The low-energy limit of AdS$_3$/CFT$_2$ and its TBA

We investigate low-energy string excitations in AdS$_3\times$S$^3\times$T$^4$. When the worldsheet is decompactified, the theory has gapless modes whose spectrum at low energies is determined by massless relativistic integrable S matrices of the type introduced by Al. B. Zamolodchikov. The S matrices are non-trivial only for excitations with identical worldsheet chirality, indicating that the low-energy theory is a CFT$_2$. We construct a Thermodynamic Bethe Ansatz (TBA) for these excitations and show how the massless modes' wrapping effects may be incorporated into the AdS$_3$ spectral problem. Using the TBA and its associated Y-system, we determine the central charge of the low energy CFT$_2$ to be $c=6$ from calculating the vacuum energy for antiperiodic fermions - with the vacuum energy being zero for periodic fermions in agreement with a supersymmetric theory - and find the energies of some excited states.


Introduction
The closed superstring spectrum on AdS 3 × S 3 × T 4 and AdS 3 × S 3 × S 3 × S 1 can be found exactly in α , in the large volume limit, by solving a set of Bethe Equations (BEs) [1,2], building on earlier integrable results of these backgrounds [3]. These algebraic equations follow from the exact worldsheet S matrix [4,5,6,7] upon making a Bethe Ansatz for the energy eigenstates. The ansatz is consistent since the worldsheet S matrix satisfies the Yang-Baxter equation. In [8] these BEs were used to determine the protected closed string states. Agreement was found with supergravity results [9,10], the calculation of which was only completed in the case of AdS 3 × S 3 × S 3 × S 1 recently [10]. In contradistinction to higher-dimensional examples, when the worldsheet theory is decompactified, the m 2 = 0 modes of AdS 3 backgrounds give rise to a gapless spectrum. This has important consequences, notably on the protected spectrum [11,8], but also on the Berenstein-Maldacena-Nastase (BMN) limit [12]. In this limit, the magnon momenta are rescaled as p → p h and h is taken large. The dispersion relation (1.1) becomes relativistic E(p) → m 2 + p 2 , (1.2) and in higher-dimensional integrable holographic models, the S matrix trivializes. Sub-leading corrections to the S matrix can be compared to α -perturbative worldsheet scattering computations. In AdS 3 integrable models, the BMN limit is more subtle. This is because massless magnons can be left-or right-moving relativistic massless modes in this limit. 2 As a result, at small momenta the all-loop massless/massless S matrix reduces to four S matrices, depending on what worldsheet chirality the scattering excitations have. 3 While left-massless/right-massless scattering becomes trivial, leftmassless/left-massless and right-massless/right-massless S matrices remain non-trivial and give rise to integrable relativistic S matrices which we will denote as S LL and S RR . 4 Direct comparison of these S matrices with worldsheet perturbative calculations is not possible: after all, massless particles of the same chirality cannot scatter with one another since both move at the speed of light. Nevertheless, viewed as an algebraic object, the S matrices are well defined.
To recapitulate, on a decompactified worldsheet the AdS 3 closed string spectrum is gapless and its small-momentum excitations are massless relativistic left-and right-movers equipped with difference-form S matrices S LL and S RR , with S LR trivial. This closely resembles the integrable description of certain CFT 2 's that arise as infra-red (IR) fixed-points of renormalization-group flows [16]. In a similar line of reasoning, we therefore conclude that the small-momentum excitations are described by a twodimensional conformal field theory, which we will denote by CFT (0) 2 . The energy spectrum of CFT (0) 2 is determined through the BEs that follow from S LL and S RR , up to wrapping corrections. When the worldsheet is compactified, Lüscher-type corrections involving exchanges of virtual particles that wrap the compact worldsheet spatial direction need to be accounted for. In integrable theories this can be done through the Thermodynamic Bethe Ansatz (TBA) [16], which in the context of integrable holographic models was found in [17,18,19]. These latter TBAs have been used to construct the Quantum Spectral Curve (QSC) [20], a powerful method for determining the exact spectrum including wrapping corrections (see for example [21] and the review [22]). Such methods are at present unavailable for the AdS 3 integrable models, also due to the presence of gapless excitations [23].
In this paper we will investigate wrapping effects on the low-momentum CFT (0) 2 states. Since S LL and S RR are relativistic, we will be able to adapt conventional methods to write down a TBA and use it to calculate the central charge of CFT (0) 2 . We expect that once a complete non-relativistic TBA for the AdS 3 models is found, it should reduce at small momenta to the relativistic TBA for CFT (0) 2 that we find here. As a result, the relativistic TBA we construct here should provide guidance on the way in which massless modes should be incorporated into the complete AdS 3 TBA . This paper is organised as follows. In section 2 we derive explicit expressions for the matrix parts of S LL and S RR in the relativistic limit. In sections 3 and 4 we show how in the BMN limit, the massless dressing factor [2] reduces to the well-known dressing factor found by Zamolodchikov and Zamolodchikov [24]. In section 5 we formulate the TBA and use it to compute the central charge of CFT (0) 2 , as well as the energies of the first excited states. We conclude in section 6 and present some of our technical findings in appendices.

Massless R matrix
Worldsheet excitations on AdS 3 × S 3 × T 4 with RR flux have mass m 2 = 1 or m 2 = 0. Both types of excitations transform in short representations of the centrally-extended su(1|1) 4 c.e. algebra of symmetries that commute with the Hamiltonian [4,5]. The structure of the central extensions is such that su(1|1) 4 c.e. ∼ = su(1|1) 2 c.e. 2 . As a result, short representations can be written as tensor products of two short representations of su(1|1) 2 c.e. , and for the most part we will focus on this smaller algebra. In this section we begin by reviewing the su(1|1) 2 c.e. algebra, its massless short representations, as well as the S matrix for scattering two such excitations. 5 We then review the relativistic limit of the massless S matrix and finally we summarize how the above structure can be understood in terms of the quantum super-Poincaré algebra introduced in [25].

