Quick Review on Superstrings

Transcription

1 Quick Review on Superstrings Hiroaki Nakajima (Sichuan University) July 24, prespsc2016, Chengdu

2 DISCLAIMER This is just a quick review, and focused on the introductory part. Many important topics are not covered (some of them are briefly mentioned), for example conformal field theory (OPE, bosonization, vertex op, modular invariance, etc.) string duality (S,T,U-duality, open-closed duality) D-branes M-theory AdS/CFT, etc. 1

3 REFERENCES (books) Green-Schwarz-Witten Superstring theory vol. 1, 2 Polchinski String Theory vol. 1, 2 Becker-Becker-Schwarz String Theory and M-theory Dine Supersymmetry and String Theory Ibáñez-Uranga String Theory and Particle Physics Blumenhagen-Lust-Theisen Basic Concepts of String Theory West Introduction to Strings and Branes, etc. As well as these books, there many good lecture notes in arxiv. 2

4 CONTENTS PART 1: First quantization of bosonic sting and its spectrum PART 2: First quantization of fermionic sting RNS formalism GSO projection and spacetime supersymmetry The five superstring theories PART 3: Low energy SUGRA and CY compactification supergravity as effective field theory of superstring Calabi-Yau compactification 3

5 1. First quantization of bosonic string 1-1. bosonic string Action of a point particle: (mass) (length of worldline) S p.p. = m dτ η µν dx µ dτ dx ν dτ Action of a string: (string tension) (area of worldsheet) S NG = 1 2πα d 2 σ det (Nambu-Goto action, - ) 4 ( ) X η µ X ν µν σ α σ β

6 The Nambu-Goto action contains the square root, which is a bit hard to study. It is convenient to rewrite the action by introducing new variables: worldsheet metric h αβ (τ, σ) Polyakov action S P = 1 4πα d 2 σ hh αβ X µ X ν η µν σ α σ β Note that h αβ (τ, σ) is not dynamical and its equation of motion gives T αβ 2 ( X µ X µ α σ α σ β 1 ) 2 h αβh γδ Xµ X µ σ γ σ δ = 0 By substituting the above into the Polyakov action, it is reduced to the Nambu-Goto action. 5

7 Local symmetries of Polyakov action 1. diffeomorphism τ τ (τ, σ), σ σ (τ, σ) 2. Weyl transformation h αβ (τ, σ) Λ(τ, σ)h αβ (τ, σ) Using these symmetries, we can gauge-fix h αβ (σ) as h αβ (τ, σ) = η αβ = ( ) conformal gauge This name comes from the fact that the gauge-fixed action S g.f. = 1 4πα d 2 σ η αβ X µ X ν η µν σ α σ β is still invariant under the conformal symmetry. 6

8 Conformal symmetry σ + f + (σ + ), σ f (σ ), σ ± = τ ± σ Here f ± are arbitrary functions. Remark The generators of the conformal transformation is T αβ itself. the conformal gauge, the constraint T αβ = 0 becomes In T ++ = 1 α +X µ + X µ = 0, T = 1 α X µ X µ = 0 In string theory, conformal symmetry plays a role of the gauge symmetry, since it is a residual symmetry of (diffeo.) (Weyl) after the gauge fixing. 7

9 Equation of motion for gauge-fixed action ( ) 2 τ 2 2 σ 2 X µ = Solution = (left mover) + (right mover) 2 σ + σ Xµ = 0 X µ (τ, σ) = X µ L (σ+ ) + X µ R (σ ) Boundary condition ( < τ <, 0 σ l) [ ] σ=l Xµ σ δxµ σ=0 X µ σ δxµ X µ σ=l σ δxµ = 0 σ=0 Here we have assumed that b.c. for temporal direction is trivial. 8

10 1. periodic b.c. closed string X µ (τ, σ) = X µ (τ, σ + l) 2. Neumann b.c. open string X µ σ Xµ (τ, 0) = (τ, l) = 0 σ 3. Dirichlet b.c. open string attached to D-brane δx µ (τ, 0) = δx µ (τ, l) = 0 9

