Introduction to Orientifolds.

Transcription

1 Introduction to Orientifolds

2 Overview

3 Orientability in Quantum Field Theory: spinors S R(2π) ψ = ψ

4 Orientability in Quantum Field Theory: spinors (S R(2π) ) 2 ψ =+ ψ S R(2π) ψ = ψ

5 Orientability in Quantum Field Theory: spinors S Λ γ µ S 1 Λ = Λ ν µ γ ν Λ µ ν S R(2π) 2π S R(2π) ψ = ψ S P Λ ν µ x x (S R(2π) ) 2 ψ =+ ψ S P ψ = p ψ (S P ) 2 ψ = p 2 ψ p 2 =?

6 Orientability in Quantum Field Theory: spinors S 2 P = ±1 ( ψl 0 ) + ( 0 ψ R ) = ψ Dirac

7 Perturbation theory anomalies G ψ L ψ L ν b µa λ c p, ν,b; k, λ,c µ j µa 0 = g2 8π 2 ɛαβµν p α k β A abc SU(2) A U(1) : A =3 ( 2 1 3) + (0 1) = 0 3

8 Perturbation theory anomalies G ψ L ψ L ν b µa λ c p, ν,b; k, λ,c µ j µa 0 = g2 8π 2 ɛανβλ p α k β A abc SU(2) A U(1) : A =3 ( 2 1 3) + (0 1) = 0 3 G U(1)) : π 5 (G) 0

9 Orientability and extended objects

10 Orientability and open strings

11 Orientability and open strings mathworld.wolfram.com Möbius strip

12 Orientability and closed strings

13 Involutions of the (worldsheet) torus τ τ = it, t R τ τ I(z) =1 z

14 Involutions of the (worldsheet) torus τ Make a Möbius strip: τ = it, t R I(z) =1 z τ τ

15 Involutions of the (worldsheet) torus τ = it, t R C τ B B A C

16 Involutions of the (worldsheet) torus τ = it, t R C τ B B A C z S (z,τ) ( z τ, 1 τ )

17 The "crosscap" C τ B B A C P π/t =: s S (z,τ) ( z τ, 1 τ )...

18 The "crosscap" C τ B B A C π/t =: s

19 Topology I (rubber) hole crosscap Möbius strip Klein bottle χ =2 2h b c Cylinder (annulus) handles crosscaps boundaries

20 S-matrix of unoriented strings g χ s 1 gym 2 tr F 2 (φ)

21 String endpoints and gauge charges N; k; ij k n i N 3 =1 j

22 String endpoints and gauge charges Ω N; k = ω N N; k ω N =( 1) 1+α m 2 =0 ω N Ω N; k; ij = ω N N; k; ji? Ω N; k; ij = ω N γ jj N; k; j i γ 1 i i

23 Gauge group from parity constraint Ω N; k; ij = ω N γ jj N; k; j i γ 1 i i i, j =1... n ω = +1 γ T = ±γ SO(n) or Sp(n) n

24 Partition functions Z(q) Tr (q L0 q L 0 ) q = e 2πiτ τ q nn = n=0 ( N=0 n=0 q nn ) = 1 1 q N 1 N=0 (1 qn ) = 1 q 1/24 η(q) η

25 Partition functions Z(q) Tr (q L0 q L 0 ) ϑ q = e 2πiτ Tr α,β (q L 0 )= ϑ[ α/2 β/2 ](0, τ) η(τ) =: Z α β (τ) τ e πiβ e πiα