The exact massless R matrix
The centrally extended su(1|1) L × su(1|1) R algebra has non-zero commutators 5 Since all m 2 = 0 short representations are isomorphic to one another, we will write all the expressions using only the so-called ρ L (m = 0) representations [5]. In order not to clutter the notation, we will drop the subscript L from most expressions. Note that the labels L and R are not related to worldsheet chirality.
where on the right-hand sides we have the four central elements. 6 A representation of (2.1) on a boson-fermion doublet {|φ , |ψ } takes the form Up to an overall dressing factor, the R matrix R is given by This form of the R matrix is fixed by compatibility with the centrally extended su(1|1) 2 symmetry , with Π the graded permutation on the tensor-product algebra Π(a ⊗ b) = (−) |a||b| b ⊗ a. The coproducts are specified as follows: Since p appears on the rhs above, we will also require ∆ N (p) = p ⊗ 1 + 1 ⊗ p . (2.8) The above coproducts provide a prescription for how the symmetry algebra acts on two-particle states, in such a way that it is a representation of (2.1). R satisfies the Yang-Baxter equation and braiding unitarity: To describe the scattering of massless AdS 3 modes, R needs to be multiplied by a suitable dressing factor, which we will denote by Φ, whose form is determined by a crossing equation [2] up to CDD factors.
Dressed in this way and evaluated in the physical region of momenta, R represents (up to a permutation of the outgoing particles) the physical S matrix, scattering particles 1 and 2 -with momenta p 1 and p 2 , respectively. 6 This algebra is in fact the conventional N = 2 supersymmetry algebra in 1+1 dimensions upon identifying and has appeared in relation to integrability before, for example in [26]. Our central extensions P and K correspond to 2∆W and 2∆W * -see for instance equation (2.1) of [26]), where the algebra is specialised to a massive relativistic dispersion relation). We would like to thank Paul Fendley and Matthias Gaberdiel for discussions related to this point.

The relativistic limit of the massless R matrix
In investigating worldsheet S matrices it is useful to consider the relativistic, or near-BMN regime In this limit it is well known that S matrices describing the scattering of massive excitations become proportional to the identity, and sub-leading terms can be matched to perturbative worldsheet scattering processes (α corrections) [13]. Similarly, the S matrices for mixed massive/massless scattering trivialise in this limit. 7 The relativistic limit of massless/massless scattering is more subtle [2] because it depends on the relative sign of the momenta of the two excitations. When p 1 > 0 and p 2 < 0, or vice versa, to leading order in the S matrix is proportional to identity with sub-leading perturbative corrections, much as in the massive case. On the other hand when p 1 , p 2 > 0 or p 1 , p 2 < 0 the S matrix remains non-trivial as → 0. It is this novel behaviour of the massless worldsheet S matrix in the relativistic limit that is the main focus of this paper.
In the relativistic limit (2.9), with p > 0, the su(1|1) 2 c.e. generators are With p 1 , p 2 > 0, the R matrix (ignoring for the moment the scalar factor) reduces to 8 R|φ ⊗ |φ = |φ ⊗ |φ , (2.12) Introducing the relativistic rapidity the R matrix takes the difference form (2.15) We will denote by R both the non-relativistic R matrix, and its relativistic limit, since it should be clear from the context which R matrix we mean. 7 Because of complications related to regularising massless particles in loops, matching to perturbative computations remains an outstanding challenge [15]. 8 Similar expressions can be found when p 1 , p 2 < 0.

The q-super-Poincaré algebra and boosts
In [25], an algebraic reformulation of the results summarised in section 2.1 was given in terms of two copies of a 1+1 dimensional q-deformed super-Poincaré algebra. Each copy satisfies the following relations: where µ ≡ 4 h 2 , (A, B) = (L, L), (L, R), (R, L), (R, R) and the boost operators act as The (suitably normalised) quadratic Casimir is given by The massless representation is characterised by the vanishing of the Casimir eigenvalue (massless dispersion relation). The coproduct for the boost operator, say, J L reads (cf. [27]) The result of [28] were used to introduce a geometric picture in the scattering problem. The equations with and From (2.20) we can write an integral formula for the R matrix, in terms of the line-integral over any given (suitably differentiable) curve γ(λ) : [0, 1] → B: where Π s is the graded permutation operator acting on two-particle states as Π(|v ⊗|w ) = (−) |v||w| |w ⊗ |v , and P exp denotes the path-ordering of the exponential. 10 The starting point of integration is chosen to reproduce the property R(p, p) = Π s . The putative connection Γ M is locally flat (pure gauge), since its curvature F M N is vanishing: Including a dressing factor Φ in the R matrix modifies (2.20) in a straightforward way The R matrix undergoes crossing when one of the momenta leaves the physical region and was not discussed previously. We analyse this effect on the above differential equation and connection in the next section.

Relativistic limit of the q-super-Poincaré algebra
We conclude this summary, by considering the effect of the relativistic limit on the q-super-Poincaré algebra. The boost operators become equal to one another and we denote them by b The coproduct reduces to

28)
i.e. the R matrix, which satisfies ∆ N (J)R = 0 [28], has to become of difference form in the strict relativistic limit. This is indeed the case, as we saw explicitly in equation (2.14). Notice also that ∆(J) and ∆ op (J) become coincident in the relativistic limit.
In the relativistic limit, the covariant derivatives in equation (2.20) reduce to with One can verify that Equivalently, in terms of rapidities θ M , we have with (2.33) 10 The sign in the exponent of (2.23) is justified since we extracted Πs in front for convenience, and one has Just as the R matrix (2.14), the connection A M is also of difference form.
Let us remark that equation (2.31) would be rather hard to detect starting from the strict relativistic case, but it emerges quite naturally when deriving it from the q-Poincaré algebra. As a matter of fact, because of the difference-form imposed by ∆(J)R = 0, both conditions (2.31) coincide with the single ordinary differential equation which can be immediately integrated to is the Gudermannian function. By explicitly working out (2.34), we obtain which can be seen to coincide with (2.14)

Dressing factor and Crossing
In this section we discuss the crossing symmetry that is used to determine the R matrix dressing factor.
We begin by explaining how crossing is implemented in the geometric formulation of the R matrix reviewed in section 2.3. We then show that the massless dressing factor found in [2] reduces to the famous Sine-Gordon scalar factor that enters the S matrix for solitons and anti-solitons [24].

