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## "Generalized symmetries in condensed matter", John McGreevy, 2023

*Reviewed February 24, 2024*

*Citation:* McGreevy, John. "Generalized symmetries in condensed matter." Annual Review of Condensed Matter Physics 14 (2023): 57-82.

*Web:* https://arxiv.org/abs/2204.03045

*Tags:* Physical, Expository, SPT/SETs

This paper discusses generalized "q-form" symmetries in condensed matter systems.
Abstractly, a q-form symmetries are summarized as
*global symmetries for which the charged operators are of
space-time dimension q.* That is, symmetries which
are comprised of transformations whose domain is q-dimensional.
When q=0, this means that the symmetry acts on points. Hence,
a 0-form symmetry is a standard gauge symmetry. When q is top
dimensional, this is a global symmetry. q-form symmetries were
introduced in

> Kapustin, Anton, and Ryan Thorngren. "Higher symmetry and gapped phases of gauge theories." Algebra, Geometry, and Physics in the 21st Century: Kontsevich Festschrift (2017): 177-202.

and clarified in

> Gaiotto, Davide, et al. "Generalized global symmetries." Journal of High Energy Physics 2015.2 (2015): 1-62.

One of the key philosophical takeaways from q-form symmetry is that the Landau paradigm
isn't that far off. In particular, Landau said that states of matter are classified by
symmetry breaking. This doesn't work for topological phases, since there is no local
operator which commutes with the Hamiltonian (no "symmetry") but yet there
is still non-trivial order. One way to reckon with this is to throw away the
Landau paradigm. Another is to consider more generalized symmetries, in which
can we find that topological order IS still classified by symmetry breaking.

One subtelty emphasized by the author is that not every symmetry can be gauged
- there can be an anomaly which prevents this from happening. Two different phases
can have the exact same symmetries, but have different anomaly. Including this,
we get the generalized Landau paradigm: phases of matter are classified by
their symmetries and their anomalies.

These sorts of generalized symmetries have also found great importance
in high energy physics circles. Are good review is found here:

> Cordova, Clay, et al. "Snowmass white paper: Generalized symmetries in quantum field theory and beyond." arXiv preprint arXiv:2205.09545 (2022).