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in d = 11 and d = 10 The first equality shows that the integral does not depend on the metric. The second one Felix Br¨ummer keeps general covariance manifest (here the metric and Hodge ∗ are those of M, so ∗Ω is a scalar).

1 p-forms A familiar example where p-form notation can sometimes be useful is electromagnetism in d = 4. These are geometric animals which will frequently There is a 1-form A with components Aµ (the gauge appear, so we start by giving a (somewhat sloppy) potential), which is locally determined only up to overview of p-form notation. A p-form in d ≥ p a U(1) gauge transformation dimensions is an antisymmetric tensor field with p indices. In particular, 0-forms are scalars and 1- A → A + d Λ (6) forms are vectors. All operations on a p-form Ω with Λ any 0-form. It gives rise to a 2-form F = can be defined in terms of its components Ω µ1...µp d A with components F (the field strength). F is in some local coordinates. What we need is µν invariant under gauge transformations of Eq. (6) by Eq. (2). The kinetic action in some 4d space-time • Addition: Two p-forms are added by adding S is their components. 1 Z 1 Z √ • Exterior differentiation: The d operator turns S = − F ∧∗F = − d4x −g F F µν . (7) 2 4 µν a p-form into a (p + 1)-form. It acts as S One may couple an external particle of charge q (d Ω) = (p + 1) ∂ Ω . (1) νµ1...µp [ν µ1...µp] (an “electron”) to the gauge field. This is done Here (as always) [ ] denotes weighted antisym- by integrating the 1-form A over the particle’s 1- metrization. Note that dimensional worldline C ⊂ S: 1 Z Z d d (anything) = 0 by symmetry . (2) S = − F ∧ ∗F + q A + ... (8) 2 S C • The wedge product: Given a p-form Ω and a Alternatively one could group all charged particles q-form Ω0, one may form a (p + q)-form Ω ∧ Ω0 into a current density 1-form J: whose components are 1 Z Z 0 (p + q)! 0 S = − F ∧ ∗F + A ∧ ∗J + ... (9) (Ω ∧ Ω )µ1...µp ν1...νq = Ω[µ ...µp Ω . 2 p! q! 1 ν1...νq] S S (3) In this formulation Maxwell’s equations are very • Hodge dualization: The ∗ operator turns a p- short: form into a (d−p)-form in d-dimensional space: d F = 0, d ∗F = ∗J. (10) √ −g The homogeneous equation, or “Bianchi identity” (∗Ω) =  ν1...νp Ω . µ1...µd−p p! µ1...µd−p ν1...νp is as usual a direct consequence of working with a (4) gauge potential. In p-form language it follows from Here g = det(gµν), and the  symbol with all F = d A and Eq. (2). lower indices is defined to be ±1 or 0 as usual, with indices raised by the inverse metric gµν. 2 in d dimensions • Integration: A p-form can be integrated over a p-dimensional manifold. In local coordinates, The d-dimensional Clifford algebra can be defined Z Z p in analogy to the four-dimensional case. It is repre- Ω = d x Ω0 1 ··· (p−1) 0 d−1 M sented by d Dirac matrices Γ ,..., Γ such that Z (5) p √ = d x −g (∗Ω) . {Γµ, Γν} = 2 ηµν 1 . (11)

