CHANG-TSE HSIEHTachikawa Y.Yonekura K.2022-06-302022-06-30202200103616https://www.scopus.com/inward/record.uri?eid=2-s2.0-85125519641&doi=10.1007%2fs00220-022-04333-w&partnerID=40&md5=fe55fcb1a3e66bed6eda0a6d9a59af39https://scholars.lib.ntu.edu.tw/handle/123456789/614623Chiral and non-chiral p-form gauge fields have gravitational anomalies and anomalies of Green-Schwarz type. This means that they are most naturally realized as the boundary modes of bulk topological phases in one higher dimensions. We give a systematic description of the total bulk-boundary system which is analogous to the realization of a chiral fermion on the boundary of a massive fermion. The anomaly of the boundary theory is given by the partition function of the bulk theory, which we explicitly compute in terms of the Atiyah–Patodi–Singer η-invariant. We use our formalism to determine the SL (2 , Z) anomaly of the 4d Maxwell theory. We also apply it to study the worldvolume theories of a single D-brane and an M5-brane in the presence of orientifolds, orbifolds, and S-folds in string, M, and F theories. In an appendix we also describe a simple class of non-unitary invertible topological theories whose partition function is not a bordism invariant, illustrating the necessity of the unitarity condition in the cobordism classification of the invertible phases. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.journal article10.1007/s00220-022-04333-w2-s2.0-85125519641WOS:000763266600001