11 (Fourier) mode expansion of closed string (l = 2π) X µ = x µ + α p µ τ + i α 2 n 0 1 n (αµ ne in(τ+σ) + α µ ne in(τ σ) ) Here x µ, p µ : center of mass position and momentum (zero modes) α µ n, α µ n : amplitudes for oscillation modes Mode expansion of open string with Neumann b.c. (l = π) X µ = x µ + 2α p µ τ + i 2α n 0 (Dirichlet b.c. : p µ = 0, cos sin) 10 1 n αµ ne inτ cos nσ

12 Doubling trick We can extend the domain of σ from 0 σ π to 0 σ 2π as { X µ (τ, σ) for 0 σ π X µ (τ, σ) = X µ (τ, 2π σ) for π σ 2π The extended X µ s satisfy the closed string boundary condition. However they contain only a single mover, of course. This is called the doubling trick. 11

13 Quantization for constrained system 1. Old covariant quantization (Gupta-Bleuler formalism) 2. Light-cone (or non-covariant) quantization 3. BRST quantization Here we adopt the old covariant quantization. 12

14 Canonical commutation relations [ X µ (τ, σ), P ν (τ, σ ) ] = iη µν δ(σ σ ), P µ (τ, σ) = 1 2πα τx µ [ X µ (τ, σ), X ν (τ, σ ) ] = [ P µ (τ, σ), P ν (τ, σ ) ] = 0 Commutation relation for Fourier modes Here [x µ, p ν ] = iη µν, [α µ m, α ν n] = [ α µ m, α ν n] = mδ m+n,0 η µν α µ n>0, αµ n>0 : annihilation op. αµ n<0, αµ n<0 : creation op. α µ 0 = αµ 0 = α 2 pµ (closed) α µ 0 = 2α p µ (open) 13

15 Fock vacuum for left mover p µ 0; 0 = 0, α µ n 0; 0 = 0 for all n > 0 Fock space { {N µ,n }; k } {Nµ,n }; k = D 1 µ=0 (α n) µ N µ,n exp(ik µ x µ ) 0; 0 n=1 Remark The Fock space contains the states with negative norm, i.e. ghosts. example: α 1 0; 0 k 2 = 0; k 0; k = 1 14

16 Conformal symmetry and physical state conditon Mode expansion of generators L n e inσ+, L n e inσ T ++ = n L n = 1 2 T = n : α µ n mα mµ :, Ln = 1 2 : α µ n m α mµ : m m The symbol : : denotes normal ordering. In fact, only L 0 and L 0 are affected by the ordering as L 0 = 1 2 αµ 0 α 0µ + α µ nα nµ, L0 = 1 2 αµ 0 α 0µ + α µ n α nµ n=1 n=1 15

17 Virasoro algebra [ Lm, L n ] = (m n)lm+n + c 12 (m3 m)δ m+n,0 The constant c is called central charge, explicit calculation gives c = η µν η µν = D (dimension of spacetime) Physical state conditon (L 0 a) phys = L n phys = 0 for n > 0 The intercept a represents the normal ordering ambiguity of L 0. 16

18 Explicit calculation of central extension [L m, L n ] = 1 4 = 1 4 = 1 2 [ ] αm k α k, α n l α l k,l [ kα m k α n l δ k+l,0 + kα m k α l δ k+n l,0 k,l ] + (m k)α n l α k δ m k+l,0 + (m k)α l α k δ m k+n l,0 ] [kα m k α n+k + (m k)α m+n k α k k Now it is clear that the ordering ambiguity may happen only when m + n = 0. Otherwise it gives (m n)l m+n by shifting the label of the first sum by k k n. 17

19 m + n = 0 case [L m, L m ] = 1 ] [kα m k α m+k + (m k)α k α k 2 = 1 2 k k : α m k α m+k : k k m 1 k [ α µ m k, α ] m+k,µ + 1 (m k) : α k α k : + 1 (m k) [ α µ k 2 2, α ] kµ k k 1 [ = 2mL 0 + δ µ µ ] k(m k) 2 k m 1 k 1 = 2mL 0 + D (m 1)m(m + 1) c = D

