While reviewing some basic field theory, I once again encountered the Bianchi identity (in the context of electromagnetism). It can be written as $$\partial_{[\lambda}\partial_{[\mu}A_{\nu]]}=0.$$ Here, $A_\nu$ is of course the electromagnetic potential. This formula is immediately reminiscent of the Jacobi identity: $$[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0.$$
This is even clearer in general relativity, where we have $$\nabla_{[\lambda}R_{\rho\sigma]\mu\nu},$$ which we can rewrite, remembering the definition of the Riemann tensor in terms of the commutator of covariant derivatives, as $$[[\nabla_\lambda,\nabla_\rho],\nabla_\sigma]+[[\nabla_\rho,\nabla_\sigma],\nabla_\lambda]+[[\nabla_\sigma,\nabla_\lambda],\nabla_\rho]=0. $$ This all looks like there should be some profound connection here, but I'm incapable of pinpointing it. Maybe one of the experts here can make this more precise? I'd love to get to know more about this. Any comments are much appreciated. I'd also be grateful if someone could suggest (more) appropriate tags to use.
Answer
I) The proofs of both the first (algebraic) Bianchi identity and the second (differential) Bianchi identity crucially use that the connection $\nabla$ is torsionfree, so they are not entirely consequences of the Jacobi identity. Proofs of the Bianchi identities are e.g. given in Ref. 1.
II) The second Bianchi identity may be formulated not only for a tangent bundle connection but also for vector bundle connections.
III) The Lie bracket in the pertinent Jacobi identities is the commutator bracket $[A,B]:=A\circ B -B\circ A$. The Jacobi identity follows because operator composition "$\circ$" is associative.
IV) In the context of Yang-Mills theory and EM, the second Bianchi identity follows because the gauge potential $A_{\mu}$ and the field strength $F_{\mu\nu}$ may be viewed as (part of) a covariant derivative and corresponding curvature tensor, respectively.
References:
- M. Nakahara, Geometry, Topology and Physics, Section 7.4.
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