and background gauge fields must be flat. This restriction reflects the kernel we are using necessitates additional terms to accommodate non-zero curvature of the frame-field and gauge connections. 5 Restricting our attention to the case where (2.7) hold, (2.6) shows the Y s act as compensator fields for target space diffeomorphisms. Paralleling the U (1) discussion, we conclude the Y s parametrize the difference in space-time frames. We may therefore define target space coordinates as In these coordinates, the target space vielbein is This follows the notation used in [8]. Our interpretation of the Y a 's as compensator fields for diffeomorphism originally provided the backbone for deriving deformed classical actions in [11]. Further, the interpretation (2.8) is actually necessary for a proper definition of the path integral. It will inform our treatment of an important zero-mode in section 4.

An intuitive derivation of the kernel from an infinitesimal analysis
Having discussed at length the various symmetries and redundancies of the kernel, we provide further intuition via a derivation along the lines of [5]. We will only explicitly write equations for the 1 flow. All other terms behave similarly. The reader should envision all the below manipulations are happening for all three sectors at once. The analysis begins by solving the flow equation (1.4) to first order in δ 1 . We use the fact the currents can be thought of as the response to a change in the corresponding background field, 6 to write the formal expression (2.11) This expression, while intuitively appealing, is not entirely well-defined, because of the two coincident functional derivatives.
[5] therefore suggested performing a Hubbard-Stratonovich transformation leading instead to The crucial step here was to absorb the linear terms generated by the Hubbard-Stratonovich into a change in the partition function. An infinitesimal step along the flow translates to an integral over small fluctuations of the vielbeins and gauge fields relating base and target space variables. Comparing with our kernel, we can identify f = e base + δe an B = A base + δA.
There is as yet no hint of the compensator fields Y a , α andα. Conversely, there is no division by the volume of gauge transformations and diffeomorphisms. We can incorporate them into this analysis by noting the integral (2.12) is Gaussian. Its value is therefore entirely controlled by the saddle point, These saddle points have the special property that Because the integral is Gaussian and therefore completely controlled by the saddle point, we may as well restrict our integration to variations satisfying (2.14). We add Lagrange multipliers imposing them as constraints. These constraints being linear in the fields, they do not affect the fluctuation determinant around the saddle either. This leads to which is clearly the infinitesimal version of (1.7). To be clear, some choices have been made here in the normalisation of the Lagrange multipliers. Finally, we note this object now exhibits gauge-invariance under both diffeomorphisms and U (1) transformations. Therefore, we should divide by the volume of those groups.

Deformed Classical Action from the Kernel
In this section, we use the classical limit of the kernel (1.7) to derive the classical action of the deformed theory on R 2 . This derivation follows from the interpretation, motivated in section 2, that the α,α, Y fields parametrize the difference between the coordinate systems and U (1) frames of the target and base spaces. We use the equations of motion derived from the action. This gives us the saddle-point values of these fields. We then perform the corresponding U (1) and coordinate transformations on the fields of the seed theory, and evaluate the full action in terms of these transformed fields.
We begin with a review of the procedure in the case of the TT deformation, and then move on to our new case of interest.

Review of the TT case
Since this section's line of reasoning closely parallels that used for the TT deformation, we briefly review the relevant arguments. We will be very schematic; a detailed explanation can be found in [11], but the idea originates in [9,10].
The path integral version of a TT -deformed theory [7,8] closely resembles our path integral. The difference lies primarily in the fact their integral runs only over the vielbeins e and the diffeomorphism compensators Y a , with action Once evaluated on-shell, this takes the form of the TT operator. 7 λ is a dimensionful coupling with dimensions of (length) 2 . It parametrizes the TT flow. Integrating by parts, it is easy to check the fields X a ensure the flatness condition µν ∂ µ e a ν = 0, or in form language de a = 0. The saddle-point equations that follow from the action (3.1) read with φ(σ) the fields of the seed theory. These equations are linear in X. In principle, this is the solution for X in terms of φ(σ). This is however not the form we are after. Since we want to perform coordinate transformations on the fields, where X * is shorthand for any tensor transformations the field must undergo, 8 it is rather cumbersome to have the coordinate transformation written in terms of the base space field φ. Instead, we rewrite the base space stress tensor in terms of the target space fieldφ(X) to find where X * φ (X) is the base space field written in terms of the target space fields.In principle, the LHS involves X a while the RHS involves X µ . These are related as in (2.9). We take the special case of R 2 , where the vielbein relating them is and so they effectively coincide. We can write the deformed classical action as the sum of the original action and the kernel action, reformulated as living on the target space (3.6) [9,11] only considered the cases where φ(σ) =φ(X), so that the "X * " acted as identity. We will similarly restrict to these cases.

