Heterotic AdS_3/CFT_2 duality with (0,4) spacetime supersymmetry

We discuss the AdS_3/CFT_2 duality of a heterotic three-charge model with (0,4) target space supersymmetry. The worldsheet theory for heterotic strings on the AdS_3 x S^3/Z_N x T^4 near-horizon geometry was constructed by Kutasov, Larsen and Leigh in [hep-th/9812027]. We propose that the dual conformal field theory is given by a two-dimensional (0,4) sigma model arising on the Higgs branch of an orbifolded ADHM model. As a non-trivial consistency check of the correspondence, we find that the left- and right-moving central charges of the infrared conformal field theory agree with those predicted by the worldsheet model. Moreover, using the entropy function formalism, we show that to next-to-leading order the central charge can also be obtained from an alpha'-corrected supergravity theory.


Introduction
Recently, several authors [1,2,3,4,5,6,7] have studied the possibility of an AdS 3 /CFT 2 duality for the fundamental heterotic string. Heterotic strings are dual to type I D1branes whose low-energy effective field theory is expected to be conformally invariant. The dual near-horizon geometry of the heterotic string should therefore contain an AdS 3 factor. This was confirmed in [8] (see also [9]) in which an AdS 3 × S 2 factor was found in a N = 2, d = 5 R 2 -corrected supergravity solution corresponding to heterotic strings in five dimensions.
In general, heterotic string setups may contain additional charged objects such as NS5-branes and Kaluza-Klein monopoles. Such setups generically have (0, 4) target space supersymmetry. Recently, it has been found that in the absence of some or all of these additional charges the target space supersymmetry is enhanced to (0, 8) [2,4,5,6] (see also [10]). Such theories are expected to be very different from those with only (0, 4) supersymmetry. For one thing, there are no linear superconformal algebras with more than four supercurrents. Indeed, it has been argued in [5,6] that the global supergroup of the boundary CFT is Osp(4 * |4), whose affine extension is given by a nonlinear N = 8, d = 2 superconformal algebra. For another, it is not clear if these theories possess unitary representations.
In this paper we take a step back and address the construction of a heterotic AdS/CFT duality with only (0, 4) target space supersymmetry. For this we revisit a heterotic threecharge model previously studied by Kutasov, Larsen and Leigh (KLL) in [11]. The setup consists of p fundamental strings embedded in the worldvolume of N ′ NS5 branes and N Kaluza-Klein (KK) monopoles. In [11] KLL work out the worldsheet theory for string theory on the corresponding near-horizon geometry AdS 3 × S 3 /Z N × T 4 . The worldsheet CFT turns out to be essentially the product of an SL(2) WZW model and a "twisted" SU(2) WZW model corresponding to the asymmetric orbifold S 3 /Z N . In contrast, not much is known about the dual conformal field theory on the boundary of the AdS 3 space.
The first part of this paper is therefore devoted to the construction of the dual twodimensional boundary conformal field theory. We first apply heterotic/type I duality to map the three-charge configuration to an intersection of p D1-branes and N ′ D5-branes plus N KK monopoles in type I string theory. In the absence of any KK monopoles the low-energy effective theory corresponds to Witten's ADHM sigma model of Yang-Mills instantons [12], as shown by Douglas in [13]. To also include KK monopoles, which have a C 2 /Z N near-core geometry, it is natural to construct a Z N orbifold theory of the massive ADHM sigma model. (Refs. [14,15] also use an orbifold construction to obtain the boundary CFT dual to type II string theory on Our proposal is that the sought-after boundary conformal field theory arises on the Higgs branch of the orbifolded ADHM model, which corresponds to the bound state phase of the D-brane setup. We will perform a consistency check for the proposal by the following line of reasoning. Lambert has shown in [16] that, even though the ADHM model is classically not conformal, it is ultraviolet finite to all orders in perturbation theory. There is no renormalisation group flow, and anomalous conformal dimensions are absent [16]. The conformal Higgs branch theory can therefore be obtained by integrating out the massive degrees of freedom in the ADHM model [17]. Moreover, the central charges of the Higgs branch theory can be determined by counting the massless degrees of freedom of the ultraviolet theory. In other words, they are given by the dimension of the instanton moduli space of the ADHM model. Repeating these steps for the orbifold version of the ADHM model, we determine the central charges of the low-energy theory of the three-charge model and match them to those predicted by the worldsheet theory.
The second part of the paper is devoted to the construction of a higher-derivative correction of the near-horizon supergravity solution of the KLL setup. In fact, for a dual setup a full solution of the N = 2 off-shell completion of four-derivative supergravity in five dimensions was constructed already in [8]. Here we will use six-dimensional corrections to the heterotic string action [18,19] and employ the entropy function formalism [20,21] to find the corrected near-horizon geometry. To first order, the latter correctly reproduces the expected central charges of the boundary CFT via the Brown-Henneaux formula [22].

Heterotic AdS 3 /CF T duality
In this section we review the supergravity solution of the heterotic three-charge model of [11] and the corresponding worldsheet model. Readers familiar with Ref. [11] may wish to proceed directly to the discussion of the boundary conformal field theory in section 3.

Three-charge model for heterotic strings
We consider heterotic string theory compactified on S 1 × T 4 which we take along the directions {x 5 } and {x 6 , x 7 , x 8 , x 9 } respectively. In particular, following [11], we study the following brane setup: • p fundamental strings F1 infinitely stretched in the x 1 direction, • N ′ NS5-branes wrapped around the T 4 and infinitely stretched along x 1 , • N KK monopoles wrapped around T 4 and extended in x 1 .
We can depict this configuration schematically in the following table: From a 5-dimensional spacetime point of view this configuration looks like an infinitely stretched string in the x 1 direction, which preserves (0, 4) supersymmetry, i.e. it is nonsupersymmetric in the left sector and contains four supercharges in the right sector. Let us recall the classical solution as given in [11]. The metric is given by with the following harmonic functions Here we use spherical coordinates (r, θ, ϕ) for the directions (x 2 , x 3 , x 4 ). The corresponding gauge fields and the dilaton read 3) The quantities P 5 , P K , Q are related to N ′ , N, p by where R is the asymptotic radius of the S 1 , V the volume of the torus and Φ 10 (∞) the asymptotic value of the dilaton.
In the near-horizon limit r → 0, the metric (2.1) reduces to where we have defined r ′ by In [11] this metric was interpreted as describing the space with AdS radius and six-dimensional string coupling Obviously, string theory on this background is weakly-coupled for N ′ ≪ p. Note that so far we have only discussed an uncorrected supergravity solution, i.e. a solution to an action at the two derivative level. In section 4 we will address the question of how to modify (2.5) in the presence of higher derivative interactions.

