Ivancevic_Applied-Diff-Geom

126 Applied Differential Geometry: A Modern Introductionbut does not imply the dual Bianchi relation, i.e., Yang–Mills relation 15∗(d ∗ F + A ∧ ∗F ) = J.15 Recall that the Yang–Mills (YM) Lagrangian density L YM is a functional of thevector potential fields A i µ, where the internal index i ranges over {1, · · · , n}, where n isthe dimension of the gauge group, and µ is a space–time index (µ = 0, · · · , 3). The fieldtensor derived from these potential fields is (see, e.g., [Pons et. al. (2000)])F i αβ = Ai β,α − Ai α,β − Ci jk Aj αA k β ,where Cjk i are the structure constants of the gauge group Lie.density is consequently given byThe YM LagrangianL YM = − 1 4p|g|Fiµν F j αβ gµα g νβ C ij ,where C ij is a nonsingular, symmetric group metric and g is the determinant of thespace–time metric tensor (in a semi–simple Lie group, C ij is usually taken to be Cit s Ct js ;in an Abelian Lie group, one usually takes C ij = δ ij ).The derivatives of L YM with respect to the velocities of the configuration–space variables,Ȧ i α give the tangent–space functions ˆP iα corresponding to the phase–space conjugatemomenta:ˆP iα = ∂L YM= p |g|Fµνg j αµ g 0ν C ij .∂Ȧi αThe Legendre map FL is defined by mapping ˆP α i to P α i in the phase–space. Because ofthe antisymmetry of the field tensor, the primary constraints are0 = ˆP i 0 = ∂L YM∂Ȧi 0= p |g|F j µνg 0µ g 0ν C ij .A generator of a projectable gauge transformation thus must be independent of Ȧ i 0 .An infinitesimal YM gauge transformation is defined by an array of gauge fields Λ iand transforms the potential byWe denote this transformation byδ R [Λ]A i µ = −Λ i ,µ − C i jk Λj A k µ.δ R A i µ = −(D µΛ) j ,where D µ is the Yang–Mills covariant derivative (in its action on space–time scalars andYM vectors). Under this transformation, the field transforms asδ R F i µν = −Ci jk Λj F k µν ,where we work to first order in Λ i and use the Jacobi identityC i jl Cl mn + Ci ml Cl nj + Ci nl Cl jm = 0.The YM Lagrangian L YM is invariant under this transformation provided that the groupmetric is symmetric,C k li C kj = −C k lj C ki (which is if C ij = C s it Ct js ).