Higher Berry phase of fermions and index theorem

Abstract

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

Heterotic global anomalies and torsion Witten index

“…Thus the target space anomalies are classified by (IΩ B ) D+2 (pt), where D = dim X is the dimension of the target space. 15 It satisfies the 14 We do not necessarily restrict our attention to massless fermions to take into account the possibility of coupling space anomalies of the type discussed in [51][52][53][54]. When we vary parameters in P of the theory T , some massive modes may become massless at some points of P and they can produce anomalies.…”

Continuous generalized symmetries in three dimensions

“…In fact, gauging a U(1) (p) symmetry gives rise to a dual U(1) (d−3−p) symmetry. Crucially, the resolution for both of these problems relies on a recent generalization of the notion of 't Hooft anomaly that has been put forward in [20][21][22] (see also [23][24][25] for similar constructions which have been referred to as higher Berry phases). 6 The traditional anomaly paradigm can in fact be extended to analyze how the partition function of a given quantum field theory can depend on background scalar fields varying over spacetime.…”