Lift to M-theory
In order to understand why the supergravity solution (2.5) is expected to receive α ′ corrections, we now determine the central charges of the boundary CFT. We begin by mapping the heterotic setup to M-theory compactified on CY 3 = K3 × T 2 . For this, we first dualize to type IIA theory, from where (after additional S and T dualities) we may lift to M-theory -see appendix A for details. We obtain the following setup of M5 branes: 0 1 2 3 4 5 6 7 8 9 10 Our convention will be that the internal T 2 is spanned by the directions {x 5 , x 10 } while the K3 resides in {x 6 , x 7 , x 8 , x 9 }.
A general method for determining the central charges of the low-energy effective theory on M5-branes wrapping a 4-cycle in a Calabi-Yau three-fold CY 3 is given in [23]. The lowenergy effective field theory is given by a two-dimensional (heterotic) sigma model with the M5-brane moduli space as target space. The left-and right-moving central charges c L,R of this sigma model are given by where c IJK are the intersection numbers of CY 3 , and p I is the (magnetic) charge of the M5-brane wrapping the Ith 4-cycle [23]. The product c 2 · p contains the second Chern class of CY 3 . 1 Let us apply these formulae to the present case 2 and identify Denoting the single modulus of the T 2 by p 1 , the only non-vanishing intersection numbers are c 1ij = c ij , where c ij is the intersection matrix for K3. For p M5-branes wrapping K3, c 2 · p = c 2 (K3)p = 24p [8], and (2.9) provides the central charges Since D = 0, this three-charge model preserves only (0, 4) supersymmetry [23]. For N = N ′ = 0, we have D = 0 and (c L , c R ) = (24p, 12p). These are the central charges of the (0, 8) low-energy effective field theory describing a stack of p heterotic strings.
Let us compare the central charges c L,R with that obtained from the supergravity solution by applying the Brown-Henneaux formula [22], N is Newton's constant in three dimensions. Substituting the AdS radius (2.8) of the uncorrected supergravity solution into (2.12), we get as was already found in [11]. We notice that (2.13) agrees with (2.11) only to leading order in the charges. The reason for the absence of the subleading term in (2.13) is the fact that it is computed from an uncorrected supergravity solution. Taking into account higher derivative terms in the action as well, one recovers the full expression (2.11), as was recently shown for a dual setup [8]. We will reproduce this result with somewhat different methods in section 4.

N = (0, 2) worldsheet theory
We now discuss heterotic string theory on the AdS 3 × S 3 /Z N × T 4 near-horizon geometry of the F1-NS5-KKM three-charge model introduced in section 2.1. The corresponding worldsheet theory has been constructed in [11], and we will only review some of its features relevant for the construction of the boundary conformal field theory. The worldsheet theory is expected to be the product of a heterotic SL(2) WZW model, a conformal field theory on S 3 /Z N and a free U(1) 4 CFT on the four-torus T 4 . As a heterotic model, the product theory is bosonic in the left-moving sector and supersymmetric in the right-moving sector. The heterotic SL(2) WZW model therefore has a bosonic affine SL(2) algebra of level k b in the left-moving sector and a supersymmetric one of level k s = k b − 2 in the right-moving sector. Accordingly, the right-moving sector is generated by three bosonic and three fermionic currents,J A andψ A (A = 1, 2, 3), while the left-moving sector contains only J A . Similarly, the right-moving CFT on T 4 is constructed from four bosonic fieldsȲ i and four fermionsλ i (i = 1, 2, 3, 4). The left-moving sector contains only the bosonic currents Y i .
In the unorbifolded case, the S 3 factor of the geometry would be described by an SU(2) WZW model with levels k ′ b and k ′ s = k ′ b + 2 in the left-and right-moving sector, respectively. The right-moving sector of the SU(2) model contains three bosonic currents, K a , and three fermionsχ a (a = 1, 2, 3). The left-moving sector has the same bosonic currents K a (a = 1, 2, 3), but again no fermions.
Let us now implement the Z N orbifold. We start from the SU(2) WZW model in which we parameterise the SU(2) group manifold in terms of the Euler angles where ξ parameterises the fibre, and θ, φ are the base coordinates. As in [11], we consider an SU(2) model at level and identify The orbifold acts asymmetrically in the near-horizon geometry. We therefore turn the SU(2) WZW model into a coset model of the type where the orbifold is embedded in SU(2) L : Z N acts on the currents as (2.18) For N > 2, the effect of the asymmetric orbifold is to break the SU(2) of the left-moving sector down to U(1), whose current K 3 is invariant under the orbifold action. 3 The consistency of the theory requires that the worldsheet central charges are (c ws L , c ws R ) = (26,15). The central charges in the right-moving sector are which adds up to c ws R = 15 provided that Similarly, for the left-moving sector we have which adds up to ten. Heterotic string theory also contains 32 left-moving current algebra fermions, i.e. 16 for each E 8 . We thus get c ws L = 10 + 16 = 26, as required. The worldsheet theory also provides some information on the boundary conformal field theory. As shown in [25], the left-and right-moving (super)Virasoro algebras of the boundary CFT can be constructed from the worldsheet affine SL(2) Lie algebra. Their central charges are (c L , c R ) = (6k b p, 6k s p) , (2.22) where, as before, k b and k s = k b −2 are the levels of left-and right-moving SL(2) algebras, and p is the number of heterotic strings. Substituting (2.15) and (2.20) in (2.22), we find the central charges which agree with (2.11) and satisfy the constraint c L − c R = 12p as also found in [11,6].
Let us finally consider the amount of worldsheet and target space supersymmetry. From the geometry we expect that the worldsheet model preserves a (0, 4) target space supersymmetry. Since T 4 is Kähler, the heterotic worldsheet CFT on T 4 has (0, 2) supersymmetry. The Kähler structure also ensures that the (0, 2) worldsheet supersymmetry leads to (0, 4) spacetime supersymmetry. The heterotic SL(2) model and the "twisted" SU(2) model separately preserve only (0, 1) supersymmetry. Only the product of both models has a chance to have (0, 2) worldsheet supersymmetry. In order to enhance N = 1 to N = 2 supersymmetry in the right sector, one must find a U(1) R current J N =2 , which is part of the N = 2 algebra. The existence of such a current is guaranteed by the fact that the orbifold is embedded in SU(2) L such that the right sector remains unaffected by it. The N = 2 U(1) R current therefore has the same structure as in the (unorbifolded) type II case, see [25].
3 Two-dimensional boundary sigma model