Generalization to JT , TJ and TT Deformations
Generalizing the above arguments to our case of interest is surprisingly straightforward. Consider a CFT whose action S 0 gives rise to two (or at least one) U (1) symmetries. Denote by J µ andJ µ the associated Noether currents. Following the steps reviewed above for the TT case, we gauge both spacetime and U (1) symmetries by coupling them to "dynamical" vielbein e a µ and gauge fields A µ andÃ µ . This promotes the action where φ collectively denotes the original matter fields in the seed CFT.
For JT and TJ, the kernel action for B =B = 0 and similarly f a µ = δ a µ is We take n,ñ to be normalized "light-like" vectors, that is σ µ again denote the base space coordinates. The total action for all the fields thus becomes We have already noted that α,α play the role of Lagrange multipliers imposing the constraints µν ∂ µ e a ν = 0. (3.8) shows the combinationsñ · X and n · X also act as Lagrange multipliers. They, in turn, enforce the vanishing field strengths for A µ andÃ µ , respectively.
The equations of motion obtained by varying (3.10) w.r.t. the gauge fields and the vielbein are We wish to obtain the classical Lagrangian for the theory defined on the plane. 9 In order to solve (3.11), we choose a gauge such that

12)
consistent with the constraints imposed by the X a , α andα fields. Now, to find the deformed action, we need to solve the system of equations (3.11). As in the TT case, they look entirely linear in X, α,α, as long as the fields are thought of as living on the base space coordinates and base space U (1) frame. However, we are interested in the solutions in terms of fields living on the target space coordinates and U (1) frame, Here, the right hand side is the base space field written in terms of target space coordinates X and U (1) transformed by the amounts α,α. In terms of thisφ, the stress tensor and the currents depend non-trivially on X, α,α, and the equation is suitably non-linear. As the transformed quantities depend on α andα, (3.11) is a system of 8 equations for 8 variables. The 8 variables are the 2 × 2 "gauge matrices" (∂ σ µ α , ∂ σ µα) and the Jacobian ∂ σ µ X a . These can be solved for algebraically.
To derive the classical deformed action in our case, we need simply to evaluate the action (3.10) on shell in terms of the gauge transformed fields living on the target space coordinates X: To see this all in action, we apply our formalism to several concrete examples in the following section.

Some Examples of JT +JT Deformations
In this section, we show how the steps outlined in the previous section work for the special case of γ = 0. We will work in complex coordinates on both manifolds, for the both space-time and target/tangent space 10 coordinates. Complex target space coordinates will be denoted by z µ ∈ (z,z) whereas we use complex coordinates w µ ∈ (w,w) for the base space. The coordinates are normalised so that The flat space vielbein is gauge-fixed to be diagonal, i.e. its non-zero components are e z w = ezw = 1. In this coordinates, the "lightlike" vectors n andñ are such that Raising the index, we find Equations (3.11) now specify the 2×2 Jacobian matrix with elements ∂z a ∂w µ and the "gauge matrices" (∂ w µ α , ∂ w µα) in terms of the currents and the stress-energy tensor defined w.r.t. the base space variables (w,w). More specifically, 10 Recall the discussion around (3.5), which explains why these two a priori different spaces are effectively identified because of the simplicity of f .

These factors of
which we will do henceforth. This redefinition is concomitant with the fact that, in this coordinate system, We now consider two particular theories, where we have concrete expressions for the stress-energy tensor and U (1) currents.

Free Scalar
As a first check for the proposal, we apply our formalism to free scalar with undeformed action By considering a real compact boson, the above theory possesses a U (1) symmetry consisting of constant shifts in field space with associated Noether current Note that both components of the current are conserved independently, when imposing the equations of motion of the undeformed theory. They could be taken as independent holomorphic and antiholomorphic currents. However, this cease to be true when the deformation is turned on. We could identify this current with either J orJ in our approach. For the sake of simplicity, we consider the case of 2 = 0. This identifies (3.23) with J ν . As first step, we promote the global shift symmetry to local one with a(σ) some function of the base space coordinates. Then the gauged action is invariant under a simultaneous shift (3.24) and the transformation We identify the X a and α fields with the corresponding target space coordinates and gauge transformations respectively. This gives the following Jacobian and gauge matrices: To make the rather abstract ideas of the general procedure as concrete as possible, we will be very explicit in this first example. Since we want to solve for derivatives w.r.t. z,z, we need to remember the chain rule in the gauge matrix above. From the resulting system , we obtain the following solutions ∂z ∂w ∂z ∂w ∂z ∂w ∂z ∂w which have to be plugged into (3.14). This becomes which nicely simplifies to following Lagrangian for the deformed theory which satisfies The factor of √ 2 is correct because of the redefinition of the s in (3.19). Alternatively, had we taken 1 = 0 (J = 0) in (3.42), the result would read now satisfying To see how the joint flow works, consider the gauged action of two scalar fields The same set of steps leads to the jointly deformed action This Lagrangian satisfies both equations (3.33) and (3.35).