20 Spurious states χ = ) (L n satisfying physical state condition (n > 0) properties orthogonal to any physical states χ phys = L n phys = 0 (n > 0) vanishing norm χ χ = L n>0 χ = 0 We require the equivalence relation for physical states phys phys if phys = phys + χ, which preserves the inner product. 19

21 No-ghost theorem The ghosts (negative norm states) can be removed from the physical Hilbert space (not only tree, but also loop level), if and only if Comment D = 26 (critical dimension), a = 1 The above can also be derived from other quantization scheme. light-cone quantization Lorentz algebra is closed if and only if D = 26 and a = 1. BRST quantization BRST charge Q must be nilpotent: Q 2 = 0, which holds if and only if D = 26 and a = 1. 20

22 Low-lying spectrum of open string (L 0 1) phys = 0: on-shell condition L 0 = α p 2 + N, N = Examples n=1 α nα µ nµ m 2 = 1 α (N 1) phys = 0; k, m 2 = 1 α : tachyon phys = ϵ µ α µ 1 0; k, m2 = 0 : massless vector All other states have positive (mass) 2 (α ) 1 and decouple from low energy physics. 21

23 Gauge invariance of masslass vector state L n phys = 0, phys phys + L n for n > 0 For massless vector state phys = ϵ µ α µ 1 0; k, the physical state conditions are reduced to k µ ϵ µ = 0, ϵ µ ϵ µ + λk µ which correspond to the gauge invariance of massless vector field µ A µ = 0, A µ A µ + µ λ The conditions imply that ϵ µ has transverse polarization only. In particular, the ghost state α 0 1 0; k is removed. 22

24 Low-lying spectrum of closed string On-shell contition and level-matching condition (L 0 1) phys = ( L 0 1) phys = 0 (L 0 + L 0 2) phys = (L 0 L 0 ) phys = 0 On-shell condition L 0 + L 0 = α p 2 + N + Ñ m2 = 1 α (N + Ñ 2) Level-matching condition (invariance under σ-translation) (L 0 L 0 ) phys = 0 N = Ñ for physical states 23

25 Examples phys = 0, 0; k, m 2 = 2 α : tachyon phys = ζ µν α µ 1 αν 1 0, 0; k, m 2 = 0 : massless tensor For the massless tensor, physical state condition is reduced to k µ ζ µν = k ν ζ µν = 0, ζ µν ζ µν + k µ λ ν + k ν λ µ For the symmetric part ζ µν = ζ νµ, the second condition becomes ζ µν ζ µν + k µ λ ν + k ν λ µ h µν h µν + µ λ ν + ν λ µ This is the gauge symmetry of graviton! (at linearized level) 24

26 The physical state condition implies that the transverse modes ζ ij of the polarization tensor are only physical, which can be decomposed as graviton ζ ij = ζ ji, δ ij ζ ij = 0, B-field dilaton ζ ij = ζ ji ζ ij = ζδ ij All states other than tachyon or massless have positive (mass) 2 (α ) 1 and decouple from low energy physics. 25

27 2. First quantization of fermionic string 2-1. RNS formalism Bosonic string does not contain fermions in the spectrum. There are two or more ways to extend the theory to include fermions: Ramond-Neveu-Schwarz(RNS) fermionic string Green-Schwarz(GS) superstring Pure-spinor formalism [Berkovits], etc. Here we adopt RNS formalism. 26

28 Worldsheet action (supersymmetric extension of Polyakov action) [Brink-Di Vecchia-Howe], [Dezer-Zumino] S = 1 8π Here [ 2 d 2 σ e α hαβ α X µ β X µ + 2i Ψ µ ρ α α Ψ µ 2 i α χ αρ β ρ α Ψ µ β X µ 1 ] 4 χ αρ β ρ α Ψ µ χ β Ψ µ Ψ µ : worldsheet Majorana spinor and spacetime vector e a α : 2D vielbein(zweibein), η ab e a αe b β = h αβ, e = det e a α χ α : 2D gravitino, ρ a : 2D gamma matrices, ρ α = e α aρ a 27

