On $T\bar{T}$ deformations and supersymmetry

We investigate the"$T\bar{T}$"deformations of two-dimensional supersymmetric quantum field theories. More precisely, we show that, by using the conservation equations for the supercurrent multiplet, the $T\bar{T}$ deforming operator can be constructed as a supersymmetric descendant. Here we focus on $\mathcal{N}=(1,0)$ and $\mathcal{N}=(1,1)$ supersymmetry. As an example, we analyse in detail the $T\bar{T}$ deformation of a free $\mathcal{N}=(1,0)$ supersymmetric action. We also argue that the link between $T\bar{T}$ and string theory can be extended to superstrings: by analysing the light-cone gauge fixing for superstrings in flat space, we show the correspondence of the string action to the $T\bar{T}$ deformation of a free theory of eight $\mathcal{N}=(1,1)$ scalar multiplets on the nose. We comment on how these constructions relate to the geometrical interpretations of $T\bar{T}$ deformations that have recently been discussed in the literature.


Introduction
The deformations arising from the composite "TT " operator [1], constructed out of the (Hilbert) stress-energy tensor, have recently attracted much attention. These irrelevant deformations are solvable, in the sense that they affect the theory's energy spectrum in a simple way. For a state |n of energy H n and zero momentum, a TT deformation of parameter α acts as where O = det[T ab ] is the TT composite operator and R is the volume of the theory. This means that the energy of a state |n in the α-deformed theory in volume R 0 is the same as that of the same state in the undeformed theory, but in volume R = R 0 + αH n . For integrable theories, such a deformation is identified with a "CDD factor" [2] in the factorised S matrix [3,4]. This CDD factor appears to be closely related to (effective) string theories on flat space [4][5][6][7][8] and stringy Wess-Zumino-Witten models [9][10][11], and more generally with string theories in uniform light-cone gauge [9,[12][13][14]. 1 The holographic interpretation of such deformations has also been directly investigated [16][17][18][19][20], as well as their relation with two-dimensional (Jackiw-Teitelboim [21,22]) gravitational theories [23][24][25][26].
One aspect that has been perhaps overlooked so far is how TT deformations affect a supersymmetric theory. This is a rather natural question given that two-dimensional supersymmetric theories are of great interest, both in their own right and in the context of string theory and holography. In this short note we aim to address this question.
If we think of the deformation as arising from a CDD factor that yields the differential equation (1.1), it is natural to conclude that it should behave well with respect to supersymmetry. States related to each other by the action of supersymmetry will have the same energy and momentum. As long as the flow of eq. (1.1) is non-singular, such degeneracies should be preserved. This seems to imply that a supersymmetric theory would generically remain so after a TT deformation.
However, to have complete control over the deformed theory-rather than on its spectrum only-it is important to construct the TT operator, which we will simply denote in the paper as O = det[T ab ], and determine how it behaves with respect to supersymmetry. It is easy to see that, even for a very simple (free) theory, det[T ab ] is not invariant under the supersymmetry variations of the undeformed theory. In this paper we will argue that it is possible to define a deformed TT operator that is supersymmetric with respect to the supersymmetries of the undeformed theory and is equivalent to O, up to total derivatives and terms that vanish when using the conservation equations of the supercurrents (which hold on the mass shell).
More specifically, we will consider N = (1, 0) and N = (1, 1) theories, and show that O can be constructed as a supersymmetric descendant of an appropriate composite operator. For instance in the N = (1, 0) case we have

2)
up to terms that vanish on-shell, where Q + is the supercharge and O − (σ) a suitable fermionic operator. This is sufficient to guarantee that so that the deformed action remains supersymmetric. The fact that eq. (1.2) is corrected by terms that vanish on-shell is inconsequential. Indeed we will argue that such terms do not affect expectation values, so that the two operators on either side of eq. (1.2) can be defined by point splitting, and will generate the same exactly solvable flow as eq. (1.1). We derive eq. (1.2) and discuss these ideas for N = (1, 0) and N = (1, 1) supersymmetry in section 2. In section 3 we provide a concrete example by discussing in detail the deformation of the free N = (1, 0) supersymmetric action for a scalar multiplet. A particularly interesting feature of TT deformations is that they appear to be naturally related to string theory. This was first observed for bosonic strings in flat space [4][5][6][7][8]. In ref. [9] it was argued that such a relation is more general, and should affect any string theory quantised in uniform light-cone gauge [12][13][14], see also [15]. Indeed, in ref. [9] such relation was exploited to study the integrable structure of AdS 3 string backgrounds with Neveu-Schwarz-Neveu-Schwarz background fluxes. 2 In section 4 we argue that this relation holds also for supersymmetric string theories, and we show explicitly that this is the case for Green-Schwarz superstrings in flat space. In addition, we comment on how our construction, which is based on "uniform" light-cone gauge [12][13][14], relates to other approaches to the TT deformation. We conclude with a brief discussion in section 5, and collect a number of technical results in the appendices.
Note added. The pre-print [43], which appeared on the arXiv shortly after this pre-print, also addresses the relation between TT deformations and supersymmetry. The results presented there are compatible with the discussion of our sections 2 and 3.