1 A representation of the Lorentz algebra so(d−1, 1) It turns out that one may consistently impose is then furnished by this condition on Dirac spinors if d ≡ 0, 1, 2, 3, 4 mod 8. One may impose it on Weyl spinors only if i Σµν = − [Γµ, Γν] . (12) d ≡ 2 mod 8. 4 From these considerations we obtain the num- In d = 2, a possible choice of Dirac matrices is ber of real degrees of freedom f in the minimal representation in d dimensions:  0 1   0 1  Γ0 = , Γ1 = . (13) d 2 3 4 5 6 7 8 9 10 11 12 −1 0 1 0 f 1 2 4 8 8 16 16 16 16 32 64 From this one can iteratively construct the Γµ for even d: 3 Why d = 11 is special  −1 0  Γµ = Γeµ ⊗ , (µ = 0, . . . , d − 3), 0 1 Using the above table we can deduce that the  0 1   0 −i  maximal dimension for is d = 11. Γd−2 = 1 ⊗ , Γd−1 = 1 ⊗ , 1 0 i 0 The argument goes as follows: For arbitrary d we d−4 (14) can always compactify on a torus T to d = 4, which preserves all . Supersym- where Γeµ are the Dirac matrices in d−2 dimensions, metry transformations are generated by spinorial and 1 is the 2d/2−1 × 2d/2−1 unit . The di- supercharges. A d = 4 supercharge has f = 4 de- mension of the Dirac representation is thus 2d/2, grees of freedom; a d = 11 supercharge has f = 8·4 and there are 2d/2+1 real degrees of freedom con- degrees of freedom, i.e. it corresponds to 8 distinct A tained in a 2d/2-dimensional complex Dirac spinor. supercharges {Q }, or N = 8 SUSY, after torus compactification to d = 4. A hypothetical d ≥ 12 5 The analogue of γ in d = 4 is supercharge would correspond to at least N = 16 SUSY in d = 4. We will show that such a large Γ = id/2−1Γ0Γ1 ··· Γd−1 . (15) number of d = 4 supercharges would lead to 4d Γ has the properties fields with spin > 2, and comment on why this is unacceptable. 2 1 µ µν (Γ) = , {Γ, Γ } = 0, [Γ, Σ ] = 0 . (16) In d = 4 the supercharges are written as two- component complex Weyl spinors which satisfy the The eigenvalues of Γ are ±1. Dirac spinors in even algebra (in the absence of central charges) d are reducible since they admit a decomposition into two Weyl spinors of opposite Γ eigenvalues, or A µ A {Qα , QβB˙ } = 2σ ˙ Pµ δB, chiralities: The Dirac representation splits into two αβ Weyl representations of dimension 2d/2−1. {Q, Q} = {Q, Q} = [P,Q] = [P, Q] = [P,P ] = 0 (19) If d is odd, one obtains the Clifford algebra by taking the Dirac matrices of the (d−1)-dimensional Here Pµ is the four-momentum, α and β˙ are Weyl d Clifford algebra and adding Γ = Γ. The Dirac rep- spinor indices, A and B label the supercharges, and (d−1)/2 resentation of the Lorentz algebra is then 2 - we have suppressed all indices in the second line. dimensional and irreducible. Consider now a massless supermultiplet, P 2 = In certain dimensions it is possible to fur- 0. In a reference frame where P = (−E, 0, 0,E), ther halve the independent degrees of freedom of the first line of Eq. (19) becomes a spinor by imposing the Majorana condition  4 E 0  {QA, Q } = δA . (20) ζ∗ = Bζ (17) α βB˙ 0 0 B

A where B is some product of Dirac matrices satisfy- The operators Q2 and Q2˙A are represented by zero ing in this frame since they anticommute with every- µν −1 µν∗ A B Σ B = −Σ . (18) thing. The operators Q1 lower the helicity of any

2 1 state by 2 .A N = 8 massless multiplet contains a Here = d A3 is the U(1) field strength 4-form. spin-2 field (a ); its two polarisation states The Chern-Simons term is gauge invariant up to |h+2 i and |h−2 i with helicities ±2 are related by a total derivative, since under a gauge transforma- A repeatedly acting with the Q1 : tion Eq. (22) it transforms as 1 8 Z |h−2 i ∼ Q1 ··· Q1 |h+2 i . (21) A3 ∧ F4 ∧ F4 A ficticious massless multiplet of N = 16 Z Z → A ∧ F ∧ F + d Λ ∧ F ∧ F (24) SUSY in d = 4 would have to contain a spin- 3 4 4 2 4 4 4 field. It is often stated that interacting theo- Z Z = A ∧ F ∧ F + d (Λ ∧ F ∧ F ) . ries with fields of spin > 2 cannot be consistently 3 4 4 2 4 4 coupled to , although recently some excep- The last equality follows from the product rule and tions may have been discovered (involving an infi- from d F = 0 nite number of fields, and on curved backgrounds). 4 Barring such exotic constructions, the maximal su- The terms in the action involving Ψ are some- persymmetry in d = 4 would be N = 8, and the what more complicated. They are however also maximal dimension for supersymmetry d = 11. fully fixed by local supersymmetry and gauge in- variance. The above arguments can also be applied, with slight modifications, to massive multiplets, and generalized to the case non-vanishing central 5 Type IIA in d = 10 charges.