General remarks
In this section we discuss the two-dimensional (0, 4) conformal field theory living on the boundary of the AdS 3 space. Our starting point is the heterotic brane setup introduced in the previous section. We first T-dualize in x 5 to go from E 8 × E 8 to SO(32) heterotic string theory and then use heterotic/type I duality in order to obtain the following type I brane configuration: Let us consider the type I setup in detail. Since the heterotic/type I duality involves a strong-coupling transition, the heterotic F1 and NS5-branes naturally map to D1 and D5-branes. Moreover, since we are dealing with a type I string theory we are also required to introduce 32 D9-branes and perform an orientifold projection. In order to understand the contribution of the KK monopoles, we recall that the approximation of the near-core region of N KK monopoles is a C 2 /Z N orbifold. This instructs us to study a Z N orbifold in the directions x 2,3,4,5 of the D1-D5-D9-brane theory.
We will start out our construction by reviewing the low-energy effective theory of the type I D1-D5-D9 intersection. In the absence of any KK monopoles this theory was shown in [13] (for p = 1) to be equivalent to Witten's ADHM model of Yang-Mills instantons. In section 3.2 we will review the model for p > 1 as constructed in [26]. In section 3.3 we will include the effect of the KK monopoles by orbifolding the ADHM model. Subsequently, in section 3.4 we discuss its instanton moduli space and determine the central charges of the Higgs branch theory.

Spectrum of D1-D5-D9 and the ADHM model
Let us briefly recall some basic facts. Spacetime fermions arise in the Ramond sector, and spacetime bosons in the Neveu-Schwarz sector. If the boundary conditions on both ends of the string are the same, then the worldsheet fermions of the R sector have integral modes, and those in the NS sector half-integers. If the boundary conditions are different, the additional signs introduced exchange the moddings, which also changes the ground state energy of the sector. In particular, the NS ground state energy in the case of N DN mixed boundary conditions is given by whereas the ground state energy in the R sector is always zero.
Let us now discuss the strings stretching between the various types of branes.

1-1 strings
In the NS-sector, the massless modes form a ten-dimensional vector A µ ab , the Chan-Paton indices running over a, b = 1, . . . , p. Considered as an object on the D1, it splits into a 2d vector A µ ab and 8 scalars b i ab . The orientifold projection Ω maps A µ ab → −A µ ba . We are thus left with the gauge bosons A µ [ab] in the adjoint of the gauge group SO(p). On the other hand, the vertex operator of b picks up no sign under Ω, as it contains no derivative along the boundary. This leaves 8 bosons b i (ab) in the symmetric representation of SO(p) which we group in a pair of 4 bosons, b AY (ab) and b A ′Ã′ (ab) . In the R-sector, the GSO projection restricts to modes which are invariant under Γ := Γ 0 . . . Γ 9 , where Γ µ denotes the fermionic zero modes. To obtain the action of Ω, note that the fermionic modes ψ 2 , . . . , ψ 9 reflect from the boundary with an extra minus sign, so that they pick up an additional minus sign under exchange of right and left movers. Ω thus acts on massless fermions as Ω = −Γ 2 Γ 3 . . . Γ 9 . The massless spinors thus must satisfy the two conditions The first condition simply states that ψ is in the 16 of SO (1,9). To obtain the worldsheet behaviour of ψ, we need to decompose 16 into representations of SO(1, 1) × SO(8), which gives 16 = 8 ′ + ⊕ 8 ′′ − , where 8 ′ , 8 ′′ are the two spinor representations of SO(8) and ± denotes the chirality with respect to SO(1, 1). The second condition in (3.2) then states that ψ (ab) transforms as 8 ′′ − , and ψ [ab] as 8 ′ + . The ψ (ab) are the right-moving superpartners of the b (ab) . Due to the D5-branes each 8 decomposes into a pair of 4's of the SO(4)'s. Following [13], these will be denoted by . The left-moving fermions ψ + [ab] are antisymmetric and therefore do not appear in the case of a single D1-brane.

1-5 strings
The analysis of this sector has been performed in [27]. Since N DN = 4, the ground state energy is also zero in the NS-sector, so that there appear both bosons and fermions. In total, we obtain bosons (32), and their right-and left-moving fermionic superpartners χ Am − a and χ Y m + a . The index m runs over m = 1, ..., 2N ′ .

1-9 strings
Since N DN = 8, the ground state energy of the NS-sector is strictly positive, so that there are no bosons. In the R-sector there are two massless modes Γ 0 , Γ 1 . The GSO projection eliminates one of them, leaving only the left moving mode. We thus obtain 32p left-moving fermions λ M +a , where M = 1, ..., 32 is the Chan-Paton index of SO(32).