The TT operator as a supersymmetric descendant
In this section we will provide some general arguments showing that the TT operator of theories possessing two-dimensional (2D) supersymmetry is itself supersymmetric. Specifically, we will see that it is the supersymmetric descendant of a composite operator constructed out of elements of the supercurrent multiplet.
We will restrict our analysis to the case of two-dimensional N = (1, 0) and N = (1, 1) supersymmetry. Moreover, we will assume that the supersymmetric QFTs in consideration are Lorentz invariant and can be coupled to two-dimensional off-shell supergravity, and to simplify our analysis we shall work in superspace. The study of 2D (1, 0) and (1, 1) supergravity in superspace was largely developed in the 1980s and we refer the reader to the following works and references therein for details [28][29][30][31][32][33][34][35][36]. Under our assumptions, the description of the supercurrent multiplet will simplify: the supercurrent and trace multiplet will coincide with the variational derivatives of the supersymmetric matter action with respect to the supergravity prepotentials, evaluated on a Minkowski superspace background (see [37,38] for general reviews on the construction of supercurrents in superspace along this line). This is the supergravity analogue to the definition of the Hilbert stress-energy tensor as the functional derivative of a matter theory minimally coupled to a background metric (vielbein). We leave for future work the extension of our arguments to more general supercurrent multiplets derived in the spirit of the analysis of, e.g. refs. [39][40][41][42].

The
We start by introducing the N = (1, 0) superspace and the structure of the supercurrent multiplet. In light-cone coordinates, a flat 2D N = (1, 0) superspace is parametrised by (see appendix A.1 for our definition of σ ±± ) with ϑ + a real Grassmann coordinate. The spinor covariant derivatives and supercharges are given by and obey the anti-commutation relation Given a (1, 0) superfield F(ζ) = F(σ, ϑ + ) its supersymmetry transformations are given by Note also that if F (σ) is the operator defined as the ϑ = 0 component of the superfield F(ζ), F (σ) := F(σ, ϑ + )| ϑ=0 , then its supersymmetry transformations are such that (2.5) We will indicate by Q + the supersymmetry generator acting on a component operator and distinguish it from Q + , which is a linear operator acting on superfields. The description of general N = (1, 0) supergravity-matter systems in superspace has been developed in full detail three decades ago. See [30][31][32][33][34][35][36] for an (incomplete) list of references. We will restrict our attention to a Lorentz invariant Lagrangian matter system coupled to linearised off-shell Poincaré supergravity. In this case, linearised supergravity can be described by three unconstrained prepotential superfields: the last one being a Lorentz scalar. Their linearised supergravity transformations are where K ±± (ζ) and K D (ζ) are unconstrained and parametrise linearised superdiffeomorphisms and super-Weyl transformations, respectively. The supergravity multiplet described here is equivalent to N = (1, 0) conformal supergravity coupled to a scalar conformal compensator multiplet. The component fields of the prepotentials are where h ++++ and h −−−− being the traceless components of the linearised metric while h is its trace, ψ −−− and ψ + are the gravitini while the field ρ +++ is pure gauge. Now consider a general Lorentz invariant matter system coupled to N = (1, 0) supergravity. Its action expanded to first order in the supergravity prepotential is Assuming that the equations of motion for the matter are satisfied, after some integrations by parts the variation of the action under arbitrary superdiffeomorphism transformations takes the form Imposing that the action is invariant then leads to the following supercurrent conservation equations: The on-shell variation of the action (2.9) under arbitrary super-Weyl transformations is If the matter system is superconformal, which will not be our main interest in our paper, this variation should vanish for arbitrary K D , which implies that are the components of the Q + -supersymmetry current. By using (2.11), it is straightforward to show that these operators satisfy the correct conservation equations (2.15)

