Semiclassical quantization of the mixed-flux AdS$_3$ giant magnon

We obtain explicit formulas for the eight bosonic and eight fermionic fluctuations around the mixed-flux generalization of the Hofman-Maldacena giant magnon on AdS$_3 \times$S$^3 \times$T$^4$ and AdS$_3 \times$S$^3 \times$S$^3 \times$S$^1$. As a check of our results, we confirm that the semiclassical quantization of these fluctuations leads to a vanishing one-loop correction to the magnon energy, as expected from symmetry based arguments.


Introduction
An important aspect of the AdS 5 /CFT 4 correspondence [1] is integrability, a hidden symmetry present both on the N = 4 super Yang-Mills gauge theory side [2][3][4] and AdS 5 × S 5 type IIB superstring theory side [5][6][7][8][9] of the duality. Interactions in a quantum integrable theory reduce to a series of diffractionless two-body scattering processes, and in the decompactified worldsheet limit the spectrum is solvable using a Bethe Ansatz [10,11]. Therefore, the main object of interest in (the planar limit of) AdS/CFT is the S-matrix, encoding these two-body scatterings of elementary excitations, or magnons [12]. AdS 5 /CFT 4 has a psu(2, 2|4) symmetry, and the subalgebra leaving the vacuum invariant is su(2|2) 2 . The off-shell, centrally extended version of this residual algebra, su(2|2) 2 c.e. fixes the S-matrix up to an overall phase [13,14], which then can be calculated from the so-called crossing symmetry [15][16][17][18][19]. These algebraic arguments also determine the magnon dispersion relation to be = 1 + 4h 2 sin 2 p 2 , (1.1) where p is the magnon momentum, and the effective string tension h is related to the string tension α and AdS radius R on the string theory side, and the 't Hooft coupling λ on the gauge theory side by . (1.2) Solitons are particle-like solutions of integrable field theories, whose dynamics can be captured by a small number of collective degrees of freedom. Quantization of these collective coordinates [20][21][22][23] provides a window into regimes of the quantum theory not directly accessible to perturbation methods. The giant magnon of Hofman and Maldacena [24] is a soliton of the integrable AdS 5 × S 5 worldsheet sigma-model [25], whose spacetime image is an open string uniformly rotating around an axis of an S 2 ⊂ S 5 , stretched between two points on the equator. In fact, the worldsheet momentum p of the giant magnon is the angular distance between these two points, and its dispersion relation = 2h sin p 2 agrees with the large h limit of (1.1). An R × S 3 generalization of this solution, the dyonic giant magnon [26], has the dispersion relation = J 2 + 4h 2 sin 2 p 2 , (1.3) where J 2 is the second angular momentum on S 3 . Upon semiclassical quantization J 2 takes integer values, and we recover the exact disperison relation (1.1).
There are a number of calculations one can perform to check that the giant magnon is indeed the large coupling limit of the elementary excitation of the quantum theory. A semiclassical analysis of the worldsheet scattering of dyonic giant magnons [27] shows that their 1-loop S-matrix agrees with the Hernandez-Lopez phase [16], and also that the 1loop correction to the giant magnon energy vanishes. From an algebraic perspective the magnon is a BPS state of the su(2|2) 2 c.e. superalgebra, and accordingly, must be part of a 16 dimensional short multiplet [14]. As a consequence the giant magnon should have eight fermionic zero modes, as Hofman and Maldacena argued in [24]. These zero modes were explicitly constructed by Minahan [28], starting from the quadratic fermionic part of the Green-Schwarz action expanded around the giant magnon. Quantizing these modes he was also able to reproduce the odd generators of the residual algebra. Subsequently, building on Minahan's work, an explicit basis of the magnon's fluctuation spectrum was found by Papathanasiou and Spradlin [29], once again confirming that the dispersion relation receives no corrections, by showing that the 1-loop functional determinant vanishes.
With 32 supercharges AdS 5 /CFT 4 has the maximal amount of supersymmetry possible for any 10 dimensional string theory, but integrability has proven to be a powerful tool in understanding other, less supersymmetric AdS/CFT dualities too. One example is AdS 4 /CFT 3 , the duality between ABJM Super Chern-Simons and type IIA string theory on AdS 4 × CP 3 with 24 supersymemtries [30], however, for the rest of this paper we focus on AdS 3 /CFT 2 , and in particular two 1 backgrounds with maximal supersymmetry allowed for such geometries (16 supercharges). One of them is AdS 3 × S 3 × T 4 , where the radii of AdS 3 and S 3 are equal, and the other one is AdS 3 × S 3 × S 3 × S 1 , where the AdS radius R and the radii of the two 3-spheres R ± satisfy [31] 1 Setting the AdS 3 radius to one, this geometry can be parametrized by an angle ϕ (1.5) and in fact