Quantum Corrections in String Compactifications on SU(3) Structure Geometries

We investigate quantum corrections to the classical four-dimensional low-energy effective action of type II string theory compactified on SU(3) structure geometries. Various methods previously developed for Calabi-Yau compactifications are adopted to determine - under some simple assumptions about the low-energy degrees of freedom - the leading perturbative corrections to the moduli space metrics in both alpha' and the string coupling constant. We find - in complete analogy to the Calabi-Yau case - that the corrections take a universal form dependent only on the Euler characteristic of the six-dimensional compact space.


Introduction
In this paper we consider type II string theory in backgrounds M 1,3 × X 6 , where M 1,3 is a four-dimensional (d = 4) Lorentzian space-time and X 6 is a compact six-dimensional manifold. Demanding that the effective theory in M 1,3 admit eight unbroken supercharges (N = 2 in d = 4) constrains X 6 to be a manifold with SU (3) structure [1,2]. 1 Manifolds with SU (3) structure admit a globally defined nowhere-vanishing spinor and as a consequence the structure group of the tangent space is reduced, meaning that it can be patched using only an SU (3) subgroup of SO (6). If in addition the spinor is covariantly constant the Levi-Civita connection has SU (3) holonomy, and in this case X 6 is a Calabi-Yau manifold.
The N = 2 low-energy effective Lagrangian of such backgrounds has been computed in refs. [2,[8][9][10][11][12][13][14][15][16] at the string tree level and for "large" manifolds X 6 assuming a suitable Kaluza-Klein reduction. For Calabi-Yau manifolds this is straightforward in that the massless modes are in one-to-one correspondence with the cohomology of X 6 , and the resulting N = 2 supergravity is ungauged with no potential [17]. In the generalized case the corresponding analysis is much harder and can only be performed if there is a suitable hierarchy of the low-energy modes. In this case the effective action is a gauged N = 2 supergravity with a potential which stabilizes (some of) the moduli. However, the existence of a supersymmetric vacuum is by no means guaranteed.
The low-energy effective Lagrangian is corrected by higher-order α ′ terms as well as by string loops (g s corrections). Both types of corrections can be parametrized by a scalar field (or a modulus) in the low-energy effective action. Higher-order α ′ corrections arise when the size of the manifold is comparable to the string length and thus they are controlled by the volume modulus of X 6 . String loop corrections on the other hand are counted by g s with the dilaton being the corresponding modulus. For Calabi-Yau compactifications both types of corrections have been partially computed, and we review some of these results in the following. However, for generalized compactifications little is known about these corrections primarily due to the fact that such compactifications are generically off-shell and so have no direct worldsheet description.
The purpose of this paper is to investigate what can be said about α ′ and g s corrections in the specific case of SU (3) structure compactifications. We will not focus on a particular background but rather attempt to constrain the generic form of the possible corrections, under some simple assumptions. We also do not consider any non-trivial NSNS or RR fluxes, or more general SU (3) × SU (3) structures (see footnote 1).
We find -given there is an expansion in terms of a finite set of low-energy fields - 1 More generally, viewed as generalized geometries [3][4][5], N = 2 supersymmetry requires an SU (3) × SU (3) structure on the generalised tangent space [6][7][8][9] or even more generally an SU (6) structure on the exceptional generalised tangent space [10]. that the leading corrections to the kinetic terms take a universal form, dependent only on the Euler characteristic of X 6 . These forms are the same as in the restricted class where X 6 is Calabi-Yau, and are derived using only simple extensions of the arguments employed in that case [18][19][20][21][22][23]. Generically, SU (3) structure compactifications admit fewer Peccei-Quinn (PQ) symmetries than Calabi-Yau compactifications, since some of the corresponding moduli acquire a mass from the gauging of the N = 2 supergravity. Remarkably, we find that the leading kinetic energy corrections still have the full set of PQ symmetries, even under shifts of the massive moduli.
