Exact solutions in supergravity

Transcription

1 Exact solutions in supergravity James T. Liu 25 July 2005 Lecture 1: Introduction and overview of supergravity Lecture 2: Conditions for unbroken supersymmetry Lecture 3: BPS black holes and branes Lecture 4: The LLM bubbling AdS construction Taipei Summer Institute Lecture 1

2 Lecture 1 Outline A quick introduction to supersymmetry The supersymmetry algebra Local supersymmetry as supergravity Eleven dimensional supergravity Field content, Lagrangian and transformation rules Ten dimensional supergravities The low energy limit of superstring theory A look at IIA, IIB and N = 1 supergravity JTL Lecture 1 1

3 Supersymmetry Supersymmetry relates bosons with fermions Fermion Boson or δ(fermion) = (Boson)ɛ, where ɛ = spinor parameter of transformation δ(boson) = ɛ(fermion) For example, in the Wess-Zumino model (on-shell formulation) L = 1 2 A2 1 2 B2 1 2 iχγµ µ χ (two real scalars A, B and one Majorana fermion χ) δa = 1 2 iɛχ, δb = 1 2 ɛγ 5χ, δχ = 1 2 γµ µ (A iγ 5 B)ɛ JTL Lecture 1 2

4 Invariance of the action For the Wess-Zumino model, we can check δl = δl B + δl F = µ A µ δa µ B µ δb iχγ µ µ δχ = 1 2 iɛ µ (A + iγ 5 B) µ χ 1 2 iχγµ µ γ ν ν (A iγ 5 B)ɛ = 1 2 iɛ (A + iγ 5B)χ 1 2 iχ (A iγ 5B)ɛ = 0 (up to integration by parts) Furthermore, the commutator of two transformations yields [δ 1, δ 2 ]A = 1 2 ξµ µ A [δ 1, δ 2 ]B = 1 2 ξµ µ B [δ 1, δ 2 ]χ = 1 2 ξµ µ χ 1 4 γµ ξ µ γ µ µ χ where ξ µ = iɛ 2 γ µ ɛ 1 JTL Lecture 1 3

5 The supersymmetry algebra An extension of the Poincaré algebra In D = 4: M µν = Lorentz generators P µ Q i α = Translations = supertranslations where i = 1,..., N with [M µν, M ρσ ] = i(η µρ M νσ η µσ M νρ + η νσ M µρ η νρ M µσ ) [M µν, P ρ ] = i(η µρ P ν η νρ P µ ) [M µν, Q i α ] = 1 2 i(γ µν) αβ Q β i {Q i α, Qj β } = 2(γµ C) αβ δ ij P µ + C αβ X [ij] + (γ 5 C) αβ Y [ij] The central charge Z = X + iy commutes with all generators and plays an important role in extended supersymmetry JTL Lecture 1 4

6 Extended supersymmetry For D = 4 there is an upper limit on N Consider massless representations of supersymmetry (given by helicities) N = 1: (0, 1 2 ) Wess-Zumino; (1 2, 1) vector; (3 2, 2) graviton N = 2: ( 1 2, 0, 1 2 ) hypermatter; (0, 1 2, 1) vector; (1, 3 2, 2) graviton N = 4: ( 1, 1 2, 0, 1 2, 1) vector; (0, 1 2, 1, 3 2, 2) graviton N = 8: ( 2, 3 2, 1, 1 2, 0, 1 2, 1, 3 2, 2) graviton Spins less than or equal to 2 give a restriction and N = 8 must include gravity N 8 in D = 4 JTL Lecture 1 5

7 Local supersymmetry is supergravity In fact, gravity shows up naturally once we impose local supersymmetry ɛ ɛ(x) Consider the action of two supersymmetries [δ 1, δ 2 ](field) = [ɛ 1 Q, ɛ 2 Q](field) = ξ µ (x) µ (field) + where ξ µ (x) = iɛ 2 γ µ ɛ 1 This is a general coordinate transformation, and the gravitino ψ µ (x) acts as a spin- 3 2 gauge field JTL Lecture 1 6