Free Scalar + Free Fermion
Let us repeat the JT deformation based on the current (3.23) but now in the presence of a Dirac field (decomposed in terms of its left/right moving components). The Fermion contributes to the deformation only through the stress-energy tensor. The undeformed Lagrangian reads The Jacobian remains that of (3.27). The gauge transformation however receives new contributions from the fermion energy-momentum tensor (3.40) After solving and inserting into the on-shell action, one finds the deformed Lagrangian As a check, this Lagrangian satisfies Equation (3.33) as well.
We conclude by adding that deformations by currents associated to U (1) fermionic phase transformations, as well as combinations of both bosonic and fermionic currents, straightforwardly fit into the presented framework.

Adding TT
We can of course follow the same procedure for the full three-parameter flow including TT . The main reason we turned this third term off in the previous section was for simplicity, and a slight conceptual novelty.
We highlight the limits 1 , 2 → 0 and γ → 0 do not commute. Non-commutativity of the order in which one deforms the original theory was previously reported in [19]. Considering 1 , 2 → 0 for finite γ, the above kernel would give a deformation of the JJ type. The JJ deformation is marginal and can be dealt with more traditional quantum field theoretical tools. In this article, we will also consider the opposite order of limits. That is, to obtain the TT deformation alone, it is clear from (3.14) that we need to take 1 , 2 → 0, γ → ∞, keeping γ 1 2 fixed.
We could repeat the steps of previous section to obtain the Lagrangian for the joint JT , TJ and TT deformed theory. We spare the reader the intermediate steps, and report instead only the starting point and final results. The computations are straightforward if albeit tedious.
The equations for the Jacobian matrix in complex coordinates are modified to while the last two lines of (3.18) remain the same. We turn to the Dirac fermion for concreteness.

Free Fermion
The undeformed Lagrangian for a free Dirac fermion reads This theory contains two U (1) currents which act independently on the ψ and χ fields as a phase transformations: ψ → e −iα ψ and χ → e −iα χ. The complex conjugate spinors transform with opposite phases. The undeformed action coupled to the gauge fields is given by, with associated currents Note that (J R )J L is naturally a (anti-)holomorphic current. In contrast to the example of the compact boson, the currents associated to these symmetries transform as scalars under the coordinate transformation and are left invariant by the U (1) transformations. This in fact simplifies the computations. They in fact preserve this property along the flow. We consider the deformation given by turning on both parameters 1 and 2 to arbitrary values. This identifies J L with J and J R withJ.
Equations (3.42) and (3.14) jointly give L =ψ∂ψ +χ∂χ + 1ψ ψ(χ∂χ) − 2χ χ(ψ∂ψ) − γ 1 2 ψ ∂ψχ∂χ −ψ∂ψχ∂χ (3.46) It is not difficult to check the above Lagrangian satisfies the following joint flow equations for any value of 1 , 2 . It is worth mentioning that, for the deformed theory, both the left and right currents acquire a non-chiral and chiral components respectively,J L = 2ψ ψχχ and J R = − 1ψ ψχχ. However, by the Grassmanian nature of the fermionic fields, both vanish when multiplied by any component of the stress-energy tensor. They thus do not appear in the flow equation. By setting different parameters to zero, we recover multiple cases of interest. For example, setting 2 = 0 we recover the example of the Dirac fermion under the JT deformation, considered in [4]. It was also noted there the J current remained chiral all along the flow. Taking instead γ → 0, one recovers the JT + TJ deformation discussed in section 3.2.

