Non-Abelian and gravitational Chern-Simons densities

Non-Abelian and gravitational Chern-Simons densities Tigran Tchrakian School of Theoretical Physics, Dublin nstitute for Advanced Studies (DAS) and Department of Computer Science, Maynooth University, Maynooth, reland Fourth Summer School on High Energy Physics and Quantum Field Theory, August 2016, Yerevan, Armenia

Plan Chern-Simons densities Chern-Simons densities of Abelian gauge fields Chern-Simons densities of non-abelian gauge fields Gauge transformation of Chern-Simons densities Einstein-Cartan formulation of gravity Equivalence with Metric formalism p-einstein-hilbert systems in d-dimensions p-einstein equations Gravitational Chern-Pontryagin densities Gravitational Chern-Simons densities Chern-Simons gravity

Abelian Chern-Simons densities Abelian curvature of Abelian connection A i F ij = i A j j A i Definition of Abelian (gauge invariant) Chern-Pontryagin density in 2n + 2 dimensions Ω CP = ε i1 j 1 i 2 j 2...i n+1 j n+1 F i1 j 1 F i2 j 2... F in+1 j n+1 = ε µ1 ν 1 µ 2 ν 2...µ n+1 ν n+1 µn+1 ( Aνn+1 F µ1 ν 1 F µ2 ν 2... F µnν n ) = µn+1 Ω µn+1 implies definition of Abelian (gauge variant) Chern-Simons density in 2n + 1 dimensions Ω CS = ε i1 j 1 i 2 j 2...i nj nj n+1 A jn+1 F i1 j 1 F i2 j 2... F inj n Ω CS is not a total divergence and leads to the variational equations which is gauge covariant. ε i1 j 1 i 2 j 2...i nj nj n+1 F i1 j 1 F i2 j 2... F inj n = 0

Non-Abelian (na) Chern-Simons densities non-abelian curvature of Abelian connection A i F ij = i A j j A i + [A i, A j ] Definition of non-abelian (gauge invariant) Chern-Pontryagin density in 2n + 2 dimensions Ω CP = ε µ1 ν 1 µ 2 ν 2...µ n+1 ν n+1 Tr F µ1 ν 1 F µ2 ν 2... F µn+1 ν n+1 = µn+1 Ω µn+1 which is a total divergence and likewise implies definition of non-abelian (gauge variant) Chern-Simons density in 2n + 1 dimensions. For n = 1, D = 3, it is ( Ω (1) CS = ε ijktr A k F ij 2 ) 3 F if j. For n = 2, D = 5, it is Ω (2) CS = ε ijklm Tr A m ( F ij F kl F ij A k A l + 2 5 A ia j A k A l ).

For n 3 there are multiple distinct definitions for the CS density, each characterised by the number of traces in the definition. For n = 3, D = 7, there are two possibilities; a definition with a single trace and another one with double trace. These are Ω (3) CS = ε ijklmnp Tr A p (F ij F kl F mn 4 5 F ijf kl A m A n 2 5 F ija k F lm A n + 4 5 F ija k A l A m A n 8 ) 35 A ia j A k A l A m A n, ( Ω (3) CS = ε ijklmnp Tr A p F mn 2 ) 3 A ma n (Tr F ij F kl )

As in the Abelian case, here too the equations of motion of (the usual) non-abelian CS densities are gauge covariant. For the examples listed above they are for d = 3, 5 respectively, and for d = 7 ε ijk F ij = 0 ε ijklm F ij F kl = 0 ε ijklmnp F ij F kl F mn = 0 ε ijklmnp (Tr F ij F kl ) F mn = 0

Gauge transformation of non-abelian CS densities Abelian CS densities in all dimensions transform as Ω CP Ω CP + Ω Non-Abelian CS densities also transform with a total divergence term, plus a winding number term featuring the group element α µ = µ g g 1. n d = 3, 5 explicit expressions for this are Ω (2) (2) CS Ω CS = Ω(2) CS 2 3 ε λµνtr α λ α µ α ν 2ε λµν λ Tr α µ A ν Ω (3) (3) CS Ω CS = Ω(3) CS 2 5 ε λµνρσtr α λ α µ α ν α ρ α σ [ ( +2 ε λµνρσ λ Tr α µ A ν F ρσ 1 ) 2 A ρa σ + (F ρσ 12 A ρa σ ) A ν 1 2 A ν α ρ A σ α ν α ρ A σ ]