When compactifying d = 11 SUGRA on S1 and 4 d = 11 supergravity truncating to the massless spectrum, one obtains type IIA SUGRA in d = 10. The number of real supercharges is still 32, contained in two Majorana- In d = 11 there is a unique supergravity theory; it Weyl spinors of opposite chiralities. The field con- contains 32 real supercharges, or equivalently a d = tent is 11 Majorana spinor supercharge. The gravitational multiplet is the unique supersymmetry multiplet. • the graviton and its gravitino superpartner, It contains • a real scalar φ and its dilatino super- • the graviton, partner ψ, • a gravitino Ψ, • a 2-form gauge potential B2 with field strength H = d B , • and a 3-form A3 . 3 2

• a 1-form gauge potential C1 and a 3-form There is a U(1) gauge symmetry under which the gauge potential C3 with field strengths F2 = gravitino is charged and for which A3 is the gauge d C1 and F4 = d C3. Alternatively, one may field. It transforms as dualize the field strengths to obtain F6 = ∗F4

A3 → A3 + d Λ2 (22) and F8 = ∗F2, with “magnetic” gauge poten- tials C5 and C7, such that d C5 = F6 and with Λ2 any 2-form, analogous to Eq. (6). d C7 = F8. The formulations in terms of ei- The bosonic action up to two derivatives con- ther C1 and C7 are equivalent, and similarly tains the usual Einstein-Hilbert term along with a for C3 and C5. kinetic term and a Chern-Simons term for A3, • There is a version, so-called massive type IIA, 1 Z √ in which also a 0-form field strength G is al- S ⊃ d11x −g R 0 2 κ2 lowed. G0 does not descend from any gauge 1 Z 1 Z potential and does not appear in dimensional + F ∧ ∗F − A ∧ F ∧ F . 2 κ2 4 4 12 κ2 3 4 4 reduction (but it does appear in theory). (23) We will not discuss this theory any further.

3 The origin of these fields can be understood C3, C5, and C7 respectively. For instance, the ac- from torus compactification (cf. the very first sem- tion in the presence of a 2- of tension T and inar on Kaluza-Klein theory). C1 originates from with world-volume W contains a term the gravitational multiplet in d = 11, just as the Z original KK theory had a massless photon 1-form. S = T C3 + ..., (29) The dilaton we are also familiar with; essentially, W it parameterizes the radius of the circle. B2 and the analogue of Eq. (8). C3 descend from the A3 of the d = 11 SUGRA: B2 1 is the zero mode of A3 with one leg along the S direction, while C3 is the zero mode of A3 with all 6 Type IIB in d = 10 legs in the uncompactified dimensions.

The gauge variation for C3 takes a slightly non- There is a second maximal supergravity in ten di- standard form: mensions which cannot be (directly) obtained from d = 11. It contains 32 real supercharges in two C1 → C1 + dΛ0, Majorana-Weyl spinors of the same chirality. The B2 → B2 + dΛ1, (25) field content of the gravitational supermultiplet is C3 → C3 + dΛ2 + Λ0 ∧ H3 • the graviton and the gravitino, with Λ0,1,2 arbitrary. It is convenient to define the invariant field strength • a real scalar dilaton φ and its dilatino super- partner ψ, Fe4 = F4 − C1 ∧ H3 . (26) • a 2-form B2 with field strength H3 = d B2, F is not a closed form, but satisfies a non-standard e4 • a 0-form C , a 2-form C and a 4-form C with Bianchi identity: 0 2 4 field strengths F1 = d C0, F3 = d C2 and F5 = d C4. dFe4 = −F2 ∧ H3 . (27)

