Supergravity 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 Spinors 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 spinor 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 supersymmetry 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 supersymmetries. 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 matrix. 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 F4 = d A3 is the U(1) field strength 4-form. spin-2 field (a graviton); 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 gravity, 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 dilaton 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 .