# Degree of freedom of gravitino

Please tell me how to count the degree of freedom of gravitino on the mass-shell? I read http://arxiv.org/abs/1112.3502, but I can't understand it. How about supervielbein?

haushofer
Take a look at Van Proeyen's online notes on SUGRA, or his book with Freedman. There it is really explicitly explained, both of-shell and on-shell :)

The Vielbein's DOF's are easy (of-shell): it's a matrix, so it has D^2 components in D dimensions. Local Lorentz transformations make you subtract 1/2*D*(D-1) components from it, leaving you with 1/2*D*(D+1) components for the metric. Which is the right amount for a symmetric tensor like the metric.

Take a look at Van Proeyen's online notes on SUGRA, or his book with Freedman. There it is really explicitly explained, both of-shell and on-shell :)

The Vielbein's DOF's are easy (of-shell): it's a matrix, so it has D^2 components in D dimensions. Local Lorentz transformations make you subtract 1/2*D*(D-1) components from it, leaving you with 1/2*D*(D+1) components for the metric. Which is the right amount for a symmetric tensor like the metric.

dear haushofer,I want to obtain the DOF of gravitino, but I just found the final answer at Supergravity by Freedman and Van Proeyen..I know gravitino has 2^[d/2](d-1) components in the off-shell formalism (i use local susy gauge invariance)..but I don't understand how to obtain DOF of gravitino in the on-shell formalism? (with E.O.M ##\gamma^\mu\psi_\mu=0## ) The answer is 2^[d/2]/2(d-3). Moreover I find at the book of SUGRA by West witch superveilbein (not vielbein) has 8*8*8 DOF (and general coordinate transformations and super local lorentz transformations subtract 8*8+8*6 components ) in 4-dim, but how to count these DOF?

haushofer