Abstract
We show that massless Kaehler-Dirac (KD) fermions exhibit a mixed gravitational anomaly involving an exact $U(1)$ symmetry which is unique to KD fields. Under this $U(1)$ symmetry the partition function transforms by a phase depending only on the Euler character of the background space. Compactifying flat space to a sphere we learn that the anomaly vanishes in odd dimensions but breaks the symmetry down to $Z_4$ in even dimensions. This $Z_4$ is sufficient to prohibit bilinear terms from arising in the fermionic effective action. Four fermion terms are allowed but require multiples of two flavors of KD field. In four dimensional flat space each KD field can be decomposed into four Dirac spinors and hence these anomaly constraints ensure that eight Dirac fermions or, for real representations, sixteen Majorana fermions are needed for a consistent interacting theory. These constraints on fermion number agree with known results for topological insulators and recent work on discrete anomalies rooted in the Dai-Freed theorem. Our work suggests that KD fermions may offer an independent path to understanding these constraints. Finally we point out that this anomaly survives intact under discretization and hence is relevant in understanding recent numerical results on lattice models possessing massive symmetric phases.
Highlights
The Kähler-Dirac equation gives an alternative to the Dirac equation for describing fermions in which the physical degrees of freedom are carried by antisymmetric tensors rather than spinors
These tensors transform under a twisted rotation group that corresponds to the diagonal subgroup of the usual (Euclidean) Lorentz group and a corresponding flavor symmetry
In flat space the KählerDirac field can be decomposed into a set of degenerate Dirac spinors but this equivalence is lost in a curved background since the coupling to gravity differs from the Dirac case
Summary
The Kähler-Dirac equation gives an alternative to the Dirac equation for describing fermions in which the physical degrees of freedom are carried by antisymmetric tensors rather than spinors. Argue that there may be another natural interpretation for this degeneracy—it is required in order to write down anomaly free, and consistent, theories of interacting fermions These constraints arise because the partition function of a massless free Kähler-Dirac field in curved space transforms by a phase under a particular global Uð1Þ symmetry with the phase being determined by the Euler character χ of the background. In even dimensions a reduced Kähler-Dirac field, in the flat space limit, can be decomposed into 2D=2 Majorana spinors and we learn that consistent interacting theories in even dimensions possess 2D=2þ2 Majorana fields These fermion numbers agree with a series of anomaly cancellation conditions associated with certain discrete symmetries in dimensions two and four—see Table I and [3,4]
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