Semiclassical spectrum of a Jordanian deformation of $AdS_5 \times S^5$

We study a Jordanian deformation of the $AdS_5 \times S^5$ superstring that preserves 12 superisometries. It is an example of homogeneous Yang-Baxter deformations, a class that generalises TsT deformations to the non-abelian case. Many of the attractive features of TsT carry over to this more general class, from the possibility of generating new supergravity solutions to the preservation of worldsheet integrability. In this paper, we exploit the fact that the deformed $\sigma$-model with periodic boundary conditions can be reformulated as an undeformed one with twisted boundary conditions, to discuss the construction of the classical spectral curve and its semi-classical quantisation. First, we find global coordinates for the deformed background, and identify the global time corresponding to the energy that should be computed in the spectral problem. Using the curve of the twisted model, we obtain the one-loop correction to the energy of a particular solution, and we find that the charge encoding the twisted boundary conditions does not receive an anomalous correction. Finally, we give evidence suggesting that the unimodular version of the deformation (giving rise to a supergravity background) and the non-unimodular one (whose background does not solve the supergravity equations) have the same spectrum at least to one-loop.


Introduction
We currently know various examples of deformations of σ-models that preserve integrability, see [1] for a review. Important representatives are the λ-deformation [2][3][4], the inhomogeneous deformation (also called η-deformation) [5][6][7][8], and the class of homogeneous Yang-Baxter deformations [9,10]. These are being studied with various motivations, for example to clarify a possible link between integrability and renormalisability of the σ-models [11][12][13]. When these deformations are applied to string σ-models, they lead to deformations of the corresponding target-space backgrounds. By now, the conditions when they give rise to backgrounds that solve the supergravity equations are well understood [14][15][16][17][18][19]. Interestingly, they were also reformulated in the language of Double Field Theory (see e.g. [19][20][21][22]), which in turn can be used to identify their α -corrections, at least to the first order (i.e. two loops in the σ-model) in the α -expansion [23], see also [24][25][26] for the generalisation to other solution-generating techniques. One reason why they have attracted so much attention is that they can be used to deform a very large class of string backgrounds, including the maximally symmetric ones with AdS factors, which were paramount in the development of the AdS/CFT correspondence. The possibility of deforming such backgrounds (e.g. AdS 5 × S 5 or AdS 3 × S 3 × T 4 ) while preserving the underlying integrability is an exciting direction to identify possible generalisations of the AdS/CFT correspondence, with new gauge theories that are potentially exactly solvable (in the same sense as N = 4 is exactly solvable in the large-N limit).
In this article, we will focus on homogeneous Yang-Baxter deformations (and for simplicity we will always omit the prefix "homogeneous" from now on). This is a class of integrable deformations that can be understood as a generalisation of TsT deformations (which in the Double Field Theory language would be called β-shifts) [27]. While TsT deformations [28][29][30] can be applied whenever we have two commuting isometries of the "seed" undeformed background, Yang-Baxter deformations generalise this possibility to the non-abelian case. The central ingredient in the construction is a constant and antisymmetric R-matrix that must solve the classical Yang-Baxter equation on the Lie (super)algebra of isometries. When restricting to supercosets (for example the one of AdS 5 × S 5 ), this family of deformations has a rich list of possibilities, and everything is known about their supergravity embedding. In particular, the deformed background (which in general can have all NSNS and RR fields non-trivially turned on) is a solution of the type IIB supergravity equations if and only if the R-matrix solves a very simple and algebraic "unimodularity" condition [14,31]. 1 On the other hand, not much has been done to extend the methods of integrability, that were developed for the undeformed AdS 5 × S 5 case, to the Yang-Baxter deformed models. 2 For the very restricted subclass of "diagonal" TsT deformations (i.e. combinations of TsT deformations that involve only the Cartan isometries of AdS 5 × S 5 ), the solution to the spectral problem is understood in terms of a simple deformation of the Bethe equations and Thermodynamic Bethe Ansatz [36,37] or Quantum Spectral Curve [38]. But as soon as one goes beyond this special subclass, various obstructions do not yet allow for the application of the integrability techniques. This happens already for the rather simple class of "non-diagonal" TsT deformations (i.e. involving at least one non-Cartan isometry) [39].
A natural strategy to try to extend the methods of integrability to the full class of Yang-Baxter deformations is to do what worked in the case of diagonal TsT deformations, where one replaces the study of the spectral problem of the deformed σ-model with that of an undeformed model which, however, has twisted boundary conditions on the worldsheet. The two σ-models (the deformed one and the undeformed yet twisted one) are equivalent on-shell, i.e. there is a map that relates solutions of the σ-model equations of motion on the two sides. In the case of TsT deformations, the twisted boundary conditions for the alternative picture are very easy to write down, essentially because they are linear in terms of the fields [29,30,40]. For more generic Yang-Baxter deformations, writing down the equivalent twisted boundary conditions is more complicated. Previous expressions were written in terms of a path-ordered exponential, and the non-localities that this introduces were making it impossible to make any progress [41][42][43]. This problem was solved in [44], where the twisted boundary conditions were rewritten in terms of the convenient degrees of freedom, namely those of the twisted model itself. While the twisted boundary conditions can lead to complicated non-linear relations among the fields-a fact which is related to the non-abelian nature of the deformations-the expressions are local.
Among the methods of integrability that we can apply, the classical spectral curve has the right balance of complexity and computational power. The curve is defined as the Nsheeted Riemann surface obtained from the eigenvalue-problem of a monodromy matrix of size N × N [45]. Its power was originally applied in the context of string theory in AdS 5 × S 5 by reformulating the construction of classical string solutions as a Riemann-Hilbert problem [46][47][48][49][50], as all their information is encoded in terms of cuts and poles on the Riemann surface. It gained even more traction after it was realised that the classical curve can also be used to obtain 1-loop information related to these classical solutions [51][52][53][54][55]. This information is retrieved by adding microscopic cuts and poles to the Riemann surface, which will behave as quantum fluctuations around the classical solution. The classical spectral curve was soon generalised from AdS 5 ×S 5 to other AdS d backgrounds, see [56] and references therein, as well as deformed backgrounds, such as the flux-deformed AdS 3 × S 3 [57][58][59] or the Schrödinger background obtained from TsT [60].
In this article, we will use the equivalence between deformed models and undeformed yet twisted model to apply the method of the classical spectral curve and its semiclassical quantisation to a particular example of a Yang-Baxter deformation of AdS 5 × S 5 of non-abelian type. It belongs to the so-called class of Jordanian deformations, which make use of an sl(2, R) subalgebra of the Lie algebra of superisometries [9,10,61,62]. Importantly, the simplest version of Jordanian deformations (i.e. the one needing just the sl(2, R)) is non-unimodular. Therefore, the deformed background does not lead to a solution of the type IIB supergravity equations, but rather to one of the modified supergravity equations of [63,64]. It is however possible to cure this problem by constructing "extended" Jordanian deformations [65] that exploit a superalgebra of isometries containing sl(2, R). They lead to unimodular R-matrices, and to backgrounds that are solutions of the standard type IIB supergravity equations. Notice that the unimodular and non-unimodular versions share the same background metric and Kalb-Ramond field. 3 Interestingly, both cases are closely related to non-abelian T-duality [62,66,67]: a non-vanishing deformation parameter can be rescaled away in the background by a simple coordinate transformation, and the solution is equivalent to the background resulting from doing non-abelian T-duality on the two-dimensional Borel subalgebra of sl(2, R) (or of its supersymmetric extension if considering the unimodular case).
The article is organised as follows. In section 2, we review the construction of Yang-Baxter deformations of semi-symmetric space σ-models, and we specify the particular Jordanian deformation which we will study. After displaying its deformed metric and Kalb-Ramond field, we identify in section 2.2 the residual bosonic isometries that survive. In section 2.3 we then propose a coordinate system for which we explicitly identify the time coordinate. We prove that this system provides global coordinates of the deformed spacetime for every value of the deformation parameter by analysing its geodesic completeness. In section 3 we derive a particular pointlike string solution of the deformed σ-model, which can be interpreted as the analog of the BMN solution of AdS 5 × S 5 .
In section 4, we transfer from the deformed Yang-Baxter picture to the undeformed model with twisted boundary conditions. We review the results of [44] regarding the on-shell equivalence in section 4.1 and identify the symmetries of the twisted model in 4.2. In section 4.3, we show how the twist of the Jordanian model can be simplified. The transformation of our BMN-like solution to the twisted model is then performed in section 4.4, where we pay particular attention to the gauge ambiguities of the supercoset. Additionally, in this section we analyse also solutions of the twisted model in more generality and show, under certain assumptions, that they lead to extended string solutions with profiles involving Airy functions.
We start the program of the classical spectral curve and its quantisation in section 5 with a review of the construction of the classical curve in terms of the quasimomenta of the (general) twisted supercoset model on P SU (2, 2|4). We show in section 5.1 how the asymptotics of the quasimomenta give rise to local Cartan charges, involving the energy, of the symmetry algebra of our Jordanian twisted model. We then construct the algebraic curve for our BMN-like solution in section 5.2. In section 6, we study the semi-classical quantum corrections to the corresponding BMN-like quasimomenta by applying the recipe of [51,55] to our twisted case. We identify the frequencies of all possible excitations whose resummation, which we perform in section 6.3, gives the one-loop correction to the energy of our string solution. In section 6.2 we furthermore show that the charge identifying the twist of the boundary conditions is protected-it does not receive any anomalous correction-at one-loop. Finally, in section 7, we discuss how our results regarding the classical spectral curve and its semi-classical quantisation are insensitive to whether we consider the unimodular or non-modular version of our Jordanian deformation.
We end in section 8 with a conclusion and outlook. In appendix A, we show how the global coordinates defined in section 2.3 are related to the embedding coordinates of (undeformed) AdS. For completeness, we discuss in appendix B why the geodesic incompleteness of the Poincaré coordinate system persists along the deformation. In appendix C we identify the possible Cartan subalgebras of the residual isometry algebra of the deformed model. In appendix D we present the details regarding the calculation of the frequencies of excitations presented in section 6. We cross-check these results in appendix E by computing explicitly, in the picture of the deformed model, the frequencies of the quadratic bosonic fluctuations around the classical solution. 3 In the unimodular case these are supplemented by a dilaton and RR fluxes that solve the type IIB equations. In the non-unimodular case the RR fields are replaced by fields that do not satisfy the standard Bianchi identities, and the background is supplemented by a non-dynamical Killing vector that enters the modified supergravity equations.