29 Local symmetries 1. diffeomorphism 2. Weyl transformation (weight: [X µ ] = 0, [ψ µ ] = 1/2, [e a α] = 1, [χ α ] = 1/2) 3. local supersymmetry (set α = 2) δx µ = ϵψ µ, δψ µ = iρ α ϵ( α X µ Ψ µ χ α ), δe a α = 2i ϵρ a χ α, δχ α = α ϵ 4. super Weyl transformation δx µ = δψ µ = δe a α = 0, δχ α = iρ α η 28

30 Using those symmetries, we can choose the gauge e a α = δα a, χ α = 0 conformal gauge But we still have to concern the constraints T αβ = 2 α αx µ β X µ + i 2 Ψ µ ρ (α β) Ψ µ (trace) = 0, J α = 1 2 ρβ ρ α Ψ µ β X µ = 0 29

31 gauge-fixed action S g.f. = 1 4π d 2 σ [ ] 1 α ηαβ α X µ β X µ + i Ψ µ ρ α α Ψ µ worldsheet supersymmetry (α = 2) δx µ = ϵψ µ, δψ µ = iρ α α X µ ϵ equation of motion + X µ = ψ µ = + ψµ = 0, Ψ µ = ( ψ µ ψ µ ) boundary condition [ψ µ δψ µ ψ µ δ ψ µ ] σ=l σ=0 = 0 30

32 NS and R boundary conditions Closed string (for ψ µ, similar for ψ µ ) ψ µ (τ, σ + 2π) = { ψ µ (τ, σ) R sector ψ µ (τ, σ) NS sector Open string ψ µ (τ, 0) = ± ψ µ (τ, 0) ψ µ (τ, π) = ψ µ (τ, π) (+): R sector, ( ): NS sector doubling trick for ψ µ ψ µ (τ, σ) = { ψ µ (τ, σ) for 0 σ π ψ µ (τ, 2π σ) for π σ 2π 31

33 Mode expansion ψ µ (τ, σ) = ψ r µ exp[ ir(τ + σ)] r Z+1/2 n Z ψ µ n exp[ in(τ + σ)] NS sector R sector Quantization { ψ µ (τ, σ), ψ ν (τ, σ ) } = 2πη µν δ(σ σ ) { ψ µ r, ψ ν s } = η µν δ r+s,0, { ψ µ m, ψn ν } = η µν δ m+n,0 Here ψ µ r>0, ψµ n>0 : annihilation op. ψµ r<0, ψµ n<0 : creation op. ψ µ 0 : zero modes (exists only in R sector) 32

34 NS Fock states (N (f) µ,r = 0, 1 for fixed µ and r) D 1 µ=0 r>0 (ψ µ r) N (f) µ,r 0 NS, where ψ µ r 0 NS = 0 for all r > 0 R Fock vacuum (nonzero modes are similar to NS sector) 1. Take a vacuum state s.t. ψ µ n 0 R = 0 for all n > Act ψ µ 0 s to the vacuum, e.g. ψµ 0 0 R, ψ µ 0 ψν 0 0 R, etc. 3. These states satisfy the vacuum condition also vacuum states 4. These vacuum states are the representation of the Clifford algebra { ψ µ 0, ψν 0} = η µν spacetime spinor 33

35 Clifford algebra and spinor representation Γ µ 2ψ µ 0 = { Γ µ, Γ ν} = 2η µν gamma matrices creation-annilation ops. (suppose D = 2k + 2, I = 0, 1,..., k) algebra γ 0 ± = 1 2 ( ±Γ 0 + Γ 1), γ I ± = 1 2 ( Γ 2I ± Γ 2I+1) for I 0 { γ I +, γ+ J } { = γ I, γ J } { = 0, γ I +, γ } J = δ IJ Clifford vacuum 0 γ I 0 = 0 for all I 34

