The algebra of anomaly interplay

Abstract

We give a general description of the interplay that can occur between local and global anomalies, in terms of (co)bordism. Mathematically, such an interplay is encoded in the non-canonical splitting of short exact sequences known to classify invertible field theories. We study various examples of the phenomenon in 2, 4, and 6 dimensions. We also describe how this understanding of anomaly interplay provides a rigorous bordism-based version of an old method for calculating global anomalies (starting from local anomalies in a related theory) due to Elitzur and Nair.

Omega vs. pi, and 6d anomaly cancellation

“…[59,60]. The present authors also offered a condensed explanation of the method in [34]. Unless stated explicitly otherwise, all cohomology groups in this appendix have coefficients in Z/2.…”

“…The exponentiated η-invariant can thence be computed by a variety of methods, for example using the 5d mod 2 index theorem again. Alternatively, one can embed the SU(2) connection inside a large group whose 5 th spin bordism group vanishes, such as U(2) or SU(3), and then extend the bundle to a bounding 6-manifold and use the APS index theorem to relate η to the integral of a degree-6 anomaly polynomial [33,34]. For the isospin j representation of SU(2), the calculation yields…”

“…But there are in general global anomalies, as can be seen for example from e.g. Ω Spin 3 (BZ/2) ∼ = Z/8, corresponding to a mod 8 valued anomaly in a 2d theory with a unitary Z/2 symmetry [34][35][36][37]. However, this anomaly cannot technically be seen from the homotopy group π 2 (Z/2), which vanishes because any basepoint preserving map must map the whole S 2 to a fixed base point in Z/2 by continuity.…”

Global anomalies on the Hilbert space

“…4 While detecting the anomalies induced by transformations connected to the identity of a Lie group G is textbook material, the detection of global anomalies, which includes anomalies for all discrete symmetries, is more subtle [34,35]. 5 The approach is often indirect, for example by embedding some global anomalies into a perturbative ones (see for example the recent work [44] and references therein).…”

Comments on the Atiyah-Patodi-Singer index theorem, domain wall, and Berry phase

“…Recently, non-perturbative generalization of anomaly inflow has been discussed in terms of the eta-invariant [6][7][8]. It has been applied to high energy physics and condensed matter physics [9][10][11][12][13][14][15][16][17][18][19]. The APS index theorem above describes a special case of anomaly inflow [20].…”