Conformal theory of MacDowell-Mansouri type Micha Szczachor - PowerPoint PPT Presentation

Conformal theory of MacDowell-Mansouri type Michał Szczachor Capstone Institute for Theoretical Research capstone-itr.eu Andrzej Borowiec Institute of Theoretical Physics www.ift.uni.wroc.pl September 12, 2019 Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 1 / 22

MacDowell–Mansouri action 3 � F IJ ∧ F KL ǫ IJKL S MM = (1) 4 G Λ M It is possible to show that S MM is equivalent to Palatini action. S Pal = 1 ( e i ∧ e j ∧ R kl − Λ � 6 e i ∧ e j ∧ e k ∧ e l ) ǫ ijkl (2) 2 G M Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 2 / 22

BF action � M 4 B IJ ∧ F IJ − β 2 B IJ ∧ B IJ − 1 2 ǫ IJKLM v M ∧ B IJ ∧ B KL . S = (3) where I, J = 0 . . . 4 . 0 · · · α � T v M = � 2 where α ≈ 10 − 120 Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 3 / 22

BF action. Motivation The construction base on two term. One is a topological term which generates the topological vacuum and second term are breaking symmetry down to Lorentz symmetry. This kind of construction has 3 adventiges: The Lagrangian is quadratic in fields, where the Palatini formalism 1 is trilinear. The presented modification will introduce Immirzi parameter. 2 It allows for introducing dynamical degrees of freedom as a 3 perturbation around topological vacuum. Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 4 / 22

BF action conti. β 2 B ij ∧ B ij − α � M 4 B ij ∧ F ij − 2 ǫ ijkl ∧ B ij ∧ B kl . S = (4) where i, j, · · · = 0 . . . 3 . Note that α � = β . If α = β then it leads to self–dual gravity. The physical meaning of constatnts is: ℓ 2 = Λ 1 G Λ G Λ γ α = β = (5) 3(1 − γ 2 ) 3(1 − γ 2 ) 3 Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 5 / 22

BF action conti. � α 2( α 2 − β 2 ) P ( ω ) + 1 β S = S H +Λ + M 4 ( 4( α 2 − β 2 ) E ( ω ) − β NY ( ω, e ) ) (6) where S H +Λ = − 1 Gǫ ijkl ( R ij ∧ e k ∧ e l − Λ 3 e i ∧ e j ∧ e k ∧ e l ) − 2 Gγ R ij ∧ e i ∧ e j (7) and E ( ω ) is Euler P ( ω ) is Pontryagin NY ( ω, e ) in Nieh-Yan invariants. Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 6 / 22

Conformal algebra [ D , K i ] = i K i , (8) [ D , P i ] = − i P i , (9) [ K i , P j ] = − 2 i ( η ij D − M ij ) , (10) [ M ij , K k ] = − i ( η ki K j − η kj K i ) , (11) [ M ij , P k ] = − i ( η ki P j − η kj P i ) , (12) [ M ij , M kl ] = − i ( η ik M jl + η jl M ik − η jk M il − η il M jk ) , (13) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 7 / 22

Initial action � M 4 B IJ ∧ F IJ − β 2 B IJ ∧ B IJ − 1 2 ǫ IJKLMN v MN ∧ B IJ ∧ B KL . S = (14) where I, J = 0 . . . 5 .  0 0  · · · · · · . . ... . .  . .    v MN = . ... α   .  .   2  − α   0 0 · · · 2 where α ≈ 10 − 120 Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 8 / 22

Initial action conti. 1 2( M ijkl ) − 1 F ij ∧ F kl + 1 � β ( F i 4 ∧ F i 4 − F i 5 ∧ F i 5 − F 45 ∧ F 45 ) S = M 4 (15) where M ij kl = ( βδ ij kl + α 2 ǫ ij kl ) F ij M ij = ( R ij − 1 1 + 1 1 ∧ f j 2 ∧ f j ℓ 2 f i ℓ 2 f i 2 ) M ij (16) F 45 D = (1 ℓ dφ − 1 ℓ 2 f j 1 ∧ f 2 j ) D (17) F i 4 R 1 i = (1 1 − 1 ℓ D ω f i ℓ 2 φ ∧ f i 2 ) R 1 i (18) F i 4 R 2 i = (1 2 − 1 ℓ D ω f i ℓ 2 φ ∧ f i 1 ) R 2 i . (19) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 9 / 22

