Chapter 5 Fourdimensional String and M-theory Compactifications In the last two chapters we discussed N = 2 supergravity and its black hole solutions. Since supergravity is not consistent as a quantum theory, it must be embedded into a larger consistent theory. The most promissing candidate is string theory, and in this chapter we review how four- dimensional N = 2 supergravity arises as the low energy effective theory of compactifications of ten-dimensional string theories and of eleven-dimensional M-theory. 5.1 Type II String Theory on a Calabi-Yau Threefold One way to obtain four-dimensional N = 2 supergravity from string theory is by compactifying type II string theory on a complex-three-dimensional Calabi-Yau manifold. Such spaces are Ricci-flat K¨ahler manifolds with vanishing first Chern class.1 Since they have holonomy group SU(3) they posess Killing spinors and the number of supercharges of the four-dimensional theory is 1/4 of the number of supercharges of the ten-dimensional theory. In the following we review some facts about Calabi-Yau compactifications of string theory. For a more complete account we refer to the reviews [147, 148, 149]. The four-dimensional theory has a massless spectrum that consists of the N = 2 supergravity multiplet plus a model-dependent number of abelian vector multiplets and neutral hypermulti- plets. A hypermultiplet contains two Weyl spinors and four real scalars as its on-shell degrees of freedom. The kinetic term of the hypermultiplet scalars is a non-linear sigma-model with a target space that is restricted to be quaternionic by local N = 2 supersymmetry [150]. Local N = 2 supersymmetry forbids neutral couplings between vector and hypermultiplets [15]. Since the string compactification for generic moduli only contains gauge-neutral fields, the total mod- uli space factorizes into a product of the special K¨ahler manifold of vector multiplet moduli and the quaternionic manifold of hypermultiplet moduli. This factorization breaks down at special points in the moduli space, where the Calabi-Yau manifold becomes singular in such a way that the moduli space metric and the low energy effective action are singular, too.2 The most popular case of such a singularity is the conifold singularity [151]. The conifold point and other more complicated singular points in the moduli space are at finite distance from regular points. The fact that string backgrounds can become singular under finite changes of the parameters was long considered as a severe problem. Then it was observed by Strominger that the conifold singularity could be physically explained by the presence of a charged hypermultiplet that be- comes massless at the conifold point [152]. The additional massless state corresponds to a type IIB threebrane, which is wrapped on the three-cycle which degenerates at the conifold point. 1 6 2 We also include the condition h1;0 = 0 in order to exclude from the definition the six-torus T and K3 × T , where K3 denotes the K3 surface. 2There exist milder singularities, such as orbifold singularities and flop transitions, which do not give rise to singularities of string theory. 81 82CHAPTER 5. FOURDIMENSIONAL STRING AND M-THEORY COMPACTIFICATIONS When the additional state is taken into account, all physical quantities behave smooth at the conifold point. Subsequent work generalized this to other types of singularities. In particular it was shown that more complicated singularities correspond to multicritical points in the scalar potential of N = 2 supergravity coupled to charged matter [153]. At these points one has the option to go to different branches of the theory, such as the Coulomb and Higgs branch, which we discussed in chapter 2. The corresponding geometric mechanism is that a singularity can be resolved in several non-equivalent ways. The transition between branches of the scalar potential corresponds to a topological phase transition, i.e. a change in the topology of the Calabi-Yau manifold. In such transitions physics, or more precisely the low energy effective action is smooth. We refer to [149] for a review and references. In the following we will only consider generic compactifications where all fields in the Lagrangian are gauge-neutral and the moduli space factorizes into a vector multiplet and a hypermultiplet part. In type IIA compactifications the numbers NV ,NH of vector and hypermultiplets are given by NV = h1,1 and NH = h2,1 +1, (5.1) where h1,1,h2,1 are Hodge numbers of the Calabi-Yau manifold. Part of the scalar fields are geometric moduli of the internal space. The h1,1 vector multiplet moduli describe deforma- tions of the K¨ahler structure and of the