Discussions on Dai-Freed Anomalies
Abstract: In both field theories of high energy physics and field theories of condensed matter theories anomalies have been imposing constraints and bringing up new theories. While the perturbational and local anomalies from triangle diagrams are well-developed, we follow and review two works leading to non-perturbational global anomalies both related with Atiyah-Patogi-Sieger (APS) index theorem. In the context of topological fieldtheories the APS index theorem imposes an η-invariant which adds a global anomalous e−iπη/2 to the path integral from each fermions on the boundary. In the case of Pin+ (4) this will require a multiple of 16 Majorana fermions for the global anomaly to cancel. On the other hand with Dai-Freed theorem, global ’t Hooft anomalies concerned to G symmetry group on 4-dimensional spin manifold which is also classifying space BG are studied. The existence of anomalies are related to the triviality of spin bordismgroup Ω Spin5(BG), where examples with SU(2), GSM and Zn are studied with Atiyah-Hirzebruch spectral sequence method and representation of Z n rings. Further there are extensions of spin groups from Pin, SpinZ4 and Spinc , where the anomaly cancellationconstraint of 16 fermions per generation from Spin Z 4 (4) is related to that of Pin + (4) in topological superconductor. A possible candidate of Z4 charge named B − L charge or X charge from several commutative breaking patterns of Spin(10) GUT is briefly discussed with Pati-Salam model and R ⊗ C ⊗ H ⊗ O.
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