8 Simple supergravity in D = 4 Constructed by Freedman, van Nieuwenhuizen and Ferrara and Deser and Zumino in 1976 To lowest order in fermions e 1 L = R iψ µγ µνσ ν ψ σ with δψ µ = µ ɛ δe µ α = 1 4 iɛγα ψ µ We can check e 1 δl = (R µν 1 2 gµν R)δg µν + iψ µ γ µνσ ν δψ σ = 1 2 i(rµν 1 2 gµν R)ɛγ µ ψ ν + iψ µ γ µνσ ν σ ɛ = 1 2 i(rµν 1 2 gµν R)ɛγ µ ψ ν iψ µγ µνσ R νσλη γ λη ɛ = 1 2 i(rµν 1 2 gµν R)ɛγ µ ψ ν i(rµν 1 2 gµν R)ψ µ γ ν ɛ = 0 JTL Lecture 1 7

9 A general coordinate transformation In addition, we verify explicitly that the commutator of two supersymmetries yields a general coordinate transformation [δ 1, δ 2 ]e µ α = 1 4 i(ɛ 2γ α µ ɛ 1 ɛ 1 γ a µ ɛ 2 ) = µ ξ a where ξ a = 1 4 iɛ 2γ a ɛ 1 This implies that [δ 1, δ 2 ]g µν = 2 (µ ξ ν) = µ ξ ν + ν ξ µ which is the expected result Closure of the algebra on fermions is more involved, as it involves (fermi) 2 terms in δψ µ JTL Lecture 1 8

10 An upper limit on spacetime dimension In four dimensions, we found the restriction N 8 We can also consider the restriction on spacetime dimension D Need to match fermions and bosons in D dimensions Bosons Fermions scalar 1 Dirac spinor D/2 complex vector D 2 Dirac gravitino 1 n-form 1 graviton 2 ` n D 2 (D 1)(D 2) 1 2 (D 3) 2 D/2 complex Eventually the fermion degrees of freedom grow too large For spins less than or equal to 2, the maximum is D = 11, N = 1 JTL Lecture 1 9

11 Eleven dimensional supergravity The maximum dimension for (conventional) supergravity is eleven A relatively simple theory with field content g MN 44 metric A MNP 84 3-form potential ψ M 128 gravitino This can be related to the D = 4, N = 8 theory by dimensional reduction Kaluza-Klein reduction on T 7 In fact, the arguments for maximum N and maximum D are closely related JTL Lecture 1 10

12 D = 11 supergravity: the setup Eleven dimensional supergravity was constructed in 1976 by Cremmer, Julia and Scherk For fermions coupled to gravity, introduce a vielbein and spin-connection M, N, P,... = curved indices, A, B, C,... = flat indices Work in signature ( + +) with {Γ A, Γ B } = 2η AB, ψ M = ψ T M C 1, C 1 Γ A C = Γ T A The spin-connection is given by ω M AB = ω M AB (e) + K M AB with contorsion K M AB = 1 16 (ψ NΓ M ABNP ψ P + 4ψ M Γ [A ψ B] + 2ψ A Γ M ψ B ) JTL Lecture 1 11

13 D = 11 supergravity: the Lagrangian The supergravity Lagrangian is given by e 1 L 11 = R(ω) 1 2 4! F 2 MNP Q ( 1 4!4!3! εmnp QRST UV W X F MNP Q F RST U A V W X ) 1 2 iψ MΓ MNP N ( 1 2 (ω + ˆω))ψ p 1 8 4! i(ψ MΓ MNP QRS ψ N + 12ψ P Γ QR ψ S )( 1 2 (F + ˆF )) P QRS where the supercovariant quantities are ˆω AB M = ω AB M ψ NΓ ABNP M ψ P ˆF MNP Q = F MNP Q iψ [MΓ NP ψ Q] The bosonic Lagrangian is simple (in a form notation) L 11 = R F (4) F (4) F (4) F (4) A (3) JTL Lecture 1 12

14 D = 11 supergravity: equations of motion The above Lagrangian gives rise to the equations of motion R MN (ˆω) 1 2 g MNR(ˆω) = 1 12 ( ˆF MP QR ˆFN P QR 1 8 g MN ˆF 2 ) Γ MNP ˆ N (ˆω)ψ P = 0 M (ˆω) ˆF MNP Q ( 1 4!4! εnp QRST UV W XY ˆFRST U ˆFV W XY ) = 0 where the supercovariant derivative is ˆ M = M (Γ M NP QR 8δ N M ΓP QR )F NP QR Often we are only concerned with the bosonic equations R MN = 1 12 (F MP QRF N P QR 1 12 g MNF 2 ) d F (4) F (4) F (4) = 0 JTL Lecture 1 13