5-5 strings, 5-9 strings
The analysis of the remaining sectors has been performed in [13]. Since their field content is not very important in what follows, we only cite the results. The 5-brane fields form a Sp(2N ′ ) gauge theory, a hypermultiplet in the antisymmetric representation with scalar component X AY [mn] , and "half- We summarise the results by listing the relevant fields in the following table (see also [26]): We have not listed fields coming from 5-5 and 5-9 strings, since here we are only interested in the case of vanishing instanton size which corresponds to setting the 5-9 fields to zero (see [28,17]). Moreover, the 5-5 fields X AY mn denote the position of the D5-branes in the transversal space, which we treat as parameters of the low energy theory. 4 The Lagrangian describing the low-energy physics of the type I D1-D5-D9 intersection can now be written in terms of the fields of table 3.1. For p ≥ 1, it is convenient to divide the Lagrangian into three parts, where L kin contains the kinetic terms for all fields in table 3.1, and L pot describes their potential. In general, L pot contains Yukawa couplings of the type bψ + ψ − and D-terms for the scalars b. For details, see ref. [26]. The Lagrangian L int describes the interaction of 1-1 with 1-5 string modes and is given by [13,26] where the trace is taken over the SO(p) indices. As first found in [13] for p = 1, this Lagrangian corresponds to Witten's ADHM model [12] describing an Sp(2N ′ ) instanton with instanton number one. It is believed that for p ≥ 1 the ADHM model describes the moduli space of Sp(2N ′ ) instantons with instanton number p.

Field content of the orbifold theory
Let us now include the effect of the KK monopoles in the ADHM model. This requires us to consider the D1-D5-D9 intersection at the origin of a C 2 /Z N orbifold acting along x 2,3,4,5 . Following Refs. [29,30,31], we start with pN D1-branes intersecting 2N ′ N D5branes and 32N D9-branes in flat space and take the corresponding ADHM Lagrangian with gauge group U(Np)×U(2NN ′ )×U(32N) as the parent theory. 5 The ADHM orbifold theory is then obtained by projecting out the degrees of freedom which are not invariant under the Z N orbifold group. 6,7,8,9 ) remain unaffected by the orbifold. The origin of x 2,3,4,5 is the only fixed point of the orbifold. The orbifold therefore acts on the fields of the ADHM model as follows (gauge indices suppressed): Here g 1 (ω), g 5 (ω), g 9 (ω) denote the usual embeddings of the Z N orbifold group in the gauge groups U(Np), U(2NN ′ ) and U(32N), respectively. We choose a basis such that the embedding matrices have the block-diagonal form g i (ω) = diag(½, ω½, ω 2 ½, ..., ω N −1 ½), where ½ denotes a p × p, 2N ′ × 2N ′ and 32 × 32 unit matrix for i = 1, 5, 9, respectively.
We observe that all fields carrying an index Y transform non-trivially under the orbifold group, i.e. the transformation law contains an additional factor ω Y . Substituting the embeddings g i (ω) into (3.8), we get the following transformation behaviour in component form: where Y = ±1. The fields invariant under the orbifold action (3.9) are thus Another important question concerns the gauge groups and representations under which these fields transform. Due to the Ω-projection of type I string theory, this issue is more intricate than in type II theories and will now be discussed at length.

Type I effective action
The type I effective theory is obtained by imposing, in addition to the Z N orbifold projection, the orientifold Ω [29]. Let us denote the embedding of Ω into the gauge groups U(Np), U(2NN ′ ) and U(32N) by g 1 (Ω), g 5 (Ω) and g 9 (Ω), respectively. A generic (scalar) field y then transforms under worldsheet parity according to while an element U of one of the above gauge groups satisfies Here t denotes the transpose and g is one of the embeddings g 1 , g 5 , g 9 .
To determine g, we have to solve various consistency conditions [29]. The first condition is (3.12) We choose the phase χ(ω, Ω) = 1 which then implies that only g(Ω) i,N −i is non-vanishing. 6 A second condition requires with some phase factor χ(Ω) = ±1. To reproduce the standard type I action, which has an SO(32) gauge group for the D9-branes, we choose the phases χ(Ω) = +1, −1, +1 for g = g 1 , g 5 , g 9 , respectively. 7 The solutions of (3.13) can be brought into the form g 1,9 (Ω) 0,0 = ½ , where ½ is the corresponding p × p, 2N ′ × 2N ′ or 32 × 32 unit matrix. For even orbifolds, we have in addition g 1,9 (Ω) N/2,N/2 = ½ , Let us now determine the unbroken gauge groups from (3.11). We distinguish between even and odd orbifolds: • even N For g = g 1 , the gauge group of the D1-branes is while for the D5-branes, it is (3.16) • odd N For g = g 1 , we get the gauge group while for g = g 5 , it is The effect on the matter fields is as follows. For the b AY , equation (3.10) reads For N even, this relates one half of the fields to the other half, but gives no additional constraints. The same holds true for the fermions ψ YÃ ′ + and ψ A ′ Y − . If N is odd, there is the additional condition The b A ′Ã′ are subject to (and also b A ′Ã′ N/2,N/2 if N is even) are symmetric. Again, the situation is exactly as described above such that the corresponding fermionic modes, ψ AÃ ′ − 0,0 and ψ A ′ A + 0,0 (and ψ AÃ ′ − N/2,N/2 and ψ A ′ A + N/2,N/2 for N even), are in the symmetric and anti-symmetric representation, respectively.
We omit the corresponding relations for the φ A ′ m i,i , as they again only relate half of the fields to the other half [29].