Metric, covariant derivative and curvature Covariant vector V M and contravariant vector V M are related by the space(-time) metric g MN V M = g MN V N, V M = g MN V N, g MN = g NM, g MK g KN = δ N M The signature of the metric can be chosen to be Euclidean, Minkowskian, etc. For gravity, we choose Minkowskian. The covariant derivative of V M is defined as M V N = M V N + Γ MN K V K in terms of the Christoffel symbol Γ MN K The metric tensor is covariantly constant K g MN = 0. The Riemann curvature tensor is defined as R MNK L = ( [M N] ) K L = [M Γ N]K L + (Γ MK Γ N L Γ NK Γ M L ) and using the Leibniz rule it satisfies the Bianchi identity L R MNR S + (cycl. L, M, N) = 0.

Frame vector formalism of gravity: Einstein-Cartan theory Covariant frame-vector φ a and contravariant vector φ a are related by the flat metric η ab φ a = η ab φ b, φ a = η ab φ b, η ab = η ba, η ac η bc = δ b a (Since η ab is constant, henceforth ignore co- and contra-variace of frame indices.) The covariant derivative of the (co- or contra-variant) frame vector is defined in terms of the the spin-connection ω ab M as D M φ a = M φ a + ω ab M φb and the corresponding curvature is R ab MN = (D [MD N] ) ab = [M ω ab N] + ωac [M ωcb N] and applying the Leibniz rule the Bianchi identity is D L RMN ab + (cycl. L, M, N) = 0.

The dynamical quantities replacing the metric tensor are the Vielbeine e a M and ema (or e M a ) with one spacetime and one (flat) frame index, such that e a M = g MNe N a, e M a = g MN e a N, and satisfying the orthonormality properties e M a e N a = g MN, e a M ea N = g MN, e a M en a = δ N M, ea M em b = δa b. The covariant derivative of the Vielbein e a M is the spin connection ω ab M The Torsion T a MN D M e a N = Me a N + ωab M eb N acting only on the frame index b. is the antisymmetrised covariant derivative T a MN = D [Me a N].

Equivalence with metric formalism The spacetime-vector φ M and frame-vector φ a are related by φ M = e a M φ a, such that the covariant derivative M φ N and (D M φ) a are related by M φ N = e a N (D Mφ) a. This results in the expression for the Christoffel symbol, Γ MN K = e K a D M e a N which results in the covariant constancy of the metric K g MN = 0, AS REQURED! When the Christoffel symbol is symmetric Γ MN K = Γ NM K Γ [MN] K = e K a T a MN = 0 T a MN = 0. This is called a Levi-Civita connection.

p-einstein-hilbert systems in all d-dimensions Employing the Vielbein field em a the definitions of the gravitational curvature and torsion are RMN ab = (D [M D N] ) ab φ b = ( M ωn ab NωM ab M ωcb N ωac N ωcb M ) φb TMN a = D [MeN] a = MeN a NeM a + ωac M ec N ωac N ec M, with M = 1, 2,..., d ; a = 1, 2,..., d. Splitting the indices on the Levi-Civita symbols as ε M 1M 2...M 2p M 2p+1...M d and ε a1 a 2...a 2p a 2p+1...a d d-dimensional p-einsten-hilbert (p-eh) Lagrangians are L (p,d) EH = ε M 1M 2...M 2p M 2p+1...M d e a 2p+1 M 2p+1 e a 2p+2 M 2p+2... e a d M d ε a1 a 2...a 2p a 2p+1...a d R a 1a 2 R a 3a 4 M 3 M 4... R a 2p 1a 2p M 2p 1 M 2p For p = 0 this is a total divergence, for p = 1 it is the usual Einstein-Hilbert Lagrangian in d-dimensions, for p = 2 it is the usual Gauss-Bonnet Lagrangian in d-dimensions, etc.