A Jordanian-deformed background 2.1 The deformed σ-model and the background fields
In order to set up the notation, let us review the construction of the homogeneous Yang-Baxter deformation of a string σ-model on a semi-symmetric space. The starting point is a Lie supergroup G with a corresponding Lie superalgebra g that admits a Z 4 -graded decomposition, i.e. g = ⊕ 3 i=0 g (i) such that [[g (i) , g (j) ]] ⊂ g (i+j mod 4) . Here we are using [ [·, ·]] to denote the graded bracket on g. The action of the string σ-model on the supercoset G \ G (0) can be written as where J = g −1 dg ∈ g is the Maurer-Cartan form constructed from the group element g(τ, σ) ∈ G that depends on the worldsheet coordinates τ, σ. The linear operatord : g → g is a linear combinationd = 1 2 P (1) + P (2) − 1 2 P (3) of projectors P (i) , which by definition project onto the subspace g (i) . 4 Moreover, STr denotes the supertrace on g, which we use to obtain an ad-invariant graded-symmetric non-degenerate bilinear form on g. The worldsheet indices α, β in the action are contracted with the projector Π αβ (±) = 1 2 ( |h|h αβ ± αβ ), where h αβ is the worldsheet metric and τ σ = − στ = −1. Finally, we use √ λ 4π to denote the string tension. To construct the homogeneous Yang-Baxter deformation of the above action, one needs a linear operator R : g → g that is antisymmetric with respect to the supertrace STr(Rx y) = −STr(x Ry), ∀x, y ∈ g, (2.2) and that solves the classical Yang-Baxter equation ( is the metric on g and K AB its inverse, we can define the dual generators T A = T B K BA that satisfy 5 STr(T A T B ) = δ B A . We can then equivalently think of R as r = − 1 2 R AB T A ∧ T B ∈ g ∧ g where we use the graded wedge product x ∧ y = x ⊗ y − (−1) deg(x) * deg(y) y ⊗ x, and R A B = K AC R CB , so that Rx = STr 2 (r(1 ⊗ x)) in which STr 2 denotes the supertrace on the second factor of the tensor product. Naturally, we only consider R-matrices that do not mix even and odd gradings, meaning that where a, b are indices of even (bosonic) generators whileα,β are indices of odd (fermionic) generators.
Notice that (2.2) implies that R ab is antisymmetric while Rαβ is symmetric. With these ingredients the deformed action is now given by [8] where R g = Ad −1 g R Ad g (with Ad g x = gxg −1 ) and η is a deformation parameter. To have a real action, we take η ∈ R and R AB antihermitian, so that R ab is real while Rαβ has imaginary components.
The space defined by the image of the R-matrix, f = Im(R), will play a central rôle in our computations. We can check that it is a subalgebra of g as a consequence of the CYBE. We will denote its corresponding Lie group by F . Later we will use indices I, J for generators T I ∈ f, and indices i, j when restricting to the bosonic subalgebra. Using the metric on g induced by the supertrace, we can also define the dual to f, which we denote by f * . It is a subspace spanned by T I = T A K AI such that STr(T I T J ) = δ J I . So far we have reviewed the main ingredients -in particular, eqs. (2.2) and (2.3) -that make the action (2.4) an integrable deformation of the seed supercoset action (2.1) [6][7][8]. See later for more details. In general, extra conditions are necessary in order to make sure that the deformed model can be interpreted also as a consistent string σ-model. Since we are deforming a σ-model on a semi-symmetric space, the necessary and sufficient condition for the background fields of (2.4) to satisfy the type IIB supergravity equations, is that R also solves the unimodularity condition [14] When this is not satisfied, the background fields solve the more general equations of "modified supergravity" [63,64].
In this article, we are interested in deformations of the superstring on AdS 5 × S 5 and thus we will take g = psu(2, 2|4). The original supercoset is then P SU (2, 2|4) \ (SO(1, 4) × SO(5)). 6 Furthermore, we want to study deformations that are not interpretable as simple TsT transformations. This implies that we must deform the AdS factor of the background. Therefore, it will be sufficient to state only the commutation relations for the so(2, 4) ∼ = su(2, 2) conformal algebra, which is a subalgebra of g and corresponds to the isometries of AdS 5 . Given the Lorentz indices µ, ν = 0, . . . , 3 in 3 + 1 dimensions, we will use the Lorentz generators M µν , the translations p µ , the special conformal transformations k µ , and the dilatation d that close into the following commutation relations with the Minkowski metric η µν = diag(−1, 1, 1, 1).
An interesting and rich class of deformations that deform AdS are given by Jordanian ones [9,10,61,62]. Here we choose to work with 7 [65] 6 We will not need to give the full set of (anti)commutation relations of this superalgebra, nor provide an explicit matrix realisation of its elements. Conventions for these can be found in various places in the literature, see e.g. [68]. 7 Comparing to [65] one should rescale the deformation parameter there as η → −η/2. (2.9) The supercharges Q 1 , Q 2 complete the (anti)commutation relations to an N = 1 super Weyl algebra in one dimension. In this article, we will not need to specify them further, and more information can be found in [65]. For our convenience we have also introduced the parameter ζ. It is worth to stress that r solves the CYBE only for ζ = 0 or ζ = 1. When ζ = 0 the purely bosonic r-matrix is not unimodular, and it gives rise to background fields that solve only the modified supergravity equations; when ζ = 1, we have an additional "fermionic tail" such that r is also unimodular, and therefore one can obtain a standard supergravity background [14,65]. In this article we will find it interesting to consider both cases and compare them.
Let us identify the bosonic part of (2.4) (i.e. when setting fermions to zero) with Obviously, the background metric G mn and the Kalb-Ramond field B mn do not depend on ζ. In order to find an explicit expression, we can parameterise the group element as g = g a g s where g a ∈ SO(2, 4) parameterises the AdS spacetime in the undeformed limit, and g s ∈ SO(6) the sphere factor. We take (2.10) which gives 9 with x ± = (x 0 ± x 1 )/ √ 2 and ds 2 S 5 the metric of the undeformed 5-sphere. The background is supported by an F 3 RR-flux, which vanishes for η → 0, 10 and by the undeformed F 5 RRflux of AdS 5 × S 5 . Furthermore, the dilaton remains constant, Φ = Φ 0 , in contrast to many other known deformations. Let us recall that it is only for ζ = 1 that one can find this set of Ramond-Ramond fields and dilaton that complete the background to a standard supergravity one.
Notice that when η = 0, this deformation parameter can be reabsorbed by a simple coordinate redefinition, which is however singular in the η → 0 limit. This property is in fact a consequence of the fact that the Jordanian deformation is related to a non-abelian T-duality transformation of the original background [66,67].
the spacetime. The deformed action (2.4) is invariant only under a subset of these global transformations, in particular those which are preserved by the operator R At the level of the algebra, this condition translates to where TĀ denotes a generator of a (super)isometry. The subset span(TĀ) ⊂ g forms a subalgebra by means of the Jacobi identity. Using the above condition it is easy to check that the nonunimodular deformation (ζ = 0) preserves 16 superisometries, while the unimodular deformation (ζ = 1) preserves 12 superisometries. In the non-unimodular case, since the R-matrix acts trivially on the sphere, all of the so(6) isometries will be preserved by the deformation, and its Noether currents correspond as usual to the Maurer-Cartan forms g −1 s dg s , see e.g. [68]. In the unimodular case, instead, some of the so(6) isometries are broken by the additional fermionic tail of the R-matrix. This is reflected, for example, at the level of the RR fields, which are not invariant under so (6). Nevertheless, classical string configurations are insensitive to this fermionic tale, and at that level one can work as if there is so(6) invariance. More generally, in the deformed model the Noether currents corresponding to the residual (super)isometries are given by 11 JĀ ± = STr(J ± TĀ), where the worldsheet one-form A = A + dσ + + A − dσ − , here written in terms of the worldsheet lightcone coordinates σ ± = 1 2 (τ ± σ), is defined as From these Noether currents one may also define the Noether charges For a generic σ-model with background metric and Kalb-Ramond field, the Killing vectors kā associated to bosonic isometries can then be found from solving (see e.g. [69]) where L kā G = 0 and L kā B = dωā, with L the Lie derivative and ωā an arbitrary one-form.
Since the deformation only acts non-trivially on AdS, we will focus on this subsector for most of the remainder of this paper.

Global coordinates
In view of the applications of the Jordanian deformation of AdS 5 × S 5 in holography, it is important to identify global coordinates for the background for generic values of η. As it is well known, the Poincaré coordinates are globally incomplete already for the undeformed AdS spacetime. Thus, our first task is to find an appropriate coordinate transformation. A first obvious attempt is to transform to the usual global coordinates of AdS and then check if they remain global coordinates also when η = 0. However, we find that the deformed metric in these coordinates is complicated and does not exhibit any of the manifest residual isometries. We have therefore not pursued this coordinate system further, as it would not be useful for practical purposes.
In fact, similar issues were found in [70], which analysed the geometry of Schödinger spacetimes Sch z . Such geometries possess an anisotropic scale invariance (t, x i ) → (λ z t, λx i ) characterised by a critical exponent z > 1, where the case z = 1 corresponds to usual AdS. A global coordinate system different from the usual global AdS coordinates was then obtained for z = 2. Conveniently, the Jordanian deformed background (2.11) is extremely reminiscent of the Sch 2 case and, in fact, exhibits the same anisotropic scale symmetry. 14 It is therefore conceivable that a good candidate for a global coordinate system of our Jordanian background coincides with that found in [70]. The transformation from polar Poincaré coordinates to the global ones X m = (T, V, P, Θ, Z) is obtained from 15 (2.20) We can cover the full spacetime if we take the coordinates in the ranges T, V ∈] − ∞, +∞[, Θ ∈ [0, 2π[ and P, Z ∈]0, +∞[. The only coordinate that we allow to be periodically identified is Θ, with period 2π. These ranges have been derived from the relation between these coordinates and the embedding coordinates of AdS, which we discuss in appendix A. In the new coordinates the background fields (2.11) then read (2.21) Clearly, this coordinate system will be useful for practical purposes, since it manifestly displays three translational isometries, namely in T , V , and Θ. Because the Cartan subalgebra of the algebra of residual isometries is three-dimensional (see appendix C) this is in fact the maximal number of manifest translational isometries that a coordinate system could display. Explicitly, the Killing vectors introduced earlier read Following the above proposal, in the rest of this section, we will prove that the coordinate system X m = (T, V, X 2 , X 3 , Z) with (X 2 , X 3 ) = (P sin Θ, P cos Θ) gives rise to a geodesically complete spacetime.

On the generators of time translations
Before performing the full analysis of geodesic completeness of the deformed spacetime, let us first identify the possible Hamiltonians (generators of time translations) within the isometry algebra k in (2.18). We will look for generators of this kind that are part of the Cartan subalgebra of k, because in that case one can find an adapted coordinate system where the generator of time translations simply shifts one of the coordinates, which can be identified as time. where we leave a possible α > 0 coefficient for later convenience.
Identifying the possible generators of time translations is now a very simple task. One can construct a generic linear combination of the Cartan generators and demand that the corresponding Killing vectork m has a strictly negative norm in every region of spacetime. Notice, however, that p 0 + p 1 and M 23 are not timelike and, being central elements, it is easy to remove their contribution by a spacetime-coordinate redefinition. We can therefore simply compute the norm of the Killing vectors associated to the two possible Cartans of the sl(2, R) subalgebra of k. We find where in the latter case we took α = 1/2. While other choices of α > 0 are possible, we make this choice here to match with the expression (2k m (2) − 2k m (3) + k m 4 )∂ m = 2 √ 2∂ T found above for the generator of T translations. Whilst in case (I) the norm of the Killing vector is not everywhere negative (set e.g. T = 0), in case (II) we do have a strictly timelike Killing vector for generic values of the coordinates and of the deformation parameter η. The relevant Cartan subalgebra is thus that of class (II) with α = 1/2. Concluding, the coordinate T will be our preferred global time coordinate and the Hamiltonian is 16 Interestingly, when taking the undeformed limit η → 0, the above Hamiltonian does not reduce to something equivalent to the BMN generator of time translations [71], because it is possible to check that there is no inner automorphism of SU (2, 2) that relates H to p 0 + k 0 .

Analysis of geodesic completeness
In this section we will analyse whether the coordinate system X m = (T, V, P, Θ, Z), which describes Sch 2 spacetimes globally, are also well-defined global coordinates for the Jordanian deformation. This analysis is necessary because when comparing the Sch 2 metric, given in eq. (3.18) of [70], and the Jordanian metric, given in (2.21), we see that the Jordanian case has an additional term P 2 4Z 6 dT 2 , which may lead to pathologies. There are typically two common ways to analyse whether coordinates are global or not. First, one can try to find the embedding in a higher dimensional spacetime 17 and argue whether or not the embedding coordinates cover the full hypersurface. Another possibility is to consider geodesics γ(t), parameterised by a geodesic parameter t ∈ R, and investigate geodesic completeness of the coordinate system. We will follow the latter method. Before doing so explicitly, let us recall the criterions concerning geodesic (in)completeness and singular spacetimes.
A coordinate system is called geodesically complete if all geodesics γ(t) are defined for all values of the geodesic parameter t ∈ R, within the ranges of coordinates that parameterise the manifold. A spacetime is said to be geodesically incomplete if there exists a finite value t 0 of the geodesic parameter such that the geodesic γ(t 0 ) hits an extremal value allowed by the range of spacetime coordinates, and the geodesic γ(t) cannot be extended to t > t 0 . Hitting an (apparent) metric singularity would not be considered pathological -and therefore would not classify as geodesic incompleteness -if the geodesic "bounces back" (i.e. if it can be continued to allowed ranges of the spacetime coordinates at later t) or if the singularity is reached only in the limit t → ∞. 18 Rather than solving the five second-order geodesic equations for X m (t), we can make use of the five residual Killing isometries to perform a single integration and simplify the calculations 16 In the literature, in similar setups, sometimes the Hamiltonian is identified with d − M 01 ∼ h. While this is of course possible when considering the complexified algebra (where any choice of Cartan can be brought to d − M 01 , see e.g. (C.1)), here we insist on respecting reality conditions. Notice that the square of d − M 01 has a positive trace, and thus cannot correspond to a timelike Killing vector. In fact, if we take for the sake of the argument the case of group manifolds, the Killing vectors can be identified by k m a = Tr( m is the inverse of m b and dX m a µ Ta = g −1 dg. When moving to the identity of the group (i.e. at g = 1), the norm k m a kam of the Killing vector reduces to Tr(TaTa). Therefore, a necessary condition for ka to be timelike is that Tr(TaTa) < 0. 17 In the case of undeformed AdS5, this is of course R 2,4 , which is invariant under the conformal algebra SO (2,4). We show the relation between the coordinate system X m = (T, V, P, Θ, Z) and the AdS embedding coordinates in appendix A. 18 When at least one geodesic ends in a point p of the manifold, then p is a singularity. In this case, one should further study whether p is a physical singularity or just a coordinate singularity. It is only in the latter case that one speaks of geodesic incompleteness. To distinguish the nature of the singularity, one typically computes curvature scalars or tidal forces. The point p is an honest curvature singularity when at least one of these become infinite at p. If all of them remain finite, and the geodesics appear to end at p, then the coordinate system is considered to be incomplete. A fully general criterion about whether or not an apparent singularity is honest, however, does not exist. Fortunately, we will not find such singularities and therefore we will not need to perform this analysis.
considerably. In particular, recall that for each Killing vector field k m a , there is an independent charge Qā = k m a G mnẊ n , (2.26) which is conserved along the geodesics. 19 In other words, since the full system of geodesic equations can be rewritten asQā = 0 and we have as many Killing vectors as independent coordinates, the geodesics can be analysed by studying the system (2.26) rather than the usual geodesic equations. To do so, it will be useful to introduce the conserved quantities related to the manifest translational isometries in T , V and Θ, 20 (2.27) An interesting feature now emerges when analysing the system (2.26): instead of five first order differential equations for X m (t), one finds that (2.26) decouples into four first order differential equations and one algebraic relation among the coordinates, which restricts the geodesics to a hypersurface. We choose to decouple (2.26) aṡ subjected to the algebraic relation, which we will call the "hypersurface equation", In fact, this equation defines a circle in the (P, Z) plane with a varying (but bounded) radius depending on T (t), i.e. P (t) 2 + Z(t) 2 = R 2 (T (t)). 21 In addition, there is another useful constant of motion along the geodesics, namely with = 0, −1, 1 for null, timelike, or spacelike paths. Substituting the system (2.28) into (2.30) leads, in general, to a complicated differential equation forŻ subjected to the hypersurface equation.
Let us now analyse potential pathological regions of the deformed background. Notice that we can completely exclude the coordinates T, V, Θ from our analysis, as translations in these directions are manifest isometries and, thus, will not give rise to pathologies. From the background 19 Here the dot refers to derivation with respect to the geodesic parameter t. 20 The relation to the previously defined Noether charges is Q2 = 1 4 2 2QT + 2 √ 2QV + Q4 and Q5 = QΘ. 21 There are a number of conditions that we can derive for the values of the charges such that R 2 ≥ 0, which should be interpreted as reality conditions for Z and P . They should be further subjected to a condition ensuring also that T is real when T remains constant along the geodesic. fields (2.21) we see the only regions that can potentially be pathological are Z, P = {0, +∞}, as the (inverse) spacetime metric appears to blow up there.
First, let us consider the regions Z → ∞ or P → ∞, in which case the hypersurface equation plays an important role in the analysis. In particular, the hypersurface bounds the geodesic motion in Z and P , and forces both to a maximal absolute value, |Z|, |P | ≤ |R|, which is always finite for a finite value of the conserved charges Qā and for any finite value of t ∈ R. The only exception to this is when the charge Q V = 0, for which the dependence of P and Z drops out of (2.29) (equivalently, the radius R becomes infinite). However, we can argue that also in this case there is no pathology. First, for Q V = 0 the hypersurface equation forces T (t) to be a constant in t, which is consistent with its geodesic equationṪ = −Q V Z 2 . Hence, the coordinate time does not evolve in the geodesic parameter. This should therefore be an extreme spacelike path, as we also see by analysing the equation forŻ obtained from (2.30). Remarkably, for Q V = 0, this equation in fact greatly simplifies toŻ 2 (t) = Z(t) 2 which is immediately solved by where A is an integration constant. Reality conditions thus tell us that paths with Q V = 0 are only possible when they are spacelike or null (the latter being trivial in that case). The spacelike paths can in fact reach Z = ∞ (as well as Z = 0, see also later), but they do so only at an infinite value of the geodesic parameter t. We can thus conclude that Z → ∞ is not a pathological region. Concerning the behaviour at P → ∞, one finds that taking Q V = 0 would lead to paths that are not defined due to reality conditions. Given that for generic Q V the motion in P is bounded, we can again conclude that also the region P → ∞ is not pathological. In fact, even though P can become very large for small Q V , the region P → ∞ will never be reached by any geodesic in finite t.
Next, let us consider the Z → 0 region and analyse the small Z behaviour. Note that we can just focus on the system T, P, Z (the solutions for Θ and V will be given once solutions for T, P, Z are found). We will write Z(t) = λẐ(t) and expand the equations around λ = 0 up to O(λ), unless stated otherwise. From the first equation in (2.28) and (2.29), we will find that T and P are constant in t up to linear order in λ. In particular 22 with T 0 the constant value of time. 23 Now let us analyse the equation forŻ obtained from (2.30) and (2.28) for small Z. Up to subleading order in λ, where 2 and T (t = 0) = 0, we find which one can analyse order by order in λ. The leading term thus giveṡ Recall that P ∈]0, ∞[. 23 Note that if T0 is such that R 2 (T0) = 0, then (2.29) implies that P (t) will have a linear in λ contribution, namely P (t) = −Ẑ(t) 2 λ + O(λ 2 ). However, due to reality conditions, the latter is only possible whenẐ = 0 and thus P = O(λ 2 ). Hence, this is consistent with (2.32) at R 2 (T0) = 0. which in general would give an imaginary solution for Z, i.e.
. This means that we cannot get arbitrarily close to Z = 0 except if (i) η = 0 (the undeformed limit, in which case the spacetime is geodesically complete, see also appendix A), (ii) Q V = 0 (a case which we already analysed in generality above and where it was found that geodesics can reach Z = 0 only in an infinite amount of the geodesic time), and (iii) R 2 (T 0 ) = 0 (in which case P = O(λ 2 ) andẐ = 0 (see also footnote 23) and thus all the coordinates are constant in t). Finally, we must analyse consistency with the equation forṖ given in (2.28), which can be interpreted as RṘ − ZŻ = PṖ . From the leading solution for P in (2.32) we know thatṖ = O(λ 2 ). Therefore also PṖ = O(λ 2 ) and we can rewrite RṘ − ZŻ = PṖ as . This is consistent with what we have found previously, i.e. eq. (2.34), requiring We can thus conclude that the geodesics will never reach Z = 0 by an evolution in t and that this region is not pathological. Finally, analysing the region P → 0 using a similar strategy does not give us a conclusive answer. However, this is of course only the apparent pathology of polar coordinates: neither the metric nor its inverse is singular for X 2 = X 3 = 0.
The arguments above prove that the coordinate system X m = (T, V, X 2 , X 3 , Z) is globally well-defined for any value of the deformation parameter. We will however prefer to work in terms of X m = (T, V, P, Θ, Z), in which the shift isometry in Θ is manifest. This concludes our discussion and, combined with the previous section, shows that the coordinate T is our global time coordinate.