internal part of the stringy B-field. Both data can be conveniently combined into the so-called complexified K¨ahler structure. Among the 4h2,1 real hypermultiplet moduli 2h2,1 describe deformations of the complex structure whereas the other 2h2,1 moduli come from the fields of the Ramond-Ramond sector. There is one additional hy- permultiplet which is always present, even when the Calabi-Yau manifold has a unique complex structure, h2,1 = 0. It is called the universal hypermultiplet because of its independence from Calabi-Yau data. One of its scalars is the dilaton φ, whose vacuum expectation value is related hφi to the four-dimensional IIA string coupling by gIIA = e . In addition it contains the stringy axion, which is obtained from the space-time part of the B-field by Hodge duality, and two scalars from the Ramond-Ramond sector. In type IIB compactifications the role of vector and hypermultiplets is reversed, (B) (B) NV = h2,1 and NH = h1,1 +1. (5.2) By mirror symmetry type IIB string theory on a Calabi-Yau manifold is equivalent to type IIA string theory on a so-called mirror manifold [154, 155], see [156, 149] for a review. A Calabi-Yau manifold and its mirror are related by exchanging the roles of the complex structure moduli and (complexified) K¨ahler moduli spaces.3 In particular the Hodge numbers are related by e e h1,1 = h2,1 and h2,1 = h1,1. Due to mirror symmetry we can restrict our attention to one of the two type II theories. For our purposes it is convenient to consider the type IIA theory. The part of the effective Lagrangian that we are interested in is the vector multiplet sector which is encoded in the function F (X, Ab). By expansion of the function, X∞ F (X, Ab)= F (g)(X)Abg , (5.3) g=0 one finds the prepotential F (0)(X) which describes the minimal part of the Lagrangian and the coefficients F (g≥1)(X) of higher derivative couplings of the form C2T 2g−2. In string perturba- tion theory one can compute on-shell scattering amplitudes and obtain an effective action. This action has ambiguities because string perturbation theory has only access to on-shell quanti- ties. Therefore terms in the effective action are only known up to terms which vanish on-shell. For example the difference between the square of the Riemann tensor and the square of the 3The precise statement of mirror symmetry is that IIA and IIB string theory in the corresponding string backgrounds are equivalent. String backgrounds are defined in terms of conformal field theories. It is important to take into accoung stringy α0 corrections of the classical geometry. Moreover one has to include regions of the moduli space, so-called non-geometric phases, which do not have a geometrical interpretation in terms of a Calabi-Yau sigma model. 5.1. TYPE II STRING THEORY ON A CALABI-YAU THREEFOLD 83 Weyl tensor vanishes in a Ricci-flat background and therefore string perturbation cannot decide whether the curvature squared term with coupling F (1)(X) involves the Riemann or the Weyl tensor. We can invoke the off-shell formalism of chapter 3 to identify this term as the square of the Weyl tensor. Conversely, the effective supergravity Lagrangian of chapter 3 needs the function F (1)(X) as input from string theory, because supergravity is not a consistent quantum theory. As discussed in chapter 3 the relation between the auxiliary T -field and the graviphoton is complicated in the off-shell formulation. In [157] a relation is found by requiring that the effective action reproduces the on-shell scattering amplitudes. In this context the graviphoton is defined through its vertex operator. The string computation involves the physical scalars zA rather then the sections XI .In order to rewrite the couplings F (g)(X) we introduce the holomorphic sections XI (z)by 1 I K(z,z) I X =mPlancke 2 X (z) , (5.4) where K(z,z)istheK¨ahler potential. Note that we have restored the Planck mass in order to be able to do dimensional analysis later. We go to special coordinates zA by imposing X0(z)=1,XA(z)=zA.NextweusethatF(g)(X) is homogenous of degree 2 − 2g and define (g) g−1 −(1−g)K (g) F (z)=i[mPlanck] e F (X) . (5.5) In the language of chapter 3 the quantity F (g)(z) is a holomorphic section of L2(1−g), whereas F (g) is a covariantly holomorphic section of P2(1−g). (g) In order to display the dependence of F (X) on the string coupling gS,wehavetoreplace the Planck mass by the string mass mString = mPlanckgS. This yields (g) − 2−2g −2+2g (1−g)K F (g) F (X)= imStringgS e (z) . (5.6) Next we use that the string coupling is given by the vacuum expectation value of the dilaton and that in type II compactifications the dilaton sits in a hypermultiplet.