The TT operator
Now that we have described the structure of the supercurrent multiplet, we are ready to show that the TT operator, which takes the form is a supersymmetric descendant. Defining the superfields T and T ++++ as and using eq. (2.11), the superfield is such that

(2.19)
Using the conservation equations (2.11), the previous expression becomes (2.20) This implies that, up to total derivatives, the following equality holds where we have defined In other words, O − (σ) is the supersymmetric primary field of the multiplet containing the operator O(σ). As we anticipated in the introduction, this shows that the TT deformation is manifestly supersymmetric, since It is also rather interesting to write down the deformed action, with deformation parameter α, in terms of the original action S 0 ≡ S of eq. (2.9) and of the operator O(σ) defined above: where the ellipses refer to higher order terms in α. In other words, the TT deformation can be reabsorbed in a shift of the prepotential superfields; this will be the case also for N = (1, 1) theories, as we will see below. This points towards a geometric interpretation of such a deformation in terms of the supergravity, reminiscent of the observations of refs. [23,25,26], which would be interesting to explore further.

Point splitting
The composite operator defined in eq. (2.18) is well-defined via point splitting, as proven in ref. [3]. Indeed, consider the point-split superfield operator By expanding it in powers of the fermionic coordinates ϑ and ϑ , it is straightforward to check that each component is of the form where s and s label the spins of the operators and As shown in ref. [3], bilocal operators like (2.26) are free of short-distance (non-derivative) divergences. Moreover, their expectation value is independent of the separation σ − σ . Consequently, the limit ζ → ζ in (2.25) defines a composite superfield operator that is unique up to total derivative terms. In particular, the integrated operator is well-defined and manifestly preserves supersymmetry. We also notice that the operator above is identical to the usual TT deformation only up to terms that vanish upon using the supercurrent conservation equations that hold onshell. This however is of no consequence when we consider expectation values of (2.28) in arbitrary states, since the equations of motion are valid inside correlation functions up to contact terms. In more detail, the relevant descendant of (2.25) reads (2.29) When we take its expectation value, Ward identities imply that the second and third line vanish up to contact terms of the form δ 2 (σ − σ ). However, since the expectation value is independent of the separation σ − σ , these contact terms do not contribute. The third and fourth lines are total derivatives and do not contribute either in states with well-defined energy and momentum.