The paper is organized as follows: In section 2 we review some basic facts about N = 2 supergravity and SU (3) structure compactifications. We also summarize our assumptions about the moduli space of low-energy fields. Section 3 then discusses the form of the leading corrections to the metrics on these moduli spaces: the prepotential for the vector multiplet fields, and the quaternionic-Kähler metric for the hypermultiplet fields. We first discuss the constraints on the metric moduli that arise from dimensionally reducing the known ten-dimensional R 4 correction terms. We then discuss the full vector and hypermultiplet moduli space metrics, first for the α ′ corrections and then for the g s corrections. Section 4 contains some concluding remarks. In order to discuss the α ′ and g s corrections we need to briefly assemble some facts about N = 2 supergravity (for a review see e.g. [24]). A generic spectrum contains a gravitational multiplet, n v vector multiplets and n h hypermultiplets. 2 The gravitational multiplet contains the spacetime metric, two gravitini and the graviphoton A 0 µ . A vector multiplet contains a vector A µ , two gaugini and a complex scalar t. Finally, a hypermultiplet contains two hyperini and four real scalars (q 1 , q 2 , q 3 , q 4 ).
The N = 2 supersymmetry enforces the scalar field space M locally to be a product where M v is a 2n v -dimensional manifold spanned by the complex scalars t i , i = 1, . . . , n v , contained in n v vector multiplets, while M h is a 4n h -dimensional manifold spanned by the real scalars q u , u = 1, . . . , 4n h , in n h hypermultiplets. Thus their sigma-model Lagrangian is of the form where g i is the metric on M v while h uv is the metric on M h . In general, isometries on M v and M h can be gauged, so that D is an appropriate covariant derivative which includes the couplings of the charged scalars to the vector bosons. Only if the theory is gauged can it admit a non-trivial potential V , which furthermore is completely determined by the choice of gauging.
Supersymmetry further constrains M v to be a special-Kähler manifold, so that the metric can be written as [25,26] Both X I (t) and F I (t), I = 0, 1, . . . , n v , are holomorphic functions of the scalars t i , and F I = ∂F /∂X I is the derivative of a holomorphic prepotential F (X) homogeneous of degree two. Furthermore, it is possible to adopt a system of 'special coordinates' in which X I = (i, t i ) (see e.g. [26] for further details).
M h is similarly constrained to be a quaternionic-Kähler manifold, which means its holonomy group is of the form Sp(1)×H with H ⊂ Sp(n h ) [27,28]. There is a special class of quaternionic-Kähler manifolds which arise in string tree-level effective actions known as 'special quaternionic-Kähler manifolds' M 4n h SQK . These manifolds can be viewed as a 2n hdimensional torus fibred over a special-Kähler base, of the form [SU (1,1) is a special-Kähler manifold of dimension 2n h − 2. This relation between special-Kähler and quaternionic Kähler manifolds is known as the 'c-map' [29] SU (1, 1) The metric on M 4n SQK takes an explicit form known as the Ferrara-Sabharwal metric, which reads [30] where φ and σ span the SU (1, 1)/U (1) factor of (2.4), ξ I andξ I are real coordinates of the torus fibre, and z a are complex coordinates on the special Kähler base M 2n h −2

SK
. 3 The metric g ab (z,z) on M 2n h −2 SK satisfies (2.3) and thus can be characterized by a holomorphic prepotential G(z). 4 The couplings N IJ (z,z) are given in terms of G as

6)
3 In (2.5) we display the ungauged metric. If some of its isometries are gauged the corresponding ordinary derivatives are replaced by appropriate covariant derivatives. 4 For type IIA, we denote the coordinates and prepotential of M 2n h −2

SK
in the hyper-sector by z a and G, respectively, in order to distinguish it from the coordinates t i and prepotential F in the vector multiplet sector. For type IIB the rôles of t i and z a and F and G will be reversed. with z I = (i, z a ). In type II string theory reduced on, in particular, a Calabi-Yau manifold, φ and σ are identified with the four-dimensional dilaton and universal axion, respectively, while ξ I andξ I are identified with four-dimensional scalar fields arising from the RR sector.