BMN-like solution of the σ-model
In this section, we give the derivation of a particular pointlike string solution to the equations of motion of our Jordanian deformation. As we will show in section 5, it will lead to relatively simple quasimomenta. For us it will be the analog of the BMN solution valid in AdS 5 × S 5 , and we will call it a BMN-like solution.
For a general bosonic string configuration X m (τ, σ), recall that the equations of motion of the action can be written in conformal gauge as which is a generalisation of the geodesic equation of a point particle. Here, Γ m nl are the usual Christoffel symbols, and H mnl are the components of the torsion 3-form H = dB. Interestingly, it can be checked that these equations evaluated for the Jordanian deformation admit a consistent truncation on x 2 (τ, σ) = x 3 (τ, σ) = 0, or equivalently P (τ, σ) = Θ(τ, σ) = 0, 24 for any value of the deformation parameter.
Let us now consider the following σ-independent ansatz on the coordinates: with a m and b m constant variables which will be determined upon imposing (3.1). Furthermore, we will work on the consistent truncation P (τ, σ) = Θ(τ, σ) = 0, equivalently we set a P = a Θ = b P = b Θ = 0. Since there is no σ-dependence, this pointlike string will not couple to the Kalb-Ramond two-form, and its equations of motion are therefore simply the geodesic equations. Solving them for the variables a m and b m while imposing that fields are real singles out the solution 25 while the other variables remain free. Because of the manifest isometries in T , V and Θ we can however set b T = b V = b Θ = 0 without loss of generality. In summary, our pointlike string solution thus evolves as This solution is in fact exactly the same as the one studied for the Schrödinger Sch 5 × S 5 background (at least, in the deformed AdS subsector) in [39]. The reason is that on the consistent truncation P = 0, the Jordanian metric given in (2.11) coincides precisely with the Schrödinger metric. Given that this solution is pointlike, the isometry charges defined in (2.26) will coincide with the Noether charges defined in (2.16). In terms of (2.26) and (2.27), we get Note that these charges are independent of the deformation parameter. As a consistency check one can furthermore verify that the hypersurface equation (2.29) is satisfied, as it should.
The solution we described here is further subjected to the Virasoro constraints which will couple the deformed AdS with the undeformed sphere subsector. For Yang-Baxter deformations the Virasoro constraints, obtained by varying the action (2.4) with respect to the worldsheet metric, can be written as [8] STr where we defined the worldsheet projections A α (±) = Π αβ (±) A β of the one-form A defined in (2.15). In conformal gauge, the Virasoro constraints would read STr(A (2) ± ) = 0, and parametrising the group element as before as g = g a g s , we can identify an AdS and sphere contribution as where we used that the supertrace is defined as the trace in the AdS algebra while it is minus the trace in the sphere algebra, and where A a± and A s± are the projections of A on the AdS and sphere subalgebra respectively. On our solution, (3.4) the Virasoro constraints then require the following relation between sphere and AdS variables: (3.8) 25 When one does not impose the consistent truncation P (τ, σ) = Θ(τ, σ) = 0, but still assumes the pointlike ansatz (3.2), there is one other real solution namely aZ = aP = 0, aV = − while the other variables remain free. Carrying the analysis of this solution through (again using the isometries to set bT = bV = bΘ = 0) one will find the associated quasimomenta to be quite complicated and we will therefore not consider this possibility further.

Map to an undeformed yet twisted model
Starting from this section, for simplicity we specify our discussion to the non-unimodular case. We therefore consider a deformation generated by an R-matrix as in (2.7) with ζ = 0. We remind that this case gives rise to background fields that do not satisfy the type IIB supergravity equations, but only the "modified" supergravity equations. This study will be however preparatory to the unimodular case, to which we will go back in section 7.

Review of the twist
Homogeneous Yang-Baxter deformed σ-models are known to be on-shell equivalent to undeformed yet twisted models, see e.g [42,43]. This is due to the fact that it is possible to identify the Lax connections of the two models, while achieving a map of the equations of motion of the two sides. In other words, there is an on-shell relation between the σ-model on the η-dependent deformed background and the σ-model on the undeformed one, and there is a one-to-one map of the solutions to the equations of motion of the two sides. This map, however, implies that if we choose to have periodic boundary conditions on the worldsheet for the deformed model (which may be motivated by the choice of studying closed strings on the deformed background), then we must generically have twisted boundary conditions for the undeformed one. This situation is a generalisation of what happens for TsT transformations, where the twisted boundary conditions can be derived and written explicitly in a local and linear way [29,30]. From now on, we will use tildes on all objects of the undeformed yet twisted model. For example, if g ∈ G denotes the group element used to construct the periodic Yang-Baxter model, we will useg ∈ G to denote the group element of the η-independent twisted model. The map between the two sets of degrees of freedom can be written as where F ∈ F is called the "twist field" and is responsible for translating between the two formulations. Recall that F is the Lie group of f = Im(R). In these expressions we are also allowing for a possible gauge transformation, implemented as a right-multiplication by h ∈ G (0) . This possibility is always present in the (super)coset case, given the local G (0) -invariance of the σ-model action. The twist field F is fixed by demanding that the Lax connections of the deformed and undeformed models can be identified (see also (5.1)). This amounts to require that the "modified current" A ± of the deformed model defined in (2.15) can be identified with the Maurer-Cartan currentJ ± =g −1 ∂ ±g of the undeformed model, up to a possible gauge transformation h, so that It is important to remark that the Virasoro constraints are compatible with the on-shell identification. In fact, it is easy to see that using (4.2) the Virasoro conditions for the deformed model written in (3.6) give STr J α(2) = 0, which are precisely the Virasoro conditions for the undeformed model. This is actually a crucial point to claim on-shell equivalence in the case of a string σ-model.
The fact that g has periodic worldsheet boundary conditions now indeed implies thatg is not periodic in σ ∈ [0, 2π] but rather satisfies the following twisted boundary conditions where W = F −1 (2π)F(0) is called the twist. 26 Both the twist W and the twist field F are gaugeinvariant under the right-multiplication by h. Moreover, it is clear that it is always possible to use a compensating gauge transformation to fix h = 1 in (4.1). In this sense, only W controls the physically relevant twisted boundary conditions, and we can talk about a left twist only. At the same time, we want to include the possibility of having a non-trivial h = 1 because it may be important in certain calculations. It becomes crucial, for example, if we want to insist on using the same parametrisation for g andg in terms of coordinates. 27 The reader may worry that the need of determining h will introduce unnecessary complications in the calculations, but in section 4.4.1 we will explain how to efficiently by-pass it and, if needed, easily derive the relevant h.
By identifying the Lax connections, it is in principle straightforward to write down an expression for F and W in terms of the degrees of freedom g of the deformed model, see [42,43]. In addition to being an unnatural expression for the twisting of the undeformed model, that solution is however written in terms of a path-order exponential. It is therefore plagued by non-localities that make it unusable for practical purposes. In [44] this problem was solved by rewriting the solution for W in terms of the degrees of freedomg instead. Depending on the choice of the R-matrix, the twisted boundary conditions may correspond to complicated nonlinear relations among the coordinates evaluated at σ = 2π and σ = 0 (a fact which is related to the non-abelian nature of the deformation) but W itself takes a simple expression in terms ofg that only involves single (as opposed to nested) integrals. Importantly, W is constant and, therefore, it is written in terms of conserved charges of the twisted σ-models. These do not need to be Noether charges, as they may correspond to some hidden symmetries. We refer to [44] for a more detailed explanation on how to construct W from a given R-matrix, where the derivation was done for the case of the PCM as well as that of (super)cosets on (semi)symmetric spaces. In fact, the generalisation from the former to the latter only involves some decorations of the formulas with certain linear combinations of projectors on the Z n -graded subspaces of g (with n = 2, 4). In particular, the derivation of the Jordanian twist of section 4.3 of [44] is still valid also in the supercoset case, and it is therefore given by 28 which takes values in the Lie subgroup F , and where we have the following expressions for the conserved charges As anticipated, the above expression differs from the one valid in the PCM case (which can be found in [44]) just by the insertion of the linear operatordg−1 = Adgd Ad −1 g , whered was defined 26 Notice that although g,g, h, F depend on both worldsheet coordinates τ and σ, when writing the boundary conditions we will omit the explicit τ -dependence and write down only the values that σ takes, to have a lighter notation. 27 For example, as we will do later, this happens if we parametrise g as in (2.10) in terms of coordinates {x 0 , x 1 , ρ, θ, z} andg in the same way but with coordinates {x 0 ,x 1 ,ρ,θ,z} (or equivalently in terms of the corresponding global coordinates under the transformation (2.20)). In this case the on-shell identification condition requires h = 1. A gauge transformation can be used to reabsorb h in g org, but that would imply that the parameterisation of the coset representative would not be given anymore by (2.10). 28 Compared to [44], here we use the notation Q = log QA and q = QB.
in section 2.1. Note that also Y is gauge-invariant under the local right-multiplication ofg by h. The above formula for the Jordanian twist W will be our starting point for the construction of the classical spectral curve of our model, as well as for the computation of the one-loop quantum corrections to its spectrum.

Manifest symmetries of the twisted model
Before continuing with the study of the Jordanian deformation and the corresponding twisted model, it is useful to analyse the manifest symmetries of the latter and understand how they are related to those of the former, which were discussed in section 2.2. The following discussion is valid for the whole family of homogeneous Yang-Baxter deformations, so we do not need to specify to the Jordanian case. We give the presentation for the supercoset, but the same considerations can be made already in the simpler PCM setup.
Let us defineQ where 29Ĵ In the undeformed and periodic (i.e., W = 1) supercoset σ-model, (4.7) are the conserved Noether charges and (4.8) the Noether currents corresponding to the (left) G invariance of the action [68]. In fact, the Noether charges Q of the deformed model defined in (2.16) agree withQ in the limit η → 0. In the twisted case (W = 1),Q in (4.7) is not necessarily a conserved charge. In fact, although the Noether current is still conserved (∂ αĴ α = 0) simply as a consequence of the equations of motion, the twisted boundary conditions may in general break the constancy in time ofQ, since where (4.3) was used. In particular, if we consider the projection ofQ along a given generator It is now clear that if some TÃ ⊂ T A commutes with the whole subalgebra f, thenQÃ is a conserved charge for the twisted model. Note that span(TÃ) forms a subalgebra of g. For the specific Jordanian deformation we consider in this paper, we have that the manifest bosonic symmetries are Importantly, if TÃ commutes with f, then it is also a symmetry of the R-matrix and of the deformed model because it automatically solves (2.13). 30 Therefore, manifest symmetries of the twisted model correspond to isometries of the YB model.
It is worth stressing that the opposite is not always true: one may have an isometry of the YB model satisfying (2.13), which is not a manifest symmetry of the twisted model because T A does not commute with f. In our case of a Jordanian R-matrix, an example of this kind is given by e = (p 0 + p 1 )/ √ 2. In fact, e is a symmetry of the R-matrix and therefore an isometry of the YB model, but it does not commute with f, soQ e = STr(Qe) is not a conserved charge (in particular W −1 eW = e). 29 In conformal gauge, it readsĴτ = Adg J (2) Notice that when T A commutes with f, the expression for the charges of the twisted model agree with those of the Noether charges of the deformed one, and they are related by the on-shell map (4.2). In fact, let us start from where J was defined in (2.14), so that Q A = STr(QT A ) is a conserved Noether charge of the deformed model already defined in (2.16) if T A satisfies (2.13). Then it is straightforward to see that According to the discussion above, all manifest symmetriesQÃ can be written as isometries QĀ of the deformed model, but not all isometries QĀ can be written as a manifest symmetryQÃ, so that schematically {QÃ} ⊂ {QĀ}. Because of the classical equivalence between the two models, we expect that the isometries of the deformed model that are not manifest symmetries of the twisted model should correspond to a more general family of hidden symmetries, which is always there for integrable models.