the ϕ → 0 limit covers the AdS 3 ×S 3 ×T 4 geometry too, once the blown up sphere is compactified on a torus. The type IIB supergravity equations allow these backgrounds to be supported by pure Ramond-Ramond (R-R) flux, pure Neveu-Schwarz-Neveu-Schwarz (NS-NS) flux, or mixed R-R and NS-NS fluxes where q ∈ [0, 1] andq = 1 − q 2 . While the pure NS-NS theory can be solved using a chiral decomposition [32][33][34], no such method exists when the R-R flux is turned on, instead, it is believed that the solution will be given in terms of integrable methods 2 , as both the pure R-R [39][40][41] and mixed-flux [42] theories were shown to be classically integrable. The AdS 3 × S 3 × T 4 backgrounds with pure R-R and pure NS-NS fluxes arise as near horizon limits of the D1/D5 and F1/NS5 brane systems, respectively. Historically, the mixed-flux background has been thought of as the near-horizon limit of bound states of D1/D5-and F1/NS5-branes, but it was recently shown that the same worldsheet action arises in a pure NS-NS theory with an R-R modulus turned on [43]. Integrable structures have been identified in the CFT 2 dual to AdS 3 ×S 3 ×T 4 strings [44], and there has also been promising progress in understanding the CFT 2 dual of string theory on AdS 3 × S 3 × S 3 × S 1 [45][46][47][48][49][50]. 1 There is a third maximally supersymmetric AdS 3 background, AdS 3 × S 3 × K3. It should be possible to apply integrable methods to this background, at least in the orbifold limit of K3, and then it would be interesting to see what the effect of turning on the blow-up modes is. 2 Although it is worth noting that there have been attempts to understand the mixed-flux theory using the hybrid formalism of Berkovits, Vafa and Witten [35][36][37][38].
Similarly to the AdS 5 /CFT 4 duality, the symmetry algebra can be used to determine both the S-matrix and the all-loop magnon disperison relation [51][52][53] where h = √ λ 2π only in the classical string limit, and in general h will receive quantum corrections. The excitations are of mass m = 1, 0 for AdS 3 ×S 3 ×T 4 and m = 0, sin 2 ϕ, cos 2 ϕ, 1 for AdS 3 × S 3 × S 3 × S 1 . The mixed-flux AdS 3 × S 3 × T 4 dyonic giant magnon was found by Hoare, Stepanchuk and Tseytlin [54], with the dispersion relation where E is the spacetime energy and J 1 , J 2 are two angular momenta on the S 3 . They also noted that upon semiclassical quantization J 2 takes integer values, and the lowest J 2 = 1 gives an exact match to the quantum dispersion relation (1.7). Just like in the AdS 5 case, there are a number of semiclassical checks on these string solutions. The 1-loop worldsheet S-matrix has been determined from multi-soliton scattering states in [55] in agreement with the finite-gap calculations of [56]. The 1-loop correction to the magnon energy can also be calculated from the algebraic curve [57], or directly from the GS action [41,58]. The off-shell residual symmetry algebras of AdS 3 × S 3 × T 4 and AdS 3 × S 3 × S 3 × S 1 are the centrally extended psu(1|1) 4 [51,53,59,60] and the centrally extended su(1|1) 2 [61,62], and the BPS magnon must transform in 4 and 2 dimensional short multiplets of these superalgebras, respectively. Therefore, the mixed-flux magnon on AdS 3 × S 3 × T 4 and AdS 3 × S 3 × S 3 × S 1 should have 4 and 2 fermion zero modes. We found these zero modes in [63], and showed how they can be used to construct the odd generators of the residual algebras. Our objective here is to find the complete spectrum of fluctuations around the AdS 3 giant magnon. Throughout, we will only consider the stationary magnon, a subclass of solutions we identified as the mixed-flux generalisation of the HM giant magnon in [63]. The rest of this paper is structured as follows.
In section 2 we first review the mixed-flux stationary giant magnon on AdS 3 × S 3 × S 3 × S 1 , then write down the spectrum of small bosonic fluctuations around the classical solution. Although the perturbation equations are rather complicated, one can construct explicit solutions algebraically using the dressing method, which we adapt to be more suited to the fluctuation analysis. In section 3 we find the fermionic fluctuations, closely following the methods developed in [28,63] extended to non-zero angular frequencies. Using the symmetries of the system and an explicit kappa-fixed ansatz, the full 2 × 32 component spinor equations are reduced to a 4 dimensional system, which we can solve explicitly.
Finally in section 4 we read off the stability angles of the fluctuations, and use them to evaluate the 1-loop functional determinant around the soliton background, following the method of Dashen, Hasslacher and Neveu [64]. We find that, in agreement with our expectations based on the superalgebra, the leading order quantum correction vanishes. We conclude in section 5 and present some of the lengthier or more technical details in the appendices.