SU (3) compactifications
Let us now briefly discuss the notion of an SU (3) compactification. Recall first the moduli that appear when the compactification space X 6 is a Calabi-Yau manifold. One finds that the scalar fields in vector-and hypermultiplets arise from h (1,1) deformations of the complexified Kähler form J + iB and from h (1,2) deformations of the complex structure, or equivalently deformations of the holomorphic 3-form Ω. Since these moduli can be varied independently, their moduli space is locally a product with each component being a special-Kähler manifold. Their respective Kähler potentials are given, prior to α ′ and string loop corrections, by [17] K J = − ln SK is the special-Kähler base in the c-map (2.4). The full space M h also includes the dilaton and axion plus 2h (1,2) + 2 scalars from the RR sector which parameterize the torus fibre. In type IIB compactifications the situation is reversed and one has M Ω = M v while M J = M 2h (1,1) SK is the special-Kähler base. The dilaton and axion plus 2h (1,1) + 2 scalars from the RR sector again complete M h .
In this paper we will be considering the weaker case where we only assume that X 6 admits an SU (3) structure [1]. This means we can still find a globally defined fundamental 2-form J and an almost complex structure with a corresponding globally defined complex (3, 0)-form Ω. However, they are now no longer closed, so that and one says that the intrinsic torsion of the SU (3) structure on X 6 is non-zero. This lack of integrability means that the space of generic SU (3) structures on X 6 is infinitedimensional, though it nonetheless still decomposes as a product of special-Kähler manifolds [8,9], with Kähler potentials given by (2.8). In the corresponding 'low-energy' effective action the intrinsic torsion plays the role of gauge charges and/or mass terms. As a consequence, the four-dimensional supergravity is gauged.
For a realistic theory, we would like to keep only a finite-dimensional moduli space of SU (3) structures, as we had for the Calabi-Yau case. This would require identifying some light fields in the Kaluza-Klein spectrum of all possible deformations of J and Ω. In this paper we will simply assume such an expansion exists and then investigate the consequences for quantum corrections to the corresponding effective action. Our assumptions are (see [2,8,9,12] for details): where there is a domain of the moduli space where the masses of the moduli t i and z a are light compared to the generic Kaluza-Klein modes. In contrast to the Calabi-Yau case, the lack of integrability (2.9) means that the basis forms ω i and the dual four-formsω i as well as α I and β I are no longer necessarily harmonic; these forms constitute a closed set under the Hodge star and also under the exterior derivative, namely where e iI and p I i are constant matrices.
2. There are no additional light spin-3/2 fields other than the N = 2 gravitini (this corresponds to an absence of moduli of type (1, 0) with respect to the almost complex structure [8,9]).
3. We can consider an expansion around the large-volume limit (the moduli space manifestly includes the volume modulus V since this is a rescaling of the real parts of t i , but we in addition require that the moduli are light in the large-volume region).
The SU (3) structure implies that the effective field theory in four dimensions is N = 2 supersymmetric. Given our assumptions, the moduli space is again a product of a moduli space M J of deformations of J + iB and of a moduli space M Ω of deformations of Ω, where the volume modulus V is part of M J . Furthermore, prior to including α ′ and string loop corrections, each component is a special-Kähler manifold with Kähler potentials again given by (2.8).
The RR potentials are assumed to admit an expansion in the same bases, namely α I and β I in type IIA and ω i ,ω i and the volume form ε in type IIB, so that In Calabi-Yau compactifications the metric on the quaternionic manifold M h takes the Ferrara-Sabharwal form (2.5) to leading order. Related to the torus fibre the metric has 2n h + 1 (perturbative) Peccei-Quinn shift symmetries which read (2.14) These symmetries arise from large gauge transformations of the type II p-form fields C p , that is, shifting them by constant multiples of the (closed) basis forms ω i etc. They are broken to discrete shift symmetries by non-perturbative space-time physics but nevertheless imply a perturbative non-renormalization theorem in that perturbative corrections of the hypermultiplets only occur at one-loop but not beyond [18-20, 22, 23].
By contrast, for SU (3) structures the basis forms are generically no longer harmonic, so in general the shift symmetries (2.14) are broken and the corresponding scalar fields are massive. Nonetheless, simple dimensional analysis shows that the leading-order calculation of the kinetic terms does not see the derivatives of the basis forms and so the metric on the hypermultiplet space still takes the Ferrara-Sabharwal form (2.5). At higher order in α ′ and g s one would naively expect that corrections to the metric see the fact that the Peccei-Quinn symmetries are broken.