Free Boson
We conclude our classical check of the kernel with one final example. We return to the free massless boson and consider its TT + JT deformation. By choosing J µ to be the current associated to constant shifts of φ, as in (3.23), and settingJ µ = 0 identically, we end up with the following deformed Lagrangian which nicely satisfies the flow equations

The Quantum Partition Function
We now probe our proposed kernel's validity at the fully quantum level. To do so, we wish to explicitly compute the path integral over base space torus geometries and gauge connections: where in the above expression we have imposed (2.7) and rewritten the action after having performed an integration by parts to make manifest the constraints imposed by the Y a and α,α integrals.
In the pure TT case, [8] found their path integral over gravitational degrees of freedom localized to an integral solely over global modes. We find that both the gravitational and the gauge degrees of freedom localize similarly in our case. We follow [8] closely.
Before delving into technical details, let us outline the three main steps in our computation.
• The first is standard: we need to avoid overcounting diffeomorphism and U (1) gauge equivalent configurations. For diffeomorphisms, we accomplish this by writing a general vielbein in terms of a Weyl rescaling, a local Lorentz transformation (an SO(2) rotation in our Euclidean setup) and a diffeomorphism of some fixed reference vielbein: e a (σ) = e Ω(σ) e φ(σ) a bê b ξ . The change of variables from e to Ω, φ and ξ is accompanied by an important Jacobian, the (diffeomorphism) Fadeev-Popov determinant. We can import the result of [8] here. Similarly, we can decompose the gauge field as where the gauge-invariant A H encode the holonomies as The Jacobian for change of variables from A to A H , g and χ gives the other U (1) Fadeev-Popov determinant. The same obviously holds forÃ.
• The second step will be to understand precisely the constraints arising from the integrals over the Y 's and α,α. Beyond their important role in localizing the path integral to minisuperspace, we determine the additional functional determinants they contribute to the path integral.
• Finally, we will need to treat field zero modes with great care. [8] already showed how the zero-mode integral for the Y a gave an important factor of the area of the target space torus. We will find that the range of integration for the analogous holonomies of the gauge fields A,Ã has equally important consequences.

A Note on the Gauge Symmetries
Before we begin the main computation, we need to elucidate a slightly subtle point about the diffeomorphism and U (1) gauge symmetries. We will find that these two gauge symmetries consist of only those transformations connected to the identity. This has an important impact on the range of integration for the moduli and holonomies. First, consider the U (1) gauge symmetry of A, whose transformations are given by (2.3), reproduced here for readability, δA µ = ∂ µ g, δα = g. The important thing is that the symmetry depends on the ability to absorb the gauge transformation into α. We stress this point because α is single-valued on the torus. Therefore, this is a symmetry iff g is also single-valued. Now, consider the gauge field configuration In usual U (1) gauge theories, since the gauge group is compact, the condition on the gauge transformations is e i q 2π g(s) = e i q 2π g(0) . (4.6) Because of this, g itself need not be single-valued, With this condition, the gauge field configuration in (4.5) is gauge-equivalent to 0, and therefore the holonomies are compact -as they should be, given the compactness of the gauge group. However, in our case, we have In other words, the restriction to gauge transformations connected to the identity cause the holonomies to be valued not in the group itself but in its universal cover -which is R for U (1). The story for the other U (1) is of course identical. The analogous restriction on the diffeomorphisms imply the moduli τ are valued not in the fundamental domain but in the entire upper half plane; see [8] for a more detailed discussion.

Path integral measures and Fadeev-Popov Determinants
We finally begin our computation, focusing on relevant path integral "measures". 11 Our discussion parallels Polchinski's original computation of the Polyakov string torus path integral [32].
For an n-dimensional manifold with coordinates x i , the invariant measure is in other words, the measure depends on g ij , which in turn can be defined through the inner product on small variations of the coordinates. In infinite dimensions, we cannot be so explicit. Rather, we define it it implicitly via fixing the value of a Gaussian path integral. The first step therefore requires defining an inner product on infinitesimal variations of the fields living in the tangent space at a given point on the field space manifold. We choose (δα, δα) e = s −2 δα ∧ * δα = s −2 (det e)δαδαd 2 σ . (4.10) The factors of s have been arbitrarily inserted so that the 'field-space metric' is dimensionless. In other words, it is an arbitrary length scale chosen to cancel the factors of length arising from the integration measure d 2 σ. These inner-products are diffeomorphism invariant and depend on the base space metric variables in the form of the vielbeins e a µ used to raise and lower indices. In particular, this implies that the path integral measure for the vielbeins will be non-linear. We have not explicitly written the inner products forÃ andα, since they are the same as the last two equations in (4.10).
Denoting by Ψ whatever field we are interested in, we then implicitly define the measure at a given point in field space by requiring To compare with the familiar finite dimensional case, this would give: which would define the measure Dx = det(g(x)) π n d n x In the infinite-dimensional case, we can absorb factors such as π via local counterterms, so those will not be of much importance [32]. 12 This might all seem overkill at first sight. In fact, it greatly simplifies the the computation of the Fadeev-Popov determinants and informs our treatment of the zero-modes.

