Anomaly for Majorana fermion?

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?

• 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. – Meng Cheng Mar 27 '15 at 21:15
• 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 – qc2014 Mar 27 '15 at 21:23
• @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 ? – FraSchelle Mar 29 '15 at 10:08
• @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. – Meng Cheng Mar 29 '15 at 21:25
• @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? – qc2014 Mar 31 '15 at 21:02