In summary, our assumptions imply that at leading order -just as in the Calabi-Yau case -in type IIA one has M J = M v while M Ω is the base of a special quaternionic-Kähler manifold M h . In type IIB the situation is reversed and one has M Ω = M v while M J is the base of M h . In both cases the special quaternionic manifold M h includes the dilaton and axion.
3 Perturbative α ′ and g s corrections In this section we determine the structure of the perturbative α ′ and g s corrections in the class of SU(3) structure compactifications defined above. The key point is that they are very strongly constrained by the N = 2 supersymmetry which by construction survives the compactification. In essence we find that many of the usual arguments that apply to Calabi-Yau compactifications go through in this case too.
First note that if the low energy spectrum contains no massive spin-3/2 multiplets, N = 2 supersymmetry enforces the split (2.1) into locally independent special-Kähler and quaternionic moduli spaces, which thus has to persist including perturbative and nonperturbative α ′ and g s corrections. Since in both type II compactifications the dilaton is part of a hypermultiplet, the component M h receives quantum corrections, whereas M v is uncorrected and thus "exact" already at the string tree level. 5 α ′ corrections on the other hand enter in an expansion in terms of the volume modulus V which resides in M J . Therefore, in type IIA α ′ corrections appear in M v whereas in type IIB they correct M h . This situation is identical to the situation for Calabi-Yau compactifications and is summarized in Table 1.
yes no no yes g s no yes no yes Table 1: Structure of α ′ and g s corrections in type IIA and type IIB.
We first show that the splitting (2.1), together with knowledge of the form of the α ′ and string loop R 4 corrections to the ten-dimensional effective action, strongly constrains the form of the leading corrections to the metrics on M v and M h . We then consider the dependence of the NS B-field and RR moduli, using analogues of the standard Calabi-Yau compactification arguments to investigate how first the perturbative vector-multiplet α ′ corrections and then the perturbative g s corrections are constrained.

Constraints from Kaluza-Klein reduction
Let us first focus on the leading perturbative corrections both in α ′ and g s which arise from higher-derivative R 4 terms in the ten-dimensional effective action. These are given by string tree-level and one-loop terms of the form (see e.g. [19,32]) where ϕ is the ten-dimensional dilaton, ζ is the Riemann zeta function, and R M 1 M 2 N 1 N 2 denotes the Riemann tensor, ǫ is the totally antisymmetric Levi-Civita tensor, and t 8 is antisymmetric in each successive pair of indices and symmetric under the exchange of any two pairs. Given an antisymmetric tensor M M N , the t-tensor contraction reads We also have where R N 1 N 2 is the curvature 2-form.
Note that the supersymmetric completion of these terms will include, in the NSNS sector, higher derivative objects built from curvatures, H = dB and derivatives of the dilaton, which would be relevant if for instance one was interested in backgrounds with non-trivial H-flux. However, although recently some significant progress has been made [33], less is known about the exact form of these terms.
The corrections in the four-dimensional effective action induced by these higherderivative terms can in principle be derived by performing a Kaluza-Klein reduction. At the two-derivative level they lead to corrections of the Einstein term and the two scalar field metrics just as in the Calabi-Yau case [19,21,22]. Let us start by considering the Einstein term; since t 8 t 8 R 4 never involves contractions of indices on the same Riemann tensor, any correction to the Einstein term must come from εεR 4 [19]. More precisely, the only contraction that gives a term with only two space-time derivatives is where P Q in (3.2) are four-dimensional space-time indices, and the two other space-time indices on each ε contract the same Riemann tensor. (If they contract different Riemann tensors one obtains higher-derivative scalar field terms.) Integrating over the internal space, we obtain where R is the four-dimensional Ricci scalar and χ(X 6 ) the Euler characteristic of X 6 . Thus we see that for SU (3) structures, the correction to the Ricci scalar term comes only from εεR 4 and is proportional to χ exactly as for Calabi-Yau compactifications.