Quiver theory
So far we have determined the spectrum of fields that survive the orientifold projection along with the gauge groups of the world-volume theories of the various branes. It remains to determine the representations under which the matter fields transform. In fact they are given by The gauge groups and matter content of the theory can now be encoded in a quiver diagram, see figure 3. Each node in the inner circle corresponds to a gauge group G 1 j (D1-branes), while an outer node represents a gauge group G 5 j (D5-branes). In principle, there are also nodes corresponding to SO(32) gauge groups (D9-branes). The latter are not needed for the interaction Lagrangian and are therefore not shown in figure 3.1. The fields b A ′Ã′ j,j , ψ AÃ ′ − j,j , ψ A ′ A + j,j transform under a single gauge group and are represented as brown circles. The bifundamentals b AY j,j+Y , ψ A ′ Y − j,j+Y , ψ YÃ ′ + j,j+Y (shown as black lines), φ A ′ m j,j , χ Am − j,j (green lines), and χ Y m + j,j+Y (blue lines) connect different nodes. We have omitted bifundamentals connecting the outer nodes. These are generated by 5-5 strings which decouple at low-energies, as already discussed earlier.
We may now write down the corresponding quiver Lagrangian which descends from the ADHM Lagrangian in flat space, Eq. (3.3). Upon projecting out the degrees of freedom which are not invariant under the orbifold, we obtain L = L kin, quiv + L pot, quiv + L int, quiv (3.22) with the quiver interaction and, similarly, L kin, quiv and L pot, quiv are the projections of L kin and L pot in (3.3), respectively. The range of summation over j and Y is restricted by the Ω projection. For instance, for N even, consider again the quiver diagram shown in figure 3.1. Each Yukawa coupling corresponds to a triangle in the quiver diagram. The field identifications of the previous section introduce a kind of reflection axis, which vertically divides the quiver in two parts. The SO(p) gauge groups at j = 0, N/2 lie on the Z 2 reflection axis. Due to constraints such as (3.19), each field on the right hand side of the axis is identified with one on the left hand side. In (3.22) we therefore sum only over j = 0, ..., N/2 and set Y = +1 at j = 0 and Y = −1 at j = N/2, Y = ±1 otherwise. The gauge groups are chosen as in (3.15) and (3.16). For N odd, the Lagrangian is constructed in a similar way.

Higgs branch theory
In this section we investigate the infrared fixed point theory of the ADHM quiver model (3.22). This theory will be interpreted as the boundary conformal field theory dual to the worldsheet theory described in section 2.3. For its construction, we first have to choose a vacuum solution which sets the potential of (3.22) to zero. Inspecting the term m 2 b 2 φ 2 in (3.22), we find two different possibilities for the scalars b AY and φ A ′ m and their vacuum expectation values b AY and φ A ′ m [12]: 9 • Coulomb branch: b AY = 0 and φ A ′ m = 0 On the Coulomb branch the D1-branes are transversely displaced from the D5branes with b AY proportional to the distance. In this case the φ A ′ m become massive.
• Higgs branch: b AY = 0 and φ A ′ m = 0 On the Higgs branch the D1-branes and D5-branes form a bound state with φ A ′ m proportional to the binding strength between the two. In this case the b AY become massive.
In the following we are interested in the situation where all branes form stacks located at the orbifold fixed point. We will therefore consider the Higgs branch of the theory. In principle, we could now proceed as in [17] and integrate out all massive modes of the quiver theory. As in [17], this would lead to a (0, 4) sigma model whose target space is the instanton moduli space M of the ultraviolet theory. The actual construction would be along the lines of [17] and involves a non-trivial gauge field F pq mn jj which is defined in terms of the bifundamentals φ A ′ m jj . Although straightforward, we will not do this explicitly here. Instead we only determine the left-and right-moving central charges of the infrared theory and compare them to those expected from the dual worldsheet model.
As outlined in the introduction, our strategy to find these charges is as follows. The ADHM quiver model is classically not conformally invariant, but ultraviolet finite such that there is no renormalisation group flow. This follows from the fact that the one-loop diagrams cancel, and all higher loop diagrams are finite [16]. The massless fields of the quiver model therefore do not acquire anomalous conformal dimensions and contribute to the central charges of the infrared conformal field theory. This allows us to determine the left-and right-moving central charges of the infrared conformal field theory from the number of massless modes in the ultraviolet quiver theory.

Number of massless modes for N even
We begin by counting the massless degrees of freedom in the case of even orbifolds: First, there are the bifundamental fields (φ A ′ m a ) j,j descending from 1-5 strings and their left-and right-moving fermionic partners (χ Am − a ) j,j and (χ Y m + a ) j,j+1 . These fields are not constrained by any D-term relations and thus contribute 2 · N · 2N ′ · p = 4NN ′ p scalars and an equal number of left-and right-moving fermions. The 5-1 string modes are related to the 1-5 modes by the Ω reflection and therefore do not contribute any additional massless modes.
Second, consider the bosons (b AY ab ) j,j+Y which are massive on the Higgs branch. Since the theory has (0, 4)-supersymmetry, we know immediately that an equal number of rightmoving fermions (ψ A ′ Y − ab ) j,j+Y has to obtain mass. However, since only non-chiral fermions can be massive, it follows that also all left-moving (ψ AA ′ + ab ) j,j become massive. The mass terms for the latter arise due to couplings of the type ψ + χ − φ in (3.22). This sector thus has no massless modes.

Number of massless modes for N odd
Much of the above analysis carries over to odd orbifolds. The fields (φ A ′ m a ) j,j again contribute 4NN ′ p massless bosonic degrees of freedom and an equal number of left-and right-moving fermions. For j = 0, the (b A ′Ã′ ab ) j,j of the U(p) gauge groups are eliminated by D-terms, and for j = 0 (b A ′Ã′ (ab) ) 0,0 give 4p degrees of freedom. Note that we only have one SO(p) gauge group and we therefore get only half as many massless degrees of freedom from these fields as required.
Since we are on the Higgs branch, all the (b AY ab ) j,j+Y become massive, except for the fields (b A+ ab ) (N −1)/2,(N +1)/2 and (b A− ab ) (N +1)/2,(N −1)/2 shown by red arrows in figure 3.2. These fields are special and essentially take on the role played by the second SO(p) gauge group in the even case. By (3.20) these particular b AY fields and their superpartners ψ − are symmetric fields with 4p(p + 1)/2 components each, while the corresponding left-moving fermions ψ + are antisymmetric fields with 4p(p−1)/2 components. From the type II theory we know that the only other left-moving fermions, the χ + , remain massless. We can thus only form 4p(p − 1)/2 Yukawa terms so that of the ψ − , 4p(p + 1)/2 − 4p(p − 1)/2 = 4p remain. By supersymmetry, the same number of bosons b must remain massless. The total number of bosonic degrees of freedom is thus again 4NN ′ p + 8p (for N > 1), the same as in the even case.
In the degenerate case N = 1 there is one SO(p) gauge group, but no bifundamentals b AY of the type described above. We therefore get only 4NN ′ p + 4p bosonic massless degrees of freedom, in agreement with the unorbifolded ADHM model.