The N = (1, 1) case
The N = (1, 1) case is an obvious generalisation of the previous case. We refer the reader to [29] for 2D N = (1, 1) off-shell supergravity and its description in terms of unconstrained prepotentials. Compared to the (1, 0) case of the previous subsections, superspace is now parametrised by an additional fermionic coordinate, so that ζ M = (σ ++ , σ −− , ϑ + , ϑ − ). The covariant derivatives and supercharges are defined as (2.30) and the anticommutators read For a Lorentz invariant Lagrangian matter system, N = (1, 1) supersymmetry implies the existence of two pairs of superfields, (J +++ (ζ), J − (ζ)) and (J −−− (ζ), J + (ζ)), which encode the supercurrents and the stress-energy tensor. These two pairs describe respectively a (1, 0) and (0, 1) supercurrent multiplet. By assuming that our matter system can be minimally coupled to off-shell (1, 1) conformal supergravity together with an unconstrained scalar compensator, it can be shown that J ± (ζ) are expressed in terms of a real scalar current J (ζ) as In fact, the N = (1, 1) unconstrained prepotential superfields are: and possess the following linearised supergravity transformation rules (see [29] for more detail) (2.34) Here K ±± (ζ) and K D (ζ) are unconstrained and parametrise linearised (1, 1) superdiffeomorphisms and super-Weyl transformations, respectively. The supergravity prepotentials comprise various pure gauge components together with the following fields where h ++++ and h −−−− are the traceless components of the linearised metric while h is its trace, ψ ±±± and ψ ± are the gravitini. Now consider a general Lorentz invariant matter system coupled to N = (1, 1) supergravity. Its action expanded to first order in the supergravity prepotential is Assuming that the equations of motion for the matter are satisfied, by imposing the invariance of the previous action under arbitrary linearised (1, 1) superdiffeomorphisms transformations (2.34) it is straightforward to obtain the following conservation equations for the associated (1, 1) supercurrent multiplet If the matter system is superconformal, the invariance of the action (2.36) under arbitrary super-Weyl implies the extra condition It is clear that J ±±± and J belong to a supercurrent multiplet where J plays the role of the supertrace. The components of the stress-energy tensor can be defined as and, due to (2.37), they satisfy Following the same arguments of the N = (1, 0) case, it is easy to prove that the composite superfield is well-defined up to terms that vanish upon using the conservation equations and up to total derivative terms. Moreover, it is straightforward to show that This shows that the TT operator can be constructed as a supersymmetric descendant of an N = (1, 1) supersymmetric multiplet, so it preserves the full N = (1, 1) supersymmetry algebra. Just like in the N = (1, 0) case, it is interesting to write down the deformed action, with deformation parameter α, in terms of the original action S 0 ≡ S of eq. (2.36) and of the operator O(σ) defined above. At first order in α we obtain: In order to illustrate the ideas discussed above, let us consider in some detail the simplest supersymmetric setup, which is a two-dimensional theory with N = (1, 0) supersymmetry. The free action for a N = (1, 0) real scalar multiplet is This is invariant under the supersymmetry transformations It is important to note that the previous supersymmetry transformations close off-shell. This is a feature that the (1, 0) case makes particularly simple compared to higher supersymmetric cases where one might have to introduce auxiliary fields to close supersymmetry off-shell. The previous real scalar multiplet is equivalently described in N = (1, 0) superspace by a real unconstrained superfield ϕ(σ, ϑ + ) such that 3) The free action (3.1) is equivalently written in superspace as We want to consider a TT deformed action S α with parameter α such that where the stress-energy tensor is computed out of S α itself. There are several ways to do this. For instance, one could construct the deformation order by order in α, following ref. [4]. At each given order, the stress-energy tensor will be a polynomial in ∂ ±± X and ψ + ∂ ±± ψ + . This, together with the Grassmannian nature of ψ + , leads to the simple ansatz where we have accounted for the fact that α has dimension [α] = (−1, −1) and introduced the dimensionless combination Imposing (3.5) in terms of the Noether stress-energy tensors then leads to a system of ordinary differential equations for A(x), B(x) and C(x), much like in ref. [44]. Relegating the details to appendix B we have which we want to solve with initial conditions at small x The differential equation for A(x) is solved by a bosonic action of Nambu-Goto form; solving the remaining equations we get It is easy to verify that, using the supersymmetry variation (3.2), whenever α = 0. In fact, it is not difficult to construct the unique Lagrangian with bosonic part L bos,α = A(x)/α which is invariant under (3.2). As we show in appendix B, imposing supersymmetry leads to a very simple differential equation, which for our bosonic Lagrangian A(x) is solved by It is also simple to show that in superspace the resulting manifestly supersymmetric action takes the form (3.14) In fact, such an action can be straightforwardly generalised to a manifestly N = (1, 1) supersymmetric action: where Φ is an unconstrained real superfield describing the off-shell N = (1, 1) scalar multiplet, Φ = X −iϑ + ψ + −iϑ − ψ − +iϑ + ϑ − F . Notice that this depends on the real auxiliary field F (σ). We will consider in more detail N = (1, 1) theories in the next section restricting to the case where F (σ) is integrated out and N = (1, 1) supersymmetry is realised on-shell. Let us comment on our results. Note that, the difference ∆ between the two actions, S α and S susy , amounts to the difference between the TT operator constructed out of the Noether tensor and the supersymmetric descendant of the operator O − considered in section 2, cf. eq. (2.18). Hence we expect ∆ to vanish on-shell, which is easy to verify. Indeed, the equations of motion for the fermion ψ + are identical for S α and S susy , essentially due to the ratio C(x)/B(x) being the same in the two cases, and take the form From this fact it immediately follows that the fermionic pieces of both S α and S susy vanish when imposing the same equation of motion for ψ + . Hence, the difference ∆ vanishes when we impose eq. (3.16). One might wonder whether our conclusion is special to the Noether stress-energy tensor, or whether it holds more generally. We start by observing that for the case at hand the Hilbert stress-energy tensor takes the same form as the Noether one; technically, this is because the spin-connection drops out for the real fermion ψ + . Nonetheless, it is still possible to define an improved stress-energy tensor, i.e. one that is manifestly symmetric (and, for the free theory, traceless). In that case too we find that the resulting TT -action can be expressed in terms of coefficients A(x), B(x), C(x) as Despite the rather different form, this action once again yields the same equation of motion for ψ + , so that again the deformations are equivalent on-shell. Note that, though only S susy is invariant under the supersymmetry transformations of the free theory, eq. (3.2), both S α and S impr α can be shown to be invariant under a deformed set of supersymmetry transformations. This is consistent with the results of the previous section that point out that only deformations based on O − will manifestly preserve supersymmetry while other deformations will do so only up to terms vanishing on-shell.