For the scalar kinetic energy corrections, we note that εεR 4 necessarily has four spacetime indices and thus cannot contribute. One can, however, get a correction from the t 8 t 8 R 4 terms [19,21,34]. We will denote these corrections to the metrics on the M J and M Ω moduli spaces as δg J and δg Ω , respectively, and the leading order, uncorrected metrics as g 0 J and g 0 Ω . Inspecting the terms in (3.1), we can now write the corrected IIA and IIB Lagrangians in the string frame as [22] where v i = Re t i and c is proportional to χ(X 6 ). (Note that since we only keep the real parts of t i only the real part of g 0 J and δg J appear.) As we have seen, the Ricci scalar correction is identical to that in the Calabi-Yau case. Hence, we can use the standard Calabi-Yau result to write c = 2 (2π) 3 χ(X 6 ) . (3.7) Note that in (3.6) we do not explicitly display the kinetic term of the dilaton and a mixing term of the form ∂ µ ϕ ∂ µ ln V between the dilaton and the volume. The tree-level coefficent of the latter has been computed for Calabi-Yau compactifications in [21,22] but the one-loop correction has only been inferred indirectly in [22] by insisting that the split (2.1) continues to hold at the loop level. For the generalized compactifications considered here we adopt a similar strategy in that we do not compute the mixing term but instead also assume that it is present with exactly the right coefficient to ensure the split (2.1). For this reason the following analysis ignores contributions from dilaton/volume mixing. We also return briefly to this issue in the next subsection. The next step is to Weyl-rescale to the Einstein frame and then expand in terms of the four-dimensional dilaton e 2φ = V −1 e 2ϕ and the inverse volume V −1 , which parametrize loop and α ′ corrections, respectively. 6 To leading order, one finds Recall that in type IIA the v i (which in particular include the volume V ) are scalars of M v and the z a are scalars of M h , with the reversed assignment in type IIB. Supersymmetry implies that there can be no g s correction in M v and no α ′ correction in M h . This is achieved to this order for both type IIA and type IIB if and only if δg Ω = c g 0 Ω , Re δg J = −c Re g 0 J . (3.9) The original form (3.6) thus reads and we see that supersymmetry implies that the corrections to the moduli space metrics are proportional to the tree-level metrics. The question remains as to whether the imaginary part of δg J might have a correction that violates this form. However, we will argue in the next section that this is not the case. Furthermore, we see that all corrections are proportional to χ. This result indeed holds for Calabi-Yau compactifications, but here we have shown that this property is more general and has to hold for any compactification manifold X 6 with SU (3) structure.
This result has implications for mirror symmetry of SU (3) structure manifolds. If type IIA compactified on X 6 is equivalent to type IIB compactified on the mirror manifoldX 6 , then we have g Ω (X 6 ) = g J (X 6 ) , g J (X 6 ) = g Ω (X 6 ) . (3.11) Inspecting (3.10), we see that this can only be true for the corrected metrics if Thus, if mirror symmetry generalizes to SU (3) structures, the two manifolds must have opposite Euler characteristic, just as in the case of Calabi-Yau manifolds. 7

Perturbative α ′ corrections
In the previous subsection we constrained the geometric corrections arising from the reduction of R 4 -terms. However, we did not consider higher-derivative couplings of the NS B-field as they are not yet completely known [33]. This in particular meant we were unable to restrict the correction to the imaginary part of g J . Let us now address that issue and also investigate the form of the full set of perturbative α ′ corrections to the prepotential.
In the reduction, the light modes of B combine with the deformation of J, as in (2.10), to form complex scalar coordinates t i on M J . The N = 2 prepotential F that determines the special-Kähler metric depends holomorphically on t i and is given in the large volume limit -so as to match (2.8) -by with κ ijk real constants and we take X 0 = i and X i = t i . (In Calabi-Yau compactifications the κ ijk are the classical intersection numbers; more generally they are related to the basis forms ω i [8,9].) In order to determine or constrain perturbative α ′ corrections, we have to determine the sub-leading corrections to F . Expanding in large t i we have generically whereF contains non-perturbative corrections (i.e. instanton corrections) together with possibly negative powers of t i . Here we use the fact that each power in the α ′ expansion comes with a volume factor V −1/3 . From (2.8) we see that V is cubic in t i where K 0 is the leading order Kähler potential computed from F 0 , and hence the corrections to F 0 are in descending powers of t i . Note that this expansion in V −1/3 strictly defines F as a sum of functions homogeneous under the rescaling t i → µt i , that is where F i (t) scales as µ 3−i and F np (t) is the non-perturbative correction. Thus in (3.14) we are really extracting only the polynomial parts of each F i (t) in the expansion. We will return to this point below.