Bosonic sector
In this section we review the mixed-flux AdS 3 stationary magnon, and solve for its bosonic fluctuations using a similar approach employed to study the AdS 5 magnon in [29]. We consider the case of the AdS 3 × S 3 × S 3 × S 1 background in our calculations, and comment on how the AdS 3 × S 3 × T 4 modes can be obtained at the end of the section.
The conformal gauge bosonic action can be written in the form with AdS 3 and S 3 ± components where η ab = diag(−1, +1), the embedding coordinates Y ∈ R 2,2 , X ± ∈ R 4 are enforced to lie on the unit-radius surfaces by the Lagrange multipliersΛ, Λ ± , and the Wess-Zumino term is defined on a 3d manifold B such that its boundary is the worldsheet ∂B = M. The equations of motion need to be supplemented by the conformal gauge Virasoro constraints Taking scalar products of (2.4) with Y, X ± , it follows from (2.3) and that the Lagrange multipliers take the classical values (2.7)

The stationary giant magnon
The classical solution we consider for the rest of this paper is the stationary mixed-flux giant magnon where the scaled and boosted worldsheet coordinate is The parameter u, restricted to u ∈ (−q,q), can be regarded as the velocity of the magnon. The worldhseet momentum p ∈ [0, 2π) is not a Noether charge of the action, rather a topological charge of the soliton, corresponding to the longitudinal distance between the two endpoints of the magnon on the equator of S 3 + (Z 2 = 0). The parameters further satisfy This is a special case of the dyonic mixed-flux magnon, which was first constructed in [54] for the AdS 3 × S 3 × T 4 background. The stationary magnon was identifed in [63] as the mixed-flux equivalent of the Hofman-Maldacena magnon [24], as compared to the more general AdS 5 dyonic magnon of [26]. The dispersion relation 3 bears witness to this analogy, to be compared to the similarly simple E − J 1 = 2h sin p 2 for the q = 0 HM magnon. The Lagrange multipliers (2.7) evaluate to the classical values (2.13)

AdS 3 fluctuation spectrum
Let us now determine the spectrum of fluctuations around the mixed-flux magnon (2.8), starting with the AdS 3 bosons. We denote the perturbed solution by where Y is the classical solution, δ 1 and the perturbationỹ ∈ R 2,2 is bounded. Substituting into the equation (2.4) and expanding to first order in δ (note thatΛ also receives corrections) we get the perturbation equation whereK µ is as in (2.4) and Furthermore, to preserve the norm (2.3), the perturbation must be orthogonal to the classical solution Y µỹ µ = 0. (2.17) These equations have one massless and two massive solutions. To get the massless perturbation we make the ansatz for which (2.15) reduces to the free wave equation (2.19) satisfying the massless dispersion relation ω 2 = k 2 . The remaining two massive solutions lie in the transverse directions (ỹ 0 =ỹ 1 = 0) of AdS 3 , automatically satisfying (2.17). A simple plane-wave ansatz gives Note that this is the small p, fixed k = hp limit of the mixed-flux AdS 3 × S 3 × S 3 × S 1 dispersion relation [53] ± = (m ± qhp) 2 + 4q 2 h 2 sin 2 p 2 . (2.21) with mass m = 1.

S 3 − fluctuation spectrum
where K − i is as in (2.4) and which needs to be supplemented by X − ix −i = 0 to preserve the norm. Just like on AdS 3 , these equations admit a massless solutioñ 25) and two perturbations of mass m = sin 2 φ (2.26)

S 3 + fluctuation spectrum
For the S 3 + perturbed solution we write 27) and also introduce the complex coordinates so that the perturbed S 3 + component of (2.8) can be written as The equations of motion for the S 3 + fluctuations read and to preserve the embedding norm These equations have two different classes of solutions. Firstly, there are the zero modes, representing collective coordinates of the magnon. The BMN limit fixes the orientation of the magnon in the (X + 1 , X + 2 ) plane, but there is a rotational freedom in traverse coordinates (X + 3 , X + 4 ) leading to the zero mode Furthermore, the magnon breaks the x-translation symmetry of the BMN vacuum, leading to the zero mode (2.34) These two normalizable zero modes are presented for completeness, but will not play any further role in our analysis. The solutions we are interested in are plane-wave fluctuations of the form where f (Y) is a bounded profile that is stationary in the magnon's frame. The equations are too complicated for us to find solutions by substituting the plane-wave ansatz into (2.30), we need to look for another strategy. The authors of [29] suggest using the dressing method [65][66][67] to construct the scattering state of a magnon and a breather, only then to expand this solution in the breather momentum to find the fluctuation as the subleading term. We find, instead, that it is simpler to apply the dressing method to the perturbed BMN vacuum, i.e. the point-like string moving along the equator together with fluctuations like (2.25)-(2.26), which results in the perturbed magnon. The details of this calculation can be found in appendix A, here we just present the solutions. As further confirmation of the validity of our approach, we show in appendix B that applying our method in the ϕ = q = 0 limit we recover the expected subset of the AdS 5 × S 5 fluctuations found in [29]. The massles plane-wave solution is given by with Herez i are not the complex conjugates of z i , rather 4 To preserve the (relative) simplicity of the formulas we considerx + i to be complex themselves. Real solutions to (2.30) can be readily obtained by taking the real parts of these fluctuations.
The two massive modes both have m = cos 2 ϕ. One of them is (2.42)

