1. Introduction Palatini formalism vierbein e a µ spin connection ω ab µ Lgrav = e (R + Λ). 16πG R µνab µ ω νab ν ω µab ω µac ω νcb + ω νac ω µcb, e =

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1 Chiral Fermion in AdS(dS) Gravity Fermions in (Anti) de Sitter Gravity in Four Dimensions, N.I, Takeshi Fukuyama, arxiv: Prog. Theor. Phys. 122 (2009)

2 1. Introduction Palatini formalism vierbein e a µ spin connection ω ab µ Lgrav = e (R + Λ). 16πG R µνab µ ω νab ν ω µab ω µac ω νcb + ω νac ω µcb, e = det(e µa ). (e a µ, ω ab µ ) Poincaré cf. Poincaré gauge theory, 3D Chern-Simons gravity, BF gravity, Ashtekar formalism, 1

3 (Anti) de Sitter Gravity (MMSW Gravity) MacDowell and Mansouri 77, West 78, Stelle and West 79, Fukuyama 83 e µ a ω µ ab multiplet ω µ AB = { ab ωµ if A = a, B = b, a5 ω µ a e µ if b = 5, A, B = 1, 2, 3, 4, 5, a, b = 1, 2, 3, 4. ω µ AB : SO(2, 3)(anti de Sitter ) SO(1, 4)(de Sitter ) SO(1, 3) AdS(dS) gravity 4 SO(2, 3) or SO(1, 4) break Einstein 2

4 metric g µν Cosmological Constant: Λ 1 l 2 l: SO(2, 3) = negative, SO(1, 4) = positive 3

5 Weyl, Majorana fermion SO(2, 3), SO(1, 4) Weyl fermion SO(1, 4) Majorana fermion SO(2, 3) Majorana fermion action Kugo, Townsend 82 4D AdS(dS) gravity Weyl, Majorana fermion 4

6 4D AdS(dS) gravity Weyl, Majorana fermion SO(2, 3) or SO(1, 4) Dirac fermion SO(1, 3) Weyl fermion, SO(1, 3) Majorana fermion 5

7 2. (Anti) de Sitter Gravity in Four Dimensions 4D spacetime SO(2, 3) or SO(1, 4) ω µab metric compensator field (Higgs ) Z A = Z A (x) σ(x) SO(1, 3) SO(2, 3) (AdS) A field strength R µνab takes the form R µνab = µ ω νab ν ω µab ω µac ω νcb + ω νac ω µcb. We construct an SO(2, 3) invariant action 6

8 AdS Gravity Sgrav = = d 4 xlgrav ( ) [ ( ) d 4 xϵ ABCDE ϵ µνρσ ZA 1 il 16g 2 R µνbc R λρde { (ZF ) ] 2 +σ(x) 1} D µ Z B D ν Z C D ρ Z D D σ Z E, il g is a coupling constant and l is a real constant. The equation of motion for Z A is (Z A ) 2 = l 2. 7

9 If we take a solution breaking the SO(2, 3) symmetry this breaking derives the vierbein e µa, Z A = (0, 0, 0, 0, il), D µ Z A ( µ δ AB ω µab )Z B = { iωµa5 l e µa ifa = a, 0 ifa = 5, Lgrav takes the Einstein gravity form Lgrav = µ C µ e 16πG ( R + 6l ) 2. Here, µ C µ is the topological Gauss-Bonnet term. G is the gravitational constant derived from 16πG = g 2 l 2. 8

10 SO(1, 4) (ds) We construct an SO(1, 4) invariant action ds Gravity Sgrav = = d 4 xlgrav ( ) [ ( ) d 4 xϵ ABCDE ϵ µνρσ ZA 1 l 16g 2 R µνbc R λρde { (ZF ) ] 2 +σ(x) 1} D µ Z B D ν Z C D ρ Z D D σ Z E. l The equation of motion for Z A is (Z A ) 2 = l 2. We break the SO(1, 4) group to 9

11 the local Lorentz group SO(1, 3) as This breaking leads to Z A = (0, 0, 0, 0, l). D µ Z A = ( µ δ AB ω µab )Z B = { ωµa5 l e µa ifa = a. 0 ifa = 5. Lgrav takes the form Lgrav = µ C µ e 16πG ( R 6l ) 2. 10

12 3. Gamma Matrix Gamma Matrix Γ A SO(1, 3) γ A, SO(2, 3) γ (AdS) A, SO(1, 4) γ (ds) A Dirac (Pauli) basis {Γ A, Γ B } = 2δ AB, Γ A = Γ A. γ T A = { γa if A = 2, 4, 5, γ A if A = 1, 3. 11