Inserting (3.14) into (2.3) one obtains where So we see that the real parts of α ij , β i and γ actually do not enter the Kähler potential and for our purpose may be set to zero without loss of generality. 8 Computing the metric from (3.17) we find where we abbreviated We would like to determine the values for a ij , b i and c. Recall that for Calabi-Yau compactifications there is a perturbative Peccei-Quinn symmetry B → B + λ i ω i , which implies that g J ij has an isometry t i → t i +iλ i for constant real λ i . This is very constraining, implying in particular that a ij = b i = 0. In fact, the PQ symmetry together with the large volume limit is enough to also prove that no negative powers of t i can appear in F and that the perturbative prepotential can only be F (t) = F 0 (t) + γ with γ constant. 9 Now let us turn to the general SU (3) structure case. In general the PQ symmetry is a priori not present since the basis forms are not necessarily closed. However, as discussed in the previous sub-section, the leading string corrections to supergravity, including those involving B, appear at order α ′3 . Thus all leading corrections to g J ij are suppressed by a factor of V −1 and hence by cubic powers in t i . This means, simply by counting powers of t, that we expect a ij = b i = 0 and in fact, somewhat surprisingly, the leading order corrected metric still has the PQ symmetry.
To argue this more directly, recall in the previous section we did not compute the B-dependence of the corrections, and hence we have to compare (3.19) with (3.10) for Im(t i ) = 0. Inspecting (3.19) and keeping only the first three subleading terms we see that any non-vanishing a ij induces corrections to the real part of the metric even for Im(t i ) = 0. Since such corrections are not present in (3.10) we conclude that a ij = 0 should hold for SU (3) structure compactifications. Similar terms do appear for non-vanishing b i , but they are sub-leading, so to this order we cannot really use them to constrain the form of F . However, for the b i terms we can use a different argument. Keeping the Im t i dependence, we see that, by simply expanding the e K factors, there is a second-order correction to the real part of the metric that has only a linear dependence on Im t i of the form ib i (t −t) i . Considering first type IIA, the GSO projection implies that the NSNS sector of the tendimensional effective theory is invariant under the combined action of spacetime parity and reversing the sign of B. However, the terms κ ij and κ i in (3.19) come from t 8 t 8 R 4 which is parity even. Since reversing the sign of B reverses the sign of Im t i we see that the linear term is forbidden by the discrete symmetry and we conclude b i = 0. For type IIB, the NSNS sector is invariant under reversing the sign of B alone and hence again we conclude b i = 0.
Thus, in summary, we indeed have a ij = b i = 0 and g J ij = −6 e K κ ij + 9 e 2K κ i κ j , with e −K = κ ijk (t +t) i (t +t) j (t +t) k + c . (3.21) This result might seem somewhat surprising as we just argued that the PQ isometries are not present in SU (3) structure compactifications, yet the metric given in (3.21) does have them. This is related to the fact that the corrections for Re t i determined in (3.10) have this universal form and together with N = 2 supersymmetry imply the absence of subleading polynomial corrections in the prepotential. In that sense the PQ isometries can be viewed as "accidental" symmetries. (Of course they are broken in the potential already at leading order [8].) The leading order correction of the metric is -exactly as for Calabi-Yau compactifications -its c-dependence. Expanding e K to first order in c/V results in where g 0 J ij and K 0 are the leading order metric and Kähler potential computed from the cubic prepotential F 0 given in (3.13) and The first term yields indeed the desired correction (3.9) while the second term does not match that expectation. However, exactly as for Calabi-Yau compactifications this term is related to the dilaton/volume mixing and reabsorbed in an appropriate α ′ corrected definition of the four-dimensional dilaton [21,22]. 