26 SO(n) or Sp(n) Tadpoles Z C,0 n 2 ds η(is/π) 8 [ Z0(is/π) 0 4 Z1(is/π) 0 4] }{{} 0 π/t =: s s 16 + O(e 2s )

27 Tadpoles Z C,0 n 2 ds η(is/π) 8 [ Z0(is/π) 0 4 Z1(is/π) 0 4] }{{} 0 s 16 + O(e 2s ) 1 p 2 = 0 ds e sp2 s 1 p =

28 SO(n) Tadpoles Z C,0 n 2 n n n

29 SO(n) Tadpoles Z C,0 n 2 ds η(is/π) 8 [ Z0(is/π) 0 4 Z1(is/π) 0 4] }{{} 0 s 16 + O(e 2s ) 1 p 2 = 0 ds e sp2 s 1 p =

30 Unoriented partition functions Ω = +1 Z(q) Tr (q L0 q L 0 ) Tr ( 1+Ω q L0 q L 0 ) ( i j = 1 2 Tr (ql0 q L 0 )+ 1 2 Tr (ΩqL0 q L 0 ) + ) j i

31 Tadpoles s Z M,1 ±2 5 2n 0 ds (16 + O(e 2s )) Tr(Ωq L 0...) 1 p =

32 Tadpole cancellation Z C,0 n 2 ds η(is/π) 8 [ Z0(is/π) 0 4 Z1(is/π) 0 4] }{{} Z M,1 ±2 5 2n 0 0 s 16 + O(e 2s ) ds (16 + O(e 2s )) Z K, ds (16 + O(e 2s )) 0 n ± 2 5 2n =(n ± 32) 2

33 Tadpole cancellation ( ( 2 = + =0 SO(32)

34 Tadpole cancellation ( ( 2 + =0

35 Tadpole cancellation and anomalies Tr a (t 6 ) = (n 32)Tr v (t 6 ) + 15Tr v (t 2 )Tr v (t 4 ) SO(32)

36 Tadpole cancellation anomaly cancellation!

37 T-Duality R α R n w R α R

38 T-Duality X(z, z) =X L (z)+x R ( z) p L = n R + wr α p R = n R wr α X (z, z) =X L (z) X R ( z)

39 Worldsheet parity / spacetime parity X (z, z) =X L (z) X R ( z) Ω : X (z, z) X ( z, z) Ω = ΩP Ω

40 Worldsheet parity / spacetime parity X (z, z) =X L (z) X R ( z) Ω : X (z, z) X ( z, z) Ω = ΩP Ω

41 What is an orientifold plane?

42 Effective descriptions: D-branes = V (φ) φ V eff

43 Effective descriptions: D-branes N 1 ds 2 = Z(r) 1/2 η µν dx µ dx ν + Z(r) 1/2 dx m dx m e 2Φ = Z(r) (3 p)/2 Z(r) = 1 + ρ7 p ρ gn r 7 p ; r2 = x m x m

44 Effective descriptions: D-branes N 1

45 Effective descriptions: O-planes themselves

46 Atiyah-Hitchin metric ds 2 =(abc) 2 dt 2 + a 2 σ b 2 σ c 2 σ 2 3 bc =: w 1, ca =: w 2, ab =: w 3 w 3 (t) = π 6 (E 2(t)+ϑ 4 3(t)+ϑ 4 4(t)) t 0 = 1

47 Negative Tension in Quantum Gravity? + =0

48 Summary so far i j + =0

49 Application I: MSSM orientifolds The Spanish Quiver b (u,d) u c e +! c d c (e-,!) c Chiral SU(3) x SU(2) x U(1) spectrum: a 3(u, d) L + 3u c L + 3d c L + 3(e, ν) L + 3e + L Y massless d

50 Application II: "KKLT" orientifold z z z z z 5 5 = c y1 1.2 x x x

51 Application III: Cosmological singularity in string theory {

52 Future work: Green's function method g 2 YM 1 2 tr F (φ) V eff T 1,1

53 Condensed matter "Applied string theory"

54 Ohm's law at strong coupling? E B σ 1 σ SO(8)

55 History: symmetry reduction in AdS/CFT AdS 5 S 5 (SO(6))

56 Symmetry reduction in AdS/CFT S 5 RP 5

57 Summary

58 Summary