The p-einstein equations Zero Torsion: The Levi-Civita connection is not an independent field since T a MN = D [Me a N] = 0 εmn... ω ab M eb N = εmn... M e a N can be inverted to give a closed form expression for the spin connactin ωm ab ωab M [e, e 1, e]. The p-einstein equations follow from the variation of the free-standing Vielbeine in the p-eh Lagrangians L (p,d) EH. (Note that the curvature dependant terms are total divergence.) Non-zero Torsion: n this case the p-eh Lagrangians must be varied separately w.r.t. the spin-connection yielding the Torsion equation, e.g. varying L (2,5) EH w.r.t. ωpq 2ε LMNRS ε pqabc RRS bc T MN a = matter current L

The p-einstein-hilbert systems in d-dimensions, for = 1, 2, 3 are L (1,d) EH = ε M 1M 2 M 3...M d e a 3 M 3 e a 4 M 4... e a d M d ε a1 a 2 a 3...a d R a 1a 2 L (2,d) EH = ε M 1M 2 M 3 M 4 M 5...M d e a 5 M 7 e a 6 M 6... e a d M d ε a1 a 2 a 3 a 4 a 5...a d R a 1a 2 R a 3a 4 M 3 M 4 L (3,d) EH = ε M 1M 2...M 7...M d e a 7 M 7... e a d M d ε a1 a 2...a 7...a d R a 1a 2 R a 3a 4 M 3 M 4 R a 5a 6 M 5 M 6 with the resulting Einstein equations E M d a d = 0 ε M 1M 2 M 3...M d e a 3 M 3 e a 4 M 4... e a d 1 M d 1 ε a1 a 2 a 3...a d R a 1a 2 = 0 ε M 1M 2 M 3 M 4 M 5...M d e a 5 M 7 e a 6 M 6... e a d 1 M d 1 ε a1 a 2 a 3 a 4 a 5...a d R a 1a 2 R a 3a 4 M 3 M 4 = 0 ε M 1M 2...M 7...M d e a 7 M 7... e a d 1 M d 1 ε a1 a 2...a 7...a d R a 1a 2 R a 3a 4 M 3 M 4 R a 5a 6 M 5 M 6 = 0 R M d a d (2p) 1 2p R(2p) em d a d = 0 R M d a d (2p) = R M d N d (2p) e N d a d, R(2p) = R a d M d (2p) e M d a d the p-ricci tensor and the p-ricci scalar.

Gravitational Chern-Pontryagin densities: two choices Passage from non-abelian gauge (YM) fields to gravity is achieved by replacing the YM connection and curvature in evaluating the Traces defining the (YM) CP densities: A M = 1 2 ωab M γ ab, F MN = 1 2 Rab MN γ ab, a = 1, 2,... D ; where γ ab are the generators of SO(D) γ ab = 1 4 [γ a, γ b ] D = 2n and γ a are the Dirac matrices in D = 2n dimensions. Since D = 2n is even, there exists a chiral matrix γ D+1, γ 2 D+1 = 1.

n evaluating the Traces there are two choices, leading to two types of gravitational CP (GCP) densities, Type- without, and Type- with, γ 2n+1 C (n) = ε M 1M 2...M 2n 1 M 2n R a 1a 2... R a 2n 1a 2n M 2n 1 M 2n Tr γ a1 a 2... γ a2n 1 a 2n C (n) = ε M 1M 2...M 2n 1 M 2n R a 1a 2... R a 2n 1a 2n M 2n 1 M 2n Tr γ D+1 γ a1 a 2... γ a2n 1 a 2n. C (n) C (n) are called (gravitational) Chern-Pontryagin densities, are called Euler-Hirzbruch densities. t follows from the prperties of the Clifford algebras that: C (n) = 0, for odd n C (n) = ε M 1M 2...M 2n 1 M 2n ˆδ a n+1...a 2n a 1 a 2...a n R a 1a 2... R a n 1a n M n 1 M n, for even n C (n) = ε M 1M 2...M 2n 1 M 2n ε a1 a 2...a 2n 1 a 2n R a 1a 2... R a 2n 1a 2n M 2n 1 M 2n, for all n.