10 At this point it is worth recalling that, strictly speaking, for SU (3) structure compactifications we cannot exclude negative powers of t i inF , or put another way, non-polynomial terms in the homogeneous functions F i in (3.16). Their appearance would signal unusual power-like singularities in the variables t i and one might expect their absence also in the SU (3) structure case. For the leading α ′ corrections we can address this question directly. First, as for the α and β terms above, since the leading string corrections to supergravity, including those involving B, appear at order α ′3 , we can immediately argue that F 1 and F 2 vanish, since they correspond to α ′ and α ′2 corrections. The α ′3 correction F 3 must reproduce the appropriate correction Re δg J = −c Re g 0 J when Im t i = 0. We have already seen that the polynomial form F 3 = γ does exactly this. With a little work, one can show that this is the only allowed solution given F 3 is homogeneous. Thus to this order there can be no non-polynomial terms. When considering the higher order corrections F i with i > 3, it is interesting to note that, just by power counting, at each order in α ′ the higher derivative corrections to the ten-dimensional effective action can only include finite powers of H = dB, and gauge invariance prohibits direct dependence on B. Assuming the complex moduli t i are not redefined by α ′ corrections, this implies that g J can have at most polynomial dependence on Im t i at each given order in α ′ . Together with the homogeneity of the F i functions this would be enough to argue that all higher order α ′ corrections vanish andF includes only non-perturbative corrections.
To summarize, for any SU (3) structure compactifications we have argued that the α ′ -corrected prepotential on M J has the form (3.24) Before we turn to the hypermultiplet sector, let us note that the perturbative α ′ corrections vanish for manifolds with χ(X 6 ) = 0. This is consistent with the arguments of [35]: for such X 6 there is necessarily an additional SU(2) structure and the compactification can be viewed as a spontaneously broken N = 4 theory. The N = 4 supersymmetry then forbids any perturbative corrections, consistent with (3.24). 11

Perturbative g s corrections
As we discussed above, for Calabi-Yau compactifications the zero-modes ξ,ξ of the RR potentials C p together with the axion σ in the hypermultiplet moduli space admit 2n h + 1 (perturbative) Peccei-Quinn shift symmetries given in (2.14). In SU(3) compactifications the situation is more involved since generically some of the scalar fields become massive, or in other words the number of the zero modes is reduced, because the basis forms are no longer harmonic. As discussed in [2,8,9,12], a subset of the shift symmetries (2.14) turns local in that c I ,c I become space-time dependent and appropriate couplings to the gauge fields are induced. N = 2 supersymmetry in turn demands a non-trivial scalar potential which lifts some of the flat directions corresponding to the "non-zero" but light modes mentioned above. An argument along the lines of ref. [36] further shows that these gauged isometries survive after including perturbative and non-perturbative corrections.
The number of gauged isometries depends on the specific structure of the non-trivial torsion, that is, the constants e iI and p I i in (2.11). Nevertheless, one can determine that for any SU(3) compactification at least n h of the isometries in (2.14) survive perturbatively, and only the question which are gauged depends on the details of X 6 [9]. To review this argument, note that in this case, for the RR scalars in (2.13), d(ξ I α I +ξ I β I ) = (dξ I )α I + (dξ I )β I + (ξ I e iI +ξ I p I i )ω i , (3.25) This means that the RR field strengths depend explicitly on the combinations ξ I e iI +ξ I p I i in type IIA and ξ i e iI and ξ i p I i in type IIB. Thus for IIB it is clear that at most one loses the n h − 1 isometries ξ i → ξ i + c i . For type IIA, one notes that d 2 = 0 implies e iI p I j − p I i e jI = 0 and hence the vectors Z i = (e iI , p I i ) span an isotropic subspace of the 2n h -dimensional symplectic space spanned by α I and β I . Thus there can be at most n h linearly independent Z i and hence at most n h combinations ξ I e iI +ξ I p I i that appear explicitly in the RR field strengths and hence have broken PQ symmetry. This result implies that all hypermultiplets can be dualized to tensor multiplets, where the scalars which transform as in (2.14) are replaced by dual antisymmetric tensors [37]. 