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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)."

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.

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.

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)."

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 connection between "the gauge symmetry $G$ variables acted on the links" and "the symmetry $G$ acted on the vertices", 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.

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)."

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.

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)."

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 can further use the idea of Dijkgraaf-Witten theory, and the connection between "the gauge symmetry $G$ variables acted on the links" and "the symmetry $G$ acted on the vertex", then we may argue your statement is an "if and only if" statement.

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)."

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 connection between "the gauge symmetry $G$ variables acted on the links" and "the symmetry $G$ acted on the vertices", 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.