Bosonic modes in AdS
In addition to the fluctuations we found above, there is of course the massless S 1 mode However, in a proper quantization of AdS 3 × S 3 × S 3 × S 1 string theory the sigma-model action (2.1) would need to be supplemented by ghosts, cancelling the massless AdS 3 mode (2.19), and also a combination of the massless S 3 ± modes (2.25), (2.36), corresponding to the S 3 + × S 3 − leg of the BMN geodesic. These are analogous to the longitudinal modes in light-cone gauge, and in our semiclassical analysis we will simply omit them [68,69].
In summary, the AdS 3 × S 3 × S 3 × S 1 magnon has two massless modes (one on the flat S 1 and another one perpendicular to the BMN angle on S 3 + × S 3 − ), two m = 1 fluctuations on AdS 3 , two m = cos 2 ϕ modes on S 3 + , and two m = sin 2 ϕ modes on S 3 − , all with the dispersion relations ω 2 = (m ± qk) 2 +q 2 k 2 . (2.44)

Bosonic modes in AdS 3 × S 3 × T 4 string theory
Taking the ϕ → 0 limit of AdS 3 × S 3 × S 3 × S 1 blows up the S 3 − factor, which we can recompactify on a T 3 to get the AdS 3 × S 3 × T 4 geometry. In this limit the AdS 3 and S 1 fluctuations are unchanged, the S 3 + modes take the same form but become m = 1, while on S 3 − the massless mode becomes the one unaffected by the ghosts, and the two m = sin 2 ϕ modes become massless T 4 modes. In summary, the AdS 3 × S 3 × T 4 magnon has four massless, and four mass 1 bosonic fluctuations.

Fermionic sector
In this section we solve for the complete fermion fluctuation spectrum around the mixedflux stationary magnon (2.8). Our approach will be mostly based on [63], but rather than normalizable zero modes, we will be looking for solutions with plane-wave asymptotes. The leading order (quadratic) action for fermion fluctuations around a general bosonic string solution X µ (t, x) is given by [70] The ϑ I are two ten-dimensional Majorana-Weyl spinors, σ IJ 3 is the Pauli matrix diag(+1, −1), and ρ a are projections of the ten-dimensional Dirac matrices Note the difference in notation compared to the previous section, X µ are now the curved space coordinates of AdS 3 × S 3 × S 3 × S 1 , and not coordinates of a flat embedding space.
For the remainder of this section we use Hopf coordinates, where the only non-constant components of the stationary magnon are along µ = t, θ + , φ + 1 , φ + 2 , φ − 1 corresponding to the tangent space components A = 0, 3, 4, 5, 7, respectively. The covariant derivative is where ω AB µ is the usual spin-connection, and the contracted 3-form fluxes are (3.5)

The fluctuation equations
The equations of motion for (3.1) are We proceed by changing variables to the more natural scaled and boosted worldsheet coordinates (2.9) of the magnon Here we defined the mixing operators and fermion derivatives (3.11) The full Green-Schwarz superstring has a local fermionic symmetry (κ-symmetry), that we need to fix for physical solutions. Noting that the operators (ρ 0 ± ρ 1 ) are half-rank, nilpotent and commute with the fermion derivatives D andD, it is clear that the projectors can be used to fix κ-gauge. Here we introduced a set of "boosted" gamma matriceŝ that simplify the notation in what follows. The kappa-fixed spinors Ψ J = K J ϑ J then satisfy (3.14) Introducing the 6d chirality projector (3.15) and, withρ 0 = −Γ 0 ρ 0Γ0 , the invertible matrix we can rewrite the equations, using the boosted gamma matrix basis (3.17) The fermion differential operators are and we define ∆ = − 1 2 Note that the only source of structural difference between the equations for AdS 3 ×S 3 ×S 3 × S 1 and AdS 3 × S 3 × T 4 is a non-zero ∆, and in fact this was our main reason to introduce the boosted gamma matrix basis. A much more detailed derivation of these equations, together with a thorough explanation of κ-gauge fixing, can be found in [63].

