Higher Berry phase of fermions and index theorem

A bstract When a quantum field theory is trivially gapped, its infrared fixed point an invertible theory. The partition function of the records response to various background fields in long-distance limit. set can include spacetime-dependent coupling constants, which case we call corresponding parameterized We study such theories arising from free Dirac fermions with mass parameters using Atiyah-Patodi-Singer index theorem for superconnections. In particular, infinitesimal modulation encoded into higher analog Berry curvature, provide general formula. curvature vanishes, still be nontrivial if there remaining torsional phase, list some computable examples.