substitution... � 1 α 2 + β 2 ( β α αδ ijkl − ǫ ijkl ))( R ij ∧ R kl − 2 ℓ 2 R ij ∧ f k 1 ∧ f l S = 4( 1 M 4 + 2 2 + 1 ℓ 2 R ij ∧ f k 2 ∧ f l ℓ 4 f i 1 ∧ f j 1 ∧ f k 1 ∧ f l (20) 1 + 1 2 − 2 2 ∧ f j 1 ∧ f j ℓ 4 f i 2 ∧ f k 2 ∧ f l ℓ 4 f i 1 ∧ f k 2 ∧ f l 2 ) βℓ 2 ( dφ ∧ dφ − 2 1 1 ∧ f 2 i − 1 1 ∧ f 2 i ∧ f j ℓ dφ ∧ f i ℓ 2 f i − 1 ∧ f 2 j ) 1 1 ∧ D ω f 1 i − 2 βℓ 2 ( D ω f i l D ω f i + 1 ∧ φ ∧ f 2 i ) 1 2 ∧ D ω f 2 i − 2 βℓ 2 ( D ω f i l D ω f i − 2 ∧ φ ∧ f 1 i ) . (21) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 10 / 22

The field equations are : 2 βℓ 3 ( D ω f a 1 ∧ f 2 a − D ω f a 2 ∧ f 1 a − d ( f a 1 ∧ f 2 a )) = 0 (22) δφ 2 : D ω (1 2 ǫ abcd e c ∧ e d + 1 γ e a ∧ e b ) = D ω (1 2 ǫ abcd f c ∧ f d + 1 δf a 1 + δf a γ f a ∧ f b ) (23) δω ab : − 1 2 ) + 1 2 Gǫ abcd D ω ( f c 2 ∧ f d 2 Gǫ abcd D ω ( f c 1 ∧ f d 1 ) (24) − 1 γGD ω ( f 2 a ∧ f 2 b ) + 1 γGD ω ( f 1 a ∧ f 1 b ) (25) 2 2 + βℓ 2 f 1[ b ∧ f 2 a ] ∧ φ − βℓ 2 f 2[ b ∧ f 1 a ] ∧ φ = 0 . (26) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 11 / 22

Changing algebra basie At that point it is more convenient to use a conformal groups isomorphism and to change the algebra base. Notice that by such treatment the vector field transform as follows 1 = 1 2( e i − f i ) f i (27) 2 = 1 2( e i + f i ) . f i (28) 1 1 Gǫ ijkl R ij ∧ e k ∧ f l + 2 ℓ 2 Gǫ ijkl e i ∧ f j ∧ e k ∧ f l = 0 . (29) D ω (1 2 ǫ abcd e c ∧ e d + 1 γ e a ∧ e b ) = D ω (1 2 ǫ abcd f c ∧ f d + 1 γ f a ∧ f b ) . (30) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 12 / 22

Action L = − 1 2 Gǫ ijkl R ij ∧ e k ∧ f l − 1 Gγ R ij ∧ e i ∧ f j 1 4 ℓ 2 Gǫ ijkl e i ∧ e j ∧ f k ∧ f l − βℓ 3 C 4 ( e, f, φ ) + ℓ 2 − 1 1 βℓ 2 S 4 ( φ ) + 4 GE 4 ( ω ) (31) 2 G P 4 ( ω ) + 2 γ 2 + 1 + γℓ 2 γG NY 4 ( e, f ) , Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 13 / 22

Action L = − 1 2 Gǫ ijkl R ij ∧ e k ∧ f l − 1 Gγ R ij ∧ e i ∧ f j 1 4 ℓ 2 Gǫ ijkl e i ∧ e j ∧ f k ∧ f l − βℓ 3 C 4 ( e, f, φ ) + ℓ 2 − 1 1 βℓ 2 S 4 ( φ ) + 4 GE 4 ( ω ) (32) 2 G P 4 ( ω ) + 2 γ 2 + 1 + γℓ 2 γG NY 4 ( e, f ) , Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 14 / 22