Measures and Fadeev-Popov Determinant for the U (1) gauge fields
In this subsection, we will calculate the Faddeev-Popov determinants needed to address the gaugeinvariance of the path integral.
The Hodge decomposition theorem guarantees we can decompose any one-form A − B as Here, we are taking g, χ to contain no constant pieces (zero-modes), since those would not contribute to A. All zero-modes of A are contained in A H . These encode the holonomies and can be written in terms of them as  with the various measures defined using the inner products appearing in (4.16). The primes denote exclusion of zero-modes. Note the dependence on the base space vielbeins via the functional determinant of the Laplacian δ ab e µ a e ν b ∇ µ ∇ ν = g in (4.13) really paramterizes the pure-gauge direction in A. We have already shown the kernel is gauge-invariant. Hence, it does not depend on g. We therefore can pull it out of the rest of the integral and need simply evaluate the ratio Dg vol(G) (4.18) This ratio is not quite unity, because of the exclusion of zero modes in the numerator. Indeed, vol(G) = Dg = Dḡ Dg where we split up a general group element g into a sum of zero-and non-zero-mode pieces, g =ḡ + g , dḡ = 0, (δg , δḡ) = 0, (4.19) the zero-mode pieceḡ is compact because of the compactness of the group U (1) (4.20) and the inner product decomposes nicely as (δg, δg) = (δḡ, δḡ) + (δg , δg ). which shows the zero mode and non-zero mode pieces are orthogonal realtive to the inner product. Using these two facts we have that

22)
A denotes, as in [8] the proper area of the base space torus. q is the unit of fundamental U (1) charge and serves as the inverse radius of the U (1) ∼ = S 1 . This gives us the explicit ratio: Of course, all the above steps are identical for the second gauge fieldÃ and the quotient by vol(G).

Measures and Fadeev-Popov Determinant for the vielbeins
This section is short. All the hard work has already been done in [8] and we can straightforwardly import their results.
To be precise, recall that any vielbein e on the torus may be written as where (. . . ) ξ means a finite diffeomorphism generated by the vector field ξ and the canonical unittorus veilbeinsê a are given byê The decomposition of the measure in this case is more involved. In fact, we will only need the Jacobian satisfying the constraints imposed by the Lagrange multipliers. On that constraint surface, we will see all non-zero modes of the vielbeins vanish. We quote the answer [8] found for later convenience: Note that in this decomposition we have also excluded ξ zero-modes, as these do not change the vielbein and would render our paramterization redundant (see [32] for more details). Since our kernel respects base space diffeomorphism invariance, the Dξ similiarly decouples. As for the U (1)'s , we again need to be careful about the ratio Dξ /vol(diff), which was also found in [8] This completes our necessary list of ingredients to proceed to the constraint integrals.

The Constraint Integrals
The next step is to perform the integrals over the compensator fields α,α, Y a . From (4.1), all three of these clearly impose δ functions. We need to evaluate the additional functional determinant prefactors they contribute to the path integral. We begin with the α andα integrals, rewriting the relevant part of their action in terms of their associated innerproduct: Integration by parts shows only the non-zero modes of α andα contribute to exponent. Indeed, we can mimick our treatment of vol(G) = Dg and split α into its zero mode and othogonal non-zero mode piece α =ᾱ + α (and similarly forα). Further remember that theᾱ,ᾱ ∈ U (1) are compact. The integrals in (4.28) thus become where the reminds us of the exclusion of zero-modes. In all these expressions, we should really be writing the vielbeins in terms of Ω, φ andê a (diff. invariance tells us we can ignore ξ), but have avoided doing so to avoid cluttering the notation even further. We now turn to the Y a integrals. The action for Y reads where we have again decomposed Y a =Ȳ a + Y a into its zero-and non-zero mode contributions. Using the approriate measures for the Y , the integral becomes (4.31) Similarly, these equations should be understood for dA = d dχ.
These three δ functions are somewhat cumbersome to work with. They involve different components and combinations of the gauge fields and vielbeins. We can tease these apart using Note that this is a 4 × 4 matrix. This means that the delta functions can be rewritten, using the defining properties of the ns, as where the factor of 4 is the determinant of the 2 × 2 matrix (ñ a , n a ).