Gravitational Chern-Simons densities: one choice These are constructed by evaluating the Traces in the YM (na) Chern-Simons densities in D 1 dimensions. The na-cs densities are extracted from the na-cp densities which are defined for SO(D) gauge group. The SO(D) gauge group is not the required group for the frame indices in D 1 = 2n 1 dimensions. Again, there are two choices, leading to two types of gravitational CP (GCP) densities, Type- without, and Type- with, γ D+1. There are two steps in this process: Evaluate the Traces yielding gravitational densities in D 1 dimensions with SO(D) frame index, Contract the gauge group SO(D) SO(D 1).

1st step: n = 2, 3, 4 Employing the index notation µ = 1, 2,... D 1, (D = 2n), list the two types Traces Ω (n) and Ω (n) for n = 2, 3, 4. Tr (2) = ε λµν δ a b ab ω ab λ Tr (2) = ε λµν ε abcd ωλ ab Tr (3) = 0 Tr (3) = ε λµνρσ ε abcdef ωλ ab [ R a b µν 2 3 (ω µω ν ) a b ] [ R cd µν 2 3 (ω µω ν ) cd ], [ R cd Tr (4) = ε λµνρστκˆδa b c d abcd ωλ ab, µνrρσ ef Rµν cd (ω ρ ω σ ) ef + 2 5 (ω µω ν ) cd (ω ρ [ RµνR cd a b ρσ R c d τκ 4 5 Rcd µνr a b ρσ (ω τ ω κ ) c d + 4 5 Rcd µν (ω ρ ω σ ) a b (ω τ ω κ ) c d 8 35 (ω µω ν ) cd ( [ Tr (4) = ε λµνρστκ ε abcdefgh ωλ ab RµνR cd ρσr ef τκ gh 4 5 Rcd µνrρσ ef (ω τ ω κ ) gh 2 5 Rcd µ + 4 R cd µν (ω ρ ω σ ) ef (ω τ ω κ ) gh 8 (ω µ ω ν ) cd (ω

2nd step: Group contraction Note that the frame indices on ω ab µ, have the range a = 1, 2,..., D. The correct range for the frame indices of gravity in d = D 1 dimensions is α = 1, 2,..., d, with d = D 1. This contraction is effected by truncating the components of the spin connection ωµ ab = (ωµ αβ, ωµ αd ) according to Ω (2) = 1 2 2! ελµν δ α β αβ Ω (2) = 0, Ω (3) = 0, Ω (3) = 0, Ω (4) = ˆΩ (4) = 0, ωµ αd = 0 Rµν αd = 0. ω αβ λ 1 2 6! ελµνρστκˆδ α β γ δ αβγδ ω αβ λ [ R α β µν 2 3 (ω µω ν ) α β ] [ RµνR γδ α β ρσ R γ δ τκ 4 5 Rγδ µνr α β ρσ (ω τ ω κ ) + 4 5 Rγδ µν (ω ρ ω σ ) α β (ω τ ω κ ) γ δ 8 35 (ω µω ν ) γδ (ω

Chern-Simons gravity Not gravitational Chern-Simons density! Extend the Ansatz used above: A µ = 1 2 ωab µ γ ab, F µν = 1 2 Rab µν γ ab, a = 1, 2,... D ; by incorporating the 2nd step above: D = 2n a = (α, D), α = 1, 2,..., d, d = D 1 = 2n 1 such that now leading to F µν = 1 2 A µ = 1 2 ωαβ µ γ αβ + κ e α µ γ αd ( ) Rµν αβ κ 2 e[µ α eβ ν] γ αβ + κ Tµνγ α αd

Substituting this Ansatz for A µ in the non-abelian SO(4) CS density of Type- in d = 2n 1 = 3 (n = 2) results in the Chern-Simons gravitational Lagrangian ( L (2) CSG = κ εµνλ ε abc eλ c Rab µν 2 ) 3 κ2 eµe a ν b eλ c and in d = 2n 1 = 3 (n = 2) ( L (3) 3 CSG = κ εµνρσλ ε abcde 4 ee λ Rab µν Rρσ cd eρe c σ d eλ e Rab µν + 3 ) 5 κ4 eµe a ν b eρe c σ d eλ e