A simpler twist
In general, there is not a unique way to write the twist W that controls the boundary conditions. We should rather think in terms of equivalent classes of twists. For example, ifg satisfies the twisted boundary conditions (4.3) and we define a new fieldg related tog by the simple field redefinitiong = ug with u ∈ G constant, then it follows that the new field satisfies the boundary conditionsg The fieldsg andg are equivalent representations of the same model in terms of different variables, so that we are free to choose whether we want to work withg org . Therefore, we should not think of the twisted boundary conditions as being identified by a unique expression W , since W = uW u −1 belonging to the same equivalence class actually describes the same physics.
Notice that if W ∈ F then in general W belongs to F = uF u −1 , in other words it belongs to an adjoint orbit of G on F . The relation to the degrees of freedom of the YB model now is g = Fu −1g h. In [44] the solution for the twist field was constructed by fixing the initial condition F(0, 0), and while the above field redefinition breaks this choice, it can be restored by the compensating field redefinition g = ug, so that we can write (4.14) Notice that in general the YB model written in terms of these new degrees of freedom g will be constructed not in terms of R but rather by the R-matrix R = Ad u R Ad −1 u . This is obviously a physically equivalent antisymmetric solution of the CYBE. 31 It is worth remarking that the Lax connections constructed out ofg andg are equal because u is assumed to be constant, and therefore also the two corresponding monodromy matrices (and not just their eigenvalues) are equal to each other.
We now want to exploit the above possibility to define a new twist W for the Jordanian deformation, that has the advantage of having a simpler expression compared to W . We start from (4.4) and we rewrite it in a "factorised" form by using the following identities coming from the Baker-Campbell-Hausdorff (BCH) formula: where we have to identify which is consistent with [h, e] = e. Now, using the above identities we can write where we defined B ≡ e s B /(e s − 1). For consistency with [A, B] = sB, we must have s = s . 32 Notice that in our case we simply have B = −qe. To conclude, the twist can be written as where W = exp(Qh), and u = exp(−qe). (4.19) This is remarkable, because the new twist is identified by the Lie algebra element h only, and all information regarding e is lost in theg -model. The above arguments also imply that the spectrum of both theg-and theg -models do not depend on the charge q, at least at the classical and one-loop level. Naively, the charge q controls the boundary conditions of the twisted model, but as we just showed, this dependence can be removed by a simple field redefinition. In the next section, we will see this independence of the spectrum on q more explicitly. Notice that in the derivation of factorising the twist there is a subtlety when Q → 0, because then some formulas diverge. We will come back to this point in section 7, where we will discuss the factorisation of the twist in a more general set-up that encompasses also the case of unimodular deformations.

Map from the classical solution of the deformed model
In this section, we return to our discussion of the particular point-like solution (3.4) of the Jordanian deformed model and show how it can be mapped to a twisted solution of undeformed 31 While the above considerations are valid for a generic u ∈ G, if u is also a symmetry of R then R = R, and both F and W are still elements of the unprimed F subgroup, see also [44]. It turns out that the u that we are about to consider is of this type. 32 In fact, as soon as we have [A, B] = sB and we define B = αB (for any non-vanishing α) while requiring that [A, B ] = s B , it is then obvious that we must have s = s. AdS 5 . This means that we need to apply the reverse logic of [44] and derive the twist field F starting from the deformed variables. As we mentioned in 4.1, its expression is then in general a complicated path-ordered exponential, but on our specific simple solution (3.4) it can however be evaluated explicitly. In this case the easiest way to obtain F is actually to work with the associated differential system, namely which is obtained from identifying the Lax connections of the two models. When using the initial condition F(0, 0) = 1, we have F ∈ F and thus in full generality we can write (4.21) Solving the differential equations (4.20) on the solution (3.4) is then rather straightforward, and after imposing F(0, 0) = 1, we obtain Note that the twist field F reduces to the identity element in the undeformed η → 0 limit, as it should.
Given the solution for F, the transformation to the undeformed variables is done in principle by employing the map g = Fgh, although this may be complicated by possible gauge transformations. In fact, in our case, if we insist on parameterising both g andg in the same way, then we cannot relate the two models if h = 1. Guessing the appropriate gauge transformation is not straightforward, if not unsatisfactory, and thus the translation between deformed and twisted variables begs a gauge-invariant procedure in the case of (super)coset models. To do this, we introduce the gauge-invariant objects where t denotes usual matrix transposition and K satisfies hKh t = K for h ∈ SO(1, 4) (see e.g. §1.5.2 of [68]). 33 Then indeed G andG are invariant under the local right-multiplications g → gh andg →gh respectively. Using these objects, the map g = Fgh can then be rewritten in the following gauge-invariant way: and also the twisted boundary conditions (4.3) can be rewritten as None of these formulae now suffer from any possible complications arising from gauge ambiguities.
Using (4.25), it is now very easy to translate to the undeformed variables. We find, forg parametrised in terms of the global coordinates, that (3.4) is mapped tõ 33 In the matrix realisation of su(2, 2|4) that is typically used (see e.g. [68]) one takes whileΘ remains free. The latter is however only a redundancy of the chosen parametrisation and we may setΘ = 0 without loss of generality. Having already determined g,g, F, the unknown gauge transformation h can now be simply derived as h =g −1 F −1 g. Furthermore, knowing F we can also obtain the twist W of the solution, i.e.
Consequently, we can identify the Jordanian charges Q and q from (4.4) giving precisely Thus, for this particular classical solution, there is no need to simplify the twist following section 4.3, because q is already zero. Translating the twisted boundary conditions forg in terms of explicit expressions for the associated coordinates, we see that all of them are periodic exceptZ, which satisfiesZ(2π) = exp(Q/2)Z(0).
There are now several consistency checks of the calculations just performed. First, one can verify that (4.26) holds, while the twisted boundary conditions forg have a non-trivial gauge transformation h in their expression. Secondly, knowing the solutiong, one can calculate the gauge-invariant object Y defined in (4.6). We find which satisfies Y (0, 0) = 0, consistent with F(0, 0) = 1. Comparing now the Jordanian charges Q and q obtained from (4.5) with the expressions (4.29) obtained directly from the twist field, we can check that the two results agree. At last, one can verify that (4.27) is a solution to the equations of motion of the undeformed σ-model.

More general classical solutions of the twisted model
We now want to explore the question of finding more general solutions of the undeformed twisted model. Although the boundary conditions will depend on the deformation parameter η, the equations of motion will not, and thus one may hope to find more solutions than (4.27). To maintain some level of simplicity, however, we will continue to work on the consistent truncatioñ P (τ, σ) =Θ(τ, σ) = 0 and assume that the global coordinate timeT evolves linearly with the worldsheet time τ , i.e. as beforeT (τ, σ) = a T τ .
On these assumptions, let us first determine the most general solution to the twisted boundary conditions (4.26) for the Jordanian twist W (4.4) in terms of the coordinatesZ(τ, σ) and V (τ, σ). It is not difficult to show that in general they must satisfỹ (4.31) As before one will have, however,g(2π) = Wg(0) implying that the group element used to construct the Lax connection of the twisted model must be dressed with a gauge transformation.
Next, we compute the equations of motion of the undeformed model, and find that they reduce to the following three differential equations: andṼ 0 ≡Ṽ 0 (σ) an unknown function of σ. Now (4.34) holds two equations, one at O(τ 0 ) and one at O(τ ), namely respectively. They can be rewritten as where the the prime denotes derivation to σ,Ṽ 0 ≡ ∂ σṼ0 andṼ 1 ≡ ∂ σṼ1 . Since all the objects involved only depend on σ, we can perform one integration to find where c 0 and c 1 are integration constants. Only the latter equation of (4.38) is now a differential equation forZ, and once we obtain its solution, the solution forṼ 0 (and consequently ofṼ ) is obtained immediately by performing a straightforward integration of the first equation of (4.38). First note that we can rewrite the expression forṼ 1 as Then the second equation of (4.38) becomes Z Z −1 = −c 1 which means we can do another simple integration to findZ with c 2 another integration constant. Here the experienced reader may recognise the Airy (or Stokes) differential equation that has as linearly independent solutions the Airy functions Ai(x) and Bi(x). Concluding, within our assumptions given in the beginning of this section, the most general solution for the variableZ of the undeformed model is before imposing any boundary condition. These are oscillatory functions that at a certain turning point become exponential. In principle, we can now also write down an explicit solution forṼ , as mentioned before, however we will refrain from doing so because it is not particularly enlightening. Nevertheless, note that, because of the additional integration performed to obtaiñ V 0 , the system has one additional integration constant (say c 3 ). Before imposing boundary conditions, the full solution thus has in total 7 free parameters, namely a T , c 0 , c 1 , c 2 , c 3 , α and β.
Imposing the twisted boundary conditions (4.31) gives three equations forZ,Ṽ 1 andṼ 0 , obtained order by order in τ . The equation forZ fixes an expression for Q while the one for V 0 fixes q. The remaining condition onṼ 1 becomes a constraint between the parameters, 34 reducing the number of free parameters to 6. Two further constraints between the parameters may arise when comparing the expressions of Q and q with those obtained from (4.5) and (4.6), as these must of course coincide for consistency. We have not looked into this further. 34 On the solution forZ andṼ1 the constraint simply readsṼ1(2π)Z(0) 2 =Ṽ1(0)Z(2π) 2 .
We close this section by discussing the special c 1 = 0 case. From (4.40) it is obvious that the most generic solution forZ then is simplỹ Depending on the sign of c 2 these are exponential or oscillatory functions. The simple solution given in (4.27), which is the one we study in the remainder of this paper, falls into the former class. It has α = 0, β = b Z and In general forṼ we then get ηa T + c 3 which seems more general than (4.27). Consistency with the boundary conditions (4.31) however sets c 0 = 0 and c 3 = q. Here we see explicitly that a possible contribution from the q charge can be eliminated by a field redefinition, which is in fact a translational isometry in the V coordinate.