Final Answer
Doing the constraint integrals exposed the inner workings of the path integral's localization to zeromodes. We need only two more ingredients before we can put it all together. First, since the global scale and rotation are not fixed, we need the measure for them. Denoting the zero-modes of Ω, φ asΩ,φ, the relevant part of the measure is Secondly, we need to deal with the det of a constant in (4.33). For this, we use the fact that the det of a constant is merely an addition to the cosmological constant and can therefore be absorbed into a choice of counterterm in Z 0 , so that we can write The measure for the holonomies, using (4.16) and the localisation to constant vielbeins, reads Finally, we parametrize the vielbeins in terms of the length vectors of the two cycles as which in turn can be parametrized in the following waȳ and analogously for the target variables. It is easy to check that We may now plug these things into the full path integral. We spare the reader the details. Instead, we simply note a few important cancellations: 1. The integral over the zero-modes of α,α, Y exactly cancel the parts the original volumes of gauge groups that failed to cancel in (4.23).
2. The scalar Laplacian determinants that arise from the gauge-fixing cancel those from the delta-functions.
After what have admittedly been many steps, the full partition function simplifies to which is s-independent and dimensionless, as it should be. This is one of the cornerstone results of this paper.

The Deformed Spectrum
To complete this non-trivial check of our proposal, we perform the resulting finite dimensional integral. We obtain the explicit form of the deformed spectrum, for different values of the deformation parameters and charges. The energy levels so obtained precisely match those in the literature, found by very different methods. We also briefly discuss the saddle point approximation to these integrals and their one-loop exactness.

The JT case (Q = 0)
We will work in complex coordinates (z,z). Our conventions may be found in the Appendix. In terms of these complex variables, the integral we wish to compute reads (4.46) Note all moduli dependence appears only in the first term of the kernel with our choice of parametrizations in (5.6).
To explicitly evaluate (4.46), we need to address the definition of Z 0 , the seed partition function. First off, we restrict our attention to seed CFTs, and hope to explore the more general QFT setting in future work. Secondly, we focus on the case where the undeformed theory has a single U (1) current coupled to h µ . Following [20], [21], we take the partition function Z 0 as the one defined via a path integral with appropriate counterterms. 13 Its dependence on hz is essentially fixed by modular invariance. More precisely, adopting the conventions of [20], we take  For simplicity, let us consider the target space configuration in wich L z = i R 2 = −Lz (i.e. L 2 = 0) along with vanishing fluxes b µ =b µ = 0. We also choose to solve the third constraint for the τ 1 variable, that isL (4.53) Putting it all together, we are left with the following integral where we have used A = R 2 τ 2 in order to cancel the prefator of the exponential. The main outcome of this computation is the right-moving deformed energy E (R) n which reads The square root branch has been chosen such that it satisfies the correct innitial condition. The above spectrum satisfies which is the analog (in our conventions) of equation (6.20) in [14]. It is satisfying to see the spectrum obtained here matches the one previously reported in the literature. For future reference, let us point out the k = 0 case of this equation can readily be solved. The h,h dependence is linear. Integration leads to a set of four constraints. Their solution localizes the base space modular parameters to the following locus which, once plugged into the partition function, leads to the simple spectrum found in [4] The last equation introduces the notion of an effective (state-dependent) radius given by R ef f = R − 1 πQ, previously described in [4]. A similar structure arises for k = 0, as explained further below.