Crossing and the q-super-Poincaré algebra
With p ∈ [0, 2π], the supercharges in the crossed region are defined as where the supertranspose of a matrix M is defined and the charge conjugation matrix as Up to a dressing factor, the R matrix for the scattering of a crossed particle with an uncrossed one is given by where p 1 is in the crossed region and p 2 in the physical region. R c satisfies The crossing equation reads Similarly to R, the crossed R matrix can be shown to satisfy with an analogous expression for J R . As in the previous section, this condition can be re-written in a more geometrical form We perform the continuation to the crossed region according to sin p 1 /2 = i | sin p 1 /2|. Integrating along a contour γ gives the following expression for the R c -matrix: where the path starts at (p, p) and ends at (p 1 , p 2 ), and the matrix Θ is defined as Including a dressing factor, which we call Ψ to distinguish it 12 from Φ -the difference being the arbitrarily chosen normalisation of (3.4) w.r.t. (2.4) -we write In fact, in order for (3.6) to be compatible with crossing symmetry, one needs to impose where the continuation to negative momenta was described in detail in [2]. 11 We have again used the fact that {Θ, Ω M } = 0 to extract the matrix Θ in front and adjust the sign of the exponent in (2.23). 12 The simple relationship between Φ and Ψ will be fixed in section 4.3.

Relativistic limit and crossing
In the relativistic limit crossing symmetry on superalgebra generators (3.1) takes the form where the crossing map reduces to the familiar relativistic one (3.14) Ignoring the dressing factor, the relativistic limit of the crossed R matrix R c (3.4) is and it satisfies the differential equation which can be solved analogously to equation (2.34).
Expanding on an idea put forward in [28], we consider the two expression for the R matrix, namely Turning to the dressing factors, the full crossing equation reads hence the dressing factors need to satisfy (3.20)

Relativistic limit of the massless dressing phase
In this section, we derive the relativistic limit of the phase for massless-massless scattering constructed in [2]. In a large-h expansion, the two leading terms in the dressing phase [2] are referred to as Arutyunov-Frolov-Staudacher (AFS) [29] and Hernández-López (HL) [30] phases, and they correspond to the O(h) and O(1) orders, respectively. Since the AFS term tends to 1 in the relativistic limit, we shall focus on the HL term in what follows. The higher order terms become trivial in the relativistic limit.
In order to solve the crossing equation, a specific path was chosen [2] along which to perform the analytic continuation of the phase from the physical region Re(p) ∈ (0, 2π) into the crossed region Re(p) ∈ (−2π, 0). Such a path in the p-plane was singled out as going from a real p ∈ (0, 2π) to −p, intersecting the imaginary axis for Im(p) < 0.
In the relativistic limit (2.9) the physical region in the q-plane is the entire half-plane Re(q) > 0, and the path used for crossing goes from a real q > 0 to −q, intercepting the imaginary axis for Im(q) < 0.
In terms of the rapidity variable θ defined in equation (2.13), the physical region is mapped into the strip Im(θ) ∈ (− π 2 , π 2 ) in the θ-plane, and the path used for crossing goes from a real θ ∈ (−∞, ∞) to θ − iπ, intercepting the lower branch cut Im(θ) = − π 2 . Below we obtain the relativistic limit of the massless dressing phase and show that it reduces to the famous scalar factor of the Sine-Gordon model obtained by the Zamolodchikovs [24].

Integral representation
The massless HL phase has the following integral representation [2] and 13 In the relativistic limit we define and take the limit h → ∞, while keeping the real part of the momenta p, and q positive. Relegating the details to Appendix A.1, we find Introducing massless rapidity variables we may write .
(4.7) 13 The function g does not depend on the choice of sign ± that enters G ± .

Redefining the integration variable
with ϑ = θ 1 − θ 2 showing that in the relativistic limit the dressing phase is of a difference form expected of a relativistic theory. The corresponding relativistic dressing factor is defined for rapidities in the physical strip as The dressing phase (4.8) takes the form of a conventional Riemann-Hilbert type integral (A.1), with a cut along the imaginary momentum axis. We may use the Sochocki-Plemelj theorem [31] to determine the value of the dressing factor after analytically continuing through the cut at Im(ϑ) = π Similarly, continuing through the cut at Im(ϑ) = − π 2 we have From these relations we can immediately deduce the crossing equations Using equations (4.10) and (4.11), and the fact that the integral (4.8) can be computed for any value of ϑ, the dressing factor on the whole rapidity plane is given by the value of the integral times the terms one picks up by crossing the cuts Above, n(ϑ) is defined in terms of the ceiling function 14 (4.14)

Comparison with Zamolodchikov's phase factor
We shall now compare the relativistic limit of the dressing factor, which we have obtained in the previous sections, with the famous scalar factor obtained by Zamolodchikov for the Sine-Gordon model (SG), and find them to agree. The Sine-Gordon scalar factor, which multiplies the scattering matrix between a SG soliton and a SG anti-soliton [24] (see [32] for a recent review) can be written as 15 (4.16) 14 The ceiling of a real number x is defined as the smallest integer greater than or equal to x, and is denoted by x . 15 One obtains this formula by setting γ = 16π ⇐⇒ β 2 = 16π 3 in formula (4.11) of [24] and redefining the rapidity variable to include a minus sign.

Expression (4.15) solves the crossing equation
which is the same as what the relativistic limit of the HL phase satisfies (4.12). Therefore, the two dressing factors can differ by at most CDD factors. We have in fact verified numerically that the formula (4.15) exactly reproduces the relativistic limit of the massless phase we derived in the previous sections.
More precisely, As far as we are aware, the integral expression (4.8) for the Zamolodchikov dressing factor has not previously appeared in the literature and is different from other known integral formulae such as those given in [33] or [34].

Comparison with the literature on 2D N = 2 theories
The connection with Zamolodchikov's phase factor is actually not the only remarkable relation with previous literature on relativistic integrable systems. The matrix R LL in the relativistic limit (2.14) coincides with a subsector of the R matrix obtained in [26]. A very similar scattering matrix was presented in equation (2.2) of [35]. This almost coincides with (2.14) apart from some crucial factors of i. These factors of i in the matrix entries are in fact associated to a difference in the setup of [35] as compared to the one under present consideration. The excitations in [35] have a fractional fermionic number, while ours are conventional bosons and fermions. This amounts to subtle differences in the coproduct of the supercharges, which ultimately determine the differences in the respective R matrices 16 . Irrespective of the difference in coproduct, the similarities arise from the fact that the algebras of symmetries do coincide and are simply the N = 2 algebra.

Thermodynamic Bethe ansatz
In this section, we provide the Thermodynamic Bethe Ansatz (TBA) equations [16] (see [36] for a recent review) restricted to the massless sector. Having established a relationship with a standard relativistic field-theory construction related to N = 2 theories, we would like to exploit this to move the first steps into the finite-size program for this sector. It will eventually be necessary to extend this framework to the whole theory in order to completely solve the model, as it was done for higher-dimensional cases [17,18,19] (see also the review [37]).
Let us get inspiration from the treatment of [33,35], where the TBA was used to obtain the Casimir energy of the 2D theory compactified on a spatial circle of length R. According to Zamolodchikov's TBA [16], one can use the asymptotic data of the scattering problem to derive integral equations for the finite-size spectrum, utilising the principle of the double Wick-rotation. This amounts to exchanging space with time, turning a problem which is periodic with period R in space and infinite time L → ∞, into one which is decompactified in space and with periodic time, i.e. at finite temperature 1 R . Thanks to relativistic invariance, we are guaranteed to be able to use the same principle of double Wick-rotation in our relativistic-limit situation. 16 We thank Paul Fendley for communication about this point.
Based on this reasoning, the ground-state energy of the original model (which is the leading contribution to the partition function at large time) can be read-off from the minimum free energy F min at large L of the doubly Wick-rotated model: For N = 2 theories, for instance, this procedure reproduces the correct central charge for the massless flows which [35] were concerned about. The massless scattering theory describes a renormalisation group flow between a UV and an IR fixed point, and the TBA computes the ground state energy at arbitrary intermediate points along the flow. This ground state (Casimir) energy then is shown to correctly approach the UV and IR CFT central charges at the two respective extrema of the flow.
The first fundamental ingredient to perform a similar analysis in our case is the formulation of a set of Bethe equations describing the large volume spectrum of the massless sector in the relativistic limit, that is the subject of the next section.

Relativistic Bethe equations
The Bethe equations can be constructed employing the tool of the transfer matrix, which is built as the trace of a string of S matrices for an ordered sequence of interacting particles. Let us briefly outline the calculation in our case.
If one considers N particles, taken to be all bosonic for the moment, on a circle of length L, interacting one with each other via an integrable scattering matrix, one is brought to impose the following quantisation conditions on the momenta: where p i is the momentum of the i-th particle on the circle is the transfer matrix, namely the trace over the auxiliary 0-th space of the monodromy matrix and S is the two-body S matrix. Equations (5.2), (5.4) and (5.5) are saying that revolving each particle around the circle of length L, while scattering all the other ones in sequence, amounts to the identity acting on an eigenstate |ψ of the transfer matrix. Normally one would exclude the same particle k in the scattering sequence, however we can include it since S db ac (0) = δ d a δ b c , which acts by effectively permuting the two scattering particle and has the result of cutting the product (5.5) precisely in correspondence with particle k, as it is needed.
From this treatment it is clear that the next task is to find the eigenstates |ψ of the transfer matrix.
For non-diagonal scattering, when the S matrix is not just a scalar but, as in our case, it does transform non-trivially the particles' internal degrees of freedom, diagonalisation is best achieved via the so-called Algebraic Bethe Ansatz (ABA) technique. One can prove that, if one constructs the tensor where E xy are the standard matrix unities, then In the supersymmetric case, we therefore now take as the appropriate definition of the monodromy matrix to be used, and switch to the supertrace. We perform the full algebraic Bethe Ansatz (ABA) for the transfer matrix resulting from such definition in Appendix A. The result is as follows: There is also a quantisation condition for the level-1 magnon momenta (level-1 Bethe equations): meaning that the level-1 magnons only interact with the K 0 frame particles (impurities on the level-1 chain), but not one with each other.
It is now clear that since one of the products is simply entirely annihilated by one of the factors being 0, specifically . We see therefore that (5.2) can be replaced by the following system of equations: quantising the momenta when the system is in the eigenstate characterised by M level-1 magnons with rapidities β i , subject to (5.12).
The same set of Bethe equations can be obtained directly by applying the relativistic limit to the all-loop Bethe equations for the massless sector [1,2]: The relativistic limit corresponds to taking the following small momentum limit on the dynamical vari- Applying this limit to the Bethe eqs. (5.15)-(5.17), we get scattering phases depending on difference of rapidities and in particular, for the right-movers (p k = e θ k ) where S(θ) is the Zamolodchikov's sine-Gordon scalar factor, as shown in Section 4.3. It is easy to check that we get exactly the same Bethe equations as (5.12) and (5.14), corresponding to the Dynkin diagram represented in Figure 2. It is this form of the quantisation condition which we will submit to the TBA analysis of section 5.2, following [38].
If we had started from the dual Bethe equations, also derived in [1,2], for the so-called "fermionic grading", in the relativistic limit we would have obtained slightly different equations for the momentum carrying node: However, the differences with respect to (5.20) will not imply any change in the procedure exposed in the next section, also since we are allowed to simply relabel the auxiliary variables: in the dual frame the roots previously labelled by +1 can be mapped to roots of type −1 and vice versa.
It is also possible to derive the dual Bethe equations directly in the relativistic limit, following section 3.2 of [2]. It is sufficient in fact to adopt the same duality transformation employed there, taken in the relativistic limit and switching off the massive roots. If one sets N 2 = N2 = 0 in P (ζ), formula (3.20) in [2], one obtains Since P is a polynomial of degree K 0 − 1 (the highest power cancels out), it must be This means that it must simultaneously happen, from (5.23) and (5.25), that The second equality can be used to convert the factor K1 k=1 in the momentum-carrying equation (5.16) in terms of the dual rootsx.
In the relativistic limit it is enough to parametrise the roots in the same way as in (5.18): set ζ = e i e u , (5.27) and let → 0. The factors of ν in (5.23) tend to 1 being exponential of momenta, then one is left with where we have used the same argument based on the auxiliary Bethe equations (5.19). One does not have a polynomial in the limit, therefore we have used the sinh function, which has the appropriate zeroes and periodicity. We can now use the fact that the limit of which can now be used to dualise the momentum-carrying equation (5.20).

Thermodynamics
Now, let us consider the thermodynamic limit of (5.19)-(5.21), whereby In this limit, the system (5.19)-(5.21) is replaced by a set of integral equations. Taking the logarithm of (5.19)-(5.21), and then applying the thermodynamic limit, amounts to introducing the density ρ 0 (θ) = ∆n ∆θ of allowed frame particle states per unit rapidity, and analogously ρ ±1 , ρ ±3 for level-1 magnons, corresponding respectively to pairs of solutions β ±n,i = z i ± iπ 2 , n = 1, 3. Actually, ρ 0 and ρ ±1 , ρ ±3 include the densities of both particles and holes and satisfy the following integral equations (see Appendix C for a derivation) where the kernels are given by Due to (5.12), the level-1 magnons are effectively free (apart from their interaction with the frame particles), then they cannot form bound states and there are only densities of fundamental particles ρ ±n appearing in (5.36), and not infinite towers of bound states densities as in [38], for instance. If we define a unique kernel φ ≡ φ − = 1 2π cosh(θ) for the interactions with and among auxiliary densities ρ ±n , then the densities equations assume an even simpler form: where we introduced the symbol * to denote the standard convolution. Now we can use the property φ 0 = φ * φ and (5.38) to simplify the equation for ρ 0 in the following way: 17 where we basically managed to get rid of the convolution of ρ 0 interacting with itself.
The procedure continues by minimising the free energy, which, as advertised at the beginning of the section, returns the ground state energy of the original model before the double Wick rotation. The free energy F in the thermodynamic limit gets two contributions: The measure term N gives an entropy factor, accounting for the combinatorics of all the possible ways the allowed states ∆n = ρ 0 (θ)∆θ -respectively, ∆m = ρ 1 (z)∆z -are filled by the available frame particles ∆ 0 = ρ r 0 (θ)∆θ -respectively, level-1 magnons with densities ∆ i = ρ r i (θ)∆θ -, namely for each species i = 0, 1. By applying Stirling's approximation of the factorial due to the large occupation numbers, one gets However, the energy turns out to receive contributions only from the frame particles, and not from the level-1 magnons, which are only contributing to the entropy: Minimising the free energy (5.41) in the thermodynamic limit, subject to the constraints (5.39), (5.38), gives a system of 10 variations, i .e. with respect to ρ r 0 , ρ h 0 , ρ r ±n and ρ h ±n , n = 1, 3. The resulting TBA equations read (see Appendix C for a derivation) where we have defined with the multi-index A = (0, ±n). In terms of the solutions of (5.45), the exact ground-state energy for right-movers is given by while the total ground-state energy reads TBA-diagram describes the ground state TBA equations for the UV limit of N = 2 super-sine-Gordon with β 2 = 16π/3 [40,35]. Therefore, we expect that our TBA will give as a result the same central charge c = 3, at least in the case with trivial chemical potentials.
In order to calculate the central charge from the TBA equations (5.45), we use the well known "dilogarithm trick" (see for example [41] for an explanation), for which it is necessary to fix the values of the pseudoenergies at θ = ±∞. Obviously, ε 0 is constant for fixed values of θ, then the equations for ε ±n reduce to At θ = +∞, in particular, the driving term ν 0 (θ) in the first of (5.45) diverges and then we have where, for simplicity, we called L 1 ≡ L ±n , since the TBA equations (5.45) tell us that all the ε ±n are equal, and we denote them by ε 1 ≡ ε ±n . In this way we get where we replaced φ * L 0 by −ε 1 , as given by the second of (5.45). This integral can be written as we obtain c = 3, as expected. Finally, taking into account that our theory contains two massless momentum-carrying roots, and then two copies of the system (5.45) represented in Figure 3, the total central charge is actually doubled to c = 6.
Actually, in this way we are calculating the ground state energy of the sector with antiperiodic boundary conditions on the fermions [42], see [18,43] for a discussion in the AdS 5 case. The ground state energy of the sector with periodic fermions, instead, is calculated by Witten's index [44], rather than the usual free energy as in (5.1). Witten's index is obtained by adding non-trivial chemical potentials to the auxiliary fermionic pseudoenergies , so that, in our case, ε ±n → ε ±n ± iπ.
A consistent solution of the ground-state TBA equations (5.45) with these chemical potentials is given by the constants ε 0 = +∞, ε 1 = 0. These yield E 0 (R) = 0 exactly, for any value of R, as expected for the vacuum energy in a supersymmetric theory with periodic fermions. More details about this solution and more general chemical potentials will be discussed in the next section.