Classical spectral curve
Knowing a Lax connection that is flat on the σ-model equations of motion allows one to apply the methods of classical integrability. While in principle it is possible to work from the point of view of the Yang-Baxter deformed σ-model, we find it more convenient to work from the point of view of the undeformed yet twisted σ-model, which is on-shell equivalent to the former. In fact, this is in analogy to how the classical and quantum spectrum of TsT deformations is obtained, see e.g. [30,36]. Here, we want to give a brief summary of facts that are valid in the undeformed and periodic case and that carry over also to the twisted one. For more details, we refer to the original literature developed in the periodic case, see e.g. [47].
The Lax connection of the twisted σ-model on the AdS 5 × S 5 supercoset is 35 which agrees with the Lax connection of the undeformed and periodic case [72,73] because boundary conditions play no role in its construction. Here,J (i) = P (i)J are projections of the currents on the Z 4 -graded subspaces of psu(2, 2|4), and γ αβ = |h|h αβ . From the Lax connection one can construct the monodromy matrix with z ∈ C the spectral parameter. In the undeformed and periodic case, the eigenvalues λ(z) of Ω(z, τ ) are conserved because L τ (2π) = L τ (0) as a consequence of the periodic boundary conditions. When considering twisted boundary conditions as in (4.3), L τ remains periodic as long as we take into account the compensating gauge transformation. Therefore, the eigenvalues λ(z) of Ω(z, τ ) are again time-independent, and encode the infinite number of conserved quantities (when expanding in powers of the spectral parameter z), rendering the model classically integrable. The eigenvalues depend analytically on z except at z = 0, ∞ (where the poles of the Lax connection imply essential singularities for λ(z)) and at the points where two (or more) eigenvalues degenerate. Instead of working with the eigenvalues, it is simpler to work with the 35 In conformal gauge, this is just L±(z) =J The Lax connection of the Yang-Baxter deformed model is obtained simply by replacingJ by A. quasimomenta p(z), defined as λ = e ip , as the essential singularities of λ at z = 0, ∞ become poles. For the quasimomenta, we use the same notation as for the undeformed and periodic case {p 1 (z),p 2 (z),p 3 (z),p 4 (z)||p 1 (z),p 2 (z),p 3 (z),p 4 (z)}, (5.3) wherep i (z) with i = 1, . . . , 4 are the quasimomenta of the (deformed/twisted) AdS factor corresponding to SU (2, 2) ⊂ P SU (2, 2|4) andp i (z) with i = 1, . . . , 4 are those of the (in this case undeformed/untwisted) sphere factor corresponding to SU (4) ⊂ P SU (2, 2|4). When two eigenvalues of the same type (i.e. bothp or bothp) degenerate, they give rise to branch points of square-root cuts, that correspond to collective bosonic excitations. When two eigenvalues of different type (i.e. onep and the otherp) degenerate, they give rise to poles, that correspond to fermionic excitations. The quasimomenta are everywhere analytic except at these bosonic branch points and fermionic poles, and at z = 0, ∞. Because of its definition, the quasimomentum has a 2πZ ambiguity, but this ambiguity is lost when considering dp, which can be thought of as an abelian differential. The spectral problem can therefore be reformulated in terms of the classification of the admissible algebraic curves with abelian differential dp. These curves can be understood as a collection of 4 + 4 sheets (one for each of the quasimomenta) connected by the above cuts and poles. We follow the usual conventions and employ an alternative useful parameterisation of the spectral parameter, taking z = 1+x 1−x so that the points z = 0, ∞ correspond to x = −1, +1.
Let us continue with the list of the facts that are valid in the undeformed and periodic AdS 5 × S 5 case and that continue to be valid also in the twisted one. The Lax connection is supertraceless, which implies that the superdeterminant of the monodromy matrix is 1. In turn, this implies the "supertraceless condition" 36 Moreover, the Z 4 automorphism of psu(2, 2|4) is still implemented by sending z → iz or equivalently x → 1/x at the level of the Lax connection. We will call this the "inversion symmetry". At the level of the quasimomenta, it implies [47] where k = (1, 2, 3, 4) and k = (2, 1, 4, 3). Notice that this permutation is related to the choice of charge conjugation matrix in psu(2, 2|4). For the sphere quasimomenta, the inversion symmetry allows for a possible shift with k = (1, 1, −1, −1) and m ∈ Z, which is related to winding around the S 5 space. For the AdS quasimomenta, this possibility is not allowed by the requirement of absence of winding for the time coordinate.
Combining the supertraceless condition, the inversion symmetry, and the Virasoro constraints, one has a synchronisation 37 of the poles of the quasimomenta at x = ±1 also in the twisted case In principle one can modify the above equation by adding a shift by 2πn with n ∈ Z. This can be removed by the freedom of shifting the quasimomenta by any integer multiple of 2π. In the periodic case one can use this freedom so that at large x the quasimomenta go like p(x) ∼ O(1/x). In the twisted case the twist can introduce additional finite terms in the strict x → ∞ limit. However, equation (5.4) remains valid because these finite shifts cancel each other in the above equation, as a consequence of det W = 1. 37 When we later study one-loop corrections, we will slightly relax this synchronisation of the poles to p(x) ∼ diag(α (±) ,α (±) ,β (±) ,β (±) |α (±) ,α (±) ,β (±) ,β (±) ) x±1 + O(x ± 1) 0 , because in that case one should consider also fermionic contributions and the tracelessness condition of su(2, 2) and su(4) of the purely bosonic classical case is no longer enforced.
The bosonic cuts and the fermionic poles must respect the inversion symmetry, which means that in both cases they must come in even number. For example, we can write the gluing condition of two quasimomenta on a cut (or more generally on a collection of cuts)Ĉ (k,l) orC (k,l) aŝ and similar equations must hold for the cuts which are the images ofĈ (k,l) andC (k,l) under x → 1/x. Similarly, if there is a fermionic pole at x * (k,l) , by inversion symmetry there will be another pole for quasimomenta at 1/x * (k,l) . Because of this property, we can declare a "physical" region as the one for which |x| > 1. The cuts in the physical region (i.e. half of the cuts in the pair of cuts connected by inversion symmetry) will be considered the "fundamental" ones.
To conclude, let us also say that we can still define the filling fractions for sphere cutsC (k,l) connecting the sheetsk andl as where the integration path is defined by a closed loop that encircles the fundamental cutC (k,l) . Similarly, we define the filling fractions for AdS cutsĈ (k,l) connecting the sheetsk andl aŝ where the integration path is defined by a closed loop that encircles the fundamental cutĈ (k,l) . In the periodic case these filling fractions are identified as the action variables of the integrable model [48], and they are taken to be quantised as integers in a semiclassical treatment [47]. In the twisted case we can do the same, because the identification of the definition of the filling fractions comes from the position of the poles of the Lax connection, and these are still at x = ±1. The filling fractions can also be written in terms of the so-called twist function [74], and this is invariant under an homogeneous Yang-Baxter deformation [42].
In the periodic case, the expansion of the monodromy matrix around z = 1 (or equivalently x = ∞) yields (at the lowest non-trivial order) the conserved Noether charges coming from the psu(2, 2|4) superisometry. In the twisted case, this picture is modified by the presence of W . In fact, after considering a gauge transformed Lax connection L σ =gL σg −1 − ∂ σgg −1 , one finds 38 which, when writing z = 1 + , leads tõ whereQ was defined in (4.7). In the periodic case (W = 1),Q is the Lie-algebra valued conserved Noether charge, and one can use a P SU (2, 2|4) automorphism to put it in the Cartan subalgebra. After doing that, the monodromy matrix is diagonal at least to order and the eigenvalues are therefore written in terms of the Cartan charges. In the twisted case, the monodromy matrix is non-trivial already at order zero in the -expansion, and the quasimomenta will depend also on the eigenvalues of W . Generically,Q on its own will not be conserved in the twisted case. As we will see in the next section, when computing the eigenvalues of the above expansion they are not necessarily written in terms of the standard Cartan charges of psu(2, 2|4): there is a "polarisation" induced by the presence of W , so that instead the quasimomenta are written in terms of the conserved Cartan charges of the algebra of symmetries of the twisted model (which we discussed in section 4.2). The symmetries of the twisted model form a subalgebra of psu(2, 2|4), and the corresponding choice of Cartan subalgebra may not be equivalent to the one of the periodic case. We will now be more explicit about the computation of the eigenvalues of the monodromy matrix around z = 1.
Let us emphasise the great advantage of making the above discussion from the point of view of the twisted rather than deformed model. In the latter case the expansion of the monodromy matrix around z = 1 would give rise to complicated non-local expressions that would be very hard to work with.

Asymptotics of the quasimomenta
To obtain the expression for the quasimomenta around z = 1, we must compute the eigenvalues of the expression (5.11). Since the Jordanian twist W given in (4.4) is diagonalisable, we can restrict our discussion to diagonalisable matrices. The explanation that we are about to give is actually valid for a generic diagonalisable twist.
The diagonalisation of (5.11) to order follows the same route as the diagonalisation of a Hamiltonian in perturbation theory of quantum mechanics. Let us rephrase that procedure in a different language. First, consider the diagonalisation of W −1 as D = S −1 W −1 S. To be as general as possible, we allow eigenvalues to be degenerate From simple reasoning in linear algebra, we know that to compute the correction to the eigenvalues of D, we must first project and then compute the eigenvalues of X [i] . If there is a block corresponding to eigenvalues all of which have multiplicity 1, then the above is equivalent to projecting on this non-degenerate block, and the diagonal part of this block will give the correction to the eigenvalues.
With this procedure, it is easy to compute the expansion of the eigenvalues. In our case, we find in the AdS sector where andQ i = STr(QT i ). The explicit expression for q given in (4.5) can be written precisely as ) which implies C 2 = 0. This fact is very important. It means that the only components ofQ that contribute to the eigenvalues of the monodromy matrix are those which correspond to the symmetries of the twisted model, see (4.10). In fact, we managed to get rid of both d + M 01 and p 0 + p 1 , which do not have this property. We remind that all of the symmetries of the twisted model are also isometries of the YB model, and in terms of the isometric charges Qā = STr(TāQ) with Q defined in (4.11) and the isometric generators Tā in (2.18), we can write Again, the combination Q2 + Q3 does not appear: it would be an isometry of the deformed model, but it is not a symmetry of the twisted model.
The appearance of a square root is archetypal in the computations of eigenvalues, and in fact it appears also in the undeformed periodic case. It is, however, an awkward feature for practical computations, especially when computing semi-classical corrections to the conserved charges of the model. Nevertheless, as in the periodic case see e.g. [46], one can use an automorphism to get rid of the square root and in fact transform the expressions to ones which only involve Cartans of the symmetry algebra. First, as done in appendix C, we identify h = 1 2 (d − M 01 ), e + = 1 2 (p 0 − p 1 ), e − = − 1 2 (k 0 + k 1 ), which close into a sl(2, R) subalgebra of the algebra of isometries k. We can construct dual generators h = 1 2 h, e ± = e ∓ spanning a dual sl(2, R) algebra. We can then construct Q sl(2,R) = Q h h + Q + e + + Q − e − = (Q1h + (Q2 − Q3)e + − Q4e − )/2. It is easy to check that 2STr[(Q sl(2,R) ) 2 ] = (Q1) 2 + (−Q2 + Q3)Q4 and thus At this point we use the classification of adjoint orbits in sl(2, R) (see e.g. appendix C). Depending on the algebra element, the adjoint action may bring it to something proportional to h, e + or e + − e − . When discussing the classical spectral curve, we always assume that we are considering classical solutions that correspond to highest weight states belonging to the same orbit as the generator of time translations H. In other words, we assume that by an inner automorphism, we can transform Q sl(2,R) to the form Q sl(2,R) = EH . In that case, one has STr[(Q sl(2,R) ) 2 ] = −E 2 /2 such that we simply get The AdS quasimomenta in the x variable now simply expand aŝ where we defined Q Θ ≡ Q5, since it is the angular momentum charge related to rotations by the angle Θ. Notice that the asymptotics of the AdS quasimomenta depend only on three conserved charges, namely the energy E and the angular momentum Q Θ at O(x −1 ), and the charge Q at order O(x 0 ), which controls the twisted boundary conditions. This situation should be compared to the three charges (energy and two spins) that appear in the asymptotics of the quasimomenta in the periodic case at O(x −1 ).

Classical algebraic curve for the BMN-like solution
Let us now consider a specific classical solution and derive the corresponding quasimomenta. We will do this for the BMN-like solution that we have constructed. We can compute the quasimomenta using either the deformed or undeformed description. On the one hand, in the deformed picture (when the solution is given by (3.4)), the Lax connection is clearly σindependent, so that the path-ordered exponential in the definition of the monodromy matrix reduces to a simple matrix exponential of L σ . On the other hand, in the twisted picture (when the solution is given by (4.27)), the solution depends on σ, making the computation potentially more involved. However, this dependence can be eliminated via a gauge transformation and leads to the same result. This is in fact not surprising, since the two models share the same Lax connection, so that the computation is bound to be the same. Given that we can ignore the path-ordering in the exponential, to obtain the quasimomenta we can then simply compute the eigenvalues of L σ itself. In terms of the spectral parameter x, we find where we defined β ≡ η 2b 2 Z , and we remind that a T and b Z are parameters that enter the classical solution, see e.g. (3.4). In the vanishing deformation limit, which now corresponds to the β → 0 limit, they reduce to the meromorphic quasimomenta associated to the BMN solution of undeformed AdS 5 × S 5 [51] and in that sense, the solution (3.4) (equivalently (4.27)) is a "BMN-like" solution. The deformation, however, introduces a cut between sheet1 and4 and a cut between sheet2 and3. Note that the inversion symmetry (5.5) is satisfied. For later convenience, let us write the quasimomenta in the physical region |x| > 1 Matching these explicit expressions with the general asymptotics of the quasimomenta (5.21) gives us the following expressions of the charges involved: with matches with (3.5) and (4.29).
In addition to the kinematics in the deformed AdS 5 space, we will take our solution to have non-trivial kinematics in the S 5 space as well. Specifically, we consider a pointlike string travelling with constant velocity around a big circle of S 5 , φ = ωτ . Here, ω is proportional to the conserved charge associated to an angular momentum on S 5 . As the S 5 part of our space is not deformed, we can borrow the quasimomenta from the usual BMN solution [51] In the solution that we are considering, the Virasoro condition (3.8) reads This implies that if want both ω and a T to be real, we must have |β| ≤ 1. Notice that, as we commented above, the Virasoro constraint can be reinterpreted as a synchronisation condition on the poles of the quasimomenta at x = ±1.

Quantum corrections to the BMN-like solution
In this section, we study the quantum corrections to the quasimomenta related to our specific BMN-like solution. Specifically, we apply the recipe presented in [51,55] to compute the one-loop correction to its energy. Despite the fact that those articles originally present this technique for undeformed AdS 5 × S 5 , it has been successfully applied also to deformed backgrounds, such as the flux-deformed AdS 3 × S 3 [57][58][59] or the Schrödinger background [60].

Corrections to the classical quasimomenta
Although we are not working on undeformed AdS 5 × S 5 , most of the road map to construct quantum fluctuations described in [51,55] applies to our deformed background. The main idea behind this computation is to introduce (quantum) excitations in the form of cuts so small that we can treat them as poles. As in the macroscopic case, if the microscopic excitation we are adding connects two sheets associated to hatted quasimomenta or two sheets associated to a tilded quasimomenta, the excitation is bosonic. If the excitation we are adding connects a tilded quasimomenta with a hatted quasimomenta, the excitation is fermionic.
These new cuts will modify the quasimomenta we computed in the previous section, but the properties that they must fulfil are restrictive enough to fully constrain how they are altered. The starting point is the fact that the corrections to the quasimomenta cannot alter the gluing condition on a cut (5.7), that is This condition is true for the macroscopic cuts associated to the classical solution, which imposes the following condition on the perturbations: A similar condition is imposed for the microscopic cuts that we are introducing to compute the quantum corrections, and this fixes the positions, x kl n , where these cuts can be placed by where the mode number n takes integer values. Notice that, due to inversion symmetry (5.5), we only focus on solutions with |x kl n | > 1. We say that an excitation is in the physical region if the position of the pole fulfils that property. Now that we know where to place the new cuts/poles, we also need information regarding their residues at x kl n . We will not discuss the explicit expression of the residues at this point, but instead will only talk about the relative sign between the residues in the two sheets. We will provide more details on the residues in appendix D. For bosonic excitations, the poles are actually infinitesimally small square-root cuts connecting two sheets. This means that the residues on the two sheets have to have opposite signs. For fermionic excitations, the poles are actual poles, and they have to have the same residue. This can be summarised as with i < j taking values1,2,3,4,1,2,3,4. Here, N ij n indicates the number of quantum excitations in the physical region that connect p i with p j and have mode number n, and α(x ij n ) is the residue of the pole associated to that excitation. A more detailed explanation on the origin of these signs can be found in [47]. This condition can be understood from a mathematical perspective as a consequence of the Riemann bilinear identity [47]. Physically, because the new poles modify the value of the filling fractions (see also footnote 40), the N ij n can be interpreted as the amplitudes of the physical modes, which are related to mode numbers through conventional string level-matching.
In addition to these pieces of information, these corrections have some important features that they inherit from properties of the classical quasimomenta. Among those, the most relevant for us are the inversion symmetry of (5.5) (6.6) and the synchronisation of the poles at ±1 δp(x) ∼ diag δα (±) , δα (±) , δβ (±) , δβ (±) |δα (±) , δα (±) , δβ (±) , δβ (±) x ± 1 + . . . . (6.7) Notice that we are using the relaxed version of the synchronisation of poles as commented on in footnote 37, because we will have to consider also fermionic excitations when computing quantum corrections.
The final piece of information that we need to compute the perturbations is their asymptotic behaviour for large values of x. This can be obtained by analysing the asymptotic behaviour of the quasimomenta (5.21). We require 40 where δ∆ is the anomalous correction of the energy and where N ij = n N ij n is the total number of poles in the physical region connecting sheet i and j. The above formula can be justified in the same way as in the undeformed periodic case. In particular, one can assume that the addition of quantum excitations produces integer shifts in the charges that appear at order 1/x in the largex asymptotics of the quasimomenta (5.21), which justifies the presence of N ij . Here we are also assuming that only the energy is allowed to receive anomalous corrections, and we will see that the results will be consistent with this. Notice that above we have not included the asymptotics ofp 2 andp 3 . The reason is that the classical asymptotic behaviour (5.21) suggests that their form is non-standard (i.e. they might be finite in the x → ∞ limit) as a consequence of the twist. We will actually not need them to compute δ∆, and we refer to [60] for a similar strategy applied in the case of the Schrödinger background. Nevertheless, we discuss the asymptotic behaviour ofp 2 in detail in section 6.2 and, importantly, we will show that the charge Q does not get any anomalous correction.
Combining all the properties we have enumerated, we possess enough information about the analytic structure of these corrections to be able to reconstruct them. As this is a tedious and sometimes repetitive process, we have relegated the details regarding this reconstruction to appendix D, while here we collect the final results. If we denote each contribution to δ∆ as δ∆ = n Ω ij (x n )N ij n , (6.9) we find the following expressions: In all the following expressions we rescale our conserved charges with a factor of √ λ, so that the reader can compare them with the previous literature more easily. We want to stress that the normalisation factor 4π √ λ can be modified in the sense that it does not affect the computation of the anomalous contribution to the energy δ∆, at the condition that we also modify the normalisation of the residues α(x ij n ) consistently. However, the natural way to fix this normalisation factor is by demanding that the correction to the filling fractions K (i,j) is given by Nij, i.e., − In this sense, the normalisation chosen for the filling fractions fixes also the normalisation of the asymptotics of δp.
The final step is to use equation (6.3) to find the value of the spectral parameter at which we have to place the microscopic cuts, x n , and substitute it in the above equations. As the process is the same for all the cases mutatis mutandis, we will illustrate it in the case13. The algebraic equationp 1 (x n ) −p 3 (x n ) = 2πn has two solutions Because a 2 T + n 2 ≥ a T ≥ a T 1 − β 2 for real values of n and |β| ≤ 1 (as required from reality conditions, see below (5.26)) the solution with minus sign has modulus smaller than one, and we will disregard it. Substituting this result into Ω13(x n ), we find The remaining contributions take the following form: Another non-trivial check that our results pass is the fact that we recover the usual BMN contributions in the undeformed limit (see, for example, [75]), i.e. Ω = −1 + 1 + n 2 ω 2 , upon using the Virasoro constraint (5.26).

