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Wish start of a good year!

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- Thread starter samuelphysics
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Wish start of a good year!

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A gauge transformation is given according to the group structure of the theory, in this case Spin(3,1). It must give to gauge independence.

Happy new year.

- #3

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In eqn.2.14 of http://arxiv.org/pdf/hep-th/0610128v3.pdf e.g. they use transformations on the Killing spinor epsilon to simplify stuff (I haven't really looked at the paper carefully). The 'gauge transformation' they talk about there is just the usual Lorentz transformation on the spinor, but this transformation depends on whether one has a cosmological constant or not (so(3,2) vs so(3,1) representations) .

i hope this helps.

- #4

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@haushofer sure your answers always do help, I just want to make sure of some stuff going on in my head as I am trying to solve KSE's for the first time.

So why do people have the right to simplify a spinor that way?they use transformations on the Killing spinor epsilon to simplify stuff

Then in the paper you cited, you're saying that in the theory they're in (i.e. N=2, D=4) SUGRA, locally they can use the Spin (3,1) gauge transformation, my question was how as a student can I know which gauge transformation I can use? (I am having the thoughts that this gauge transformation might differ from one dimension to other and from certain supegravity theory to another, I might be hugely mistaken)

- #5

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If you write down a theory like a sugra-theory, you know which symmetries you have. The specific form of a Local Lorentz transformation indeed changes per dimension, because the spinor representation differs per dimension (because the transformation is described with the Clifford algebra).

So my advice for students would be to find a textbook describing the sugra theory of your interest and check all the given symmetries and the closure of the algebra.

- #6

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I was asking because if you go back to the paper you cited, specifically, you can see that the authors (upon the use of gauge transformation) turned the original very general ##\epsilon =\lambda 1 +{\mu}^{i}e^i+\sigma e^{12}## to 3 canonical forms which are ##\epsilon=e^2##, ##\epsilon=1+\alpha e^1## and finally ##\epsilon=1+\beta e^2## (this last orbit represents the Killing spinor for the IWP metric which has a time-like Killing vector)The Killing-spinor equation is invariant under a local Lorentz transformation, so why not use this freedom to make your life a bit easier?

meanwhile the other two orbits correspond to plane-waves with null Killing vector. So, it looks like each canonical spinor that is resulting from Spin(3,1) will give us different solutions, isn't it?

- #7

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I was thinking about your question about gauge transformations. Maybe it helps to know that GR can be obtained by gauging the Poincaré algebra. The gauge transformations one has are then local Lorentz transfo's, (LLT's), local translations and general coordinate transformations (because the gauge fields have a curved index). Imposing a curvature constraint effectively 'removes' this local translation, leaving you with just gct's and LLT's. For (A)dS algebra's I'm not sure if this works, because of the curvature constraint one imposes.

For N=1 Superpoincare one can do the same; now you obtain also local SUSY-transformations and an extra gauge field: the gravitino. All the SUSY transformations can be obtained by this gauging procedure; the gauge fields are as usual in the adjoint rep. of the algebra. Beyond N=1 this becomes impossible (afaik) because the multiplets become bigger and the extra gauge fields cannot be obtained by gauging the corresponding 'extra' spacetime transformations. One can gauge the R-symmetry group for SUSY algebra's, giving "gauged SUGRA's". But the SUSY-transformations for this field are not gauge transformations of the algebra.

- #8

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Well, yes, but I also get different Killing vectors for my Schwarzschild background when I use different coordinates. A gct can then always make these components look more complex, but it's still the same vector.

Could we know how this ##\epsilon=1+\beta e^2## represents the Killing spinor for the IWP metric which has a

thank you @haushofer

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