36 spinor representation (dim. = 2 k+1 = 2 (D/2) ) s = (γ 0 +) s (γ 1 + ) s (γ k + ) s k s = (s 0, s 1,..., s k ), s I = ± 1 2 generators of Lorentz group (or spin group, more strictly) Σ µν = i 4 [ Γ µ, Γ ν] = Cartan: S I i δ I,0 Σ 2I,2I+1 = γ I +γ I 1 2 s : eigenstate of S I S I s = s I s s becomes the weight of the spinor representation. 35

37 Weyl representation Γ D+1 i k Γ 0 Γ 1 Γ D 1, (Γ D+1 ) 2 = 1, { ΓD+1, Γ µ} = 0 We can show that Γ D+1 = 2 k+1 S 0 S 1 S k Then the spinor representation s can be decomposed by the eigenvalue of Γ D+1 as s = A A A for Γ D+1 = +1 of s I = 1 2 A for Γ D+1 = 1 of s I = 1 2 is even. is odd. 36

38 Super Virasoro generators (from T αβ and J α ) NS sector L n = 1 2 : α µ n mα mµ : (2r n) : ψ n rψ µ rµ :, m r G r = m α µ mψ r m,µ R sector L n = 1 2 : α µ n mα mµ : (2m n) : ψ n mψ µ mµ :, m m F n = m α µ mψ n m,µ 37

39 super Virasoro algebra (NS sector) [ Lm, L n ] = (m n)lm+n + c 12 (m3 m)δ m+n,0, [ Lm, G r ] = ( m 2 r ) G m+r, { Gr, G s } = 2Lr+s + c 3 ( r 2 1 ) δ r+s,0 4 (R sector) [ Lm, L n ] = (m n)lm+n + c 12 m3 δ m+n,0, [ Lm, F n ] = ( m 2 n ) F m+n, { Fm, F n } = 2Lm+n + c 3 m2 δ m+n,0 38

40 Explicit calculation gives c = D D = 3 2 D Physical state condition (NS sector) (L 0 a) phys = L n>0 phys = G r>0 phys = 0, phys phys if phys = phys + L n + G r for n, r > 0 (R sector) L n 0 phys = F n 0 phys = 0, phys phys if phys = phys + L n + G n for n > 0 39

41 Here we choose a = 0 for R sector, because we have L 0 = F 2 0 and F 0 has no ambiguity of the normal ordering. Furthermore this is also consistent with the result of no-ghost theorem: D = 10 (critical dim. of superstring), a = 1/2 (NS) 0 (R) 40

42 Low-lying spectrum of open fermionic string NS sector (integer spin) m 2 = 1 2α 0; k 0 NS (tachyon) m 2 = 0 0; k ψ µ NS (vector) m 2 = 1 2α α µ 1 0; k 0 NS, 0; k ψ µ 2ψ ν NS 2 R sector (half-integer spin) m 2 = 0 0; k s R (Majorana spinor) m 2 = 1 α α µ 1 0; k s R, 0; k ψ µ 1 s R The spectrum of closed fermionic string can be similarly obtained. 41

43 2-3. Gliozzi-Scherk-Olive(GSO) projection G-parity operator ( 1) F [ ( ( 1) F = exp iπ (F : worldsheet fermion number) )] ψ rψ µ rµ 1 for NS sector r>0 [ ( 1) F = Γ 11 exp iπ n>0 ψ µ nψ nµ ] for R sector properties { ( 1) F, ψ µ (τ, σ) } = 0, ( 1) F 0 NS = 0 NS, ( 1) F s R = s (Γ 11 ) ss s R 42

44 Since ( 1) F anticommutes with all the modes of ψ µ, any states are the eigenstates of ( 1) F = ±1. Projecting out one of the eigenstates is called GSO projection. GSO projection in NS sector (n: zero or positive integer) m 2 = 1 ( α n 1 ) 2 : ( 1) F = 1 NS sector m 2 = n α : ( 1) F = +1 NS+ sector GSO projection in R sector vacuum : ( 1) F = Γ 11 = ±1, one of chirality is chosen. excited : flipping the sign by acting ψ µ n (n > 0) 43