Anomalous correction to the twist
Before computing the one-loop correction to the dispersion relation of the BMN-like solution, we want to make some comments on the asymptotic behaviour of δp 2 and its effect on the charge Q of the twist. In particular, we will show that there is no contribution to the O(x 0 ) term of the expansion. To that end, we will use the inversion symmetry (6.6) to compute δp 2 from the different δp 1 excitations that we constructed in appendix D.
Let us begin with the excitation23. Using inversion symmetry on we can check that Despite getting a non-trivial O(x −1 ) term, we can check that it actually vanishes when we impose the level-matching condition. The other three excitations associated to the S 5 have the same behaviour, as δp 1 has a similar form for all four.
The situation is very similar for both the excitations associated to14 and13. In particular, we have respectively. In both cases, the 1/x term is proportional to the pole-fixing condition (6.3) and, thus, proportional to n. This means that both the x 0 and x −1 term vanish once the levelmatching condition is imposed.
The situation for the excitations associated to13 and23 is slightly different. For these kinds of excitations, we find that respectively. The analysis in both cases is the same: the x 0 contribution vanishes automatically, while the second term in the x −1 contribution vanishes due to the level-matching condition. However, in both cases, we have a non-vanishing ∓ n 2πNn √ λ at order x −1 . The behaviour of the remaining excitations can be inferred using the different tricks described in appendix D, e.g. the composition of excitations described in D.4 or the fact that the classical solution has pairwise symmetric quasimomenta (D.26).
Let us summarise and compare our results with what we would expect from the asymptotic behaviour of the quasimomenta (5.21). First, we find that the x 0 terms of the expansion of δp 2 for large values of x vanish in all the cases. In (5.21), these terms corresponds to the conserved charge Q, associated to the twist. Hence, we find that Q does not receive any anomalous correction, δQ = 0. On the other hand, depending on the excitation we are considering, the x −1 terms either vanish (after imposing the level-matching condition) or give a constant contribution. In particular,13,24,23,24,13 and23 fall in this second category, where the two bosonic excitations contribute with −N n /2 and the four fermionic excitations with +N n /2. 41 If we try to interpret this from the lens of undeformed AdS 5 × S 5 , the appearance of, e.g., a contribution proportional to N13 n /2 in δp 2 is surprising. Due to the presence of the twist, however, the contribution of certain excitations to the asymptotics of δp 2 and δp 3 do in fact mix. Note in particular that the only non-vanishing excitations are those which connect one sheet affected by the twist with a sheet that is not affected by the twist. Comparing with (5.21) the new asymptotics suggest a shift of the spin charge Q Θ as Let us emphasise that this contribution is non-dynamical, as it only depends on the number of excitations N ij n but not on the position of the poles x ij n . Similar shifts of spin charges are found also in undeformed AdS 5 × S 5 , see e.g. [51], where the spins get corrections in which the contributions of fermionic excitations are weighted with a factor of 1/2 relative to those of bosonic excitations. It would be interesting to understand this behaviour for twisted models in more generality.

One-loop correction to the energy
Now we have enough information to compute the one-loop correction to the energy of our classical string. This is done by considering that the excitations we are introducing behave as harmonic oscillators, and computing their ground state energy. Mathematically, this means that where we have F ij = 0 for bosonic and F ij = 1 for fermionic contributions, respectively, and where E class. = −E = √ λa T (recall that we rescale charges with √ λ).
Before attempting to perform the sum over n, we should check if the series of the addends converges. We can check that it is the case, as meaning that the partial sums over the mode number have to converge, and we are allowed to compute the sum over integer numbers.
First of all, we should notice that the mode number n always appears divided by a T . We can safely assume that a T 1 because it is related to the energy of our classical solution. This allows us to approximate our sum over n ∈ Z by an integral over n ∈ R. 42 Then 33) where by x a T n we mean that we are rescaling n by a factor a T . Notice that, to get the second equality, we have used that the frequencies only depend on n quadratically and we are therefore dealing with an even integrand.
At this point, it is useful to divide our contributions into two different types Ω nested = Ω14(x n ) + Ω23(x n ) = √ 2 2 + n 2 + n 2 (n 2 + 4β 2 ) , (6.34) Ω other = −8 1 + n 2 + 4 1 + n 2 − β 2 + 2 1 + n 2 + β 2 . (6.35) We do so because the nested square root structure makes the integration of Ω nested a bit involved. Let us first address the integral of Ω other . Instead of integrating from 0 to ∞, we shall integrate only up to a positive cut-off, Λ. We do so because, while the total series is convergent, the two separate contributions Ω nested and Ω other are not. Obviously, they have divergences that later will cancel each other. After some algebra, we find The integral over Ω nested is more intricate, but it can be simplified using a change of variables inspired by the one proposed in Appendix C of [77]. In particular, after we change into a variable y(n) that eliminates the nested integral, i.e. Ω nested = 4 + y(n) 2 , we can perform the integral as lim Λ→∞ Λ 0 dn Ω nested = lim Λ→∞ Λ 0 dy 4 + y 2 y 3 + 8yβ 2 2 y 2 + 4β 2 3 , . Notice that, although we are interested in the limit of very large Λ, we have kept the next-to-leading order of the map between the two cut-offs. We have done that because the integral will diverge as Λ 2 , which means that this subleading order of the transformation will give rise to a non-trivial contribution to the integral. After evaluating the integral, and with a healthy dose of algebra, we arrive at dn Ω nested = 2Λ 2 + 2(1 + β 2 ) log(2Λ) Putting both integrals together, it is immediate to check that the divergent parts perfectly cancel, giving us As a check of our expression, we can examine two particularly interesting values of β. On the one hand, we can check that E 1−loop = 0 for β = 0. This is consistent with the fact that we recover the BMN solution in the undeformed limit, which does not receive corrections. On the other hand, in the β = 1 limit, the form of Ω nested simplifies, as we can get rid of the nested square root lim β→1 Ω nested = 4 + 2n 2 + 2 n 2 (n 2 + 4) = √ n 2 + n 2 + 4 . (6.40) Thus, in this limit all the contributions have the form of a square root and there is no need to separate the integral into two divergent contributions, avoiding the possible issues that a cut-off may introduce. When taking β → 1, the limit of the integral and the integral of the limit coincide and give E 1−loop = −a T log(8).
To end this section, we want to elaborate on our choice of cut-off. As the deformation changes the masses associated to the different quantum fluctuations differently, we may worry that an issue with the choice of regularisation similar to the one present in AdS 4 ×CP 3 may arise, see e.g. [78,79], where a unique summation prescription does not exist and one needs further information to choose the correct one. While for AdS 5 all the quantum excitations have the same mass, for the less-symmetrical AdS 4 they can take two possible values, which allows one to separate them into "heavy" or "light" modes. As the number of bosonic and fermionic heavy modes is the same, and idem for light modes, one is allowed to choose a different cut-off for each type leading to a different result. Although our situation may look similar at first glance, the mass of the excitations in our background are associated to their origin, i.e. if they are bosonic modes associated to the sphere, or to the deformed AdS space, or to fermions. Consequently, although a different regularisation scheme may exist, it would be highly non-trivial. 7 The unimodular case and the spectral equivalence So far we have considered the non-unimodular deformation, i.e. we have set ζ = 0 in (2.7). Here, we wish to show that these results can be extended to unimodular deformations (ζ = 1) thanks to the observation that even in this case the twist can be factorised as explained in section 4.3 for the non-unimodular case. First, an extended Jordanian R-matrix of the type (2.7) with ζ = 1 solves the classical Yang-Baxter equation if the odd elements satisfy the (anti)commutation relations [14,65,80] where ξ is a free real parameter. From [44], we know that for a generic homogeneous Yang-Baxter deformation the twist can be taken to be in the subgroup F , and therefore, it can be written as W −1 = exp(ηRQ), where Q is a conserved charge that takes values in the dual (with respect to the bilinear form of g) of f. That means that we can write it as where Q A are projections of Q. We will not calculate them explicitly because this will not be needed for the following argument, but notice that, when setting ζ = 1, in principle all of them can receive contributions from the fermionic degrees of freedom. In other words, when going from the non-unimodular to the unimodular case, the difference is not only the presence of the new (fermionic) charges Q 1 , Q 2 , because we can also have different expressions for Q h , Q e .
First notice that Q 1 , Q 2 are Grassmann variables. We are therefore working with the Grassmann enveloping algebra, in which case anticommutators become standard commutators. In particular, notice that if we define Z = −iη(Q 1 Q 1 + Q 2 Q 2 ), then [Z, Z] = 0, because the only non-vanishing anticommutator in (7.1) is when we take the same odd generator, but then Q 2 i = 0 because it is Grassmann. The non-unimodular twist is recovered by formally setting Z = 0.
When ξ = 0 in the commutation relations, the generators Q 1 , Q 2 do not mix and we can repeat the argument of section 4.3 to show that W = v −1 W v, where W = exp(ηQ e h). In fact, first we can write the twist in the form W −1 = exp A + s 1−e −s B , and repeat the steps in (4.18) if we now identify This has the same form as the non-unimodular twist and, as explained in section 4.3, it then follows that it is also related by a similarity transformation to W .
To prove the factorisation of the twist for generic values of ξ, we will now consider a different and even simpler method, that works both for ζ = 0 and ζ = 1 and for any ξ. For definiteness, we will provide the discussion in the ζ = 1 case. First, let us construct v = exp(α e e) exp(α 1 Q 1 ) exp(α 2 Q 2 ), (7.4) where α 1 , α 2 are Grassmann, and compute The above formula implies that Now we want to match the right-hand-side of the above equation with the right-hand-side of (7.2). We can do this only if we assume c = 0, and then and if we also require This system has always a solution except if ξ 2 = −1/4, but this possibility is excluded because it must be ξ ∈ R to respect the real form of psu(2, 2|4). To conclude, we can always solve also for α 1 , α 2 , α e , and then the twist can be put in the wanted form Let us make a comment on the assumption c = 0 made above. Given the solution for c in (7.7), there are only two possibilities in which c can vanish. The undeformed limit η → 0 is obviously not problematic: when η → 0 then W → 1 so the factorisation is trivially true. The only real worry should be the case Q e = 0 with η = 0. Let us analyse this situation in the non-unimodular case, when Q e = −Q/η and Q was given in (4.5). Then, Q e can vanish (at finite η) only if Q = 0. Given (4.4), this leads to a trivial twist W = 1, unless q compensates the zero of Q with a divergence. This is possible only if the field configuration is such that In other words, the degrees of freedom in Y e are periodic while those in Y h are not. Therefore, in this sector of the theory the non-unimodular twist cannot be put into the form exp(ch), and it is instead equal to W = exp(−we) with w = Qq finite and non-zero. This is in fact a non-diagonalisable twist, with all eigenvalues equal to 1. While the leading-order asymptotics of the quasimomenta around z = 1 (or x = ∞) would match those of the untwisted case, the next-to-leading order will change. A similar analysis is obviously valid also for the unimodular twist.
We conclude this section by arguing the spectral equivalence of the unimodular and the nonunimodular models, up to certain caveats that we are about to point out. In the sector where the twists of the unimodular and non-unimodular models are both equivalent to W = exp(ηQ e h), the fact that the classical spectra of the two models are the same is rather straightforward. First, in both cases one can implement field redefinitions as explained in 4.3 to obtain boundary conditions controlled just by W = exp(ηQ e h). As remarked above, the explicit expressions for Q e in the two cases differ, because in the unimodular case there are additional contributions from the fermionic degrees of freedom. However, these vanish when considering classical solutions, and therefore the classical spectral curves of the two models are indistinguishable. One may worry that the non-diagonalisable sector (where the twist is instead W = exp(−we)) may be problematic from the point of view of the spectral equivalence. But also in this case the twists of the two models would differ only by contributions of the fermionic degrees of freedom, that do not contribute at the classical level.
We can push this argument even to the one-loop level. In fact, one may use the method of [51,55] to compute quantum corrections to the classical spectrum, as we have done in the previous section for the non-unimodular case. The advantage of this method is that the only data that is needed to compute the one-loop shift is the classical spectral curve itself. Because the two classical spectral curves agree, the unimodular and non-unimodular models will also have the same 1-loop corrections to the spectrum. It would be interesting to understand if it is possible to make any statement beyond one-loop, since it is always possible that the spectral equivalence breaks down at higher loops. However, we want to point out that even at one-loop level, there may be extra subtleties not discussed so far. In fact, the whole argument is done under the assumption that it is possible to reformulate the Yang-Baxter deformed model as the undeformed yet twisted one. While this is certainly true at the classical level because the two σ-models are equivalent on-shell, it is possible that the equivalence breaks down at the quantum level. An anomaly may occur especially in the non-unimodular case, when the background fields do not satisfy the standard supergravity equations and the σ-model is not Weyl invariant. In the presence of such an anomaly, even if the (quantum) spectra of the two twisted models were the same, the spectra of the corresponding Yang-Baxter deformations (with ζ = 0 and with ζ = 1) may not be related in an obvious way. It would be very interesting to investigate these possible scenarios in more details.

