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Communications in

Mathematical Physics

Anomalies and Bosonization Ryan Thorngren Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, USA E-mail: [email protected] Received: 8 April 2019 / Accepted: 29 May 2020 Published online: 14 August 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: Recently, general methods of bosonization beyond 1+1 dimensions have been developed. In this article, we review these bosonizations and extend them to systems with boundary. Of particular interest is the case when the bulk theory is a G-symmetry protected topological phase and the boundary theory has a G ’t Hooft anomaly. We discuss how, when the anomaly is not realizable in a bosonic system, the G symmetry algebra becomes modified in the bosonization of the anomalous theory. This gives us a useful tool for understanding anomalies of fermionic systems, since there is no way to compute a boundary gauge variation of the anomaly polynomial, as one does for anomalies of bosonic systems. We take the chiral anomalies in 1+1D as case studies and comment on our expectations for parity/time reversal anomalies in 2+1D. We also provide a derivation of new constraints in SPT phases with domain defects decorated by p + i p superconductors and Kitaev strings, which is necessary to understand the bosonized symmetry algebras which appear.

1. Introduction A theory is said to have an ’t Hooft anomaly if it has a global symmetry G which cannot be gauged while preserving locality of interactions. Anomalies are quantized, so if we can identify an anomaly at weak coupling, it is guaranteed to hold at all energy scales [1]. This makes anomalies useful for constraining phase diagrams of condensed matter systems whose long range limit is strongly interacting. Likewise, in high energy theory, anomalies which appear in the UV theory constrain the theory at all energy scales [2,3]. Anomalies are characterized by the anomaly in-flow mechanism [4]: although we cannot gauge the G symmetry, we can often formulate these D-spacetime-dimensional theories as gauge-invariant boundary conditions for a G gauge theory in D + 1 spacetime dimensions (the “anomaly theory”). In simple situations, the anomaly theory has a vanishing coupling and a topological term, written as a density made out of the gauge

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field A:

 Sanom (A) =

ω(A).

(1.1)

D+1

In this case, under a gauge transformation A → A g , Sanom (A) begets a boundary variation  δSanom (A) = ω1 (A, g). (1.2) D

This variation characterizes how the boundary partition function (the partition function of our theory of interest) coupled to the gauge background A fails to be gauge-invariant. Equivalently, ω1 (A, g) tells us about a kind of higher projective representation of G on the Hilbert space of our theory [5,6]. In this way, Sanom (A) characterizes the anomaly. Possible anomaly theories Sanom (A) have been classified by supposing that Sanom (A) is a cobordism invariant of the auxilliary D + 1-manifold equipped with the