13 4. Dirac Fermion Fukuyama 83 Let ψ be an SO(2, 3)(SO(1, 4)) Dirac fermion. SO(2, 3) (AdS) An SO(2, 3) invariant Dirac spinor action is defined as L Dirac = ϵ ABCDE ϵ µνρσ ψ ( is AB D µ 3! iλ Z A il D µ Z B 4! where S AB 1 4i [γ(ads) A, γ (AdS) B], and λ is a mass. ) ψd ν Z C D ρ Z D D σ Z E ψ ψ γ (AdS) 5γ (AdS) 4 12

14 By the symmetry breaking Z A = (0, 0, 0, 0, il) from SO(2, 3) to SO(1, 3), L Dirac reduces to the Dirac action in the four-dimensional curved spacetime L Dirac = e ψ ( γ a e µa ) ( D µ + λ ψ, = e ψ 1 ( 2 eµa γ ad µ ) D µ γ a ) + λ ψ, ψ = ψ γ 4. where γ a iγ (AdS) 5γ (AdS) a, γ 5 γ (AdS) 5. γ (AdS) a iγ 5 γ a, γ (AdS) 5 γ 5. 13

15 SO(1, 4) (ds) In the ds gravity, we consider an SO(1, 4) invariant Dirac spinor action L Dirac = ϵ ABCDE ϵ µνρσ ψ ( Z A D µ l γ(ds) B 3! + λ Z A l D µ Z B 4! ) ψd ν Z C D ρ Z D D σ Z E which is a slightly different form from the SO(2, 3) case. Here, ψ = ψ γ (ds) 4. By the symmetry breaking Z A = (0, 0, 0, 0, l) from SO(1, 4) to SO(1, 3), L Dirac 14

16 reduces to the Dirac action in the four-dimensional curved spacetime L Dirac = e ψ where ψ = ψ γ 4 and ( γ a e µa ) ( D µ + λ ψ, = e ψ 1 ( 2 eµa γ ad µ ) D µ γ a γ (ds) A γ A. ) + λ ψ, 15

17 5. Weyl Fermion symmetry 4D Weyl fermion SO(2, 3) SO(1, 4) spinor 1, SO(2,3)(SO(1,4)) covariant 2, chiral projections 1±γ 5 2 operator P ± SO(2, 3) (AdS) Let ψ be an SO(2, 3) Dirac spinor. We introduce a projection operator, P ± 1 2 ( 1 ± ) l2 Z A γ (AdS) A Z 2 il, 16

18 which is P± 2 = P ± and P + P = 0. We define ψ ± P ± ψ. If we break the SO(2, 3) symmetry Z A = (0, 0, 0, 0, il), P ± reduces to the chiral projections P ± P ± P ± = 1 ± γ(ads) 5 2 = 1 ± γ 5. 2 Then, ψ ± becomes Weyl spinors ψ ± ψ ± ψ ± = P ± ψ, 17

19 respectively, which have definite chirality. We can construct an SO(2, 3) invariant action by modifying the action for a Dirac fermion, L Weyl = ϵ ABCDE ϵ µνρσ ψ+ ( is AB D µ 3! iλ Z A il D µ Z B 4! ) ψ + D ν Z C D ρ Z D D σ Z E The action becomes a SO(1, 3) massless Weyl fermion action by breaking the symmetry L Weyl = e (γ ψ+ a e µa ) D µ + λ ψ+ = e (γ ψ+ a e µa D ) µ ψ +, 18

20 SO(1, 4) (ds) Let ψ be an SO(1, 4) Dirac spinor. In the SO(1, 4) case, we introduce P ± 1 2 ( 1 ± l 2 Z A γ (ds) A Z 2 l ), which is P± 2 = P ± and P + P = 0. We define ψ ± P ± ψ. If we break the SO(1, 4) symmetry as Z A = (0, 0, 0, 0, l), 19

21 P ± reduces to chiral projections P ± P ± P ± = 1 ± γ(ds) 5 2 = 1 ± γ 5. 2 Then ψ ± becomes Weyl fermions ψ ±, ψ ± ψ ± = P ± ψ, respectively, which have definite chirality. We can construct SO(1, 4) invariant action by modifying the Dirac action 20

22 ( L Weyl = ϵ ABCDE ϵ µνρσ Z A D µ ψ+ l γ(ds) B 3! D ν Z C D ρ Z D D σ Z E. + λ Z A l D µ Z B 4! The action becomes an SO(1, 3) massless Weyl fermion action by breaking the symmetry L Weyl = e (γ ψ+ a e µa ) D µ + λ ψ+ = e (γ ψ+ a e µa D ) µ ) ψ + ψ +. 21