Conclusions
In this paper, we have considered a Jordanian deformation of the AdS 5 × S 5 superstring background, that preserves the (classical) integrability and at least 12 superisometries. 43 We have identified global coordinates for the deformed background, and in particular, a global time coordinate. This fact is crucial for the correct identification of the energy when considering the spectral problem from first principles, and we believe that it will be important in order to have insights into an AdS/CFT interpretation of the deformation. Importantly, we have reformulated the deformed model in terms of an undeformed yet twisted model. In this twisted picture, we were able to obtain a general class of solutions to the σ-model equations of motion written in terms of Airy functions. Our further study, however, focused on a particular simple solution which we called BMN-like. Given that the difference compared to the standard (periodic) case of AdS 5 × S 5 is only in a twist appearing in the boundary conditions, we could borrow several methods of integrability that had already been used in the past. We did this not just at the level of the classical spectral curve but also when including its first quantum corrections since, despite the twist, the methods are essentially unaltered.
The only place where the construction of the classical spectral curve is changed with respect to the standard case is in the form that the quasimomenta take at large values of the spectral parameter. In particular, we are forced to re-evaluate the ansatz for the asymptotic behaviour of the quantum corrections for some of our quasimomenta. For example, in the standard case, we can extract the contribution to the anomalous correction to the energy from any of the corrections to the quasimomenta associated to the AdS directions. However, in the deformed background we have studied, this contribution does not appear in those quasimomenta which are affected by the twist.
An interesting outcome of our results is that we find no anomalous quantum correction for the charges that control the twisted boundary conditions. The only charge that receives quantum corrections is the energy, and we interpret this as the spectral problem being well-posed. The situation is reminiscent to the β-deformation [28], where the twist appears in the form of phases that enter the Bethe equations [36], for example. Also in that situation, it is the energy of the string (or the dual anomalous dimension of single-trace operators) that receives quantum corrections, while the charges controlling the twist appear as external data, so that the Bethe equations can be solved consistently.
The advantage of working in the language of the twisted model resides also in the fact that the conserved charges that label the full spectrum admit local expressions in terms of the variables of the twisted model, while they would be non-local in the variables of the deformed one. Moreover, only (the Cartans of) the manifest symmetries of the twisted model appear as labels of the spectrum, and this is crucial because these are only a subset of the isometries of the deformed one.
In the deformed picture, we know that the deformation parameter can be reabsorbed, although this is done in such a way that the undeformed limit η → 0 becomes subtle. One can actually think in terms of two possible cases to be considered, namely η = 0 and η = 0. This is related to the fact that Jordanian deformation (at η = 0) are essentially non-abelian T-duality, without any continuous deformation parameter. However, in the twisted picture we do not see evidence for a similar interpretation, because the one-loop spectrum seems to depend continuously on the deformation parameter η through the combination β = η 2b 2 z . It would be interesting to understand this further.
Let us also comment on the fact that we were able to consider both the unimodular Jordanian deformation (giving rise to a type IIB supergravity background) and the non-unimodular one (which does not correspond to a supergravity solution). In this respect, the main message from our results is that the corresponding twisted models share the same spectrum, at least to one-loop. Unless the argument regarding the equivalence of the deformed model to the twisted one at the quantum level fails, this seems to suggest that the unimodular and nonunimodular deformations share the same spectrum at least to that order. It would be interesting to understand these subtleties further and see whether we can make any statement beyond oneloop. A first non-trivial check was obtained already in appendix E, where we found that the bosonic (AdS) frequencies obtained from the analysis of quadratic fluctuations of the deformed σ-model matches with the bosonic (AdS) frequencies obtained from the curve of the twisted σ-model. In addition, let us mention that the equivalence of the quantum spectrum between a unimodular and a non-unimodular Yang-Baxter deformation has been noted also in [83], although in that case the inhomogeneous Yang-Baxter deformation was considered, for which the reinterpretation as a twist of the boundary conditions in the undeformed model does not hold.
Our results offer a first step towards the understanding of how to tackle the spectral problem for Jordanian deformations (and more generally diagonalisable Yang-Baxter deformations) of AdS 5 × S 5 . It would be interesting to start working from the other side of the AdS/CFT duality and construct a corresponding deformation of N = 4 super Yang-Mills. The expectation is that there should be a notion of a deformed/twisted spin-chain in that case, and it is worth exploring the possibility of constructing this spin-chain directly, to obtain insights on the potentially more difficult construction of the deformation of the gauge theory.
From a broader perspective, it would be very interesting to reinterpret the more generic supergravity solution-generating techniques of [18,19] (which include also non-abelian T-duality, and are integrability-preserving) in terms of twisted models within a first-order formulation. This may open the possibility of tackling the spectral problem in this more generic class of deformed and dual σ-models. singular de investigación de Galicia accreditation 2019-2022, and project ED431C-2021/14), and by the European Union FEDER. LW is funded by a University of Surrey Doctoral College Studentship Award. Additionally, LW is grateful to IGFAE for its hospitality during his research visit under the Turing grant scheme. JMNG is supported by the EPSRC-SFI grant EP/S020888/1 Solving Spins and Strings. No data beyond those presented and cited in this work are needed to validate this study.

A Relation to embedding coordinates of AdS
In this appendix, we want to explain how the global coordinates that we use to parameterise the Jordanian-deformed spacetime are related to the embedding coordinates of AdS. This is both to offer additional geometric intuition on the coordinates used and to give an alternative argument to the fact that they are global coordinates, although here this applies only in the undeformed η = 0 limit.
Global coordinates in AdS can be obtained by setting X 0 = R cosh ρ cos τ G , X D = R cosh ρ sin τ G , X i = R sinh ρ Ω i with the constraint D−1 i=1 Ω 2 i = 1. This gives the AdS metric ds 2 = R 2 (− cosh 2 ρ dτ G +dρ 2 +sinh 2 ρ dΩ 2 ), with dΩ 2 the metric of the (D − 2)-dimensional sphere. For D > 2, the hyperboloid is covered once if we take ρ ≥ 0, 0 ≤ τ G < 2π, and the universal cover is obtained by decompactifying the global time τ G . The metric diverges at ρ → ∞, but that is not a problematic place: geodesics either reach it in an infinite time, or if they do it in a finite time (as it happens for some null geodesics) then they come back to finite ρ at later time. Therefore, the spacetime is geodesically complete in these coordinates.
) gives the metric in the so-called Poincaré coordinates. These coordinates do not cover the whole hyperboloid and in fact, as we will show in the more generic deformed case in appendix B, some geodesics are not complete in this patch (e.g. they reach z = ∞ in a finite proper time, and cannot be continued to z < ∞). In these coordinates, the boundary of AdS is at z = 0, where the metric diverges.
In the coordinates (T, V, Θ, P, Z) of (2.20), we see that the AdS metric (i.e. when setting η = 0) diverges at Z = 0, and because of the relation to the previous coordinates we can identify that as the boundary of AdS. The composition of relations between the different sets of coordinates leads to the relation of the global coordinates (T, V, Θ, P, Z) to the embedding coordinates. Setting R = 1, we find These expressions can be inverted as From these expressions, we see that we can take P > 0 and Z > 0, and moreover 44 Θ ∈ [0, 2π[ and T ∈ [0, 2π[, which in both cases are periodically identified with periods 2π. For V , we have instead V ∈] − ∞, +∞[. While τ G is an angle in the (X 0 , X 5 ) plane, T is instead an angle in the plane (X 4 − X 0 , X 1 − X 5 ). Also in this case, we decompactify it and take T ∈] − ∞, +∞[.
We will now argue that these coordinates cover the whole hyperboloid. First, P and Θ cover all the (X 2 , X 3 ) plane, with a "warping" depending on the other 4 coordinates. If we now define Y ± 0 = X 0 ± X 4 and Y ± 1 = (X 1 ± X 5 )/ √ 2, then Z and T take care of the whole (Y − 0 , Y − 1 ) plane (notice that Z is the inverse of the radial coordinate in that plane). Finally, we need to understand if we can cover the whole (Y + 0 , Y + 1 ) plane. Notice that (unlike other coordinates) V depends also on the combinations One may worry that we have just one degree of freedom (i.e. V ) to cover this (Y + 0 , Y + 1 ) plane, but in fact this is not an issue, because we should remember that we are constraining the coordinates to satisfy 1 = 1 , X 2 , X 3 have been fixed (because, for example, one solves for Y + 0 in terms of V ), then the remaining Y + 1 is uniquely identified. Notice that the hyperboloid constraint is quadratic in X M , but it is linear (for example) in Y + 1 , so when solving for Y + 1 the sign is unambiguous. Naively, an issue is present when Y − 1 = 0, because then Y + 1 is undetermined. But when Y − 1 = 0 then V = Y + 1 /Y − 0 , and we can just swap Y + 0 and Y + 1 in the previous reasoning. Then one can solve the hyperboloid constraint for Y + 0 instead. The issue remains when both Y − 0 = Y − 1 = 0, because then V, P and Z diverge. This is not problematic because Y − 0 = Y − 1 = 0 is a codimension-2 subspace, which is not an open set in spacetime.

B Geodesic incompleteness of Poincaré coordinates
For completeness, we show in this appendix that the geodesic incompleteness of the parametrisation by Poincaré coordinates persists also when turning on the deformation η. As usual, the strategy is to look for solutions to the geodesic equations that are pathological, i.e. the boundary values of the coordinates are reached in a finite amount of geodesic time.
To start, let us construct the conserved quantities Qā = k µ a G µνẊ ν along the geodesics. They are equivalent to the following equationṡ At the same time we also have the conserved quantity Let us consider null geodesics, so that = 0. Now the strategy is to look for solutions that at large z go like z ∝ (τ − τ 0 ) −A for some A > 0. These are pathological because z = ∞ is reached at finite τ . To simplify the analysis we fix some of the values of the Qā. At the same time, we must be careful and make sure that the last equation in (B.1) does not reduce to the form if that happened, the geodesic motion would be bounded and no pathological behaviour at z = ∞ would be possible. To be concrete we take It is of course possible to take more generic situations, but this will already be enough to identify pathological behaviour. In this case, (B.1) reduce tȯ Notice that θ is constant, and that a solution for x + can be found once a solution for z is given. Therefore, in the following we focus on the equations for the remaining coordinates. At this point we look for geodesic solutions that at large values of z have with α − a constant. Within this ansatz, the algebraic constraint given in the last equation of (B.4) reduces to (1 − 2α 2 − )z 2 + ρ 2 ∼ 0, which is compatible with large z, ρ only if Demanding compatibility with the equation forρ then giveṡ This is what we were seeking, since such an equation is solved by 45 The pathological behaviour of this solution is enough to conclude that the Poincaré coordinates are not global coordinates for the deformed spacetime.

C Identifying the Cartan subalgebra of isometries
In this appendix, we identify the possible Cartan subalgebras of the algebra of isometries k given in (2.18). Since M 23 and p 0 + p 1 are central elements, i.e. they commute with all generators in k, we can focus on the remaining generators which span an sl(2, R) = span(h, e + , e − ) subalgebra with [h, e ± ] = ±e ± and [e + , e − ] = 2h as is found by identifying h = 1 2 (d − M 01 ), e + = 1 2 (p 0 − p 1 ), and e − = − 1 2 (k 0 + k 1 ). Inequivalent choices of the Cartan subalgebra are then obtained by classifying the inequivalent adjoint orbits of sl(2, R) and then requiring that the representative elements have a diagonalisable adjoint action. Although this is a standard exercise, we repeat it explicitly to clarify certain comments.
Let us write a generic element of the group SL(2, R) as g = exp(c h h) exp(c + e + ) exp(c − e − ) and of the algebra sl(2, R) as x = α h h + α + e + + α − e − . For each element x of sl(2, R), one can then identify its adjoint orbit {gxg −1 }. One finds three inequivalent possibilities. First, in fact, we can transform the generic element x to the element h whilst preserving reality conditions as by specifying the element g with the coefficients if γ > 0. 46 The second case is that for γ = 0, which can be obtained when at least α + or α − are non-zero (otherwise x would also be zero). For definiteness say that α + = 0. One can then transform x into e + as Alternatively, we can take α − = 0, which would transform x into e − . However, e ± are related by an inner automorphism asḡe +ḡ −1 = −e − withḡ = exp(π/2(e + − e − )) so these two possibilities are in fact equivalent. The third and final case is that for γ < 0, which can be obtained only when both α + and α − are non-vanishing. In this case, one can transform x into e + − e − as follows: by using Thus, given a generic x ∈ sl(2, R) which we want to simplify by means of inner automorphisms, we have three inequivalent possibilities 47 To identify the possible Cartan subalgebras, we must now verify whether or not the adjoint actions ad x for the above inequivalent elements are diagonalisable. A simple computation shows that ad h is diagonalisable with eigenvalues (0, 1, −1), ad e + is not diagonalisable, and ad e + −e − is diagonalisable with eigenvalues (0, 2i, −2i). 48 To conclude, up to automorphisms, there are two possible choices of Cartan subalgebras of k, namely where we leave a possible α > 0 coefficient for later convenience. The choice e + − e − would correspond to α = 1, but in the main text we find it convenient to choose α = 1/2.

D Analytic structure of the perturbations
In this appendix, we detail how to compute the correction to the quasimomenta δp i that arise from adding small excitations to our classical solution using only their analytic properties, as described in section 6.1. Most of our computations mimic those of the sl(2, R) circular string in [51] due to the presence of a single cut in thep i quasimomenta, but the asymptotic behaviour of our classical quasimomenta and some of their corrections is very different.