45 Low-lying spectrum of open fermionic string after GSO (keeping NS+ and R+) NS sector (Tachyon and the first massive state are eliminated.) m 2 = 0 0; k ψ µ NS (vector) Physical state condition selects the transverse modes: = 0; k ψ i NS This is the 8 v representation of the little group SO(8). SO(9, 1) SO(1, 1) SO(8), v 44

46 R sector m 2 = 0 0; k s ζ s s R = 0; k A ζ A A R F 0 condition F 0 = 0 = k µ (Γ µ )ȦA ζ A = 0 Dirac equation Let us choose k µ = (k, k, 0,..., 0). Dirac operator becomes ( k µ Γ µ = k 0 Γ 0 + k 1 Γ 1 = 2kΓ 0 S 0 1 ), 2 which implies that s 0 = 1 2 and ( ) ( 1 A = 2, a, a = ± 1 ) 2, ±1 2, ±1 2, ±1 2 with even of

47 In terms of the decomposition of the representation, we have ( ) , 8 ( 12 ), 8 Here we have to choose the first term in the right hand side. R sector is similarly decomposed as ( ) , 8 ( 12 ), 8 and we must choose ( 1 2, 8 ). (Notation) without prime: with prime: chiral representation anti-chiral representation 46

48 Spacetime supersymmetry boson : 8 v from 10 of SO(9,1) (vector) fermion : 8 from 16 of SO(9,1) (MW spinor) These spectra form 10D N = (1, 0) vector multiplet and the effective action is given by the super Yang-MIlls theory: S SYM = 2 g 2 [ d 10 x Tr 1 4 F µν F µν + i ] 2 ΛΓ µ D µ Λ Here we have introduced the Chan-Paton factor to the endpoints of the open string. Note that this spacetime supersymmetry exists not only in massless level but also any massive level, that is, whole superstring theory. 47

49 2-3. The five superstring theories Type IIA superstring (closed) left mover : NS+, R+ right mover : NS+, R This projection gives boson : fermion : (NS+, NS+), (R+, R ) (R+, NS+), (NS+, R ) massless spectrum (in terms of SO(8) little group) (NS+, NS+) 8 v 8 v = = (dilaton) (2-form) (graviton) 48

50 (R+, R ) 8 8 = 8 v 56 T = (1-form) (3-form) (R+, NS+) 8 8 v = 8 56 = (dilatino) (gravitino) (NS+, R ) 8 v 8 = 8 56 = (dilatino) (gravitino) This theory has N = (1, 1) supersymmetry. 49

51 Type IIB superstring (closed) left mover : NS+, R+ right mover : NS+, R+ This projection gives boson : fermion : (NS+, NS+), (R+, R+) (R+, NS+), (NS+, R+) massless spectrum (NS+, NS+) 8 v 8 v = = (dilaton) (2-form) (graviton) 50

52 (R+, R+) 8 8 = = (0-form) (2-form) (self-dual 4-form) (R+, NS+) 8 8 v = 8 56 = (dilatino) (gravitino) (NS+, R+) 8 v 8 = 8 56 = (dilatino) (gravitino) This theory has N = (2, 0) supersymmetry. 51

53 Type I superstring (closed + open, unoriented) closed string sector = (type IIB) / (left right flip) left right flip ( α µ n, ψ µ r, ψ µ n) massless spectrum (IIB I) ( α n µ, ψ r µ, ψ n µ ) (NS+, NS+) : = 1 35 (R+, R+) : = 28 (R+, NS+), (NS+, R+) : (8 56) 2 =

54 open string sector unoriented SO or Sp gauge group (U(n) is not allowed.) anomaly cancelation SO(32) gauge group massless spectrum (8 v, 496) (8, 496) : SO(32) gauge boson and gaugino This theory has N = (1, 0) supersymmetry. 53