One-loop exactness and the chiral charge
[8] found the integral giving rise to the TT deformed torus partition function to be one-loop exact. That implies, in particular, the exact spectrum can be consistently obtained from its saddle point approximation.
We find that our (4.55) shares this remarkable one-loop exactness property. To be clear, this means it is completely dominated by its saddle point value and the determinant for quadratic fluctuations around the saddle. It thus behaves like a Gaussian integral, even though it certainly is not. This is yet another diagnostic of the flow's solubility, as seen from our path integral perspective. The saddle point approximation to the integral clarifies some important conceptual puzzles and makes contact with previous discussions in the literature. In particular, it immediately singles out a set of equations which, within a completely different approach, led to spectrum just found (4.58).
In order to proceed, let us take (4.55) which is admittedly Gaussian in hz but not in τ 2 . For sake of the argument, we include a putative "1/ " parameter in the action, which we later set back to 1: We compute the saddle point equations for τ 2 and hz respectively It remains to evaluate the fluctuation determinant thus canceling the prefactor in (4.62). The one-loop approximation thus recovers the exact expression in (4.57). The semiclassical spectrum is there actually the exact one (4.58). We may gain intuition by identifying hz| saddlem = −2π 1 E (R) n (4.66) By further making the following definition, the second saddle point equation (4.64) takes on a very suggestive form which is to be compared with the k = 0 solution for τ 2 (4.60). Combining (4.67) and (4.63) we also get Requiring all quantities depend on the dimensionless ratio 1 /R, we can recast the above expressions as two differential equations 14 We thus recover the defining equations for the energy levels. These differential equations were solved in [13], [14]. The solution for the energy is none other than (4.58) (consistent with one-loop exactness found above), together with (4.72) In order to better understand the role of Q, let us briefly comment further on equations (4.70) and (4.71). In the TT scenario, solving a single differential equation leads to the dressed energy levels.
In [1,2], that equation results from the factorization property of the deforming operator. Along with rotational invariance, the flow can be written purely in terms of the energy and momentum of a given state. Without rotational invariance, additional assumptions would be needed [6]. Even though the JT operator still nicely factorizes, there is no way of getting a sensible differential equation for the energy levels without impossing an additional constraint: the deforming current needs to be chiral. Otherwise, the flow equation involves the expectation value of its spatial component, which usually is not quantized. Requiring chirality allowed [4] to solve for the spectrum, which we recovered here for k = 0 in (4.61).
Requiring the flow to preserve chirality is in many ways too strong a constraint on the set of possible trajectories. In the more general case, defining the charge Q associated with the chiral projection of the deforming current circumvents this issue [13,14]. This identification in some sense "emulates" the chiral case. It is therefore not surprising the effective radius R ef f (4.68) depends on Q in the same way as for k = 0. The price one has to pay is having to solve two differential equations instead of one, namely (4.70), (4.71). These equations arise naturally from our approach.
The role of k along the flow thus becomes clearer. Physically, when k = 0, the chiral current develops an anti-holomorphic part, spoiling chirality at the quantum level. As k → 0, the second equation (4.71) becomes trivial, and we recover the case studied in [4]. In our path integral approach, this phenomenon is manifested by the h 2 term in the seed partition function. Without it, the integration over holonomies would lead to a simple constraint over the geometric variables. It would not appear in an additional saddle point equation.
Finally, here we show a different approach within these quantities arise naturally. So far, we have been focused on the spectrum of the deformed theory. The background fluxes b,b played no role in the previous discussion. For sake of completeness, we write down the result of the finite dimensional integral in presence of non-trivial target space holonomies. Again, we considerQ = 0, so theb fluxes still play no role. They can be absorved in theh integration variable. In fact, the same applies to the holomorphic component of b. We thus only need consider non-trivial anti-holomorphic bz. The intermediate steps being rather unenlightening, we simply report the result Z b = n e 2πiτ 1 n−2πτ 2 (2RF b,n +n) (4.73) 14 Note all the relations obatined here completely match the ones listed in [14] by taking µ there = 2π 1 and k = 1/4.
0,n + bz(bzk − Q)) (4.74) The deformed theory is not conformally invariant. We are thus not able to fix the dependence on the holonomies for the torus partition function. (4.48) no longer serves as point of comparison. Therefore, the actual physical meaning of the function F b,n multiplying τ 2 is unclear. In particular, it cannot be identified with an energy level.
However, some physical intuition can be gained by expanding the result for small values of bz. This gives n , R ef f and Q given by (4.58), (4.68) and (4.72) respectively.

General JT + TJ deformation
Consider finally the general case with non-trivial charged states for both J andJ. Denote these charges Q andQ respectively. As this computation closely parallels that of the above section, we will be brief. The seed partition function becomes Our choice of parametrization here will be different from (5.6), namely 15 n a L a z = iτ L z ,ñ a L ā z = iLz (4.77) n a L a z = iL z , n a L ā z = iτ Lz (4.78) Our kernel's action (again with b =b = 0) now reads To do the integrals, it is convenient to shift hz → hz +h z . The h z ,h z ,hz integrations now impose the following constraintsL 15 Note we are not including the 1 √ 2 factors here, as they are again absorbed by rescaling the couplings.
The prefactor arising from the delta-functions imposing the above contraints is Finally, performing the remaining hz and τ 2 integrals, we arrive at with the right-moving deformed energy quoted early on (1.11), which reads The branch of the square root has again been chosen so the deformed energy satisfy the correct initial conditions. The above spectrum precisely matches the one found in [16], [17]