Ansatz and reduced equations
To reduce the seemingly complicated (3.17) to a more manageable set of equations we will make an ansatz that reflects the symmetries of the system. Firstly, all ofΓ 012345 ,Γ 12 ,Γ 68 commute with the kappa projectors (3.12), so the kappa-fixed spinors can be written as where the eigenvalues of V J λ P ,λ 12 ,λ 68 underΓ 12 ,Γ 68 andΓ 012345 are iλ 12 , iλ 68 and λ P , respectively. Note that λ 12 , λ 68 , λ P all take values in ±1. There are multiple ways to make the above ansatz satisfy K J Ψ J = Ψ J , in [63] we chose to impose the additional constraint 5 Γ 34 V J = +iV J and found Note that kappa-fixing reduces the degrees of freedom by half, and in our ansatz this is done at the level of the projectionsΓ 34 (3.23) and (3.24) While the zero modes are time-independent in the magnon's frame ∂ S Ψ J = 0, for the S-dependence of the non-zero modes we make a Fourier ansatz As opposed to the kappa-projectors, the equations of motion (3.17) only commute witĥ Γ 12 andΓ 68 , and the solutions will not have definite chirality underΓ 012345 , unless ∆ = 0, i.e. for the AdS 3 × S 3 × T 4 background, or on theΓ 1268 = −1 spinor subspace for the AdS 3 × S 3 × S 3 × S 1 background. With this in mind, we take the general ansatz where the constant Weyl 6 spinor U , that is shared between Ψ 1 and Ψ 2 , has eigenvalues iλ 12 , iλ 68 , +i, −1 underΓ 12 ,Γ 68 ,Γ 34 ,Γ 012345 , respectively. The P − part of the solution is represented by the scalar functions f 1 , f 2 , while g 1 , g 2 correspond to the P + components.
The validity of such an ansatz is further justified by a quick counting of the degrees of freedom. A general Weyls spinor has 16 complex components, and after 4 mutually commuting projections, there is a single free component left, hence we can capture the Y-dependence with a single function f J multiplying U . Substituting (3.26) into (3.17), after a considerable amount of simplification we get 27) 6 We postpone the analysis of the Majorana condition until later, see the discussion around (3.54).

Solutions
Let us first find the solutions for ϕ > 0, i.e. for the AdS 3 × S 3 × S 3 × S 1 geometry. For p 1268 = 0 the P + componentsg 1 andg 2 decouple, and we have the two solutions g 1 = e ikY ,g 2 = 0,k = −ω, (3.33) while on the P − subspace we have the equations with the two solutionsf (3.36) The case of p 1268 = 1 is a bit more complicated, but solving the first two equations of (3.31) forg J and substituting into the second two, we get two second order differential equations forf 1 ,f 2 . The difference of those two equations is which is easily solved, and inserting the solution into the f 1 +f 2 equation we find with masses m = 0, cos 2 ϕ, sin 2 ϕ, and 1. This is not quite the expected dispersion relation, and there are two reasons why. Firstly, (S, Y) are scaled versions of the boosted worldsheet coordinates (T , X ), but more importantly, the dispersion relation (2.44) is not relativistically invariant. We therefore need to rewrite the fermion fluctuations in the form where α will be necessary to match (2.44). From (3.7) it follows that (3.42) and substituting these into (3.40) we get the expected relation provided that α = sec 2 ϕ √q 2 −u 2 λ 12 q m. (3.44) Using this transformation we can parametrize the fluctuations by their wavenumber k, and we find that for a given wavenumber there are two positive frequency, and two negative frequency solutions of each mass, m = 0, cos 2 ϕ, sin 2 ϕ, and 1. Further defininĝ we collect these solutions below.  Fermion fluctuations with m = cos 2 ϕ. These solutions live on the same subspace as the normalizable zero modes (λ P = −1, λ 12 λ 68 = 1) and are given by wherê and the constant spinor W λ satisfies the same conditions as for m = sin 2 ϕ.

Majorana condition. In a Majorana basis (Γ A ) * = −Γ A and the Majorana condition is (Ψ
To impose this condition we need to consider linear combinations of two solutions (from the same mass group) such that the wavenumbers are k and −k, the frequencies are of opposite sign (apart from the massless case), and so are the λ eigenvalues.
Noting that the dispersion relation is invariant under (k → −k, λ → −λ), and it follows that (Ψ J ) * = Ψ J will simply relate the constant spinor multipliers of the two components. We show explicitly how to construct solutions satisfying the Majorana condition in the massless case. Analogous expressions for the massive modes can also be found, but these are quite lengthy. Since they do not play any role in the subsequent analysis we do not write them explicitly here. We start with the linear combination where the two components have opposite k, ω, and λ. Its complex conjugate is and (Ψ J ) * = Ψ J as long aŝ These two conditions are equivalent, and consistent with theΓ 34 ,Γ 12 ,Γ 68 andΓ 012345 eigenvalues of V 1 + and V 2 − . We have found an explicit Majorana solution.
Solutions for AdS 3 × S 3 × T 4 . Again, this geometry corresponds to the ϕ → 0 limit, the reduced equations (3.31) decouple for the P ± subspaces, and all of the solutions are the same form as the p 1268 = 0 fluctuations above. In particular, we have four massless fermions and four massive fermions where under the operatorsΓ 34 ,Γ 12 andΓ 012345 the constant spinor V λ has eigenvalues +i, iλ and +1, while U λ has eigenvalues +i, iλ and −1, respectively. The difference compared to (3.47), (3.49) is that theΓ 68 eigenvalues of U λ , V λ are no longer constrained.