Central charges of the Higgs branch theory
From the above counting of massless degrees of freedom, we find that the moduli space of the ultraviolet theory is spanned by the 4NN ′ p fields (φ A ′ m a ) j,j and the 8p independent degrees of freedom provided by (b A ′Ã′ (ab) ) j,j (j = 0, N/2). Its dimension is therefore given by Recalling that the target space of the conformal sigma model on the Higgs branch is the instanton moduli space of the ADHM quiver model, we may now also determine the central charges of the infrared theory. For N ≥ 2 we find In conclusion, we propose the (0, 4) sigma model on the Higgs branch of the type I quiver model (3.22) as the appropriate candidate for the boundary conformal field theory of heterotic string theory on AdS 3 × S 3 /Z N × T 4 (N ≥ 2).
4 Entropy function formalism in 5-dimensional heterotic string theory

Outline
In this section we return to the construction of the near-horizon geometry of the heterotic three-charge model. The corresponding classical supergravity solution has been reviewed in section 2.1. We now wish to go beyond classical supergravity by introducing additional higher-derivative operators in the heterotic string action. Our calculations are valid only for large values of the charges N, N ′ , q. In particular, we shall only calculate the first subleading correction to the classical solution. Similar computations in a dual setup have already been performed in [8] which exploit the recently discovered N = 2 off-shell completion of the R 2 -terms in the 5-dimensional supergravity action [35]. Here, we will study the modification of the near-horizon solution (2.5) in the presence of the four derivative corrections to the heterotic string effective action at the string tree level [18,19].
We will make use of the entropy function formalism; for an introduction see e.g. [20,21] or the recent review [36]. It was originally developed for 4-dimensional AdS 2 × S 2 black holes, but it can also be generalised to geometries containing AdS p -factors with p > 2 (see e.g. [37]). We will first use the formalism to rederive the classical contribution to the central charge. In a second step we then apply it to the α ′ corrected action to obtain corrections to the central charge.
Generically, the 5-dimensional action will also contain Chern-Simons like contributions which contain the gauge fields in a non-covariant way (i.e. terms which contain the gauge potentials rather than the field strengths). We therefore cannot use the entropy function formalism in a straightforward way. Fortunately, following [38], we can circumvent this problem by considering the theory in 6 dimensions, from where we can get the 5-dimensional theory by Kaluza-Klein reduction. This approach has not only the advantage that we can reformulate the gauge Chern-Simons term in a covariant way, but it also allows us to think of the 5-dimensional 2-and 3-form field strengths as coming from the Kaluza-Klein reduction of a single 6-dimensional three-form. Since the latter is in fact self-dual, this provides us with a very compact way of dealing with the 5-dimensional fields. We will see however that the action still contains a gravitational Chern-Simons term which will require special treatment.
Throughout this section we will use the convention α ′ = 16.

Uncorrected solution
We begin by lifting the heterotic theory to 6 dimensions, where we have the following massless bosonic fields • 6-dimensional metric G (6) M N : This reduces to the 5-dimensional metric as well as to a vector field under which the black string can be electrically and magnetically charged.
• anti-symmetric tensor B (6) M N : This reduces to a 5-dimensional 2-form potential and to a dual vector field. The black string can be electrically and magnetically charged under B (6) M N .
This reduces to the 5-dimensional dilaton.
The 6-dimensional Lagrangian obtained from heterotic string theory is given by where Ω (6) M N P is the gravitational Chern-Simons 3-form. The parameter κ can be fixed as in [38], which gives the value κ = 192 for our setup. To covariantise the action, we introduce a new field C (6) M N together with its field strength Consider the new Lagrangian where ζ is some constant which will cancel out in all physical quantities. Upon exploiting the equations of motion for the auxiliary field C (6) , QRS − κΩ which we use to eliminate H (6),M N P from the original Lagrangian (4.1). We have thus replaced the 3-form field strength of the 6-dimensional Lagrangian by the (auxiliary-)field C (6) M N , which only appears through its field strength K (6) M N P . Let us comment briefly on the gravitational Chern-Simons term − ζκǫ M N P QRS K (6) M N P Ω QRS .

