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Substitution in the following supersymmetry transformation

  1. Dec 2, 2014 #1
    I was reading in this book: Supergravity for Daniel Freedman and was checking the part that has to do with Extremal Reissner Nordstrom Black Hole. He was using killing spinors (that I am very new to).

    I was understanding the theory until he stated with the calculations:

    He said that the Supersymmetry transformation (21.49) in his book is:

    $$\delta \psi_\mu^i=(\partial_\mu +1/4 \gamma^{ab}\omega_{\mu ab})\epsilon^i -1/8\sqrt{2}\kappa \gamma^{ab}F_{ab}\epsilon^{ij} \gamma_\mu \epsilon_j$$

    He then started "It is convenient to work with the chiral projections of the two Majo-rana spinors. We thus use the up/down position of the R-symmetry indices, now denoted by **(A, B=1, 2)**, to specify the chirality. Thus, for the SUSY transformation parameters,
    we have"

    $$\gamma_* \epsilon^A= P_L \epsilon^A = \epsilon^A$$ and $$\gamma_* \epsilon_A= - P_R \epsilon_AA = - \epsilon_A$$ where $$\gamma_*$$ is the usual $$\gamma_5$$ and $$P_L, P_R$$ are projection operators to define chiral parts.

    So back to the first equation that I wrote here: He substituted it with:
    $$ \delta \psi_{tA} = \partial_t \epsilon _A +1/2 e^{2U} \partial_i U\gamma^i \gamma^0 \epsilon_A -1/4 \sqrt{2}\kappa e^u \partial_i A_t \gamma^i \epsilon_{AB} \epsilon^b =0$$

    It is specifically the substitutions that I could not follow, where did the gamma matrices in the last equation come from?

    Concerning the spin conection I found them to be: $$\omega^{0i} = e^U \partial_iU e^0$$ and $$\omega ^{ij} = -dx^i\partial_jU+dx^j\partial_iU$$ and they each have 2 indices while in the supersymmetry transformation equation the omega has 3 indices. I know I am missing something and I hope you can help me understand the substitution better.
  2. jcsd
  3. Dec 2, 2014 #2


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    are the 2 of the 3 indices, spinor indices? then this can explain your problem by [itex]\omega_{\mu (ab)} \rightarrow \omega_{\mu \nu} [/itex]

    However I hope someone else [who has studied the book] can help you more
  4. Dec 2, 2014 #3
    oh what u wrote makes sense, what about the gamma matrices? We started with $$\gamma^{ab}$$ and then we had $$\gamma^{0}\gamma^i$$?
  5. Dec 2, 2014 #4


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    how does freedman define the [itex]\gamma^{ab} [/itex]?
    In general I see the gamma coming from the omega_[0,i]
  6. Dec 2, 2014 #5
    In "Our Conventions" page he used it in what follows (Please see attachment) and then he uses it without really defining it or at least I missed its definition some place.. I have not read the book I actually picked up the equation of supersymmetry transformation in order to work on the RN black hole. I could use any reference if you have one in mind in order to solve this problem.. Though this gave me an insight of what's going on, my problem was with the very changes that took place between the given equations and what happened to it after substitutions. Concerning the part where you said that you saw that the gamma coming from omega_[0,i]. May you please elaborate on this one?

    Attached Files:

  7. Dec 2, 2014 #6


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  8. Dec 3, 2014 #7
    I checked the paper you gave me, in eq (4.3.27) I noticed that when omega(0,0i) that corresponding gamma matrices split into 2 gammas with respective (0) and (i) upper indices but when omega (0,ij) this is not the case. Any clue of what's going on?
  9. Dec 3, 2014 #8


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    The spin connection is antisymmetric in ab. So the contraction with the gamma's only gives ab=0i=-i0 and ij=-ji contributions (where i=\=j), hence the factor of 2.
  10. Dec 3, 2014 #9
    Aha I see, and concerning the separation gamma matrices? Any ideas?
  11. Dec 4, 2014 #10


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    What do you mean by that? Up to a factor depending on conventions one has gamma_{0i} = gamma_0 gamma_i because of the Clifford algebra; these two gamma's anticommute.
  12. Dec 4, 2014 #11
    The Clifford algebra and all gamma matrix identities you need are introduced in chapter 3.
    This chapter also develops a lot of the spinor properties needed. Especially the Fierz (rearrangement) identity is used often in the chapters leading to the introduction of actual supergravity.

    That's about all I can help you with, I only read through it one time (up to chapter 11). I'm starting to work through the text just now.
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