Geometrical methods for string compactifications Alessandro Tomasiello These notes were prepared for my lectures at LACES 2009. It should be noted that they are a preliminary version that I initially prepared for my own reference; as such, they are bound to contain mistakes and inaccuracies. If you think something in it looks fishy, please let me know. They do contain some extra material that I did not explain in class, and some that I will explain next time. On the other hand, as of today (December 9, 2009), they are unfinished: they do not contain the last (and most important) part, about the application of generalized complex geometry to flux compactifications with RR fields. 1 Fluxes and supersymmetry We will deal in these lectures with type II supergravity. Of course we are interested in this only because it is the low–energy limit of the type II string, but for most of these lectures the stringy effects will not be taken into account, and pure supergravity will be enough. 1.1 Fields Type II supergravity comes in two flavors, IIA and IIB. Let us begin by recalling the fields of each of them. (In this subsection and the next, we will follow the conventions and language of [1], except for a few changes noted in appendix A.) The matter fields are given by the fermions1 a a ψM (gravitino), λ (dilatino). (1.1) 1 1 2 2 where a = 1, 2. In IIA, ψM and λ has chirality +, and ψM , λ have chirality −. In IIB, all have chirality +. All of them are Majorana. The forces are mediated by the bosons gMN (metric),BMN , (1.2) and by a collection of fields of the form = 1, 3, 5, 7, 9 (IIA) CM1...Mp , p = RR − fields. (1.3) = 0, 2, 4, 6, 8 (IIB) 1For notations, see appendix A. 1 The metric gMN is obviously symmetric, as a metric should be. Both BMN and the CM1...Mp are totally antisymmetric, and they should be understood as gauge potentials, generalizing the potential AM of electro–magnetism. We will often use for them the form notation 1 1 B ≡ B dxM ∧ dxN ,C ≡ C dxM1 ∧ ... ∧ dxMp . (1.4) 2 MN p p! M1...Mp M ∂ 2 Using also the exterior differential d ≡ dx ∂xM , we can introduce the field–strengths (or “fluxes”) H = dB , Fp = dCp−1 − H ∧ Cp . (1.5) These can be thought of as similar to the FMN of electro–magnetism. For example, there 0 is a gauge transformation B → B = B +dλ1, with λ1 a one–form, that leaves H invariant 0 0 (because H = dB = d(B + dλ1) = dB = H). The gauge transformations for the Cp potentials are more complicated, and they read 0 Cp → Cp = Cp + dλp−1 − H ∧ λp−3 (1.6) where obviously λk is a k–form. Another complication about the Cp is that their field strengths are related by b p c Fp = (−1) 2 ∗ F10−p . (1.7) In most other reviews of supergravity, the constraint (1.7) is solved by keeping as fun- damental fields only the field–strengths of the fields with the fewest indices. For exam- ple, [2, Vol. 2] keeps only F0, F2 and F4 in IIA, and F1, F3 and F5 in IIB. (Notice that, for IIB, part of the constraint is still with us: we still have F5 = ∗F5.) This choice of which field–strengths to keep is called “electric basis”. In these lectures, as we will see later, we will choose another electric basis, one which is better adapted to the problem of compactifying to four dimensions. Equations (1.6) might look clumsy. It is a good idea to collect them into a single object, a differential form of mixed degree. We can think of this as a formal sum: C1 + C3 + C5 + C7 + C9 (IIA) C = (1.8) C0 + C2 + C4 + C6 + C8 (IIB) In the same way, we can define F0 + F2 + F4 + F6 + F8 + F10 (IIA) F = (1.9) F1 + F3 + F5 + F7 + F9 (IIB) 2 Actually, this formula is more complicated if F0 6= 0, because that is a field–strength without a potential, but for the time being we can simply ignore this. F0 is also called “Romans mass”. 