(4.5)
Although it is not of a manifestly covariant form, we will argue below that in our specific setup the term is actually covariant. This means that after replacing H (6),M N P by K (6) M N P , (4.3) is covariant, so that we can apply the entropy function formalism. Although it will be more convenient to stay in the 6-dimensional setup, let us spell out the ansatz with which we can reduce this Lagrangian back to 5 dimensions: 5µ G 5ν , where V is the volume of the compactified x 5 -direction. The field strengths of the various forms are then given by νρ ) + cyclic permutation of (µ, ν, ρ) . (4.7) Note that after compactifying to 5 dimensions, K µνρ is no longer covariant, as it contains A (1,2) µ explicitly. In principle, one would therefore have to introduce new auxiliary fields and repeat the steps performed above (see [38]). It turns out however that this gives the same result as when we use the reduced version of (4.3) directly.
We are now in a position to compute the entropy function, which is given by where H (6),M N P is to be replaced by K (6) M N P using (4.4). In order to evaluate (4.8) we make the following ansatz for the near-horizon form of all the 5-dimensional fields involved ds 2 = g µν dx µ dx ν = v 1 (−r 2 dt 2 + r 2 dz 2 + dr 2 r 2 ) + v 2 (dθ 2 + sin 2 θdϕ 2 ), tr = e 1 , e −2Φ = λ , (4.9) where we interpret p 1 and p 2 as magnetic and e 1 as the Legendre transform of an electric charge. Using (4.6), this corresponds to the 6-dimensional configuration. (4.10) Let us now turn to the gravitational Chern-Simons term (4.5). We will argue that in our setup it is already covariant. First note that the 6-dimensional space factorizes into two 3-dimensional spaces, which we label in the following way α, β, γ = t, r, z, and a, b, c = θ, ϕ, where the metrics of the two subspaces read The situation is now almost exactly as in [38]. There, the setup was reduced to a twodimensional geometry in t, r, since all other directions were periodic and could thus be considered as compactified. In our case although the z direction is non-compact, it does not appear explicitly in any of the expressions, so that the argument carries over. The conclusion is then that (up to total derivative terms which give no contribution) (4.5) is already covariant, as was shown in [38]. We can thus directly plug the expression for Ω (6) , into (4.5) to obtain the contribution with κ = 192. A direct calculation shows however that ∆E CS only gives subleading corrections to the classical geometry. We will thus omit the Chern-Simons term as long as we consider the classical solution.
Inserting the ansatz (4.10) into the entropy function (4.8), we obtain the result (4.14) In order to find the entropy of the black hole, we have to extremise this expression. Under the assumption q 1 > 0, p 1 > 0 and p 2 > 0, the only physically acceptable extremum is Note in particular that we find the relation v 1 = 4v 2 . (4.16) We note, however, that the quantities p 1 , p 2 , q 1 are not yet physically normalised expressions. The unphysical quantity ζ still enters into the solution. We will determine the correct normalisation at the end of the next subsection.

Corrected solution
We now wish to consider corrections to the classical supergravity theory. This means that we have to include the contribution (4.13) of the Chern-Simons term. Moreover, the α ′ -corrected supergravity Lagrangian also contains higher order derivative terms which we have to take into account. We follow [38] and write down the action containing the four derivative corrections to the heterotic string effective action as M N P H (6),M N P + 2R (6) KLM N R (6),KLM N − R As in the classical case, we introduce the new field C (6) M N with field strength K (6) M N P , as defined in (4.2). As before, we modify the action M N P H (6),M N P + 2R (6) KLM N R (6),KLM N − R Reducing this Lagrangian to L (6) by using the equations of motion for C (6) M N is essentially the same as in the classical case. However, elimination of H (6) M N P is now modified due to the presence of the higher derivative terms. Indeed, the equation of motion for H  Following the classical example, we would now have to invert this equation to express H (6) M N P in terms of K (6) M N P . Since this is in general very hard, we will solve (4.19) only to first subleading order. To this end we make the ansatz H (6),M N P = H where H (6),M N P 0 is the solution from the classical equations of motion (see (4.4)). H (6),M N P 1 Let us finally normalise the charges q 1 , p 1 , p 2 and relate them to the physical quantities N ′ , p, N. Following [38] we are led to identify To first order, the solution is then given by (α ′ = 16) Note that the corrected solution still obeys v 1 = 4v 2 .
In summary, the corrected ten-dimensional near-horizon geometry is still AdS 3 × S 3 /Z N × T 4 , but now with AdS radius and six-dimensional string-coupling given by where it is understood that the sub-subleading terms can also be suppressed by powers of p. The Brown-Henneaux formula gives in the uncorrected case c class = 6NN ′ p , (4.29) while for the corrected solution we find To subleading order this agrees with (2.11).

Heterotic two-charge models
In view of a possible heterotic string duality with (0, 8) spacetime supersymmetry [5,6], it is an interesting question whether we can systematically switch off charges in the present (0, 4) duality. Clearly, the worldsheet theory for strings on AdS 3 × S 3 /Z N × T 4 requires at least one KK monopole and is not applicable for vanishing KK monopole charge. Since the KK monopoles break supersymmetry down to (0, 4) there seems to be no obvious way to generalise the model to (0, 8). Nevertheless, it is interesting to consider models with less charges such as the F1-KKM and the NS5-KKM intersection.

F1-KKM intersection (N ′ = 0)
We shall first consider a heterotic two-charge model consisting of a stack of p fundamental strings in the background of a KK monopole with charge proportional to N ≥ 2. The setup is the same as in section 2.1, but now N ′ = 0 (no NS5-branes). From (2.11), we find the central charges of the boundary conformal field theory to be (c L , c R ) = (24p, 12p). Remarkably, the central charges do not depend on the charge of the KK monopole since the leading term cubic in the charges (∝ NN ′ p) is absent. This has some interesting consequences.
Let us first have a look at the supergravity solution. Classically, the solution has a horizon of zero area leaving a naked curvature singularity at the origin. This corresponds to a vanishing Bekenstein-Hawking entropy on the classical level. It is however believed that higher-derivative corrections to the supergravity solution resolve the classical singularity leading to a finite entropy. The corrected supergravity solution presented in the previous section is valid for large NN ′ and thus cannot be applied to this case.
The heterotic worldsheet theory for this case has some peculiar features. The left sector of the CFT on the S 3 /Z N has collapsed to a trivial theory with bosonic level k ′ b = c ws L (S 3 /Z N ) = 0. The supersymmetric level corresponding to the right sector is k ′ s = k ′ b + 2 = 2, and we have c ws R (S 3 /Z N ) = 3 2 . We are thus left with a trivial theory in the left sector and three fermionsχ a (a = 1, 2, 3) in the right sector. The AdS 3 part of the geometry is described by a heterotic SL(2) WZW model with levels k b = 4 and k s = 2. The full (supersymmetric part of the) background is thus ensuring criticality, (c ws L , c ws R ) = (26,15), given that c ws L (E 8 × E 8 ) = 16. The worldsheet model also gives the correct central charges for the boundary CFT, cf. Eq. (2.23). Related heterotic models involving three fermions can be found in [1,3].
We conclude with some comments on the dual boundary conformal field theory. Removing the D5 branes in the quiver ADHM theory corresponds to the removal of the outer circle and the spikes in the quiver diagram in figure 3.1. The ADHM part of the quiver action disappears, leaving only that part of the action which corresponds to the inner circle of the quiver diagram. Nevertheless, the counting of the massless degrees of freedom in the remaining quiver theory seems to yield the correct central charges, (c L , c R ) = (24p, 12p) (for N ≥ 2). It is interesting to observe that the independence of c L,R on N is reflected by fact that varying N changes only the number of sites in the quiver diagram corresponding to U(p) gauge groups. Recall, however, that the fields of the U(p) gauge groups do not contribute to the central charges of the infrared conformal field theory. Certainly, it would be interesting to study this field theory in more detail.