15 D = 11 supergravity: transformation laws The above system has a large set of symmetries general covariance SO(1, 10) local Lorentz symmetry abelian 3-form gauge symmetry δa (3) = dλ (2) N = 1 local supersymmetry δe A M = 1 4 iɛγa ψ M δa MNP = 3 4 iɛγ [MNψ P ] δψ M = ˆ M (ˆω)ɛ Closure of supersymmetry is expressed as [δ 1, δ 2 ] = δ g.c. + δ l.l. + δ gauge + δ susy + e.o.m. JTL Lecture 1 14

16 Connection to string theory We will return to eleven dimensional supergravity But note that supergravity by itself cannot be the ultimate theory UV divergences of quantum gravity not really cured by supersymmetry May be viewed as the low energy limit of superstring theory (M-theory) Useful as an effective theory below the Planck scale String theory gives rise to a tower of massive states 1 4 α M 2 = 0, 1, 2,... (closed string) by integrating out the massive fields, we are left with a theory of massless fields spin-2 graviton supersymmetry This must be supergravity JTL Lecture 1 15

17 Ten dimensional supersymmetry There are five consistent supersymmetric string theories and three supergravities in D = 10 SUSY String Supergravity 32 Type IIA N = 2A (non-chiral) Type IIB N = 2B (chiral) 16 E 8 E 8 Heterotic N = 1 SO(32) Heterotic N = 1 SO(32) Type I N = 1 D=11 In fact, these D = 10 string theories can be related to each other and to D = 11 supergravity through dualities HE HO I IIA IIB JTL Lecture 1 16

18 Massless string states In the RNS formulation, spacetime supersymmetry is obtained form worldsheet supersymmetry and the GSO projection The open string tower of states 1 4 α M 2 Bosonic Fermionic.. 1 ψ µ 1/2 ψν 1/2 ψρ 1/2 NS +, ψ µ 3/2 NS + α µ 1 R +, α µ 1 R ψ µ 1/2 αν 1 NS + ψ µ 1 R, ψ µ 1 R ψ µ 1/2 ψν 1/2 NS, α µ 1 NS 0 ψ µ 1/2 NS + R +, R. 1 2 NS After GSO, the massless states are ψ µ 1/2 NS + and R + 8 v + 8 s Physical states of D = 10, N = 1 super-yang-mills JTL Lecture 1 17

19 Massless string states For the closed string, we take two copies (left and right movers) Type IIA (non-chiral combination) (8 v + 8 s ) L (8 v + 8 c ) R = ( ) NSNS + (8 v + 56 t ) RR +(8 c + 56 s + 8 s + 56 c ) RNS+NSR Bosonic states: (g µν, B µν, φ) NSNS + (F (2), F (4) ) RR Type IIB (chiral) (8 v + 8 s ) L (8 v s) R = ( ) NSNS + ( t ) RR +(8 c + 56 s + 8 c + 56 s ) RNS+NSR Bosonic states: (g µν, B µν, φ) NSNS + (F (1), F (3), F + (5) ) RR The NSNS sector is identical (also for the Heterotic string) JTL Lecture 1 18

20 Type IIA supergravity The IIA theory may be obtained by reduction from eleven dimensions The Kaluza-Klein idea Split your spacetime M D = M d T D d, x M = (x µ, y m ) Assume fields are independent of the internal coordinates Φ(x M ) = ϕ(x µ ) This may be generalized to non-trivial internal manifolds as well T D d X D d and Φ(x M ) = ϕ i (x µ )f i (y m ) where f i (y m ) are harmonics on X JTL Lecture 1 19

21 Type IIA supergravity: reduction from D = 11 Consider the bosonic sector of D = 11 supergravity g MN g µν, g µ 11, g A MNP A µνρ, A µν 11 These are the fields of IIA supergravity For the actual reduction, write ds 2 11 = e 2 3 φ ds e4 3 φ (dy + C (1) ) 2 A (3) = C (3) + B (2) dy This yields the ten-dimensional fields (g µν, B (2), φ) + (C (1), C (3) ) JTL Lecture 1 20

