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In 4-spacetime dimension, is there U(1) gauge field chiral anomaly associated with Majorana fermion (or I am not sure if it is equivalent, majorana representation)? Besides, I have read from several papers that there are gravitational chiral anomaly associated with both Dirac fermion and Majorana fermion. The only difference is the 1/2 factor in front of the anomaly term. Where does this difference appear in Fujikawa's chiral rotation method?

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  • $\begingroup$ The $1/2$ difference in the perturbative gravitational anomaly is true in $1+1$ dimensions. The mathematical reason is probably that Grassmann integral for Majorana fermions gives a Pfaffian while for Dirac fermions you get a Determinant, and $\text{Det}=\text{Pf}^2$. Basically Majorana fermions have "half" degrees of freedom compared to Dirac fermions. $\endgroup$
    – Meng Cheng
    Commented Mar 27, 2015 at 21:15
  • $\begingroup$ Thanks for the comment! Could you please also give me a clue or reference about what should the functional integral measure look like in the Majorana case? @Meng Cheng $\endgroup$
    – qc2014
    Commented Mar 27, 2015 at 21:23
  • $\begingroup$ @MengCheng I'm not sure this question has something to do with condensed-matter, does it ? In matter, there is no Majorana fermions, only Majorana mode (in short, the Majorana-staff exists only at zero energy, so it's not a particle as usual). I would naively guess there is no anomaly in condensed matter associated to the Majorana's physics, am I correct ? $\endgroup$
    – FraSchelle
    Commented Mar 29, 2015 at 10:08
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    $\begingroup$ @FraSchelle Majorana fermion (not zero modes) can show up in two ways: 1. Non-chiral version can arise as a critical theory, for example the critical point of the transverse Ising model in 1+1. 2. Chiral version shows up on the edge of 2d p+ip superconductor, which has gravitational anomaly. $\endgroup$
    – Meng Cheng
    Commented Mar 29, 2015 at 21:25
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    $\begingroup$ @FraSchelle Thanks for your comment! I am also interested in examples. I also have a followup question: does gravitational anomaly appears in 3D topological superconductor? If it exists, is it related to Majorana fermions on 2D surface? $\endgroup$
    – qc2014
    Commented Mar 31, 2015 at 21:02

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