With this definition the bosonic action is Motivated again by non-standard gauge trans- formation laws, it is convenient to define the com- Z 1 10 √ −2φ 2 binations S = 2 d x −g e R + 4(∂φ) 2κ10 Z Fe3 = F3 − C0 ∧ H3, 1  −2φ  − e H3 ∧ ∗H3 + F2 ∧ ∗F2 + Fe4 ∧ ∗Fe4 (30) 4κ2 1 1 10 Fe5 = F5 − C2 ∧ H3 + B2 ∧ F3 . 1 Z 2 2 − B ∧ F ∧ F . 4κ2 2 4 4 10 The field strength combination Fe5 is restricted to (28) be self-dual: Fe5 = ∗Fe5 (31) Again the Chern-Simons term in the last line is gauge invariant up to a total derivative. (a restriction which cannot be derived from an ac- tion, but must be imposed on the equations of mo- Type IIA SUGRA is the low-energy limit of tion as a further constraint). The bosonic action type IIA . Type IIA string theory reads contains several higher-dimensional objects which Z couple to, and provide sources for, the higher p- 1 10 √ −2φ 2 S = 2 d x −g e R + 4(∂φ) form gauge fields. Classically, this is analogous 2κ10 to electrons coupling to the electromagnetic gauge Z 1  −2φ field, which we described by integrating the gauge − 2 e H3 ∧ ∗H3 + F1 ∧ ∗F1 4κ10 (32) potential over the electron worldline. Type IIA  string theory contains D- with D = 0, 2, 4, 6. + Fe3 ∧ ∗Fe3 + Fe5 ∧ ∗Fe5 These are objects with D spatial dimensions and 1 Z − C ∧ H ∧ F . D + 1-dimensional world-volumes, coupling to C , 2 4 3 3 1 4κ10

4 Type IIB supergravity is the low-energy limit 7 Type I in d = 10 of type IIB string theory, which contains a fun- damental string and various D-branes of odd D Type I is not a maximal supergravity since it only coupling to the gauge fields, similar to the even contains 16 supercharges in a single Majorana- D-branes of type IIA . Weyl spinor. It is obtained by compactifying d = 1 The action exhibits an interesting symmetry 11 supergravity on S /Z2 and adding gauge de- under modular transformations. We perform a grees of freedom. There are two multiplets. The Weyl rescaling to the Einstein frame metric Gµν = gravitational multiplet contains −φ/2 e gµν and define • the graviton and the gravitino, τ = C + ie−φ(the “axio-dilaton”) (33) 0 • the dilaton and the dilatino, and • a 2-form B2 with field strength H3.

G3 = F3 − τH3 . (34) The vector multiplet contains The action Eq. (32) becomes (with the Einstein frame metric used everywhere) • a 1-form non-abelian gauge potential A1 in the of some gauge group Z √   1 10 ∂τ∂τ S = 2 d x −G R − 2 • and its gaugino superpartner. 2κ10 2(Im τ) 1 Z 1 G ∧ ∗G 1  3 3 While it would seem at first sight that this leaves − 2 + Fe5 ∧ ∗Fe5 (35) 2κ10 6 Im τ 2 much freedom to engineer theories with various Z 1 C4 ∧ G3 ∧ G3 vector multiplets and large gauge symmetries, it − 2 . 8 i κ10 Im τ turns out that the theory is in fact tightly re- stricted by cancellation. The classic re- sult of Green and Schwarz in the 1980s shows that Recall that SL (2, R) is the group of real for gravitational anomalies to cancel, the dimen- 2 × 2 matrices with determinant one. A matrix  a b  sion of the gauge group should be 496. Cancella- ∈ SL (2, ) acts on the fields as c d R tion of gauge anomalies then restricts the possible gauge groups to either SO(32) or ×E8 (the cases 248 496 aτ + b E8 ×U(1) and U(1) , while not covered by this τ → , cτ + d argument, are believed not to admit any UV com- pletion). G3 G3 → , (36) cτ + d Both SO(32) and E8 × E8 gauged supergravi- Fe5 → Fe5, ties have UV completions in terms of string theo- ries. The SO(32) can arise as the low-energy limit Gµν → Gµν . of either or SO(32) heterotic The action Eq. (35) is invariant under this trans- string theory. The E8 × E8 case is the low-energy formation. This is because limit of E8 × E8 .