12 This property was used in [23] to parameterise the possible perturbative corrections in Calabi-Yau compactifications in terms of one scalar function ∆(z). N = 2 supersymmetry alone already constrains the string loop-corrected scalar field spaces to be quaternionic-Kähler. Our assumption of SU (3) structure further implies that M h is a torus fibration over a special-Kähler manifold. However, because not all the RR isometries survive, the metric on M h no longer has to be 'special quaternionic-Kähler', or in other words the loop-corrected metric is generically not of the Ferrara-Sabharwal form given in (2.5) with merely a loop-corrected prepotential G. However, the existence of at least n h +1 unbroken translational isometries additionally constrains the form of the metric [18,20,23], and this is best described in terms of the dual tensor multiplets. For Calabi-Yau compactifications -and as we just argued, also for SU(3) structure compactifications -all hypermultiplets can be dualized to tensor multiplets, and as a consequence the constraints determined in [37,39] apply. Using the additional property that the dilaton organizes the g s expansion, we can repeat the analysis performed in [23] and arrive at the same result that the corrections to M h are determined by a single holomorphic function ∆(z) of the basecoordinates z which is homogeneous of degree zero. Explicitly, the correction to (2.5) has to be of the form [23] where we included appropriate covariant derivatives. A µ = − Im (∂ a K ∂ µ z a ) is the Kähler connection on the special-Kähler base,Ã µ = Re (∂ a ∆ ∂ µ z a ), and M IJ (z,z, φ), T IJ (z,z, φ) are quantum deformations of N IJ (defined in (2.6)) and 2 Re N IJ , respectively. All these corrections are proportional to powers of ∆ and the precise expressions of M IJ (z,z, φ), T IJ (z,z, φ) and the matrix Y IJ (z,z, φ) can be found in [23]. Here we have written z a for the coordinates on the base, corresponding to type IIA. In type IIB these would be replaced by t i .
Again, we can compare this form of the metric directly with the higher-derivative corrections to M J and M Ω we derived above. In particular, comparing with the kinetic terms for z a in IIA and the kinetic terms for t i in IIB, we see that in fact ∆ is constant, ∆ IIA = −∆ IIB = 4ζ(2) c = ζ(2) π 3 χ(X 6 ), (3.27) just as for the case of Calabi-Yau backgrounds.
We see that, again, the SU (3) structure corrections take the exact same form as in the Calabi-Yau case. In particular, although the expansion is not generically compatible with preserving all 2n h +1 Peccei-Quinn symmetries, this is not realised in the correction to the hypermultiplet moduli space -the corrected metric still preserves all the shift symmetries. Instead, the breaking is only realised in the mass terms. Furthermore, it was argued in [23] that the tensor multiplet structure and the dilaton expansion was enough to exclude any further corrections beyond one-loop. Using the same logic here, the implication is that for SU (3) structure compactifications there is a non-renormalization theorem stating that the hypermultiplet metric can only be corrected perturbatively at one-loop but not beyond.

Summary and Outlook
We have obtained the perturbative α ′ and g s corrections to the moduli space metric arising in compactifications of type II theories on six-dimensional SU(3) structure manifolds, subject to some simple assumptions about the low-energy modes. We have shown that these are of exactly the same form as for Calabi-Yau manifolds, namely they are all proportional to the Euler characteristic. For that we only needed to compute the correction to the four-dimensional Einstein term. All the other corrections can be derived from this single one using symmetry and supersymmetry arguments, which carry along from the Calabi-Yau to the general SU(3) structure case (i.e., from the ungauged N = 2 to the gauged N = 2 four-dimensional action). As a byproduct, we have shown that even though the mass terms originating from the gaugings break some of the shift symmetries, the perturbatively corrected kinetic terms actually possess all shift symmetries. Of course, a key open question is how dependent these results are on our assumptions about the low-energy modes, in particular, the absence of light spin-3/2 particles. We hope nonetheless that this work provides useful basis point for studying the form of general corrections to non-Calabi-Yau compactifications.
We also have only considered the case of SU(3) structure manifolds, without background fluxes. An obvious extension of our results would be to consider more general SU(3) × SU(3)-structure manifolds, and/or NSNS and RR fluxes. One could also try to investigate the form of the non-perturbative corrections. We hope to make progress on these issues in the near future.