Topological invariants E 4 ( ω ) = ǫ µναβ R µνij ⋆ R αβ ij (33) P 4 ( ω ) = ǫ µναβ R µνij R αβ ij (34) NY 4 ( e, f ) = 1 2 NY 4 ( e − f ) = 1 2 ∂ µ [ ǫ µναβ ( e − f ) ν I · D α ( e − f ) βI ] (35) S 4 ( φ ) = d ( φ ∧ dφ ) (36) C 4 ( e, f, φ ) = d ( f a ∧ e a ∧ φ ) (37) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 15 / 22

Topological invariants In CS notation: E 4 ( ω ) = ǫ µναβ R µνij ⋆ R αβ ij (38) P 4 ( ω ) = ǫ µναβ R µνij R αβ ij (39) NY 4 ( e, f ) = 1 2 NY 4 ( e − f ) = 1 2 ∂ µ [ ǫ µναβ ( e − f ) ν I · D α ( e − f ) βI ] (40) S 4 ( φ ) = d ( φ ∧ dφ ) , (41) C 4 ( e, f, φ ) = d ( f a ∧ e a ∧ φ ) (42) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 16 / 22

Topological invariants in CS notation E 4 ( ω ) = 32 id ( C ( + ω ) + C ( − ω )) (43) P 4 ( ω ) = 16 d ( C ( + ω ) + C ( − ω )) (44) NY 4 ( e, f ) = 1 2 dC ( e − f ) (45) S 4 ( φ ) = dC ( φ ) (46) C 4 ( e, f, φ ) = d ( − 6 R a ∧ e a ∧ φ + R ∧ φ ) . (47) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 17 / 22

Constraints and distingues frame field If one distinguishes e a as a field associated with translation generator, the solution of is 1 f a µ = − 6 R a µ + Re a µ . (48) The above assumption can be justify if the constraint 2 D ω ( e c ) = 0 ⇔ D ω ( f c ) = 0 (49) has been added. Then, indeed ω becomes to be a spin connection field e.i. ω = ω ( e, φ ) . 1 M. Kaku, P . K. Townsend and P . van Nieuwenhuizen, Phys. Rev. D 17 (1978) 3179. 2 P . K. Townsend and P . van Nieuwenhuizen, Phys. Rev. D 19 (1979) 3166. Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 18 / 22

Ation as a function of ’graviton’ L = − 1 2 Gconst · e · ( R µν R µν − 1 3 R 2 ) + L Holst βℓ 3 C 4 ( e, φ ) + ℓ 2 1 1 βℓ 2 S 4 ( φ ) + 4 GE 4 ( ω ) − (50) 2 G P 4 ( ω ) + 2 γ 2 + 1 + γℓ 2 γG NY 4 ( e ) . where Gγ R ij ∧ e i ∧ f j = 24 ∗ 5 1 L Holst ( ω, e ) = Gγ ( ⋆R ) ∧ R . (51) (52) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 19 / 22

Topological terms conti. L = − 1 4 Gconst · e · ( R µνρσ R µνρσ − 2 R µν R µν + 1 3 R 2 ) + L Holst − 1 1 1 4 Gconst · e · GB 4 ( ω, e ) − βℓ 2 S 4 ( φ ) + βℓ 3 C 4 ( e, ω, φ ) 2 G P 4 ( ω ) + 2 γ 2 + 1 + ℓ 2 4 GE 4 ( ω ) + γℓ 2 γG NY 4 ( e, ω ) . (53) The Gauss-Bonet term GB 4 ( ω, e ) = − ( R µνρσ R µνρσ − 4 R µν R µν + R 2 ) (54) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 20 / 22

Weyl action L = − 1 4 Gconst · L Weyl + L Holst + “ topological terms ” . (55) where L Weyl = C µνab C ρσcd ǫ µνρσ ǫ abcd e = R µνρσ R µνρσ − 2 R µν R µν + 1 3 R 2 · e (56) Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 21 / 22

Thank you for your attention! email: ms@capstone-itr.eu Michał Szczachor (CIfTR) Conformal BF theory September 12, 2019 22 / 22