The moduli space of hyper-K¨ahler four-fold compactifications 7 0 0 2 n Ram Sriharsha† a Department of Physics, University of Maryland J 5 College Park, MD 20742-4111 3 v 3 2 0 I discuss some aspects of the moduli space of hyper-Ka¨hler four-fold compactifications 2 1 of type II and - theories. The dimension of the moduli space of these theories is strictly 6 M 0 / bounded from above. As an example I study Hilb2(K3) and the generalized Kummer h -t variety K2(T4). In both cases RR-flux (or G-flux in -theory) must be turned on, and p M e we show that they give rise to vacua with = 2 or = 3 supersymmetry upon turning on h N N : v appropriate fluxes. An interesting subtlety involving the symmetric product limit S2(K3) i X is pointed out. r a December 1st, 2006 [email protected] † 1. Introduction Compactifications of type II strings on hyper-Ka¨hler two-folds has been much stud- ied and very well understood (see the review [1] and references therein for an excellent overview). The reason for this happy state of affairs is that any two compact hyper-K¨ahler two-folds are diffeomorphic to each other and there is essentially only one K3 surface. The moduli space of K3 surfaces can be determined precisely, and in string theory we see this simplicity as the fact that type IIA on a K3 surface is dual to heterotic strings on T4 whose moduli space is the Narain moduli space Gr(4,20). An analogous understanding of hyper-K¨ahler four-folds is lacking in literature. In fact there are still only two known examples of compact hyper-Ka¨hler four-folds even though the cohomology of a compact hyper-K¨ahler four-fold has been understood for a long time. Any treatment of compacti- fications on hyper-K¨ahler four-folds suffers from the fact that there are so few examples. Fortunately, it turns out that one can map out the moduli space of type II string theo- ries on compact hyper-K¨ahler four-folds using simple CFT arguments [2] . We review the argument that obtains the moduli space of = (4,4) SCFTs in Appendix 1 and use the N results of [2] to obtain the moduli space of type IIA /B compactifications on hyper-K¨ahler four-folds. There is an action of O(4,b 2;Z) on the moduli space of hyper-K¨ahler four- 2 − fold compactifications of type IIA, which was observed by Verbitsky as the group acting on the lattice H∗(X;Z) for an arbitrary hyper-Ka¨hler manifold. We point out that there is a simple reason why this group acts on H∗(X;Z). Somewhat surprisingly, it is possible to show that these theories have a moduli space of bounded dimension, essentially due to the fact that the topological types of hyper-Ka¨hler four-folds is bounded. It would be interesting to obtain a simple physical understanding of this fact. In section 2 the basic facts of hyper-Ka¨hler four-folds is summarized. Sections 3 and 4 work out aspects of the dimensional reduction of type II theories on hyper-K¨ahler four- folds. In most respects this is similar to the case of Calabi-Yau four-folds and we follow the paper of Gates, Gukov and Witten [3] in performing this reduction. Some of the type IIA/ -theory compactifications will be not be solutions of the 1- M loop effective action coming from string theory. We analyze this in more detail in section 5. In section 6 we work out the conditions under which the two known examples of hyper- 1 K¨ahler four-folds yield supersymmetric vacua. We also show that the symmetric product S2(K3) does not arise in the moduli space of hyper-Ka¨hler compactifications with fluxes. Though we are only talking about hyper-Ka¨hler four-folds in this paper, it appears that there are few other ways of obtaining = 3 supersymmetric theories in 3d. In N particular there is no other class of compactifications that yield = 3 supersymmetry in N 3d at weak coupling. So one may view our results as indicating that the moduli space of = 3 theories in 3d (and the moduli space of = (3,3) supergravity