Anomalies and symmetry fractionalization

qq-charges describe the possible actions of a generalized symmetry on qq-dimensional operators. In Part I of this series of papers, we describe qq-charges for invertible symmetries; while the discussion of qq-charges for non-invertible symmetries is the topic of Part II. We argue that qq-charges of a standard global symmetry, also known as a 0-form symmetry, correspond to the so-called (q+1)(q+1)-representations of the 0-form symmetry group, which are natural higher-categorical generalizations of the standard notion of representations of a group. This generalizes already our understanding of possible charges under a 0-form symmetry! Just like local operators form representations of the 0-form symmetry group, higher-dimensional extended operators form higher-representations. This statement has a straightforward generalization to other invertible symmetries: qq-charges of higher-form and higher-group symmetries are (q+1)(q+1)-representations of the corresponding higher-groups. There is a natural extension to higher-charges of non-genuine operators (i.e. operators that are attached to higher-dimensional operators), which will be shown to be intertwiners of higher-representations. This brings into play the higher-categorical structure of higher-representations. We also discuss higher-charges of twisted sector operators (i.e. operators that appear at the boundary of topological operators of one dimension higher), including operators that appear at the boundary of condensation defects.

Section: -Chargesmentioning

confidence: 89%

Generalized charges, part I: Invertible symmetries and higher representations

Bhardwaj,

Schäfer-Nameki

2024

“…As explained in Appendix C (see also Refs. [42,43]), these anomalies are in one-to-one correspondence with different fractionalization patterns of the…”

Section: Full Internal Symmetry Anomalymentioning

confidence: 93%

Symmetries and anomalies of Kitaev spin-S models: Identifying symmetry-enforced exotic quantum matter

Liu,

Lam,

et al. 2024

We analyze the internal symmetries and their anomalies in the Kitaev spin-SS models. Importantly, these models have a lattice version of a \mathbb{Z}_2ℤ2 1-form symmetry, denoted by \mathbb{Z}_2^{[1]}ℤ2[1]. There is also an ordinary 0-form \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)}×\mathbb{Z}_2^Tℤ2(x)×ℤ2(y)×ℤ2T symmetry, where \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)}ℤ2(x)×ℤ2(y) are \piπ spin rotations around two orthogonal axes, and \mathbb{Z}_2^Tℤ2T is the time reversal symmetry. The anomalies associated with the full \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)}×\mathbb{Z}_2^T×\mathbb{Z}_2^{[1]}ℤ2(x)×ℤ2(y)×ℤ2T×ℤ2[1] symmetry are classified by \mathbb{Z}_2^{17}ℤ217. We find that for S∈\mathbb{Z}S∈ℤ the model is anomaly-free, while for S∈\mathbb{Z}+\frac{1}{2}S∈ℤ+12 there is an anomaly purely associated with the 1-form symmetry, but there is no anomaly purely associated with the ordinary symmetry or mixed anomaly between the 0-form and 1-form symmetries. The consequences of these symmetries and anomalies apply to not only the Kitaev spin-SS models, but also any of their perturbed versions, assuming that the perturbations are local and respect the symmetries. If these local perturbations are weak, generically these consequences still apply even if the perturbations break the 1-form symmetry. A notable consequence is that there should generically be a deconfined fermionic excitation carrying no fractional quantum number under the \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)}×\mathbb{Z}_2^Tℤ2(x)×ℤ2(y)×ℤ2T symmetry if S∈\mathbb{Z}+\frac{1}{2}S∈ℤ+12, which implies symmetry-enforced exotic quantum matter. We also discuss the consequences for S∈\mathbb{Z}S∈ℤ.

“…[40,41]): The higher-form symmetry mixes with the lower form symmetries. If we wish to study the fractionalization [23,41,57,134] of only the lower form symmetry part of the higher group, this represents an obstruction to symmetry localization: we cannot only consider defects of the lower form symmetries, since some configuration of defect intersection necessarily produces the defects that generate the higher-form symmetry. All of these follow from the generalized Witten effect or the chargeflux attachment.…”

Section: D-group Symmetry In (D + 1)d Dijkgraaf-witten Gauge Theory W...mentioning

confidence: 99%

“…Refs. [23,[39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]. The non-invertible topological defects (simple examples being non-Abelian anyons in (2+1)D) define higher categorical (noninvertible) symmetries, as studied in e.g.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Higher-group symmetry in finite gauge theory and stabilizer codes

Barkeshli,

Chen,

Hsin

et al. 2024

Self Cite

A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we show how such gauge theories possess a higher-group global symmetry, which we study in detail. We derive the dd-group global symmetry and its ’t Hooft anomaly for topological finite group gauge theories in (d+1)(d+1) space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the [O_5] ∈ H^5(BG, U(1))[O5]∈H5(BG,U(1)) obstruction that has appeared in prior work. We also show how the dd-group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the dd-group symmetry, such as a controlled Z gate in the (3+1) D \mathbb{Z}_2ℤ2 toric code.