TT deformations and superstring theory
While in the case of a free N = (1, 0) theory we could explicitly construct a TT deformation for finite α (following the strategy of ref. [44]), doing so for a generic theory becomes rather cumbersome. We will see in this section how to exploit a link with strings in light-cone gauge, first highlighted in ref. [9], to construct the TT deformation of more general theories; in particular, we will construct the TT deformation of the free theory of eight on-shell N = (1, 1) scalar multiplets, which emerges from superstrings on R 1,9 .

Uniform light-cone gauge and TT deformations
To begin with, let us briefly review the link between strings and TT deformations. The first observation in this sense was that a relation exists between the TT -deformed Lagrangian for free bosons and the Nambu-Goto action in flat space [4]. As described in ref. [9], this relation becomes particularly transparent in the uniform light-cone gauge of refs. [12][13][14], see also the review [15]. Let us briefly review that argument.
Consider a string background with shift isometries along the time coordinate t and along the spacial coordinate ϕ. 3 We introduce the one-parameter family of light-cone coordinates 4 X + = a ϕ + (1 − a) t , The most notable cases are given by a = 0 and a = 1, where X + becomes aligned to the temporal and spatial coordinate, respectively, and the light-like choice a = 1 2 ; however we can consider any 0 ≤ a ≤ 1. Introducing the momenta canonically conjugate to t and ϕ, denoted by P t and P ϕ respectively, we find It is convenient to relate this to physical quantities by introducing the Noether charges E and J corresponding to shifts under t and ϕ. Then we have where R is the size of the closed-string worldsheet. The uniform light-cone gauge [12][13][14] is fixed by imposing where σ 0 is the worldsheet time. The name uniform emphasises that the momentum density P − is constant on the string. Equivalently, this can be seen as fixing the T-dual coordinate to X − to be X − = σ 1 [45] (see also appendix C.1). From this rather simple condition, a number of remarkable facts follow. Firstly, since X + is identified with the worldsheet time σ 0 , the Hamiltonian H of the two-dimensional model is identified with the conjugate momentum P + ; in particular In other words, for any given a the size of the worldsheet is state-dependent and fixed in terms of each state's energy E and charge J-reminiscent of what happens to TTdeformed theories. We can be more quantitative by looking more closely to the charge H. This is given by the integral of the a-dependent 5 Hamiltonian density P + = P + (a) over the volume R(a). On the other hand, by gauge invariance the physical spectrum of H cannot depend on a, so that That is to say: tuning a changes the Hamiltonian density in such a way as to precisely compensate the rescaling of R(a). Given that the a-dependence of R is precisely that of a TT deformation for a state without worldsheet momentum, 6 tuning a mimics a TT deformation of the Lagrangian. Another way to understand how this similarity arises is by thinking of TT deformations as CDD factors [2], as in refs. [3,4]. Under a shift of the gauge parameter a the worldsheet S matrix is modified by a CDD factor e i a Φ CDD of the form [15] Φ CDD (p i , p j ) = H i p j − H j p i , (4.8) so that the spectrum is unchanged; schematically, the Bethe-Yang equations take the form which is independent of a when we use that j p j = 0 and j H j = H. The form of such a CDD factor is indeed what expected for a TT deformation [3,4] and what observed for 5 The explicit form of P+(a) can be found quite straightforwardly, at least for the bosonic part of the theory, see e.g. ref. [15]; we will analyse this in more detail for superstrings in the next section. 6 This is not really a restriction, as in string theory the level-matching constraint dictates that the worldsheet momentum of physical states should vanish up to winding. Non-trivial winding sectors can also be incorporated, see ref. [10].
flat-space strings [5]. This highlights the generality of the relation between TT -deformed Lagrangians and light-cone gauge fixed strings. It is worth stressing once more that, unlike what happens when performing a TT deformation, changing the gauge parameter a does not change the spectrum. What does change the spectrum is to change the parameter a in the Lagrangian density, while keeping the volume R fixed -or vice-versa.