23 6. Majorana Fermion SO(1, 3) 4D Majorana fermion ψ M ψ M = ψ c M C ψ T M, C is the charge conjugation in SO(1, 3). If we take the Dirac (Pauli) basis, C is C = γ 2 γ 4. However, C is not covariant under either SO(2, 3) or SO(1, 4). ψ M is not consistent with the SO(2, 3) (SO(1, 4)) covariance. If a charge conjugation is defined, a Majorana fermion can be defined. 22

24 Conditions for SO(2, 3) or SO(1, 4) charge conjugation C 1. C 1 γ A C is covariant under the symmetry to be consistent with the action. C 1 γ A C = ±γ T A, is sufficient where the signatures are the same for all A. 2. B defined by Bψ M = C ψ T M must satisfy B B = 1, since a charge conjugation has a Z 2 symmetry. (B = γ 2 for SO(1, 3).) 3. C reduces to C = γ2 γ 4 by breaking the symmetry. 23

25 SO(2, 3) (AdS) C = γ (AdS) 2γ (AdS) 4. SO(1, 4) (ds) C ( Z A γ (ds) A l + ) Z2 l 2 i γ (ds) 2γ (ds) 4γ (ds) 5. l 2 24

26 SO(2, 3) (AdS) The SO(2, 3) gamma matrices γ (AdS) A are constructed as γ (AdS) a iγ 5 γ a, γ (AdS) 5 γ 5, From the condition 1, we have two candidates C 1 = γ (AdS) 1γ (AdS) 3γ (AdS) 5, C 2 = γ (AdS) 2γ (AdS) 4. C 2 = γ (AdS) 2γ (AdS) 4 = γ 2 γ 4 is equal to the SO(1, 3) charge conjugation Therefore C 2 satisfies the condition 2 and 3. Note that C = C 2 is not a 25

27 charge conjugation in the SO(2, 3) representation. fermion ψ M is defined by Therefore AdS Majorana ψ M = C ψ T M = C 2 ψt M. SO(2, 3) invariant AdS Majorana fermion action L Majorana = ϵ ABCDE ϵ µνρσ ψm ( is AB D µ 3! iλ Z A il D µ Z B 4! ) ψ M D ν Z C D ρ Z D D σ Z E Let us investigate the consistency of this action. Substituting ψ M = C 2 ψt M, to 26

28 the right-hand of the action, we obtain ϵ ABCDE ϵ µνρσ ( ψ T M( C T ) 1) ( is AB D µ 3! iλ Z A il D µ Z B 4! ) ( ) C ψt M D ν Z C D ρ Z D D σ Z E. We can easily check that = L Majorana. Thus, the definition of the charge conjugation is consistent with the action. If we break the SO(2, 3) symmetry by Z A = (0, 0, 0, 0, il), the action reduces to an SO(1, 3) Majorana fermion action in the Einstein gravitational theory in four dimensions L Majorana = e ψ M (γ a e µa ) D µ + λ ψ M. 27

29 SO(1, 4) (ds) γ (ds) A γ A From the condition 1, we obtain two candidates C 3 γ (ds) 1γ (ds) 3, C 4 γ (ds) 2γ (ds) 4γ (ds) 5. Condition 2 B B = 1: Neither C 3 nor C 4 can be defined as a consistent charge conjugation. 28

30 Now, we consider a third candidate: C 5 ( Z A γ (ds) A l + ) Z2 l 2 i γ (ds) 2γ (ds) 4γ (ds) 5. l 2 This satisfies the condition 1. Condition 2. B 5B 5 = 1 ( B 5 = ( Z A γ (ds) A l + Condition 3. C 5 γ (ds) 2γ (ds) 4 = γ 2 γ 4 = C. A ds Majorana spinor ) ) Z 2 l 2 i γ (ds) l 2γ (ds) 2 5 ψ M = C 5 ψt M. 29

31 SO(1, 4) invariant ds Majorana fermion action ( L Majorana = ϵ ABCDE ϵ µνρσ Z A D µ ψm l γ(ds) B 3! + λ Z A l D µ Z B 4! ) ψ M D ν Z C D ρ Z D D σ Z E We can prove the consistency of the action for the charge conjugation C 5 similar to SO(2, 3) case. ϵ ABCDE ϵ µνρσ ( ( ψm(c T T ) 1) Z A D µ l γ(ds) B 3! D ν Z C D ρ Z D D σ Z E. + λ Z A l D µ Z B 4! ) (C ) ψt M 30

32 We can easily check that = L Majorana. If we break the SO(1, 4) symmetry by Z A = (0, 0, 0, 0, l), the action becomes the Majorana fermion action in the Einstein gravitational theory in four dimensions L Majorana = e ψ M (γ a e µa ) D µ + λ ψ M, 31

33 7. Summary and Discussion Weyl, Majorana fermion action AdS (ds) Gravity action New mechanism to derive a chiral fermion from a nonchiral fermion Chiral symmetry and chiral anomaly Z A dynamical 32

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann