# Anomalies for not-on-site discrete gauge symmetries

If a symmetry group $G$ (let's say finite for simplicity) acts on a lattice theory by acting only on the vertex variables, I will call it ultralocal. Any ultralocal symmetry can be gauged. However, in general there are discrete symmetries that cannot be gauged. For example, Freed and Vafa in http://link.springer.com/article/10.1007%2FBF01212418 discuss how in 1+1d one needs the pullback of a certain class in $H^3(G,U(1))$ to have some trivial periods.

So, is the converse true--that if $G$ can be gauged then there is a formulation of the theory where $G$ acts ultralocally? In other words, is a symmetry with no ultralocal action necessarily anomalous?

And if so, can we see this anomaly as an explicit class in $H^3(G,U(1))$ for 1+1d theories, for example?

It seems to me like the answer is yes. If we have no anomaly and go ahead and gauge $G$, then we can put the result on a lattice where the $G$ gauge field will live on edges. These edge variables will have the flatness condition that the start and end vertex variables differ by the action of the edge variable (an element of $G$). It seems like $G$ shouldn't act anywhere else, since in some sense gauging $G$ is a type of "fat quotient" of the theory by $G$. Thus, if we take this lattice formulation and forget the gauge field, we end up again with the original theory, but now with an ultralocal action of $G$. What remains is how to quantify this anomaly in the cohomology group.

• Hi Ryan, just read your profile and thought maybe you are interested in this too ... Aug 18, 2013 at 19:11
• We also had some discussion about the topic here and here. What is really needed to (re)start is to bring a large enough group if interested people together ... Aug 18, 2013 at 20:03
• I would say after 3.5 years working on this problem that it's still open :) Mar 16, 2018 at 16:53

Your question is very interesting. I would like to mention something along the line of your question, but perhaps from another viewpoint. Recently there are some better understanding along the thinking between

(1)"whether a theory is free from anomaly (the anomaly matching condition satisfied),"

(2)"whether the symmetry of a theory is on-site symmetry,"

(3)"whether the symmetry of a theory can be gauged,"

(4)"whether the theory can exist alone in its own dimension without an extra bulk dimension,"

(5)"whether the massless modes of the theory can be gapped (opened up a mass gap) without breaking the assigned symmetry."

The insight connects to a topic in condensed matter physics, such as the intrinsic topological order and symmetric protected topological order(such as the topological insulator).

(A) In this paper: Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders, it is proposed that the anomaly can be classified by a cohomology group of $$\text{Free}[\mathcal{H}^{d+1}(G,U(1))] \oplus \mathscr{H}_\pi^{d+1}(BG,U(1))$$ The ABJ anomalies are classified by $\text{Free}[\mathcal{H}^{d+1}(G,U(1))]$, while $\mathscr{H}_\pi^{d+1}(BG,U(1))$ is beyond ABJ type, such as for discrete gauge anomaly.

In this 1303.1803, it is explained the above notions, to a certain degree (1),(2),(3),(4) are related, or even identical.

(B) In this paper: A Lattice Non-Perturbative Definition of 1+1D Anomaly-Free Chiral Fermions and Bosons, it has been shown the relation between (1),(4) and (5), i.e. the anomaly matching condition = the massless modes of the theory can be fully gapped, for a specific case that the theory has a U(1) symmetry:

$$%{\boxed{ \text{ ABJ's U(1) anomaly matching condition in 1+1D} \Leftrightarrow\\ \text{the boundary fully gapping rules of 1+1D boundary/2+1D bulk }\\ \text{with unbroken U(1) symmetry.} %}}$$

There in 1307.7480, based on this understanding, the chiral fermions on the lattice is proposed by including strong interactions. It avoids Fermion doubling problem due to the theory is not free, but interacting. A similar idea to put a SO(10) chiral gauge theory and its induced standard model on the lattice is proposed in 1305.1045.

Back to your question, you had said that $$\text{Any ultralocal symmetry can be gauged}$$ I suspect this understanding can connect to Dijkgraaf-Witten theory. It seems to me your converse statement: $$\text{if G can be gauged then there is a formulation of the theory where G acts ultralocally.}$$ would also be true. If one use the understanding that my above listed notions, (3) a theory can be gauged $\leftrightarrow$ (1) a theory is free from anomaly $\leftrightarrow$ (2) the symmetry is on-site symmetry. We suppose one can further use the idea of Dijkgraaf-Witten theory, and the correspondence between "the gauge symmetry $G$ variables acted on the links(the gauge symmetry $G$ of a gauge theory)" and "the symmetry $G$ acted on the vertices(the global symmetry $G$ of a Symmetry Protected Topological order)", in principle "$G$ acts on the links" and "$G$ acts on the vertices" are dual to each other, then we may argue your statement is an "if and only if" statement.

• Thanks for your answer, Idear. You point out some interesting references I didn't know about. SPT phases are indeed the context I've been thinking about this problem in and that any ultralocal symmetry can be gauged does connect to DW theory. Aug 19, 2013 at 18:49
• . @Idear: About your recent question about unexplained breakthroughs, I think it was a really good question. It may be on-topic on Math Overflow (which also does accept advanced physics questions; many of your questions may in fact be rather on-topic there.). Aug 25, 2013 at 12:43
• Thanks DIMension 10, are you really 14 years-old??? Dec 20, 2013 at 16:32