On 2-group global symmetries and their anomalies

Abstract

In general quantum field theories (QFTs), ordinary (0-form) global symmetries and 1-form symmetries can combine into 2-group global symmetries. We describe this phenomenon in detail using the language of symmetry defects. We exhibit a simple procedure to determine the (possible) 2-group global symmetry of a given QFT, and provide a classification of the related 't Hooft anomalies (for symmetries not acting on spacetime). We also describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory. Finally, we provide a variety of examples, ranging from TQFTs (gapped systems) to gapless QFTs. Along the way, we stress that the "obstruction to symmetry fractionalization" discussed in some condensed matter literature is really an instance of 2-group global symmetry.

Modified instanton sum in QCD and higher-groups

“…This expression shows that at the intersection of two generic codimension-2 defects U 2 , we must assign the Z N p phase, and the one-form symmetry is transmuted to the three-form symmetry. The analogous situation also appears between the 0-form and 1-form symmetry, which leads to the 2-group structure [64][65][66]. Accordingly, ours is an example of the 4-group structure in 4-dimensional QFT 2 .…”

Line operators of gauge theories on non-spin manifolds

“…then S and T become duality transformations [13,28]. In particular, for the T -transformation we get a duality between U (1) τ f and U (1) τ +1 b , that is shifting θ by 2π is equivalent to changing the spin of the monopole 2 [16,21,26,29]. Altogether, we get the identifications…”

Lorentz symmetry fractionalization and dualities in (2+1)d

“…We start with a brief review of symmetry fractionalization in (2+1)d. Consider a (2+1)d QFT with a zero-form global symmetry G and a one-form global symmetry A. Symmetry fractionalization in (2+1)d is a phenomenon where the line operators can be in projective representations of G, while local operators are in linear representations of G. More specifically, the symmetry fractionalization can be realized by activating the A background field B using the pullback of an element in H 2 (G, A) by the G background field [11]. This nontrivial background for the one-form symmetry inserts the symmetry generator -Abelian anyon-at the junction of three G defects as specified by the chosen element in H 2 (G, A) [18,[20][21][22].…”

“…Since the symmetry fractionalization activates the one-form symmetry background A, the anomaly of the one-form symmetry gives rise to an anomaly of the zero-form symmetry G through symmetry fractionalizations [11]. In the case of the Lorentz symmetry fractionalization (2.1), the one-form symmetry anomaly gives rise to the framing anomaly.…”

“…Applications to summing over the spin structures, time-reversal symmetry, and level/rank dualities are explored. The Lorentz symmetry fractionalization naturally arises in Chern-Simons matter dualities that obey certain spin/charge relations, and is instrumental for the dualities to hold when viewed as non-spin theories.1 See [9,[11][12][13] for related works in (3+1)d.…”