2 This allows to write the RR field strength in (1.5) as F = dC − H ∧ C = (d − H∧)C ≡ dH C. (1.10) Notice that 1 1 1 d2 = {d , d } = {d, d} + {d, H∧} + {H∧,H∧} = (dH) ∧ . (1.11) H 2 H H 2 2 The idea of these equalities is the same as when we compute the commutator of the derivative operator and the multiplication operator in quantum mechanics: we can imag- ine acting on a test object (for quantum mechanics, a wave function; in our case, a differential form), compute the (anti)commutator, and then eliminate the test object at the end. Equation (1.7) does not seem to simplify too much by the introduction of the formal sum (1.9). But the signs we have in (1.7) appear often enough in string theory that it pays off to introduce a symbol to avoid writing them out every time. We introduce an operator λ, defined by its action on a differential form αk of degree k: b p c λαk = (−1) 2 αk . (1.12) Then we can write (1.7) as3 F = ∗λF = λ ∗ F. (1.13) 1.2 Supersymmetry transformations We can now introduce the supersymmetry transformations of type II supergravity. They contain infinitesimal parameters a, a = 1, 2; these are fermionic, so that the supersymme- try transformations mix bosons and fermions (just like − × + = −, − × − = +). In IIA, 1 has chirality +, 2 has chirality −. In IIB, both have chirality +. Both are Majorana. Before we write down the supersymmetry transformations, we need a bit of notation. The fluxes appear in these equations multiplied by an appropriate number of gamma matrices: 1 1 1 H ≡ H ΓNP , H ≡ H ΓMNP , F ≡ F ΓM1 ... ΓMk . (1.14) M 2 MNP 6 MNP k k! M1...Mk 1 M1 Notice that the definition of Fk is very similar to the form notation Fk = k! FM1...Mk dx ∧ Mk ... ∧ dx . One can think of F k as being obtained from the k–form Fk via a map that 3The operators ∗ and λ commute in dimension ten, but not in dimension six. 3 sends dxM 7→ ΓM : 1 1 X ii ik X ii...ik Clifford map : α ≡ αi ...i dx ∧...∧dx ←→ α ≡ αi ...i γ . (1.15) k! 1 k k! 1 k αβ k k M Notice also that F k has two spinorial indices (because each of the Γ does), so it is a “bispinor”. The use of slashes to distinguish a form from its corresponding bispinor is more precise, and it is indeed essential to keep track of certain subtle signs, but it can will quickly get out of hand (and make unreadable the equations in which they appear) as soon as one applies it to more complicated forms. In what follows, we will drop the slash whenever it should not lead to confusion. They can be written as [1] 1 eφ δψ1 = D + H 1 + F Γ Γ2 , M M 4 M 16 M 1 eφ δψ2 = D − H 2 − λ(F )Γ Γ1 ; M M 4 M 16 M (1.16) 1 ΓM δψ1 − δλ1 = D − ∂φ − H 1 , M 4 1 ΓM δψ2 − δλ2 = D − ∂φ + H 2 . M 4 M M In the third and fourth equations, D = Γ DM and ∂φ = Γ ∂M φ; these definitions are in the spirit of (1.14), and, just as for those definitions, we decided to drop the slashes. Fortunately, in (1.16) we were able to assemble here all the RR fluxes in the combi- nation F from (1.9), without any extra factors. Notice also that (1.16) are valid both in IIA and IIB. If we are looking for supersymmetric solutions of the equations of motion, we can a a follow a standard strategy: set to zero the expectation values of the fermions ψM and λ . Invariance under supersymmetry then means that all the variations in (1.16) should be set to zero. This gives rise to four equations. As we will now explain, these are almost all we need. Because of the supersymmetry algebra {Q, Q} = P , one expects that the Hamiltonian can be written as some kind of square. In supersymmetric field theories, this leads to the BPS bound on the energy; in particular, one finds that supersymmetric configurations also solve the equations of motion. For example, in four–dimensional N = 1 super–Yang–Mills, invariance under super- symmetry gives the self–duality equations F = ∗F . If one recalls the Bianchi identity dF = 0 (in absence of monopoles), the equations of motion d∗F = 0 follow automatically. 4 The situation in type II supergravity is similar. The Bianchi identities that have to be imposed are (d − H∧)F = 0 , dH = 0 (almost everywhere). (1.17) The reason to specify “almost everywhere” is that there could be sources. We will see later that only some kind of sources are allowed by supersymmetry. Once one imposes (1.17), almost all the equations of motion follow automatically. Those that do not (the ones for g01) will not play any role in the rest of our lectures. In fact, we will see that, for compactifications, half of the equations in (1.17) are also a consequence of supersymmetry. That is because half of the equations for F should be thought of as true Bianchi identities; the other half are actually the equations of motion for F .