Heterotic NS5-KKM intersection (p = 0)
For completeness, we also consider the NS5-KKM intersection which can be obtained from the three-charge model of section 2.1 by setting p = 0.
Let us approach this setup from a slightly different point of view. In [5] Lapan, Simons and Strominger suggested to start from a four-dimensional monopole black hole with near-horizon geometry where R t denotes time and R φ a real line labelled by φ with linear dilaton. Decompactifying one of the compact directions, i.e. replacing R t × S 1 by a two-dimensional Minkowski space R 1,1 leads to the geometry The CFT on (5.4) is then expected to describe a monopole string in five dimensions [5].
Ref. [5] also suggested that the S 2 factor could be described by the coset model of [24].
Here, however, we deviate from the proposal of [5] and include a KK monopole charge by replacing S 2 × T 5 by S 3 /Z N × T 4 . Of course, we thereby break half of the target space supersymmetry. Heterotic string theory in the background of a five-dimensional monopole string with additional KK monopole charge is then expected to be given by the CFT on R 1,1 × R φ × S 3 /Z N × T 4 . (5.5) In fact, the thus derived background is nothing but the near-horizon geometry of the F1-NS5-KKM set-up for vanishing electrical F1 charge, p = 0. This can be seen by setting F = 1 in (2.1) and taking the limit r → 0. Heterotic string theory on the background (5.5) can be described by a linear dilaton theory with central charges c ws L (R 1,1 × R φ ) = 2 + (1 + 3Q 2 D ) , c ws R (R 1,1 × R φ ) = 3 + ( 3 2 + 3Q 2 D ) , (5.6) and dilaton charge Q D . The internal part of the geometry, S 3 /Z N and T 4 , will be described as before, see section 2.3. By criticality, the linear dilaton charge Q D is related to the bosonic level k ′ b of the S 3 /Z N theory as where k ′ b = k ′ s − 2 = NN ′ , if we assume k ′ s = NN ′ + 2. Finally, as explained in [40], there is a simple relation between linear dilaton and SL(2) models. Adding p D1-branes along the R 1,1 and taking the near-horizon limit amounts to replacing the factor R 1,1 × R φ by AdS 3 . The level of SL(2) is related to the dilaton charge by k s = 2/Q 2 D (k b = k s + 2). This leads back to AdS 3 × S 3 /Z N × T 4 , as expected.

Conclusions
We studied the AdS 3 /CFT 2 correspondence of a heterotic three-charge model with (0, 4) supersymmetry. We gathered evidence for the equivalence of the following two theories: i) E 8 × E 8 heterotic string theory on AdS 3 × S 3 /Z N × T 4 ii) the (0, 4) Higgs branch theory of a Z N orbifold of Witten's ADHM sigma model We motivated the duality by studying the low-energy effective action of a particular type I setup dual to a heterotic configuration with AdS 3 × S 3 /Z N × T 4 near-horizon geometry.
We constructed the ultraviolet theory in terms of a Z N orbifold of the ADHM massive sigma model [12] and verified that the corresponding Higgs branch theory has the correct central charges. We also found that the first-order α ′ -corrected supergravity solution correctly reproduces the (supersymmetric) central charge of the boundary conformal field theory up to terms of order O( 1 N N ′ ), cf. (4.30) with (2.11). The proposed heterotic duality obviously requires further investigation. The evidence we gave is based on the counting of the massless degrees of freedom of the ultraviolet orbifold theory. These modes are not renormalised and therefore also constitute the Higgs branch theory. Its actual construction is expected to be straightforward along the lines of [17] by integrating out the massive modes in the UV theory. This procedure will be made more complicated by the fact that the Higgs branch metric will receive α ′ corrections and seems to be divergent at the origin [17]. It would also be interesting to work out the dictionary between the chiral primaries of the boundary CFT and those of the worldsheet model [11]. The primaries of the boundary CFT will be composite operators of the massless fields of the ultraviolet ADHM quiver model. A comparison of the corresponding n-point functions should then provide further evidence for the duality. Such tests have previously been performed in the type II AdS 3 /CFT 2 duality in [41,42,43,44,45].

Appendix A Web of Dualities
In this appendix we display the various dualities leading from the heterotic theory on T 5 (along x 5,6,7,8,9 ) to M-theory on K3 × T 2 (along x 6,7,8,9 and x 5,10 ). For a review of string dualities see e.g. [39]. In order to facilitate keeping track of the various steps, we have depicted a schematic overview in the following web We start in the lower right corner with the heterotic theory as described in section 2.1. Following the first arrow to the left 10 , heterotic-type IIA duality takes us to a setup with NS5-branes, fundamental strings and KK monopoles as described in the corresponding box. Going further to the left (using the arrow labelled T 5 ), we perform a T-duality along the isometry direction of the KK monopoles (direction x 5 ), which exchanges the KK monopoles and the NS5-branes but leaves the F1 untouched. Since we have performed the T-duality only along a single direction, the setup is now in the type IIB theory. Following the next arrow upwards (labelled by S), we perform S-duality in the type IIB framework, which turns the NS5-branes and F1 into D5-and D1-branes, respectively. Next we follow the arrow labelled T 567 to the right, which represents T-duality transformations along x 5,6,7 . Since again the isometry direction of the KK monopoles is affected, they are transformed to NS5-branes, while the D1 and D5-branes are mapped to D4-branes. Since we have performed the duality transformation in an odd number of dimensions, we are back to the type IIA framework. The final arrow pointing upwards is the M-theory lift, which takes us to the setup of three stacks of M5-branes described in section 2.2.