55 Heterotic string left mover : right mover : fermionic string (10D) bosonic string (26D) We compactify 16D of the right mover and construct 10D theory. left mover : X µ L, ψµ right mover : X µ R, Xm R (m = 10,..., 25) Here the fields X m R are compactified on the torus T Λ = R 16 /Λ with the lattice Λ. 54

56 massless spectrum left mover : X µ L, ψµ 8 v 8 right mover : X µ R, Xm R 8 v (16 scalars + more) Actually, we can construct 480 scalars in addition to 16 from X m R, by choosing Λ to be the root lattice of SO(32) or E 8 E 8. Those scalars form the adjoint representation of each group (Frenkel- Kac construction), where 16 scalars correspond to the Cartan subalgebra. Then the spectrum becomes boson : 8 v 8 v = and (8 v, 496) fermion : 8 8 v = and (8, 496) 55

57 fermionic construction (focusing on low-lying states) bosonization (one chiral scalar two MW spinor) X m R λ A A = 1, 2,..., 32. boundary condition and intercept R : λ A (τ, σ + 2π) = +λ A (τ, σ), a = 1 heavy ignored NS : λ A (τ, σ + 2π) = λ A (τ, σ), a = +1 low-lying states 0 NS (m 2 = 4/α ), λ A NS (m 2 = 2/α ), λ A 2 1 λ B NS (m 2 = 0) 56

58 For right mover, NS vacuum is chosen to be ( 1) F 0 NS = + 0 NS. 0 NS : dropout by level-matching λ A NS : dropout by GSO (keeping ( 1) F = +1) λ A 1 λ B NS : remaining Then we have λ A λ B NS as lowest level state, which forms the adjoint representation of SO(32). The E 8 E 8 theory can be obtained by the following GSO projection ( 1) F 1 = ( 1) F 2 = 1 Here F 1 is the fermion number for λ A (A = 1,..., 16) and F 2 is the one for λ A (A = 17,..., 32). 57

59 Summary of five superstring theories 1. Type I 2. Type IIA 3. Type IIB 4. Heterotic SO(32) 5. Heterotic E 8 E 8 If we do not need supersymmetry, we can construct other theories by different GSO projections, for example Type 0A, Type 0B, SO(16) SO(16) heterotic, etc. 58

60 3. Low energy SUGRA and CY compactification 3-1. supergravity (SUGRA) 11D SUGRA ( M-theory) [Cremmer-Julia-Scherk] Field contents and their on-shell degrees of freedom (44+84 = 128) 11D metric G MN : ( of d.o.f.) = 11 (11 + 1) = 44 3-form A MNP : ( of d.o.f.) = 11 2 C 3 = 84 gravitino χ M : ( of d.o.f.) = (11 1 2) =

61 Action for bosonic fields (kinetic + Chern-Simons) S 11D = 1 2κ 2 d 11 x G (R 12 ) F κ 2 11 A 3 F 4 F 4 Here A 3 = 1 3! A MNP dx M dx N dx P, F 4 = da 3, supersymmetry d D x G F n 2 = F n F n δe m M = 1 4 ϵ Γm χ M, δa MNP = 3 4 ϵ Γ [MNχ P ], δχ M = D M (ˆω)ϵ (Γ M NP QR 8δ N MΓ P QR] ) ˆF NP QR ϵ 60

62 Type IIA SUGRA ( circle compactification of 11D SUGRA) Field contents and their on-shell degrees of freedom bosons: dilaton ϕ (1)+NS-NS 2-form B 2 (28)+ metric G µν (35) + R-R 1-form C 1 (8) + R-R 3-form C 3 (56) fermions: dilatino(8) 2 + gravitino(56) 2 Action for bosonic fields (string frame) S IIA = 1 2κ κ 2 10 d 10 x Ge 2ϕ ( R + 4( M ϕ) H 3 2 ) d 10 x ) G ( F F κ 2 10 B 2 F 4 F 4 61