The 1-loop functional determinant
Using the fluctuations found in the previous two sections we now calculate the leading order quantum corrections to the energy of the stationary giant magnon. We follow a similar argument in [29], which is based on well-established quantization techniques for solitons [22,23,64,71]. By energy we mean the Noether charge combination E − J 1 , where E is the conserved charge associated with translations in global AdS 3 time, while J 1 is the U(1) charge associated with rotations along the BMN geodesic. In light-cone gauge, the quantity E − J 1 can be identified with the (transverse) Hamiltonian of physical string excitations [68]. In conformal gauge the sigma-model action has to be supplemented by ghosts to cancel two unphysical bosons, however, for the purposes of our semiclassical analysis it is sufficient to simply omit two of the massless bosonic modes, as disucssed in Section 2.
A detailed calculation (building on the AdS 3 × S 3 × T 4 case discussed in [53]) can be found in [63], here we just note that the mixed-flux dyonic giant magnon has classical charges where cos 2 ϕ is the mass of the magnon and J 2 is its second angular momentum. Remarkably, this classical expression is in agreement with the exact dispersion relation of elementary excitations determined from supersymmetry [51][52][53], hence we expect no quantum corrections. The one-loop correction to the energy can be calculated as the functional determinant ln det |δ 2 S| around the classical background, and is given by where F is the fermion number operator, ν i are the so-called stability angles, frequencies of small oscillations around the classical solution, and the sum is over excitations i and wavenumbers k. For a non-static soliton, like the giant magnon, we can apply the method of Dashen, Hasslacher and Neveu [64] to calculate these stability angles. We put the system in a box of length L 1, with periodic boundary conditions x ∼ = x + L. It is clear from the form of the solution (2.8) that the system is also periodic in worldsheet time, with period T = L/u. Then, the stability angle ν of a generic fluctuation δφ can be read off from δφ(t, x). (4.4) Although we had to write the oscillations in the original worldsheet coordinates (x, t) to get the correct dispersion relations, the magnon's stationary frame (X , T ) is better suited to the analysis of stability angles. In Sections 2 and 3 we found fluctuations with oscillatory terms e ikx−iωt (4.5) parametrized by mass m and an additional eigenvalue λ = ±1, and satisfying dispersion relations Rewriting the plane-wave terms as 7 the new frequency and wavenumber satisfŷ while e iλqmγX can be absorbed into the rest of the Y-dependent solution.

1-loop correction in AdS 3 × S 3 × S 3 × S 1 string theory
For each excitation, the stability angle can be further decomposed as where ν Under the transformation (4.7) we have and it is then straightforward to read off the ν where we have defined With these, the integrand of (4.12) becomes Since E bos (m, +1)E bos (m, −1) holds for general m, (4.12) simplifies to Further noting that it is clear that the integrand is antisymmetric ink. Moreover, we have the asymptotic and taking logarithm, the integrand of (4.19) is O k −3 , hence the integral itself is bounded and well-defined. We conclude that the integral is zero, and, in agreement with our expectations, the giant magnon energy receives no corrections at one loop, providing another check on our results.

1-loop correction in AdS 3 × S 3 × T 4 string theory
On AdS 3 × S 3 × T 4 the situation is even simpler. We have two bosons and two fermions for each of the 4 combinations of m = 0, 1, λ = ±1. Paring these up, the ν (0) i contribution vanishes, while the ν (2) i terms cancel between λ = ±1 pairs. With two of the massive bosons and four of the massive fermions contributing, the integrand of (4.12) becomes which is the same as the ϕ → 0 limit of (4.17). Using (4.18) we arrive at the expected zero one-loop correction result even before integrating overk.