Superstrings in flat space as a TT deformation
Let us now apply the logic illustrated above to superstring in flat space. We will explicitly see that the gauge-fixed (and κ-gauge-fixed) Lagrangian in uniform light-cone gauge is related to the TT deformation of a free theory of eight on-shell N = (1, 1) scalar multiplets. The deformation parameter α is related to the gauge parameter a by Indeed the Lagrangian is free at a = 1/2. This theory also exhibits an so(8) flavour symmetry, which is the manifest isometry of R 1,9 that survives in light-cone gauge. The derivation of the uniform light-cone gauge-action including fermions is relatively straightforward. We follow the procedure outlined in ref. [46]; for the sake of completeness, we include some intermediate steps and technical details in appendix C. We start from the Green-Schwarz action Some definitions are in order. The worldsheet metric with unit-determinant is γ ab ; the 10-dimensional flat target-space metric is η µν , with signature (−, + · · · +). The targetspace indices are µ = 0, 1, . . . 9 while the worldsheet ones are a = 0, 1. We denote the two 10-dimensional Majorana-Weyl spinors as θ I=1,2 , while Γ µ are 10-dimensional Gamma matrices; we defineθ I = (θ I ) † Γ 0 . Further details on our conventions are collected in appendix C. We can now fix uniform light-cone gauge like in eqs. (4.1-4.4), with the identification t ≡ X 0 and ϕ ≡ X 9 for the light-cone coordinates. Additionally, we must fix the κ gauge; this can be done by setting which halves the fermionic degrees of freedom. This leaves us with eight real fermions in θ 1 , which we denote by {ψ j − } j=1,...8 , and just as many in θ 2 , which we denote by {ψ j + } j=1,...8 . With some straightforward algebraic manipulations (cf. appendix C.2) we can show that the fermions enter in the Lagrangian through bilinears of the form (θ I ) t ∂ a θ I . For this reason, let us define the short-hand notations where we have chosen the normalisation of the fields ψ ± for later convenience. A convenient way to fix uniform light-cone gauge is to perform a T-duality along X − [45] (see also appendix C.3). Denoting the T-dual coordinate as X − , we find an action of the form which we want to evaluate in the gauge 8 Note that the parameter α is related to the gauge parameter by eq. (4.10), and the index i runs over the transverse bosons, i = 1, . . . 8. Imposing these conditions and eliminating the worldsheet metric γ ab by virtue of its equations of motion, we obtain the action (4.19) The expression above is fairly involved. It is instructive to explicitly consider a few limits. First of all, there exists a limit in which the light-cone gauge-fixed Lagrangian for 7 The worldsheet indices a, b of A ab are implicitly symmetrised. 8 We have already dropped a term of the form ab ∂a X − ∂ b X − ≈Ẋ − in the expression of B in eq. (4.16) as this is a total (temporal) derivative in the gauge-fixed theory. 9 In the formula below we introduced the short-hand notation flat superstrings is free. This happens when both X + and X − are light-like, i.e. for a = 1 2 and α = 0, cf. eq. (4.1). Indeed we find (4.20) Another interesting limit is restricting to a theory of bosons only. In this case we get the non-linear Lagrangian This is a TT deformation of the free-boson action with parameter −α [4]. This indeed fits with our string-theory construction: recall that the worldsheet size in terms of α is R = R 0 + α H, see eq. (4.6). In this case, the transformation of the Lagrangian is compensating for the transformation of the radius, so in our convention this corresponds to the opposite of a canonical TT transformation. We can also consider the N = (1, 0) Lagrangian by restricting to the boson X ≡ X 1 and one single chiral fermion ψ + ≡ ψ 1 + . The action again simplifies and we find which, upon expanding −|A| over the fermions, precisely reproduces (3.6-3.10) up to changing α → −α as expected. Finally, it is easy to explicitly verify that the full action (4.19) satisfies the TT differential equation (3.5) by following the approach we outlined in the N = (1, 0) case, see appendix B.1. As we have remarked in the N = (1, 0) case, the real fermions ψ i do not couple to the spin connection so that the Noether and Hilbert stress-energy tensors coincide for this action. In addition to providing an explicit deformation of a more general free supersymmetric action associated to α = 0, this example shows that the link between string theory and TT deformations holds also when fermionic degrees of freedom are included. The precise relation requires fixing (light-cone) κ-gauge as well as the bosonic gauge.