aτ + b ∂τ ∂τ → ∂ = , cτ + d (cτ + d)2 8 Compactification (37) Im τ Im τ → . |cτ + d|2 A trivial exact solutions to the field equations is always given by flat d = 10 or d = 11 Minkowski space with all other fields vanishing. The invariance under the SL(2, Z), where a, b, c, d are , lifts to “S-duality” in For various reasons (among others, we know type IIB string theory, which might play some role only of four non-compact dimensions), it is inter- in the upcoming talks. esting to look at solutions of the form M×C, where

5 M is an m-dimensional non-compact manifold and Together with the difficulties of obtaining chiral C is a (10 − m)- or (11 − m)-dimensional compact fermions which was also mentioned in the KK the- manifold. We recapitulate some of the statements ory lecture, it seems that the KK idea cannot made in talk on Kaluza-Klein theory: be straightforwardly applied to 10-dimensional or 11-dimensional supergravity to produce a realistic • One obtains a field theory on M. model of the real world. String theory does how- ever provide ways to avoid these obstacles. • The degrees of freedom are the KK modes, Because they are of some interest for the up- i.e. (for bosonic fields) the eigenfunctions of coming talks (especially on moduli stabilization 2 the covariant Laplacians ∇ on C. and on AdS/CFT), we close with an interesting class of exact solutions of type IIB SUGRA on • Even if the metric on C (and thus the Laplace homogeneous spaces. Take M and C to be so- and Dirac operators) is not known explicitly, lutions to the d = 5 vacuum Einstein equations information about the massless spectrum can in Lorentzian and Riemannian signatures respec- be gained from topology. tively; M negatively and C positively curved. The most important examples have M =AdS , the • If C has G, there will be a 5 unique d = 5 Lorentzian manifold of topology 5 gauge symmetry with gauge group G. R and constant negative curvature. One obtains a solution to the equations of motion provided that The last point motivates a search for a solution there is also some 5-form flux in the background. with some C whose isometry group is large enough Explicitly, with m, n . . . indices on M and α, β . . . to contain the Standard Model. Note however that, indices on C: upon compactification on a product space, the cur- √ vature scalar splits as follows: Femnopq = −r −gM mnopq, √ Feαβγδε = r −gC αβγδε, R = RM + RC + ... (38) 2 Rmn = 4 r gmn, (40) 2 so the Einstein-Hilbert term in d dimensions be- Rαβ = −4 r gαβ, comes, after carrying out the integral over C, all other fields = 0 . Z 1 d √ d x −g R In the simplest example, C is C = S5 = 2κ2 d (39) SO(6)/SO(5) a 5-sphere of radius r, and M =AdS 1 Z √ 5 m with scale 1/r. This is the near-horizon geometry = 2 d x −gM (RM + Λ) + ... 2κm for a stack of D3 branes (recall that these indeed source the 5-form gauge field). Another important where Λ ∼ R R is a cosmological constant. One C C example has C = T 1,1 = (SU(2) × SU(2))/U(1). faces two options: While these solutions have no immediate relevance for 4d physics, they are extremely important in the • Calabi-Yau compactification on manifolds C AdS/CFT correspondence. with RC = 0, but whose Riemann tensor is still non-vanishing – otherwise one has just a torus compactification with too much SUSY. References These manifolds have no isometries and there- fore no gauge symmetry. • Differential forms: Nakahara, Geometry, Topology and Physics • Compactifications with large isometry groups, e.g. on homogeneous spaces C = G/H with G • SUSY in d = 4: Wess & Bagger, Supersymme- and H compact Lie groups. C then has the try and Supergravity, Chapter 2 isometry group G. However, for small com- pactification radii the cosmological constant • Everything and more: Polchinski, String The- will be unacceptably large. ory Vol. II, Appendix B

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If you think you’ve found a sign error, a missing prefactor, or a gaping hole in the reasoning, you’re probably correct.

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