Nonlinear sigma model description of deconfined quantum criticality in arbitrary dimensions

We thank the referee for pointing out this reference. This reference in general describes the embedding procedure to recover global anomaly cancellation conditions from local anomaly cancellation based on advanced bordism calculation. There are two cases where the embedding procedure could go invalid. (1) the larger group does not contain the relevant irreducible representation that causes the anomaly. As discussed in 5.3 of this reference, the symplectic Majorana multiplet, which is responsible for the 5d SU(2) anomaly, cannot be embedded in U(2). However, if one finds a suitable larger group, then this embedding procedure would still work. (2) the w2w3 anomaly or the new SU(2) anomaly characterizes the global anomaly and is written in bordism invariant. This global anomaly is still present when embedding SU(2) to U(2), also with additional local anomalies. However, the conditions for local anomaly cancellation in the Spin-U(2) theory preclude a non-vanishing new SU(2) anomaly.
In our current manuscript, we use embedding to resolve the homotopy group with discrete generators, such that the homotopy group of the larger space contains integer-valued defects, which can be written in differential forms. This embedding is not directly related to the issues in the aforementioned reference, but it is still worth mentioning. For (1), since G/K contains all the information of topological defects in both symmetry-breaking phases, it is a suitable larger space to consider and reproduce the correct topological charges. For (2), we didn't consider the high codimensional topological defects, so it is unrelated to the 5th-degree bordism group. But for the anomaly matching using the Wess-Zumino-Witten term, this Z2 torsion part is relevant to the sign of the WZW term, and could be determined from the UV theory (please see arXiv:2009.00033, arXiv:2009.04692).

1- In the current manuscript, we construct the coset G/K to describe the given DQCP. In general, the Zn-valued topological defects will become Z valued after the embedding, therefore, it can be explicitly written in the action. For the GUT case, one of the topological defects is Z2 valued, therefore, it is hard to write a differential form for such a defect. One may use Cech cohomology, but it seems hard to do analytical calculations. We embed the topological defect into a large coset G/K, such that the topological defect becomes Z valued and has an associated charge operator written in explicit differential form. We then propose the exotic DQCP among GUTs can be understood by a simple O(6) nonlinear sigma model. The benefits are (1) the duality in the original gauge theory description becomes a global symmetry in the O(6) the nonlinear sigma model, (2) the WZW term assigns phase to the linking between the topological defects, (3) one can do renormalization group analysis using this description. 2- We thank the referee for pointing out this, we are lacking of description in the previous manuscript. Indeed, the relative Chern-Simons term only characterizes the anomaly of even spacetime dimensional theory. For odd spacetime dimensional theory, the anomaly is characterized by the mixed $\theta$ term. The gauged WZW term will be an exact form then. We add necessary clarification and relevant references in the revised manuscript.