3.5 General properties of anomaly in 1+1d

3.5.1 Anomaly-BC relationship

In the above definition (3.30), \(\alpha\) depends on the choice of \(\mathcal{B}_0\). So is this projective phase a property of the boundary condition, rather than a theory? The answer is no, the cohomology class of the projective phase should be uniform over the boundary conditions of a fixed theory. To see this, we assume that the boundary conditions admits a sum so that \(\mathcal{H}_{\mathcal{B}_1\oplus \mathcal{B}_2,\mathcal{B}_R} = \mathcal{H}_{\mathcal{B}_1,\mathcal{B}_R} \oplus \mathcal{H}_{\mathcal{B}_2,\mathcal{B}_R}\). The boundary condition \(\mathcal{B}_1\oplus \mathcal{B}_2\) is like putting a qubit entangled with the boundary condition (e.g. the value of \(X(0)\) in the periodic scalar example). Then, we also want the symmetry \(U_g\) acts on \(\mathcal{H}_{\mathcal{B}_1\oplus \mathcal{B}_2,\mathcal{B}_R}\). However if the cohomology classes associated to \(\mathcal{B}_1\) and \(\mathcal{B}_2\) were different, the projective “phase” on this direct sum space is not a overall phase, but just a diagonal matrix not proportional to the identity. This breaks the assumption that the symmetry is \(G\). Therefore, the cohomology class is the property of the theory, not the boundary condition, and you can calculate it by picking any boundary condition you like.

This leads to the following theorem, first claimed by  [18] 21:

Theorem 3.1 If there is a simple boundary condition \(\mathcal{B}\) such that \(g \mathcal{B} =\mathcal{B}\) for elements of a symmetry group \(g\in G\), this symmetry \(G\) is non-anomalous, i.e. the cohomology class is trivial. (This hold for a QFT in a general dimensions.)

Indeed by the definition (3.26) \(\alpha = 0\) by choosing \(\mathcal{B}_0 = \mathcal{B}\) in such a case.22

3.5.2 Countinuous symmetry and anomalous Ward identity

3.5.3 Anomaly as the obsturction to gauging

Colloquial definition of “anomaly” might be that a symmetry has an anomaly when it cannot be (consistently) gauged, e.g. an obstruction in gauging the symmetry. Let us see how the definition we used also leads to this obstruction when \(G\) is discrete.

If the considered symmetry \(G\) has an anomaly, it does not have a simple boundary condition that is invariant under the symmetry. However, one can always have a non-simple \(G\)-invariant boundary condition of the form \[\begin{equation} \tag{3.37} G\mathcal{B} := \bigoplus_{g\in G} g\mathcal{B}. \end{equation}\] Therefore, we expect that the boundary condition \(G\mathcal{B}\) survives the gauging. However, the symmetry \(G\) does not linearly act on \(\mathcal{H}_{G\mathcal{B},G\mathcal{B}}\) (and the “projective phase” is even a matrix). Thus we cannot define a healthy notion of “gauging G” acting on this Hilbert space \(\mathcal{H}_{G\mathcal{B},G\mathcal{B}}\). 23

References

[18]
R. Thorngren and Y. Wang, Anomalous symmetries end at the boundary, JHEP 09, 017 (2021), https://arxiv.org/abs/2012.15861.
[19]
S. Hellerman, D. Orlando, and M. Watanabe, Quantum Information Theory of the Gravitational Anomaly, (2021), https://arxiv.org/abs/2101.03320.
[20]
E. Witten, The ”Parity” Anomaly On An Unorientable Manifold, Phys. Rev. B 94, 195150 (2016), https://arxiv.org/abs/1605.02391.
[21]
J. Wang, X.-G. Wen, and E. Witten, Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions, Phys. Rev. X 8, 031048 (2018), https://arxiv.org/abs/1705.06728.

  1. See also  [19].↩︎

  2. The work by  [18] is not this trivial, as their definition of the anomaly is different from us.↩︎

  3. One can try to enlarge the group into \(G'\) by including the projective phases as their group elements for all \(\mathcal{B}\). One can gauge this enlarged \(G'\), but as the projective phases are absent on \(S^1\), the new elements acts trivially on \(\mathcal{H}_{S^1}\). Gauging a symmetry trivially acting on \(\mathcal{H}_{S^1}\) means the gauged theory has a TQFT sector, in particular has degenerate vacua. This is first pointed out and utilized in  [20,21].↩︎