Joint flow with TT
Studying the joint flow of JT and TT provides our last application of the path integral representation of the deformed partition function. This amounts to taking γ = 0 in the kernel. We consider the general k andQ = 0 case. Using the parametrization in (5.6), almost identical manipulations as those described so far lead to the kernel Integrating overh z andhz gives rise to two delta functions. They allow us to immediately perform the h z and hz integrals which set There is a factor 4γ −2 comming from the delta functions. On the above solution locus, the kernel is still linear inL z . We can thus further integrate over this variable. It localizes theLz integral tō along with a 1 2 16γτ 2 prefactor. 16 In order to check the agreement with the results listed in [16] one should take qR = −Q, k = 1/4 together with the following relation between parameterŝ By defining λ = γ 1 2 , and shifting τ 1 → τ 1 + τ 1 , we arrive at Z = n A 4πλ d 2 τ (τ 2 ) 2 e R(τ 2 +iτ )(R(2λ(τ 2 −iτ )−πk 2 1 (τ 2 +iτ ))+4πQ 1 λτ 2 ) 8λ 2 τ 2 +2πin(τ 1 +τ 1 )−2πτ 2 ε 0,n (4.92) We can again perform this integral over base space torus moduli and extract the desired spectrum, giving E n = 1 2πλ R − π 1 Q2π 2 k 2 1 P n (4.93) − (R − π 1 Q) 2 − 2πλ(2ε 0,n − 4πλP 2 n ) + 4π 2 1 P n (k 1 R + 2λ(Q − πk 1 P n )) (4.94) where we have reintroduced the momentum P n = n/R and defined the coupling λ = λ + πk 2 1 (4.95) In particular, by taking the 1 → 0 limit keeping γ 1 2 fixed, the expression above reduces to the standard formula for the TT dressed energy levels displaying the usual Hagedorn behaviour for negative λ. Note the presence of such a behaviour is dictated by the sign ofλ given in (4.95). In particular, when both couplings are present and λ < 0, we find a crossover point at 1 = 2|λ|/πk. The ability of JT deformation to "remove" the Hagedorn regime has been discussed in [16,17].

Conclusion and future directions
This work presented a path integral realization of Lorentz breaking irrelevant JT and TJ deformations. We have recast their joint flow with TT in (1.7) as coupling the seed to a topological quantum gravity and gauge theory. As it was for its pure TT predecessor, our path integral kernel fundamentally translates between a base space, where the undeformed theory lives, and a target space on which the deformed theory is defined. The path integral is an integral over a restricted set of maps between the spacetime and U (1) frames of the base space -the vielbeins e a and gauge connections A,Ã -to target space ones denoted by f a , B andB. The compensator fields Y a and α,α make diffeomorphism and gauge invariance of the kernel manifest. They further serve to implement important constraints on the path integral, which ultimately make it soluble. Our proposal succesfully passed a wide variety of non-trivial checks. At the classical level, the kernel recovers the exact classical deformed actions. The procedure involves nothing more than solving an algebraic system of equations and reproduces the known expressions for the free boson and fermion. At the quantum level, we reduced the full path integral over base space torus geometries and gauge connections to a finite dimensional one. It solves the desired difussion-like equation. Most importantly, by explicitly evaluating the torus partition function for certain seed theories, we extracted the deformed spectrum along particular flows triggered by combinations of JT , TJ and TT . Our results all matched the known expressions in the literature.
In many ways, this path integral construction puts the JT and TJ deformations on similiar footing to TT 's . However, it also brings with it a plethora of new questions. First off, we saw a proper treatment of target space diffeomorphism invariance required df a = dB = dB = 0. Ultimately, we hope to engineer a kernel that generalizes away from flat vielbeins and vanishing U (1) field strengths. In [31], we will report on progress on defining a TT deformation for curved spacetimes with df a = 0. This encourages us to seek, in future work, a kernel that also accommodates dB, dB = 0. Furthermore, the one-loop exactness properties first found in [7] led the authors to conjecture the TT deformation of a general QFT might be captured as coupling to a form of 2d gravity. The one-loop exactness discovered for our kernel gives us hopes to similarly extend our formalism beyond CFT seeds. However, the expression of the seed partition function as a sum appears more involved in this case.
Finally, [25] engineered a modified TT flow (adding the "Λ 2 flow") to derive a holographic field theory dual to de-Sitter bulk geometries. We are quite curious what our more general flows, including JT and TJ, might have to say about these dS/dS holographic constructions and their novel bulk reconstruction features [30].
We close by admitting the partition function captures only a small portion of the physics lurking in the deformed theory. One of the most pressing issues to address is the structure of correlation functions. First steps have been taken in [22,23]. We are exploring the introduction of additional sources in our path integral kernel to open up another front of attack on this important problem.

Note Added
It has been brought to our attention that similar results have been independently obtained by Monica Guica and Tarek Anous. We understand their conclusions will appear shortly in a forthcoming publication. We thank them for kind correspondence on this matter. Furthermore, the torus lenghts and holonomies are L z = 1 2 (iL 1 + L 2 ) , Lz = 1 2 (−iL 1 + L 2 ) (5.2) and similarly forL andh. So the integration measure becomes d 2L d 2 hd 2h = 2 3 dL z dLzdh z dhzdh z dhz . Finally, we make the following choice for tangent space indices This is a choice of tangent space orientation; the tangent space metric is whatever it needs to be for these two vectors to be light-like. For convenience, we also redefined our couplings in section 4.3