D.1 Excitations associated to the sphere
Excitations that connect two quasimomentap i andp j are related to considering quadratic fluctuations around a classical solution for the modes associated to the sphere. Of those, the only ones that give rise to physical excitations are those that connect either1 or2 with either3 or 4. Similarly to the case described in [51], the two remaining possible combinations,12 and34, do not give rise to physical excitations.
For concreteness, here we will focus on the excitation that connectsp 2 andp 3 , which we will denote as23, and turn off any other excitation. The computations for the other three excitations are the same mutatis mutandis. Nevertheless, we will make some comments on24, and we will discuss the remaining two in section D.4. 47 It is immediate to see that these choices cannot be related by inner automorphisms, given that Tr(h 2 ) = 1, Tr(e 2 + ) = 0, Tr((e+ − e−) 2 ) = −2. 48 Notice that ade + −e − is real, only its diagonalised version has imaginary coefficients.
Let us focus first on the correction δp 2 . We know that it has to have poles at ±1 and at x23 n . Thus, we propose the following ansatz: As there will be no ambiguity, we will drop the superindex of x n and N n to alleviate our notation.
We also know that δp 2 is associated to δp 1 by inversion symmetry. Thus, we can use the information about their behaviour at large values of x to fix the residues of the poles. On the one hand, from equation (6.8), we have On the other hand, from our ansatz and inversion symmetry (6.6), we have Consistency between the two expansions gives us enough information to fix the three residues As we anticipated in section 6.1, the function α(x) does not need to be fixed beforehand. The consistency between the positions and signs of the poles with the asymptotic behaviour of the corrections is enough to fix its form as a function of the position of the pole. As expected, the function α(x) is unchanged from the undeformed case [51]. This is consistent with the fact that it is related to the weight appearing in the definition of the filling fractions (5.9). In particular, we require that where C is the integration contour that encircles the small cut we are putting in. As the expressions of the filling fractions are not affected by the deformation, neither are the residues α(x).
If we now use the condition (6.3) to find the position of the cut, we get Here, we are only interested in the solution with the plus sign, as it is the one in the physical region. Substituting the explicit expression for x n in α(x n ), we find that (D.6) gives The second term in this expression actually vanishes when we impose the level-matching condition n nN n = 0. This means that δα + = δα − , although we will not make use of this relation in the following equations. Now that we have computed δα ± , we can consider the effect that the excitation has on the quasimomenta associated to the (deformed) AdS part of our space and compute δ∆. The ansätze for these corrections are a bit more involved, as we have to consider the possibility of a shift of the branch points arising from a backreaction of the excitation. Thus, we will assume the following ansätze: where K(x) = 1 − x 2 β 2 and the functions f (x) and g(x) are functions to be determined. The second term of our ansatz is inspired by the fact that ∂ β K(x) ∝ 1 K(x) and by eq. (5.23), as the quasimomentump 4 still has to be the analytic continuation ofp 1 through a square-root cut.
Let us start by constraining the function f (x) = δp 1 (x)+δp 4 (x) 2 . As we are considering a bosonic excitation, the synchronisation of the poles at ±1 forces δp 1 (x) and δp 4 (x) to have residues with opposite signs, meaning that f (x) has no poles at ±1. In addition, for large values of x. As we have not added any new poles to the AdS directions, this means that f (x) is a holomorphic function that approaches zero at infinity. According to Liouville's theorem, the only function that fulfils these requirements is f (x) = 0. Now, we can fix the function g(x) by demanding that δp 1 has the correct properties. As δp 1 (x) only has poles at ±1, we propose the following ansatz for g(x) where we have used the synchronisation of the poles at ±1 as in eq. (6.7) to fix the residues. Matching the asymptotic behaviour of δp 1 (x) with its expected behaviour gives us Notice that δ∆ can only be real if |β| ≤ 1, which is exactly what we found from reality conditions implied by the Virasoro constraint.
We should remark that this construction is blind to having the second pole in the sheet3 or the sheet4, as we did not use that information at any point of the reconstruction. This implies that the contributions to δ∆ of23 and24 have the same form as a function of x n . We will later use the inversion symmetry to show that the other two give us the same result.

D.2 Excitations associated to the deformed AdS space
Excitations that connect two quasimomentap i andp j are related to modes associated to the deformed AdS space. Similarly to the case of the sphere, the only ones that give rise to physical excitations are those that connect either1 or2 with either3 or4. Due to the presence of the twist, the excitations associated to14 and23 behave differently to those associated to13 and 24, and thus we will treat them separately. We will consider only one case for each respective set because the other one can be obtained through inversion symmetry, as we will discuss in D.4.
Let us start with the case associated to14 and turn off any other excitation. First, note that we are not modifying the quasimomenta associated to the sphere. This means that all δp i decay at infinity as O(x −2 ). The only possible configuration with this property is the one with δα ± = δβ ± = 0, which allows us to set all the sphere corrections to zero, δp i = 0. Consequently, none of the corrections δp i have poles at ±1 due to their synchronisation.
Similarly to what happened for the excitation23, the residues at x n of δp 1 and δp 4 have the opposite value. Furthermore, for large values of x, we have that δp 1 ≈ −δp 4 ≈ 4π . If we consider the ansatz these considerations imply that f (x) = 0 and fix g(x) to (D.14) Here we have used the insight from our sphere computation to argue that the residue at x n has to be the same as the undeformed one, so we can use the function α(x) we computed above in (D.5). Matching the asymptotic behaviour of our ansatz with its expected behaviour gives us Let us consider now the case associated to13 and turn off any other excitation. Again, we are not modifying the quasimomenta associated to the sphere, implying that none of the corrections δp i have poles at ±1.
Similarly to our previous cases, we start by considering the ansatz However, in contrast to previously, δp 1 (x) has a pole at x n but δp 4 (x) does not, meaning that the function f (x) is not holomorphic and thus does not vanish in this case. Notice that, instead, δp 4 (x) has a pole at 1/x n inherited from δp 3 (x) due to the inversion symmetry. In fact, assuming that δp 3 (x) has a pole at x n with residue −α(x n ), inversion symmetry forces δp 4 (x) to have a pole at 1/x n with residue − α(xn) Similarly to the function f (x), we can assume that g(x) can be written as g(x) = a + b x−xn + c x−1/xn . These constants can be fixed by demanding that δp 1 has the properties we want, i.e. it has a pole with residue α(x n ) at x n and no pole at 1/x n . After some algebra, we get that

D.3 Fermionic excitations
Let us move finally to fermionic excitations. Excitations that connect a quasimomentump i and a quasimomentump j or vice-versa are related to fermionic modes. Here, we will focus only on the excitation associated to13 and turn off all other excitations. We will make some comments on the remaining excitations at the end of this section.
We should first focus on the corrections to the quasimomenta associated to the sphere. As we are adding an excitation to the sheet3, we will consider the following ansatz: Notice that, as we are considering a fermionic excitation, these corrections fulfil the relaxed synchronisation condition (6.7). If we impose the correct value of the residues and the correct asymptotic behaviour, we get exactly the same equations as the ones for the23 excitations (up to replacing δα ± with δβ ± and N n with −N n ). Thus, we can borrow the results we found for δβ ± and the residue α(x n ) given in (D.5) and (D.9).
For the correction to the AdS quasimomenta, we will assume the same ansatz as in the previous cases. Similarly to the case13, we cannot set the function f (x) to zero because δp 1 has a pole at x n but δp 4 does not. In addition, δp 4 has poles at ±1 from the synchronisation condition, while δp 1 does not. After imposing these restrictions, as well as the required asymptotic behaviour, we get .

(D.22)
With this information, we can now compute the function g(x) by demanding that δp 1 (x) has the correct residues at ±1 and x n , which gives us In the next section we will discuss how to obtain the contribution of14 from this one, but it is easy to see that the computation has to be exactly the same as the one presented above, giving us the same δ∆. The same happens for the excitations14 and24.
The steps to compute the contribution of23 are relatively similar. In fact, by analysing the pole structure, it is easy to reach the conclusion that the expression of δp 2 is exactly the same one as the expression of δp 1 for the13 excitation after substituting K(1/x) by K(x) and adding a constant contribution to g(x). Applying the inversion symmetry to get δp 1 and matching it with its required asymptotic behaviour, we obtain that δ∆ = n K(x n ) + K(1) x 2 n − 1 N n . (D. 25) Although at this point we would have to study also the fermionic excitations that involve either3 or4, we can argue that this is not necessary. This happens because our classical solution has pairwise symmetric quasimomentâ p 1 = −p 4 ,p 2 = −p 3 ,p 1 = −p 4 ,p 2 = −p 3 .
(D. 26) This implies that, after all the appropriate computations, we will find that Ω14 = Ω14 , Ω24 = Ω13 , Ω13 = Ω24 , Ω23 = Ω23 , (D. 27) both for Ω understood as a function of the position of the poles and as a function of the mode number n.

D.4 Composition and inversion
Up to this point, we have computed δ∆ for some selected excitations. This is enough for us, as the contribution of the other excitations can be computed using properties such as composition of poles and the inversion property.
The idea is that the procedure we used to compute δ∆ does not care about the exact position x n of the poles/microscopic cuts we are adding. Before substituting the expressions of x n obtained from (6.3) in the frequency of the corresponding excitation, we can thus think of Ω(x) as formal, off-shell, functions of x. Once we substitute x = x n the frequency becomes on-shell. We can use the off-shell expressions of Ω(x) in our advantage in two different ways. The first one is the inversion symmetry: if we consider an excitation connecting a given pair of sheets, we can move the pole associated to it to the interior of the unit circle, making it non-physical, and a new physical pole will emerge in the sheets connected to the original ones by inversion symmetry. The second one is composing two excitations that share a pole with opposite residue on the same sheet, say j, but connect to different sheets, say i and k respectively. The pole on sheet j will cancel, making the composition of the two excitations ij and jk indistinguishable from having only one excitation ik. We illustrate the latter in figure 1. In mathematical terms, these two ideas imply and Ω ij (x) ± Ω jk (x) = Ω ik (x) , (D. 30) where the ± is chosen appropriately to cancel the residue in sheet j. We will not reproduce here the derivation of these expressions, which can be found in [55]. Let us stress again that these relations between the different contributions only hold when they are understood as functions of x, not as functions of n.
Let us now compute some of the remaining frequencies using the ones we obtained above. The process is relatively similar for most of them, so we will consider only some specific cases: the excitations23,14 and14. Both23 and14 can be obtained by immediate application of the above formulas  Figure 1: Illustration of the composition rule (D.30). The fermionic frequency Ω13(x) and the bosonic frequency Ω34(y) can be composed, because the two excitations have a pole with opposite residue on sheetp3. In particular, the residue on sheetp3 vanishes in the limit y → x, giving us the pole structure (and asymptotic behaviour) associated with the fermionic excitation14. Thus, Ω13(x) + Ω34(x) = Ω14(x) off-shell.
The last one is obtained by considering first the composition of the excitations23 and24. This gives us that the excitation34 has Ω34 = 0, which is consistent with the fact that it is an unphysical excitation. This result can then be used to show, e.g., that Ω14 = Ω13 + Ω34 = Ω13.

E Bosonic sector of the quadratic fluctuations
In order to cross-check the results we obtained using the classical spectral curve method, we can compute the contribution of the bosonic excitations to the one-loop correction to the energy by considering the effective Lagrangian of small fluctuations around the classical solution we are interested in. We will perform this computation in the deformed periodic picture, rather than the undeformed twisted one. The expansion of the deformed AdS 5 sector of the Lagrangian is given by 50 L(x clas. + x) ≈ L(x clas. ) + (EOM) where, by a slight abuse of notation, we have denoted the fluctuation around the coordinates with the same symbols as the coordinates. Here, the dot and prime represent derivatives with respect to τ and σ, respectively. Notice that neither the coordinate Θ nor the Kalb-Ramond field terms contribute at quadratic order. The fact that Θ does not appear is simply a consequence of the non-canonical kinetic term that it has, which would appear at higher orders in the field expansion. 51 The next step is to compute the equations of motion of the fluctuations. We will use the fact that we do not want to spoil the periodicity condition of our classical solution by considering the following ansätze for each of the coordinates: X m = n A m cos(Ω n τ + nσ) + B m sin(Ω n τ + nσ) . (E.2) After some algebra, the equations of motion become the system of linear equations M v = 0 with 50 The term (EOM) in the expansion vanishes upon the equations of motion (EOM). 51 If we want to do the counting of degrees of freedom correctly, we should change from P and Θ to Cartesian coordinates X 2 , X 3 . However, as we are only interested in independently checking our results in appendix D, this computation is sufficient. v the vector given by v = (A Z A T A V A P B Z B T B V B P ) and M is the matrix (E.3) If we want this system of equations to have a non-trivial solution, we need the matrix of coefficients to have vanishing determinant. This gives us the condition (n 2 − Ω 2 n ) 2 (n 2 + a 2 T (1 + β 2 ) − Ω 2 n ) 2 (n 4 − 4a 2 T β 2 n 2 − 2(2a 2 T + n 2 )Ω 2 n + Ω 4 n ) 2 = 0 .

(E.4)
We can check that the solutions for Ω n to this equation, divided by Q T = −a T , perfectly match (6.19), (6.20) and (6.21) up to a constant term. Additionally, we also get solutions of the form Ω n = ±n. The modes associated to these are not physical and are cancelled by conformal ghosts [84].
Instead of repeating the same computation for the S 5 sector of the Lagrangian, we can argue that this part of the space is blind to the deformation and hence the final result has to be the same as the one for the undeformed background. In fact, we can check that which perfectly matches (6.18).
The mismatch by a constant between the contributions computed using the quadratic fluctuations and the ones computed using the classical spectral curve may be uncomfortable. However, they are also present for undeformed AdS 5 × S 5 and arise due to each method describing the perturbations around the classical solution using a different frame of reference [51].
Even though these shifts have a non-physical origin, this does not mean at all that they are harmless, as they can give rise to ambiguities. To discuss those ambiguities, first we have to distinguish between two kinds of shifts that may appear: a constant shift of Ω (usually proportional to the energy or an angular momentum), and a shift of the mode number n (usually proportional to a winding number). The ones of the first kind are completely harmless, but the ones of the second kind give rise to an ambiguity. To see that, we can compare what would be the contribution to the one-loop energy we obtain with and without performing a shift of the mode number. Although the sum ∞ n=−∞ [2Ω(x n ) − Ω(x n−m ) − Ω(x n+m )] might initially seem to vanish because the terms cancel each other after a relabelling, this is not entirely correct if we consider a partial sum up to a large enough value, Λ. Using that Ω(x n ) ≈ n for large n we see that, up to subleading orders in Λ, [Ω(x n ) − Ω(x n−m )] ≈ 2m .
(E.6) Luckily, we do not have shifts of this second kind in our problem. In fact, they are usually associated to winding in the classical solution, which ours does not possess. We therefore conclude that these shifts do not create an ambiguity in our final result for E 1−loop .