Conclusions
In this paper we found the full spectrum of fluctuations around the mixed-flux AdS 3 stationary giant magnon, the q > 0 generalisation of the Hofman-Maldacena giant magnon.
To obtain the non-trivial bosonic fluctuations, we adapted the method used in [29]. Rather than dressing the vacuum twice to get a complicated breather-soliton superposition (only then to expand in small breather momentum), we dress the perturbed BMN vacuum once, keeping terms up to subleading order throughout the calculation. The leading order term in the dressed solution is the giant magnon, so the subleading term must be its perturbation. The fermionic fluctuations are obtained as solutions of the equations derived from the quadratic fermionic action, using the formalism developed in [63], which builds on the original developments of [28] for AdS 5 . We find that all of the fluctuations can be written in the form where u is the magnon's speed on the worldsheet (2.11), and with k, ω satisfying which is the small-momentum limit of the exact dispersion relation (1.7). Furthermore, the fluctuations can be arranged into short multiplets of the residual symmetry algebras, according to mass and chirality (± sign in the dispersion relation). On AdS 3 × S 3 × T 4 there are four 4 dimensional multiplets of psu(1|1) 4 c.e. with two bosons and two fermions, while AdS 3 × S 3 × S 3 × S 1 has eight 2 dimensional multiplets of su(1|1) 2 c.e. , with a boson and a fermion each.
Finally, from the explicit form of each fluctuation we read off the so called stability angles, which sum to the one-loop functional determinant. In both of the geometries we were able to show that this one-loop determinant is zero, or in other words, the one-loop correction to the magnon energy vanishes. It is interesting to compare this result with other calculations of the one-loop correction to energies of AdS 3 string states. The expansion of the coupling h around the classical string limit is equivalent to the expansion of the energy (1.7) where the subscript 0 refers to the classical (string) values, and we see that our results translate to c = 0 for both geometries. The one-loop correction to the giant magnon energy on AdS 3 × S 3 × S 3 × S 1 with pure R-R flux was derived in [41] directly from the GS action, and in [57] from the algebraic curve. They both found that the correction is dependent on the chosen regularisation scheme, with two naturally emerging prescriptions: in the physical regularisation the cutoff is at the same mode number for all excitations, while in the new prescription the cutoff is proportional to the mass of the polarisation. The two prescriptions both give zero correction c phys = c new = 0 on the AdS 3 × S 3 × T 4 background, but differ for the AdS 3 × S 3 × S 3 × S 1 theory For the mixed-flux AdS 3 × S 3 × T 4 background the direct GS action calculation [58] shows that there is no one-loop correction, c = 0, and the same conclusion can be drawn by considering the worldsheet scattering of giant magnons [55]. Our results are in agreement with the new prescription, although it is not clear that we work in either of the regularisation schemes, as in (4.19) we have an implicit cutoff 8 on the mode numbersk in the magnon's frame. There have been recent advances in our understanding of the protected spectrum of AdS 3 /CFT 2 using integrable methods [72,73]. The protected spectrum for AdS 3 × S 3 × T 4 agrees with the older results of [74], while the AdS 3 × S 3 × S 3 × S 1 case was independently derived using supergravity and WZW methods in [75].
As we have discussed in [63], there is no stationary magnon for q = 1, hence the results of present paper do not apply in this limit. The pure NS-NS string theory has been long known to be solvable using a chiral decomposition [32][33][34], and there is now a good understanding of integrability for the microscopic excitations [76][77][78][79], but it would be interesting to see a soliton analysis on these backgrounds. In more recent developments, the CFT dual of the k = 1 WZW model, i.e. AdS 3 × S 3 × T 4 with minimal quantized NS-NS flux, has been identified as a symmetric product orbifold [49,[80][81][82].
Semiclassical methods continue to provide valuable insight into the AdS 3 /CFT 2 duality, as one can see in this paper, the analysis of fermion zero modes [63], or the calculation of 1-loop corrections to the rigid spinning string dispersion relations [83]. Where they seem to fail is the description of massless modes. In the α → 0 limit our fluctuations simply reduce to the plane-wave perturbations of the BMN vacuum, shedding no further light on the nature of the massless soliton of the theory, in agreement with our previous findings [63], and the fact that the α → 0 limit of the spin-chain fails to capture these inherently non-perturbative modes on the other side of the duality [84]. Furthermore, massless modes render a perturbative computation of wrapping corrections impossible, once the theory is put on a compactified worldsheet [85]. Instead, wrapping corrections may be computed from a non-perturbative TBA using an alternative low-momentum expansion [86][87][88].
There are two interesting directions for future research. Firstly, we would like to get a better understanding of the solitons of the q = 1 theory, their zero modes and fluctuation spectrum. Secondly, and this is a more speculative direction, one could try and describe the elusive massless modes by finding solitons of the fermionic part of the action, motivated by the fact that the massless representation's highest weight state is a fermion [59].

Acknowledgments
AV would like to thank B. Stefanski for useful discussions. AV acknowledges the support of the George Daniels postgraduate scholarship.

A Dressing the perturbed BMN string
In this section we briefly review the dressing method for the SU (2) principal chiral model [65][66][67] , and describe how it can be used when working with perturbations. We then apply this recipe to the three different perturbed BMN strings to obtain the three S 3 fluctuations of the AdS 3 × S 3 × T 4 giant magnon. The S 3 + perturbation of the AdS 3 × S 3 × S 3 × S 1 magnon can be obtained from these, simply by scaling the worldsheet coordinates by cos 2 ϕ.