Twisted theory and excited states
In the case of generic chemical potentials iγ ±n added to the fermionic pseudoenergies, our ground state TBA equations for the right-movers (5.45) become φ * (log(1 + e iγ+n e −ε+n ) + log(1 + e iγ−n e −ε−n )) ; ε ±n = −φ * L 0 , n = 1, 3 , (5.55) where γ ±n are real constants. In what follows these will be called twists for shortness sake and we shall consider the right-movers sector only.
The main motivation for us to consider such twisted version of the ground state TBA equations (5.45) is to calculate the energies of the excited states. Equations (5.55), indeed, turn out to be very similar to those studied in [42] to determine the excited states' energies of the sine-Gordon model at its N = 2 supersymmetric point, i.e. at β 2 = 16π/3, in the massless limit: the only difference is that we have two more equations for the additional auxiliary fermionic variables of type 3. Thanks to this similarity, in this section basically we shall follow the analysis performed in section 4 of [42] for the massless limit.
We shall then consider the case 18 γ +1 = γ +3 = −γ −1 = −γ −3 = γ and allow the Y-functions 18 This case corresponds also to the UV limit of a twisted version of the N = 2 super-sine-Gordon with β 2 = 16π/3, mentioned in section 5.3, with twists α F = (k + 2)α T = γ and k = 0 in the notations of [35]. This means also that more general twists are possible, and then other sectors of excited states may remain to be explored. Y 0 ≡ e −ε0 and Y ±n ≡ e −ε±n to develop zeros as γ increases. It is then useful to derive from the TBA equations (5.55) a set of functional relations connecting the Y-functions, the so-called Y-system, valid also when Y 0 and Y 1 have zeros: (1 + e iγ+n Y +n (θ))(1 + e iγ−n Y −n (θ)) , (5.56) At the level of TBA equations, instead, if Y 0 and Y ±n have zeros in the strip |Im(θ)| ≤ π/2, then they satisfy a modified set of twisted TBA equations, given by where, as in the previous section, since all the Y ±n (θ) are equal, we denote them all as Y 1 (θ), and the positions of the zeros {x j } J j=1 and {y k } K k=1 are fixed by These conditions follow from the Y-system (5.57) evaluated at the locations of the zeros, and, using (5.58)-(5.59), they can be written as integral equations We shall adopt the same conjecture of [42] about all the zeros coming from θ = −∞. Therefore, in order to understand at which values of γ they come into play, it is essential to solve the TBA system (5.58)-(5.59) at θ = −∞: with Y 0 (−∞) = e −ε0,min and Y 1 (−∞) = e −ε1,min ; n = 1, 3.
At γ = π, Y 0 (−∞) is zero, but actually Y 0 (θ) is zero at any θ. As in [42], this means that the first zero x 1 of Y 0 (θ) at γ = π enters at −∞ and goes straight to +∞, ensuring that Y 0 (θ) = 0 for any θ, and its effect is just to change the sign of Y 0 (θ), or equivalently to add a −iπ in the r.h.s. of the TBA equation for ε 0 : Also the second equation of (5.63) has to be modified accordingly, by changing sign of Y 0 (−∞) in the l.h.s.. This implies that we have to choose the solution (5.68) for the minima (ε 0,min , ε 1,min ), so that (5.65) gives ; for γ ∈ (π, 3π/2) .
These choices for the lower limits have been verified also numerically. It is then easy to evaluate the two integrals in (5.74) in terms of dilogarithms: Now, let us explain how formula (5.75) matches the results (5.66) for γ ∈ (0, π/2), (5.69) for γ ∈ (π/2, π), and gives for any value of γ ≥ π, by taking into account the behaviour of the Y-functions' zeros, that has been also verified numerically in Appendix D.1.
• For γ ∈ (3π/2, 2π), instead, K = 1, but the discrete contributions still vanish since J = J ∞ = 1 and N 1 = 0, while we need to take into account the last term of the second line in (5.75) with K = 1.
• For γ ∈ (5π/2, 3π), a second zero y 2 with N 2 = 1 enters, then the contribution from the last term of the second line vanishes, while we have to add 4π + 2γ − 6π − 2π.
• Finally, for γ ∈ (7π/2, 4π), a third zero y 3 enters, then we have to take again into account the last term of the second line in (5.75), while the discrete terms give 12π + 2γ − 10π − 6π.
Let us notice that, from γ = 2π to γ = 4π, to get the result (5.76) we have to add simply 2γ − 4π to the first two lines of (5.75).
These results have been tested by the numerical analysis discussed in Appendix D.1, see Figure 5, and let us guess that formula (5.76), that is quite similar to the well known formula for sine-Gordon at the N = 2 point [42], is valid for γ ≥ π. This let us also conjecture that the energies of some excited states belonging to the sector with periodic (antiperiodic) fermions can be calculated by E(γ, R) = 4E right (γ, R) = 2 (π−γ) 2 πR at odd (even) integer values of γ/π ≥ 2:

Conclusions
In this paper we have investigated AdS 3 /CFT 2 states whose energies are closest to the BMN vacuum.
On a decompactified world-sheet these correspond to the gapless (massless) excitations that distinguish AdS 3 /CFT 2 from higher-dimensional holographic duals. At low energies, they behave as massless relativistic left-or right-movers on the world-sheet. Remarkably, in this limit the exact worldsheet S matrix remains non-trivial. More precisely, while massless left/right, massive and mixed-mass scattering does trivialise, the scattering of massless excitations of the same worldsheet chirality is described by a non-trivial integrable relativistic S matrix.
This S matrix is essentially non-perturbative in its form: after all, relativistic excitations moving at the speed of light in the same direction cannot scatter with one another! Instead, as first proposed by Zamolodchikov [38], the S matrix should be thought of as an auxiliary algebraic tool which can be used to determine the spectrum of the gapless excitations using Bethe Ansatz methods. Since the S matrix for left/right massless scattering is trivial in the low energy limit, we further conclude, following Zamolodchikovs' approach, that the low energy spectrum should be that of a two-dimensional conformal field theory, which we have denoted as CFT (0) 2 . In order to understand the CFT (0) 2 further, we have analysed how finite-size, or wrapping, corrections enter the spectral problem in the low energy limit. Since the theory is relativistic in this limit, we were able to construct the TBA and corresponding Y-system both for the ground-state and for the excited states building on some of the original constructions in the integrable literature [45,42]. We find that the central charge of CFT (0) 2 is c = 0 (c = 6) for the sector with (anti)periodic fermions, and that some excited states energies are given by integer multiples of 2π/R. These findings, together with the targetspace supersymmetry of the spectrum in the relativistic limit, point towards CFT (0) 2 being a free CFT 2 , perhaps just a space-time supersymmetric T 4 theory. We hope to return to a more detailed analysis of the exact identification of CFT (0) 2 in the future. An outstanding problem in integrable AdS 3 /CFT 2 holography has been the challenge of incorporating finite-size effects, with perturbative calculations proving difficult due to the presence of gapless/massless excitations [46]. Our analysis shows that to overcome these obstacles one needs to adopt an essentially non-perturbative approach. We were able to do this at low energies and have shown that wrapping effects do not spoil integrability. It would be very interesting to extend our findings beyond the relativistic limit, to a complete TBA and QSC for the theory.
Throughout this paper we have focussed on the AdS 3 × S 3 × T 4 theory supported by RR flux. It would be particularly interesting to generalise our analysis to backgrounds supported by NSNS flux. In the presence of non-zero RR moduli, the exact worldsheet S matrix of this theory is known [47]. In the low energy limit the S matrix of massless modes remains non-trivial and we are currently investigating the resulting CFT (0) 2 [48]. This should also lead to a better understanding of the pure NSNS theory in the limit of zero RR modulus. Here, the non-perturbative massless S matrices S LL and S RR remain non-trivial. A careful analysis of this limit should help to provide an integrable description of the WZW theory, as well as determine the status of a recent proposal based on an almost trivial S matrix [49].
Given the non-perturbative nature of our findings, one may additionally hope to shed light on the k = 1 theory and its relation to the symmetric orbifold CFT 2 , as recently investigated in [50]. It would also be interesting to identify the role these gapless excitations play in the Higgs branch spin-chain [51]. Finally, generalising our construction to the AdS 3 × S 3 × S 3 × S 1 background supported by RR flux should be straightforward and one may also consider extending the analysis to mixed-flux backgrounds [52,6,7].
Advances in String and Gauge Theory".
No data beyond those presented and cited in this work are needed to validate this study.

A Relativistic Dressing Phase
In this appendix we collect the computational details used to determine the relativistic limit of the HL dressing factor.

A.1 Relativistic limit of integral expression for HL phase
The integral representation of the HL phase (4.1) takes the form of a Riemann-Hilbert integral schematically of the form with ϕ an analytic function. Crossing is closely related to the Sochocki-Plemelj theorem with the value of φ jumping as w goes from one side of the contour C to the other. It is helpful to split the integration interval in equation (4.1) into two In the relativistic limit the momenta p and q are small (compared to h) and so crossing can only take place for z ∈ [e −1 , 1] since we can always increase the value of h to ensure this. The integrals over z ∈ [−1, e −1 ] then involve analytic functions only and can be performed by expanding the integrands at Similarly for z ∈ [0, e −1 ] integrals we have 3 6(e − 1) 3 h 3 + epq (1 + e + e 2 )p 2 + eq 2 12(e − 1) 4 h 4 + . . . .

(A.4)
Since the integrals over z ∈ [−1, e −1 ] are trivial under crossing in the large-h limit, they give rise to a (sub-leading) CDD-factor, On the other hand, the integral over z ∈ [e −1 , 1] does contribute to crossing in the relativistic limit.
We define a new integration variable Further, for z ∈ [e −1 , 1] we can take → 0 in the integrals as this was introduced to regularize the singularity at z = 0. Then expanding the integrand gives The leading large-h term when combined with the non-integral part of the HL phase to ensure antisymmetry then gives equation (4.5).

A.2 The dilogarithm form of the HL phase
We can obtain another useful expression for the relativistic phase in terms of dilogarithms, starting from the expression for the HL phase given in [53] such that the dressing phase is expressed in terms of the function We set x = e i p 1 2 , y = e i p 2 2 , (A.11) and take the simultaneous relativistic limit In doing this, however, we make a specific choice. By relying on the fact that the final answer will have to display difference-form in the variable θ because of ordinary relativistic-invariance, we use the freedom of setting where we restrict to Im(θ) ∈ 0, π 2 .
(A. 13) This means that we are sending x to 1 from real values greater than 1, and y to 1 from values of the real part greater than 1, which also means that both x and y are consistently approaching the boundary of the region (A.8). As they do, the variable v 1 ≡ u 1 − 2 approaches the branch cut on the positive real axis from above. Specifically, Relativistic invariance will imply that the dependence we obtain in the sole variable θ will account for the whole dependence on θ 1 − θ 2 .
The leading order in the -expansion reads We now can use the fact the we are in the region Re(κ) > 0 ∪ Im(κ) > 0, and that Moreover, we set the branch cut of the logarithm on the negative real axis, with argument approaching +iπ from above and −iπ from below. Taking all of this into account, a careful analysis allows to reduce the expression (A.16) to which is manifestly a finite limit. Recalling that and by the arguments laid out earlier, formula (A.17) is the relativistic limit of the massless phase as a function of θ = log 2 − θ 1 + θ 2 = log 2 − ϑ.
Let us investigate the discontinuities of χ 0 and relate it to the relativistic crossing equation. It is convenient to continue working with the variable κ, which we now consider approaching the positive real axis from above, where it meets a branch cut of the phase (A.17). The idea is that we can continue the expression for the phase and see what values it approaches when we reach the branch cut from below.
For this, we not only need the branch cut discontinuity of the logarithm, but also that the function Li 2 (z) has a branch cut along z ∈ (1, ∞) with When this information is all put together, there is still a difference in the contribution to the jumpdiscontinuity of the phase depending on whether Re(κ) ∈ (0, 1) or Reκ > 1 as Im(κ) approaches 0. We shall focus for convenience on the region Re(κ) ∈ (0, 1), Re(ϑ) > 0. (A.20) In this region, the contributions to the discontinuity come only from the term − 1 2π Li 2 2 1−κ , and the difference of the limit from above minus the limit from below the cut is By the argument spelled out in [2], adding to the discontinuity (A.21) -continued to the crossed value of κ, i.e.
-the contribution from the functional identity obtained by repeated use of the dilogarithm-identity we obtain sum of r.h.s. of last two eqs.