theories in 2d) are N N very tightly constrained. In particular the moduli space is of finite dimension with a strict upper bound on the dimension. It has already been noted that a similar thing happens for Calabi-Yau compactifications in general, in that the dimension of the moduli space of = 2 supergravity theories in 4d for example is expected to be finite. What is perhaps N surprising is the simplicity of showing this for = 3 supergravities in 3d. In particular, N suppose we consider = 4 supergravity in 3d, we know that a class of these theories arise N via compactification on K3 K3 whose moduli space is of finite dimension. However these × compactifications do not exhaust all = 4 supergravity theories in 3d and in particular a N large class of such compactifications arise via dimensional reduction on T2 CY for which 3 × there are only indirect arguments that suggest a bound on the dimension of the moduli space. Flux compactification in -theory and type II context has a long history (see [4] M for overview and references) . However, the recent work of Aspinwall and Kallosh [5] on K3 K3 is very closely related to the analysis in this paper, and some of the techniques × used there are applied to the case of hyper-Ka¨hler four-folds here. 2. Some facts on Hyper-K¨ahler four-folds A hyper-K¨ahler 4-fold is a K¨ahler manifold with a nowhere vanishing non degenerate holomorphic 2-form ω. Then ω2 trivializes the canonical line bundle, so by Yau’s proof of Calabi conjecture, there is a unique Ricci-flat metric that respects the hyper-K¨ahler structure. The cohomology of a general K¨ahler manifold can be decomposed via Hodge decomposition. Forahyper-Ka¨hler4-fold, thenontrivialHodgenumbersareh1,1,h2,1,h3,1 2 and h2,2. However, not all of them are independent. Given any type (1,1)-form we can create a (3,1) form by wedging with ω, so that h3,1 = h1,1. Also h1,1 1 as the space ≥ is K¨ahler, so we can write h1,1 = 1 + p for some p in Z+. Furthermore, just as for a Calabi-Yau 4-fold, h2,2 is not independent. The quickest way to note this is to consider ¯ the index of the Dolbeault operator ∂ acting on the bundle E of holomorphic type (2,0) E2 2 forms. This index is given by: Ind∂¯ = ( 1)qh2,q (2.1) E2 Xq=0 − However, the index also has a purely topological character, and can be expressed via the Atiyah-Singer Index theorem as: Ind(∂¯ ) = Todd(X)Ch(Ω2,0) (2.2) E2 Z Using the standard expression for the Todd genus and Chern character, we compute: 1 Ind(∂¯ ) = (3c 2 +79c ) (2.3) E2 120 Z 2 4 where we used the fact that c = 0. Now, the Todd genus of a hyper-K¨ahler 4-fold 1 is precisely 3, and this implies a relation between c 2 and c (incidentally, c is the 2 4 4 R arithmetic genus or Euler characteristic of the hyper-Ka¨hler 4-fold X). Specifically: 1 Todd(X) = (3c 2 c ) (2.4) 2 4 720 − so that χ c 2 = 720+ (2.5) Z 2 3 X Using (2.5) in (2.3) we get a relation between the various Hodge numbers. Denoting h2,1 = 2q 1 this relation is: h2,2 = 72+8p 4q (2.6) − So the hyper-K¨ahler 4-folds are characterized by two non negative integers (p,q). 1 Here we used the fact that b3 is divisible by 4, for a hyper-K¨ahler four-fold. Incidentally, this also implies χ is divisible by 12, which is a stronger result than the one for Calabi-Yau four-folds. The Hilbert scheme of two points on K3 gives us an example where χ is divisible by 12, and not by 24, so this is the strongest result we can get. In our notation χ = 1(7+p−q). 24 2 3 3. Compactification of type IIA on Hyper-K¨ahler four-folds In this section we will describe the compactification of type IIA string theory on a hyper-K¨ahler four-fold X. In the large volume limit these compactifications can be discussed by dimensionally reducing type IIA supergravity on hyper-K¨ahler four-folds. The bosonic content of type IIA supergravity in ten dimensions is the metric g , MN an antisymmetric two-form B and dilaton φ from the NS-NS sector. The R-R sector MN gives rise to the one-form gauge field A and three form C . The bosonic action in M MNP string frame is of the form: 1 1 1 L = d10x√ g[e−2φ(R10 +4( φ)2 H2) F2 G2] (3.1) Z − ∇ − 12 − 4 − 48 Where: F = dA H = dB G = dC +A H (3.2) ∧ are the gauge invariant field strengths. The action (3.1) is of course the tree level action for type IIA string theory in ten dimensions. There are higher order terms in the effective actionthatarenotcaptured in(3.1). Forthemostparttheirstructureisnotknown. There is however an important term of the form B X where X is a particular contraction of 8 8 ∧ four powers of the Riemann tensor. This term was shown to be present in type IIA by considering scattering amplitudes in type II string theory [6] . This term leads to a tadpole for the B-field which has to be cancelled in type IIA by turning on G-flux and/ or adding N F1-strings such that: χ 1 N = G G (3.3) 24 − 2(2π)2 Z ∧ X If the Euler number of X is not divisible by 24, then the tadpole cannot be canceled by simply adding F-strings and we must turn on RR-flux G also. Of course, turning on G-flux we will typically end up breaking supersymmetry unless the G-flux happens to be primitive with respect to the P1 of complex structures on X. For the moment we will ignore these subtleties and address them in section 3. The action for type IIA supergravity is invariant with respect to 32 supercharges, 16 of which are left-chiral and 16 right-chiral with respect to the chirality operator in 10d. Upon compactifying on X, the resulting action in two dimensions possesses residual supersymmetry only if X admits a covariantly 4 constant spinor. In the case of hyper-Ka¨hler four-folds the holonomy group of X is sp(2). A generic eight dimensional spinor is in one of the two inequivalent spinor representations of spin(8) say 8 . Under sp(2) we have the decomposition: + 8 = 5+1+1+1 8 = 4+4 (3.4) + − so that there is a three-dimensional space of covariantly constant spinors on X. Via the decomposition: 16 = (8 ,+)+(8 , ), 16′ = (8 , )+(8 ,+), (3.5) + − + − − − corresponding to SO(1,9) SO(8) SO(1,1) we end up with a non-chiral two → × dimensional supergravity theory2 with = (3,3) supersymmetry upon compactifying N type IIA on X. To determine the spectrum of the resulting two dimensional theory one performs Kaluza-Klein reduction of the various fields of type IIA. As the resulting two dimensional theory is non-chiral the fermions simply arise as = (3,3) superpartners and it is enough N to count the massless bosonic degrees. These are associated to the harmonics of the various bosonic fields of type IIA. Denoting the holomorphic 2-form on X by ω, one can expand the B zero modes as: MN B = biω1,1 +bω (3.6) i X i where: ω1,1 H1,1(X) b C bi R (3.7) ∈ ∈ ∈ leading to h1,1+2 scalars. The C zero modes lead to 2h2,1 scalars and h1,1+2 vectors MNP via: C = cjω2,1 + C nω1,1 +C ω ω2,1 H2,1(X) cj C (3.8) j µ n µ ∈ ∈ X X j n 2 In (3.5) the 16 and 16′ refer to the ten dimensional Majorana-Weyl spinors of opposite chirality associated to type IIA, whereas the spinor representations of SO(1,1) are labeled by their charges under spin(1,1). 5 The metric deformations lead to 3h1,1 2 scalars gk as follows: The zero modes of the − graviton satisfy the Lichnerowicz equation which in a suitable gauge can be written as: D Dkh R hst = 0 (3.9) k ij isjt − It is easy to see that the metric variations of the form δh and δh do not mix in (3.9) so ab a¯b they can be considered separately. For every element ω1,1 one obtains a variation of the form δh so that the number of such deformations is h1,1. Similarly, given ω1,1 H1,1(X) a¯b ∈ one can construct a variation of type δh as: ab δh = ωc¯ ω1,1 (3.10) ab (a b)c¯ However if ω1,1 is proportional to the K¨ahler form then (3.10) vanishes, so that there are only 2h1,1 2 deformations of type δh so that the space of sp(2) holonomy metrics ab − on a hyper-K¨ahler four-fold has dimension 3h1,1 2. − Collecting all the matter content together we end up with h1,1 = (p+1) = (4,4) N vector multiplets containing gk,b as the scalar components, together with q = (4,4) i N hyper multiplets containing the 4q scalars cj. Even though we have only = (3,3) N supersymmetry, the matter sector arranges itself into = (4,4) multiplets, which is a N familiar fact given that any supersymmetric sigma model with = 3 supersymmetry N is automatically = 4 supersymmetric also. Of course the higher order terms in the N effective action will only be = (3,3) supersymmetric. N The supergravity sector contains the graviton, three abelian gauge fields and a scalar, along with three gravitini and three Majorana fermions. The dilaton sits in the supergrav- ity multiplet. The low energy effective action for the vector and hyper-multiplet moduli will in general be given by a = (4,4) supersymmetric sigma model. In the case of the vector N multiplets with rigid supersymmetry this sigma model is based on a target space that is hyper-K¨ahler with torsion (HKT), so we expect upon coupling to supergravity that the target space is quaternionic K¨ahler with torsion (QKT). The hyper multiplet moduli space is similarly a hyper-Ka¨hler or Quaternionic K¨ahler manifold. As the two multiplets carry scalars with different R-symmetries the moduli space factorizes just as in = 2 N 6 supergravity coupled to matter in four dimensions. Denoting the moduli space of type M IIA on a hyper-K¨ahler four-fold as: = (3.11) V H M M ×M what can be said about and ? V H M M The worldsheet description of any = (3,3) supersymmetric compactification to two N dimensions is in the form of a = 4 SCFT with small = 4 SCA and c = 12. The N N space-time moduli that sit in the (p+ 1) vector multiplets are all = 4 chiral primary N operators of this internal = 4 SCA. Since any = 4 SCA has a SU(2) SU(2) L R N N × R-symmetry this implies 3 that the moduli space has a SO(4) isometry. It turns out V M due to a theorem of Berger and Simons (see [1] for a nice discussion on the Berger-Simons result) that the smooth manifolds with SO(4) holonomy and dimension greater than 4 are only the symmetric spaces, the so called Grassmann manifolds. This leads us to identify: O(4,p+1) = (3.12) V M O(4) O(p+1) × There is a natural O(4,p+1;Z) symmetry of the moduli space which we can quotient bymaintainingtheHausdorffpropertyofthetheresultingspace. Itisnaturaltoconjecture that the U-duality group for this theory is O(4,p+1;Z). In type IIA the dilatonφ is in the = (3,3) supergravity multiplet. This implies that N the form of the moduli space is completely independent of string coupling g = eφ. For s large g , type IIA goes over to 11d supergravity which is the low energy limit of -theory. s M This means the -theory moduli space is also given by (3.12). By the same argument, M the metric on the moduli space is independent of string coupling. Given the moduli space of the form (3.12), we can take the large radius limit. The large radius limit can be determined by examining the Dynkin diagram of O(4,p+1), and it turns out that the structure of the moduli space in the large radius limit is given by: O(3,p) = R Rp+3 (3.13) + M O(3) O(p) × × × 3 Details of this standard argument are provided in appendix 1. This argument was first applied for determining the moduli space of N = 4 SCFTs by Cecotti. 7 This is what we expect in the large radius limit. In this limit we expect the met- ric deformations to be characterized by the moduli space of sp(2) holonomy metrics of fixed volume of a hyper-Ka¨hler four-fold, which is the O(3,p) factor, the R factor corre- + sponds to the trivial radial mode. The Rp+3 factor corresponds to the scalars arising from dimensional reduction of the NSNS 2-form. This provides a non-trivial consistency check. The = (3,3) supergravity coupled to matter has not been constructed in literature. N There is however the case of = (4,4) supergravity coupled to matter which has been N analysed [7]. This theory has a gauged SU(2) SO(4) R-symmetry and it has been shown ∈ that the target space parameterized by the scalars in this theory can be hyper-K¨ahler or Quaternionic K¨ahler. We expect a similar result to hold even in the case of = (3,3) N supergravity coupled to matter. That is, with = (3,3) supersymmetry, the form of the N moduli space remains non-trivial in general. This raises the puzzle as to how the CFT analysis was able to determine the local form of the moduli space as (3.12). We will resolve this puzzle in the next section. One subtlety that has to be pointed out is that there is a difference between the K3 case and the case of general hyper-Ka¨hler manifolds which affects our understanding of the moduli space. For K3 surfaces the global Torelli theorem holds, so that the moduli space of complex structures is determined by the space of periods. It is the space of periods that the supergravity analysis is sensitive to, and so is the chiral primary ring of the = (4,4) N worldsheet theory. It is not known whether a version of the global Torelli theorem holds for the higher dimensional cases. If it does not, then the choice of periods does not determine the complex structure fully. What will be lacking is some discrete data. It is known that all hyper-K¨ahler manifolds are deformations of a projective variety so they all have π = 0. 1 So it is not possible to have discrete torsion [8] in the worldsheet SCFT. I do not know what extra data the SCFT can have in this case that is not captured by the chiral primary ring. So the analysis of the moduli space in this paper is carried out modulo the discrete ambiguity arising from lack of a global Torelli like theorem. 3.1. -theory on hyper-Ka¨hler four-folds M The low energy limit of theory is 11d supergravity whose bosonic content is a M 8 graviton and a 3-form potential A, with four-form flux G. Dimensional reduction of 11d supergravity on a hyper-Ka¨hler four-fold yields a three dimensional = 3 supergravity coupled to matter. The matter multiplets are the vector N multiplet ( whose bosonic content is three scalars transforming as 3 of the SO(3) R- symmetry together a gauge field) and the hyper multiplet (which contains four scalars transformingasacomplexdoubletoftheR-symmetry). Anyactionforthehypermultiplets is automatically = 4 supersymmetric, so is the low energy effective action for the vector N multiplets(intheabsence ofG-flux). Upondimensionalreduction, weendupwitha = 3 N supergravity multiplet with a graviton, three gravitini. The matter sector consists p +1 vector multiplets (after dualizing some vectors into scalars) and q hyper multiplets. The moduli space factorizes as in the type IIA case. Upon dualizing the vectors into scalars, we expect the -theory moduli space to coincide with the type IIA case. The -theory M M moduli space will be of the form: O(4,p+1) = (3.14) 11d H M O(4) O(p+1) ⊗M × 4. Compactification of type IIB on hyper-K¨ahler four-folds The compactification of type IIB string theory on a hyper-K¨ahler four-fold X leads to a two dimensional = (0,6) supersymmetri theory in the non-compact directions. Its N low energy limit is = (0,6) supergravity coupled to matter. In this section we determine N the matter content of this theory and the moduli space. In the large volume liimit type IIB string theory in ten dimensions is well approximated by type IIB supergravity. The bosonic content of type IIB supergravity is the graviton g , the anti-symmetric two form MN B , the dilaton φ, the RR axion C, along with the RR two form A and the self-dual MN MN four-form G . Type IIB in ten dimensions has a sl (Z) action where the two forms MNPQ 2 A,B form a doublet of sl (Z) and the axio-dilaton can be combined as: 2 τ = c+ie−φ (4.1) and transforms under sl (Z) as: 2 (aτ +b) τ a,b,c,d Z ad bc = 1 (4.2) → (cτ +d) ∈ − 9