A.1 Review of the SU(2) dressing method
The S 3 part of the sigma-model action (2.1) is equivalent to the SU(2) principal chiral model with Wess-Zumino term where the left currents are J a = g −1 ∂ a g, the partial derivatives on J = g −1 ∂g,J = g −1∂ g are with respect to z = 1 2 (t − x) andz = 1 2 (t + x), and the embedding connects to the R 4 coordinates of (2.1) Note thatZ i are the complex conjugates of Z i for the real classical solution, but not necessarily for the perturbation that we will write as complex functions. The equations of motion for the action (A.1) are and starting with a solution g, the dressing method aims to find the appropriate dressing factor factor χ(z,z) such that is a new solution. The construction exploits the equivalence between the compatibility condition of the overdetermined auxiliary system One then solves the auxiliary problem for general complex spectral parameter λ such that Ψ(λ) satisfies Ψ(0) = g, (A. 8) and the unitarity condition The simplest non-trivial dressing factor is then given by with the projector We will also refer to the matrix X and scalar y In order for the dressed solution χ(0)Ψ(0) to have unit determinant, we need to introduce an additional constant phase (λ 1 /λ 1 ) 1/2 , and with this, the dressed solution becomes

A.2 Dressing the unperturbed BMN string
To set the scene and some notation, let us quickly run through the application of the dressing method to the BMN string Z 1 = e it , Z 2 = 0. We solve the auxiliary problem to find Introducing the real variables 16) the projector (A.11) becomes Pametrizing the pole as λ 1 = re i p 2 , the dressing (A.13) yields the giant magnon Furthermore, setting r =q −1 , we get the stationary magnon where (A.20)

A.3 Dressing the perturbed BMN string
To apply the dressing method to the perturbed BMN string in each step we keep terms up to first order in δ. For example The auxiliary problem can be written as and its solution Ψ 0 (λ) = Ψ BMN (λ) + δ Ψ pert (λ). (A.24) Then we expand the projector 25) i.e. 26) and the dressing factor (A.13) Finally, the dressed solution is from which we can read off the perturbation as the first order term. Let us now apply these steps to the three perturbations we found in 9 (2.25)-(2.26).

Massless fluctuation. The massless BMN perturbation is
for which the auxiliary problem has perturbations Then P pert can be calculated in terms of Finally, from we can read off the fluctuation components (after a constant rescaling)

Massive fluctuation (1). The first massive BMN fluctuation is
for which the auxiliary problem has perturbations Further substituting and using the identity one gets the magnon fluctuation (rescaled by a constant) In this appendix we compare our solutions, in the ϕ = q = 0 limit, to the fluctuations of the AdS 5 × S 5 giant magnon found in [29]. To harmonize notation, we need to write the frequency and wavenumber in the boosted worldsheet basis e ikx−iωt = e ikX −iωT , k = γ(k + uω) = csc p 2 (k + cos p 2ω ), ω = γ(ω + uk) = csc p 2 (ω + cos p 2k ), where we also used the q = 0 version of (2.11).

B.1 Bosonic fluctuations
Although in the q = 0 limit the stationary magnon reduces to the HM giant magnon, due to obvious differences in the geometry we will only match a subset of our fluctuations to a subset of the ones found in [29]. The magnon on AdS 5 × S 5 has four massive and one (unphysical) massless fluctuations on AdS 5 , and, four massive and one (unphysical) massless fluctuations on S 5 . Out of these, we will match both unphysical and four of the massive modes (two each on AdS

B.2 Fermionic fluctuations
Since AdS 5 × S 5 is supported by 5-form fluxes, while AdS 3 × S 3 × T 4 is supported by 3-form fluxes, the spinor structure of fermion fluctuations on the two backgrounds will be quite different, however, it is reasonable to expect similar functional forms. The kappa-fixed solutions (3.35), (3.37) in [29] are of the form and we can rewrite (B.7) as (B.11) in agreement with the q = 0 limit of (3.59), with the caveat that in [29] the spinors are swapped Ψ 1 ↔ Ψ 2 compared to our notation.

C Coefficients in the reduced equations of motion
Here we present the coefficients of the reduced equations (3.27), and to do so in a relatively compact form we need to introduce the shorthands With these, we have C f 1 f 1 = +N −+ + i(ω + λ 12 (1 + p 1268 tan 2 ϕ)ξ) − iλ 12 q γζ −1 p 1268 tan 2 ϕ, C g 1 g 1 = −N ++ + i(ω + λ 12 p 1268 tan 2 ϕ ξ) − iλ 12  Note that p 1268 is the eigenvalue of the ansatz with respect to the projector 1 2 (1+Γ 1268 ), and ∆ = 0 exactly when p 1268 tan ϕ = 0. In this case we see that the last block of coefficients are zero, the P ± parts of the equations decouple and we have solutions with definiteΓ 012345 chirality.