B Algebraic Bethe Ansatz
In this appendix we summarise the Algebraic Bethe Ansatz procedure needed in section 5.1. We shall follow [54].

B.1 General formulation
To begin with, we define two functions such that the R matrix (2.14) can be written as The N -fold transfer matrix T (trace of the monodromy matrix M), is given by and it is associated with the propagation of an auxiliary 0-th particle with rapidity ϑ 0 through an array of i = 1, ..., N particles with rapidities ϑ i -collectively grouped into a vector ϑ. The trace is taken in the auxiliary 0 space.
Notice that the transfer matrix we define in this appendix differs from formula (5.9)  This implies that the set of eigenvalues (which is all that it is needed for the thermodynamic Bethe ansatz -see section 5.1) will be the same for (5.9) as for the transfer matrix we will diagonalise here below 19 .
One then chooses a pseudo-vacuum, namely a highest-weight eigenvector of the transfer matrix which may serve as a starting point. A natural choice in this case is the simple N -fold tensor-product state It is not difficult to see that such a state is an eigenstate of the transfer matrix, with eigenvalue At this stage, one writes the monodromy matrix in the form where one has separated the 0-th space upfront, with written for two auxiliary spaces and N physical ones. In terms of (B.6), and using the fact that through all the B's in the M -magnon state, and accumulate an eigenvalue where we have used elementary properties of the functions a(ϑ) and b(ϑ) to cancel terms in the intermediate steps. Of course, only for X = 0 we can claim that |β 1 , ..., β M is an eigenstate. Since X collects the contributions from the first term on the r.h.s. of (B.10), one realises that it is possible to set X = 0 by requiring the level-1 Bethe equations One finally needs to multiply the eigenvalues we have found here by the product over the quantum spaces of the dressing factors, namely reinstating the correct normalisation for the R matrix (B.2).

B.2 Lowest-level eigenstates
In this subsection, we show how the Algebraic Bethe Ansatz we have performed in the previous subsection determines the transfer-matrix eigenstates in some specific example with low values of N .

B.2.1 Two physical sites
Let us begin with N = 2. It is easy to directly diagonalise the transfer matrix We find for the bosonic eigenstates The (un-normalised) fermionic eigenstates are slightly more involved: and use such solutions to construct the 1-particle eigenstates as Explicit evaluation of the B operator from the monodromy matrix gives Plugging in β = ±∞ produces exactly Finally, acting with B(∞|ϑ 1 , ϑ 2 )B(−∞|ϑ 1 , ϑ 2 ) produces a state proportional to |ψ ⊗ |ψ : we have reached the highest-weight vector, and the spectrum is complete. Given that we also recover exactly all the eigenvalues from formula (B.12), as it can be verified by explicit calculation using (B.18) and the hyperbolic-function identities.

B.2.2 Three physical sites
For N = 2 only solutions at infinity are found of the auxiliary Bethe equations, while for N = 3 one Let us define as y the solution with the + sign in the second formula of (B.25): the solution with the minus sign will therefore be equal to −iy. Correspondingly, the associated values of β differ by i π 2 . The eigenvalues (B.12) can be expressed, using the auxiliary Bethe equations which appear in the formula as a multiplier, in terms of the location of their zeroes, which are precisely the auxiliary roots.

C Derivation of the TBA equations
First, let us take the logarithm of equations (5.19)-(5.21), divide them by i and define the counting functions in a way that they give integer multiples of 2π/L when evaluated at the Bethe roots or holes: The counting functions should be conventionally defined in a way to be monotonically increasing functions; we try by defining them as follows In the thermodynamic limit the sums become integrals as 1 L i → dθρ r (θ), then definitions (C.2)-(C.3) become Let us then take the derivatives of the counting functions in their respective arguments: because of (C.7)-(C.9) we get the nonlinear integral equations (5.34)-(5.35) for the densities.

D Derivation of the excited states' energy formula
Basically, in order to derive a closed formula for the energies of the excited states, we have to solve the system of equations given by (5.58)-(5.62) and plug the solutions for ε 0 and y k into the excited states' energy formula RE right (R, γ) = K k=1 e y k − 1 2π dθ e θ log(1 + e −ε0(θ) ) . (D.1) As done in section 5.3 for the ground state, the starting trick consists in taking the first derivative of (5.58) and inverting it for e θ , so that we can plug into (D.1), where we defined L γ 1 (θ) ≡ log[1 + e iγ Y 1 (θ)][1 + e −iγ Y 1 (θ)] and took into account that the x j 's at ∞ do not contribute. Next, we can replace φ * L 0 by using (5.59), so that the second term of (D.1) Integrating by parts the last term, we get and that for γ > π the addition of −iπ to the r.h.s. of (5.58) implies the replacements 2N k + 1 → Analogously, for γ ∈ (π/2, π) we get where we recall that Y 0 (−∞) = tan(γ) 2 and Y 1 (−∞) = −1/ cos(γ), while for γ ∈ (0, π/2) where Y 0 (−∞) = Y 1 (−∞) 2 = ∞.
• For γ ∈ (π, 3π/2), we adopted the prescription discussed in section 5.4 (the additional −iπ in the equation for ε 0 (5.70)) and got perfect matching with the analytic prediction (5.71), except for γ close to 3π/2, where the numerical algorithm becomes sensitive to the approaching of a new zero.
• In order to push the numerics beyond γ = 2π, we had to consider a second zero x 2 of Y 0 (θ), with We recall that the structure of the zeros are suggested by the behaviour of the zeros of Y 0 (±∞) and Y 1 (±∞) deduced by equations (5.63) and (5.64), as discussed in section 5.4, see Figure 4.
• In particular, the zero of Y 0 (+∞) at γ = 3π suggests us that x 2 goes to +∞, but a zero of Y 0 (−∞) for the same value of γ implies that a new zero, . 20 The −2iπ is due to x 2 gone to ∞.
In summary, we solved numerically the system of equations (5.58)-(5.62) up to γ = 4π taking into account the structure of zero discussed here and in section 5.4. The corresponding results are plotted in Figure 5.