Deformation from the induced worldsheet metric
We have obtained the TT deformed Lagrangian of eq. (4.18) by integrating out the worldsheet metric γ ab in eq. (4.15) (and of course fixing uniform light-cone gauge). This means that we can think of (4.18) as the NLSM action (4.15) on a specific worldsheet metric given by which follows from the equations of motion. This is somewhat reminiscent of ref. [26] where it was argued that TT deformations can be understood in terms of a field-dependent "wordlsheet" metric. 10 To make this analogy more manifest let us write down explicitly eq. (4.15) in uniform light-cone gauge (4.4). We also restrict to the bosons for simplicity. Then where the unit-determinant metric γ ab is given by Firstly, observe that in eq. (4.24) the last term gives the matter action (the transverse fields of the string) minimally coupled to the metric. The first bracket, instead, is not covariant and implements the light-cone gauge constraints. Indeed in the limit α → 0 (where we expect a free theory) this forces γ ab = η ab . This is rather reminiscent both of the construction of ref. [23], where the TT deformation emerges from coupling the matter action to Jackiw-Teitelboim gravity, and of the more recent observation of ref. [26] that the TT transformation may be "undone" by introducing a field-dependent metric. However, the precise forms both of the action (4.24) and of the metric (4.25) differ from what considered in refs. [23,26]-despite eventually leading to the same TT -deformed Lagrangian, at least for the case of free bosons. It would certainly be interesting to explore this connection in more detail and for more general theories. It is worth remarking that, already for the case of free bosons and fermions, the form of the metric becomes significantly more complicated, as it can be seen from A ab in eq. (4.19); the same will be true when considering more general geometries. We hope to return to this in the near future.

Conclusions and outlook
We have seen that the structure of TT deformations is compatible with supersymmetry, and can be studied quite explicitly in the case of N = (1, 0) and N = (1, 1) supersymmetry. There, the operator O = det[T ab ] is a full supersymmetric descendant-in other words, a D-term-of a multiplet of composite operators free of contact terms. Our results were based on theories whose supercurrent multiplet satisfies the (1, 0) and (1, 1) conservation equations (2.11) and (2.37), respectively. It would be very interesting to extend our analysis to the case of (p, q) extended supersymmetry. A preliminary analysis of the N = (2, 2) case suggests that the strategy we have adopted of writing O as a D-term fails for the most general N = (2, 2) supercurrent which is described by the S-multiplet studied in [42]. Moreover, in the case of more supersymmetry, from a dimensional argument it seems likely that O should be the bottom component of a suitable short supersymmetric multiplet.
Therefore, a more detailed analysis of the structure of supercurrents and short multiplets is in order; we hope to come back to this in a future publication. It is also rather striking that, when coupling the action to linearised supergravity, the TT deformation takes the form of a shift of the prepontential superfields. This fits well with the proposal that these deformations can be related to Jackiw-Teitelboim gravity [23,25] and to the geometry of the two-dimensional space [26], and indeed suggests that it might be possible to extend such relations to superspace.
As a simple but non-trivial example, we have constructed the TT deformation of a free theory of eight N = (1, 1) scalar multiplets. We have done so by exploiting a map between TT deformations and (super)strings in light-cone gauge first highlighted in ref. [9]. We constructed the deformation of a free, relativistic theory from superstrings in flat space. It is natural to ask which theories might correspond to non-flat (super)string backgrounds. A first step in this direction was taken in ref. [9], where it was argued that AdS 3 backgrounds supported by Neveu-Schwarz-Neveu-Schwarz fluxes only are closely related to the deformation of a free non-relativistic theory, see also ref. [10]. The appearance of a nonrelativistic two-dimensional dynamics is rather common in uniform light-cone gauge, see e.g. the review [15]. This should not pose an obstacle to studying TT deformations, as these have been recently generalised to non-relativistic theories [47]. Hence, it would be very interesting to investigate this relation with string theory in greater detail. Another direction which deserves further attention is how our uniform light-cone gauge construction of TT deformations might relate to the construction of such deformations in terms of gravity [23,25] and background geometry [26]. We hope to return to some of these questions in the future.
A fresh look at AdS 3 /CF T 2 in Villa Garbald, Castasegna, for the stimulating atmosphere where this work was started.

