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WZW terms without anomalies: generalised symmetries in chiral Lagrangians
by Joe Davighi, Nakarin Lohitsiri
Submission summary
Authors (as registered SciPost users):  Joe Davighi · Nakarin Lohitsiri 
Submission information  

Preprint Link:  https://arxiv.org/abs/2407.20340v1 (pdf) 
Date submitted:  20240814 18:17 
Submitted by:  Davighi, Joe 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider a 4d nonlinear sigma model on the coset $(\mathrm{SU}(N)_L \times \mathrm{SU}(N)_R \times \mathrm{SU}(2))/(\mathrm{SU}(N)_{L+R}\times \mathrm{U}(1))\cong \mathrm{SU}(N) \times S^2$, that features a topological WessZuminoWitten (WZW) term whose curvature is $\frac{n}{24\pi^2}\mathrm{Tr}(g^{1}dg)^3 \wedge \mathrm{Vol}_{S^2}$ where $g$ is the $\mathrm{SU}(N)$ pion field. This WZW term, unlike its familiar cousin in QCD, does not match any chiral anomaly, so its microscopic origin is not obviously QCDlike. We find that generalised symmetries provide a key to unlocking a UV completion. The $S^2$ winding number bestows the theory with a 1form symmetry, and the WZW term intertwines this with the $\mathrm{SU}(N)^2$ flavour symmetry into a 2group global symmetry. Like a 't Hooft anomaly, the 2group symmetry should match between UV and IR, precluding QCDlike completions that otherwise give the right pion manifold. We instead construct a weaklycoupled UV completion that matches the 2group symmetry, in which an abelian gauge field connects the QCD baryon number current to the winding number current of a $\mathbb{C}P^1$ model, and explicitly show how the mixed WZW term arises upon flowing to the IR. The coefficient is fixed to be the number of QCD colours and, strikingly, this matching must be 'treelevel exact' to satisfy a quantization condition. We discuss generalisations, and elucidate the more intricate generalised symmetry structure that arises upon gauging an anomalyfree subgroup of $\mathrm{SU}(N)_{L+R}$. This WZW term may even play a phenomenological role as a portal to a dark sector, that determines the relic abundance of dark matter.
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 Provide a novel and synergetic link between different research areas.
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Current status:
Reports on this Submission
Strengths
 Clear and explicit explanation
 New type of symmetry matching between gauge theory and sigma model
Report
This manuscript studies the symmetry of a $\sigma$model with a particular topological term and its possible UV completion. Specifically, the topological term is specified by a (generalized) cohomology class of the target manifold $SU(N) \times S^2$, which can be seen as a generalization of the Wess–Zumino–Witten (WZW) term. The authors argue that the $\sigma$model possesses a continuous 2group symmetry that enforces the presence of a continuous 1form symmetry, which in turn prohibits stronglycoupled nonabelian gauge theory as its UV completion. The authors propose a UV completion involving an abelian gauge field that matches the symmetry structure.
The manuscript is wellwritten, and the argument is clear. The results are interesting, and as the authors claim, they can potentially be applied to phenomenological problems. Therefore, I recommend the publication of this manuscript after the following minor issues are addressed:
Requested changes
1. The manuscript suggests a theory with abelian gauge field as a UV completion of the $\sigma$model. However, an Abelian gauge theory is not UV complete due to the Landau pole. Could authors provide a comment on how this "UV" abelian gauge theory could be further UV completed. In particular, does the nogo theorem imply that the flavor symmetry must be broken at some scale?
2. On Page 5: "(Co)homological [22] (or (co)bordismbased [23, 24]) classifications of Wess–Zumino–Witten (WZW) terms." I believe the (generalized) cohomological perspective of WZW terms should trace back to Freed's work, which is cited as [28] but not referenced here. I recommend including this citation for completeness.
3. The statement on page 6, "This sign can be fixed by anomaly matching: the solution with a minus sign must be chosen if the symmetry group SU(2) suffers from the mod 2 global anomaly discovered by Witten [26] in the UV," requires clarification. Since the expression (2.8) is not real positive for a general $X_4$, the "solution with a minus" does not have a canonical meaning. The branch of the square root should be chosen consistently across the potential four manifolds (with backgrounds) $X_4$. A way to do this is fixing a representative $X_4^{(0)}$ for the bordism class and choosing a branch for the representative. For other $X_4$ in the same bordism class the WZW can be defined using a bordism between $X_4^{(0)}$ and $X_4$. In this particular case one can choose $X_4^{(0)}$ to have the trivial $SU(N)$ background, with which the value in Eq. (2.8) is 1, and then the sign choice for the square root of one corresponds to the $SU(2)$ anomaly.
4. On Page 9: "a topological invariant called the Postnikov class, which is the pair $(\hat{\kappa}_L, \hat{\kappa}_R) \in H^3(BSU(N)^2; U(1)) \cong \mathbb{Z} \times \mathbb{Z}$." The meaning of "$H^n(BG,U(1))$" with $G$ being continuous is ambiguous, as there can be various versions of it. The authors can either comment on this point or avoid it by writing $H^4(BSU(N),\mathbb{Z})$ instead.
5. On Page 24: When $X_\phi \neq 1$, the $\mathbb{Z}_{X_\phi}$ subgroup of $U(1)$ does not act on the scalar field $\phi$, and thus the discrete subgroup acts on $S^2$ trivially. The IR theory is not $S^2/\mathbb{Z}_{X_\phi}$target sigma model if it means the geometric quotient, and it is rather a $S^2$target sigma model coupled with $Z_{X_\phi}$ topological gauge theory. Equivalently, the IR theory can be obtained by gauging $Z_{X_\phi}$ subgroup of the $U(1)$ oneform symmetry of $S^2$target sigma model.
Recommendation
Ask for minor revision