Degree of freedom of gravitino

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Discussion Overview

The discussion revolves around the counting of degrees of freedom (DOF) of the gravitino in both on-shell and off-shell formalisms within the context of supergravity (SUGRA). Participants explore various resources and methods for understanding these concepts, including the role of the supervielbein.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks clarification on how to count the DOF of the gravitino on-shell, referencing a specific paper but expressing confusion.
  • Another participant suggests consulting Van Proeyen's notes and book for explicit explanations of the DOF both on-shell and off-shell, providing a brief overview of the DOF for vielbeins.
  • A participant mentions that the gravitino has 2^[d/2](d-1) components in the off-shell formalism and questions how to derive the DOF in the on-shell formalism, citing the equation of motion and a specific answer they found.
  • One participant asks for clarification on what aspects of Van Proeyen's treatment are unclear, noting that it explicitly addresses the on-shell case and discusses initial conditions in terms of antisymmetric derivatives.

Areas of Agreement / Disagreement

Participants express differing levels of understanding regarding the treatment of DOF for gravitinos, with some referencing specific resources while others seek further clarification. No consensus is reached on the methods or interpretations presented.

Contextual Notes

Participants mention various mathematical steps and assumptions related to local Lorentz transformations and the treatment of vielbeins and supervielbeins, but these remain unresolved within the discussion.

shooride
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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?
 
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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.
 
haushofer said:
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?
 
What is it exactly that you don't understand of the treatment of Van Proeyen? He does it very explicitly on-shell, in section 5.1. It's basically a survey of the amount of initial conditions to be specified if you rewrite the EOM in terms of an antisymmetric derivative. :)
 

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