22 Type IIA supergravity: the bosonic Lagrangian We insert the above reduction ansatz into the D = 11 Lagrangian to obtain L 11 = R F (4) F (4) F (4) F (4) A (3) L IIA = e 2φ [R 1 + 4dφ dφ 1 2 H (3) H (3) ] 1 2 F (2) F (2) 1 e 2F (4) F e (4) 1 2 F (4) F (4) B (2) where H (3) = db (2), F (2) = dc (1), F (4) = dc (3) and ef (4) = F (4) C (1) H (3) JTL Lecture 1 21

23 D = 10, N = 1 supergravity Type IIA supergravity admits a further truncation to N = 1 We obtain the gravity sector of the Heterotic string by removing the RR sector (C (1), C (3) ) 0 L N =1 = e 2φ [R 1 + 4dφ dφ 1 2 H (3) H (3) ] This theory is anomalous, but can be cured by adding an E 8 E 8 or SO(32) gauge sector The result is the effective Lagrangian for the Heterotic string L het = e 2φ [R 1 + 4dφ dφ 1 2 e H (3) e H (3) + α Tr v F (2) F (2) ] where e H (3) = db (2) 1 4 α ω Y M (3) with ω Y M (3) = Tr v (A (1) da (1) A3 (1) ) JTL Lecture 1 22

24 Type IIB supergravity The final possibility in ten dimensions (N = 2B) cannot be obtained from dimensional reduction This theory has a self-dual field strength F (5) = F (5), and hence does not admit a covariant action formulation Self-dual fields also show up in six-dimensional N = (1, 0) and N = (2, 0) supergravity However, we can write down an action that gives rise to all equations of motion except the self-dual condition (which must subsequently be imposed by hand) Recall that we have the fields (g µν, B (2), φ) + (C (0), C (2), C (4) ) JTL Lecture 1 23

25 Type IIB supergravity: the bosonic Lagrangian The IIB Lagrangian has the form L IIB = e 2φ [R 1 + 4dφ dφ 1 2 H (3) H (3) ] 1 2 F (1) F (1) 1 e 2F (3) F e (3) 1 e 4F (5) F e (5) 1 2 C (4) H (3) F (3) where H (3) = db (2), F (1) = dc (0), F (3) = dc (2), F (5) = dc (4) and ef (3) = F (3) C (0) H (3), e F(5) = F (5) 1 2 C (2) H (3) B (2) F (3) In addition, we have to impose F (5) self-duality by hand on the solution JTL Lecture 1 24

26 Supergravities in ten and eleven dimensions A summary of the four supergravities we have considered D = 11, N = 1: g MN A (3) D = 10, N = 2A: g µν φ C (1) B (2) C (3) D = 10, N = 2B: g µν φ, C (0) B (2), C (2) C (4) D = 10, N = 1: g µν φ A a (1) B (2) JTL Lecture 1 25

27 Why study supergravity? Backgrounds for string compactification Calabi-Yau manifolds, flux compactifications Low energy dynamics of string theory String solutions, black holes, D-brane probes of geometry AdS/CFT, gravity duals to supersymmetric gauge theories Phenomenology and string cosmology Also braneworlds and large extra dimensions JTL Lecture 1 26

28 Supersymmetric backgrounds Many of the exact results in string theory and AdS/CFT depend on understanding backgrounds with some fraction of unbroken supersymmetry, ie BPS configurations use of powerful non-renormalization theorems {Q, Q} P + Z M Z Single particle representations with M = Z have discretely fewer states than those with M > Z Since we do not expect representations to change discontinuously, these states are protected, and are know as BPS states Allows us to investigate connections between strong and weak coupling JTL Lecture 1 27

29 Next time In the next lecture, we will examine the conditions for unbroken supersymmetry, and will look at some examples of supersymmetric vacua preserving various fractions of supersymmetry In the third lecture, we will look at the supersymmetry of BPS branes and AdS vacua and will introduce the notion of generalized holonomy as a means of classifying supersymmetric backgrounds with fluxes In the final lecture, we will review the recent construction of bubbling 1/2 BPS solutions of IIB theory by Lin, Lunin and Maldacena (LLM) JTL Lecture 1 28