A Conventions
Below we describe our conventions.

A.2 Ten-dimensional conventions
The ten-dimensional metric is The ten-dimensional Gamma matrices satisfy and can be written as where γ i , i = 1, . . . 8 are 8-dimensional Euclidean Gamma matrices and γ 9 = γ 1 · · · γ 8 . The charge conjugation is given by Γ 0 , so that the Majorana condition forces the components of θ I to be real, andθ I = (θ I ) t Γ 0 . For the Weyl condition, we impose 1 2 (I + Γ 11 )θ = θ which for type IIB strings can be solved by where ψ i ± have 16 real entries and the normalisation is chosen for future convenience.

B Derivation of deformed N = (1, 0) actions
We collect here the derivation of some results for N = (1, 0) actions.

B.1 The Noether stress-energy tensor and deformation
We start by recalling the definition of the Noether stress-energy tensor T where Φ i are all the fields of the theory. In particular, for the free action (3.1) we have, in the light-cone coordinates (A.1) Note that this is not a symmetric traceless tensor unless we use the fermion equations of motion ∂ −− ψ + = 0. It is convenient to rewrite the definition of the Noether stress-energy tensor in terms of a linear operator Then, for the terms that make up the action (3.6) we find Using these expressions we can easily find the stress-energy tensor as a function of A(x), B(x), C(x) and their derivatives. We find that Hence, from eq. (3.5) we have and one can verify that indeed (3.10) gives a solution.

B.2 Direct construction of the supersymmetric action
It is quite easy, for this particular case, to construct an action that is invariant under eq. (3.2). In fact, following the logic of appendix B.1, we get δ δ − F (x) = iα(∂ ++ X∂ −− ψ + + ∂ ++ ψ + ∂ −− X) F (x), (B.7) while the variations of the fermion bilinears follow immediately from (3.2). After taking the variation, we get two conditions which multiply the cubic and linear fermion terms; they are, respectively, B = 2C + x C , A = B .

(B.8)
This allows us to write down the supersymmetric version of any bosonic Lagrangian given by A(x). It is easy to verify that eq. (3.13) solves these equations.

B.3 Improved stress-energy tensor
We want here to construct a stress-energy tensor for the N = (1, 0) theory which is symmetric and, for a conformal theory, traceless. We can do so by following the Belinfante procedure. Formally, we can treat ψ + as if it were a complex field, introducing fermion bilinears ψ * + ∂ ±± ψ + and coupling the theory to a curved metric and spin-connection. Then the variations of the spin-connection contribution precisely yields the needed improvement terms. In practice, this boils down to modifying the linear operator K of appendix B.1 as In this way we get to a set of differential equations for the coefficients A(x), B(x) and C(x), which read The solution of this is given by eq. (3.17).

C Flat-space Green-Schwarz action in uniform light-cone gauge
We collect here some further details on the computation of the light-cone gauge-fixed Lagrangian for Green-Schwarz strings in flat space.

C.2 Fermion bilinears
The κ gauge fixing imposes G + θ I = (θ I ) t G − =θ I G + = 0 , I = 1, 2 . (C.6) Let us observe that under this condition we can simplify a term of the formθ I Γ µ θ I , which is what appears in the Green-Schwarz action (4.11). We havē It is therefore natural to introduce the short-hands (4.14).

C.3 T-duality and uniform light-cone gauge fixing
In order to impose the gauge-fixing condition P − = 1 from eq. (4.4) without having to perform a Legendre transform, it is convenient to T-dualise the coordinate X − , or in other words to gauge the X − isometry. This can be done for a generic background by modifying the string NLSM Lagrangian L by introducing the gauge field A a : L(∂ a X + , ∂ a X − , ∂ a X i ) → L new = L(∂ a X + , ∂ a X − + A a , ∂ a X i ) + X − ab ∂ b A a . (C.8) The equations of motion for X − impose that ab ∂ a A b = 0 i.e. that the gauge connection is flat. In this way the action is unchanged, i.e. L = L new . On the other hand we have so that the equations of motion for A 0 give ∂ 1 X − = P − and we can implement the gauge fixing (4.4) by X + = σ 0 , X − = σ 1 . (C.10) Working out the T-dual action and gauge-fixing is relatively straightforward for flat space, and it can be done e.g. following ref. [46].