63 Type IIB SUGRA ( T-dual of IIA SUGRA) Field contents and their on-shell degrees of freedom bosons: dilaton ϕ (1)+NS-NS 2-form B 2 (28)+ metric G MN (35) + R-R 0-form C 0 (1) + R-R 2-form C 2 (28) + self-dual R-R 4-form C 4 (35) fermions: dilatino(8) 2 + gravitino(56) 2 Action for bosonic fields (string frame) S IIB = 1 d 10 x ( Ge 2ϕ R + 4( M ϕ) 2 1 ) 2 H κ κ 2 10 d 10 x G ( F F ) F κ 2 10 H 3 F 3 C 4

64 Type I SUGRA Field contents and their on-shell degrees of freedom bosons: dilaton ϕ (1) + 2-form C 2 (28) + metric G µν (35) + SO(32) gauge field a 1 (8 496) fermions: dilatino(8) + gravitino(56) + SO(32) gaugino (8 496) Action for bosonic fields (string frame) S I = 1 2κ 2 10 d 10 x ) G [e (R 2ϕ + 4( M ϕ) 2 12 ] F g 2 10 d 10 x Ge ϕ Tr v f

65 Heterotic SUGRA Field contents and their on-shell degrees of freedom bosons: dilaton ϕ (1) + 2-form B 2 (28) + metric G µν (35) + SO(32) or E 8 E 8 gauge field a 1 (8 496) fermions: dilatino(8) + gravitino(56) + SO(32) or E 8 E 8 gaugino (8 496) Action for bosonic fields (string frame) S H = 1 2κ 2 10 d 10 x [ Ge 2ϕ R + 4( M ϕ) 2 1 ] 2 H 3 2 κ2 10 Tr v f 2 2 g

66 3-2. Calabi-Yau(CY) compactification Parallel spinor ( unbroken supersymmetry) δχ µ = D µ (ω)ϵ + = D µ (ω)ϵ = 0 (suppose the fields other than metric = 0) compactification from 10D to 4D SO(9, 1) SO(3, 1) SO(6) : 16 (2, 4) ( 2, 4) ϵ = ϵ L ϵ + + ϵ R ϵ, ϵ R = ϵ L, ϵ = ϵ + Suppose 4D is flat: D µ (ω)ϵ = 0 m ϵ L = 0, D i (ω)ϵ + = 0 65

67 necessary condition for existense of solution [D i, D j ]ϵ + = 1 4 R ijklγ kl ϵ + = 0 muitiplying Γ j with use of Bianchi identity: R ijkl Γ j Γ kl ϵ + = 2R ij Γ j ϵ + = 0 R ij = 0 Ricci-flat sufficient condition: reduction of holonomy R ijkl SU(3) SO(6) SO(6) SU(3) : The above 1 gives the parallel spinor, which represents the unbroken N = 1 supersymmetry in 4D. 66

68 geometric interpretation: SU(3) invariant bilenear forms J ij = iϵ +Γ ij ϵ + (2-form), Ω ijk = iϵ T +Γ ijk ϵ + (3-form) properties J 2 = 1, dj = dω = 0, J Ω = 0, etc. The above implies that J is the Kähler form and Ω is the (unique) (3.0)-form. The manifolds satisfying these properties are called Calabi-Yau manifolds. CY manifolds are shown to have Ricci-flat metric (Calabi conjecture and Yau s proof). 67

69 Moduli of Calabi-Yau manifolds (6D) deformation of metric keeping Calabi-Yau condition g + δg : Kähler and Ricci-flat two kinds of deformation keeping complex structure: δg = δg a b changing complex structure: δg = δg ab = (δgā b) differential forms ω (1,1) = δg a bdz a d z b ω (2,1) = Ω abc δg c dg dc dz a dz c d z c 68

70 ω (1,1) : Kähler moduli ω (2,1) : complex structure moduli Note that they are the harmonic forms on the Calabi-Yau manifolds. Then these forms express the massless degrees of freedom after the compactification. 69