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WZW terms without anomalies: generalised symmetries in chiral Lagrangians
by Joe Davighi, Nakarin Lohitsiri
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Submission summary
| Authors (as registered SciPost users): | Joe Davighi · Nakarin Lohitsiri |
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| Preprint Link: | https://arxiv.org/abs/2407.20340v1 (pdf) |
| Date submitted: | 2024-08-14 18:17 |
| Submitted by: | Davighi, Joe |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
We consider a 4d non-linear 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 Wess-Zumino-Witten (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 QCD-like. We find that generalised symmetries provide a key to unlocking a UV completion. The $S^2$ winding number bestows the theory with a 1-form symmetry, and the WZW term intertwines this with the $\mathrm{SU}(N)^2$ flavour symmetry into a 2-group global symmetry. Like a 't Hooft anomaly, the 2-group symmetry should match between UV and IR, precluding QCD-like completions that otherwise give the right pion manifold. We instead construct a weakly-coupled UV completion that matches the 2-group 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 'tree-level exact' to satisfy a quantization condition. We discuss generalisations, and elucidate the more intricate generalised symmetry structure that arises upon gauging an anomaly-free 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.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report
The paper discusses a phenomenon where the quantized level of a Wess-Zumino-Witten coupling term of an infrared pion theory matches the structure constant (Postnikov class) of a 2-group symmetry in the ultraviolet. This is to be contrasted with the more famliar case where the level of the WZW term matches the UV chiral anomaly of an ordinary flavor symmetry. A prominent example discussed is the model with target space $SU(N) \times S^2$.
Such a 2-group symmetry structure constrains possible UV completions of the model, as it requires the UV theory to have an exact 1-form symmetry. For instance, it excludes QCD-like completions with non-abelian gauge groups.
The paper is well-written and emphasizes an interesting phenomenon, and it is recommended that the paper is published in SciPost.
Below are minor comments for authors’ consideration:
- On page 2, regarding Ref. [11], to be more precise the mixed anomaly in the pure SU(N) Yang-Mills theory at $\theta = \pi$ is nontrivial only for even N. For odd N, no such mixed anomaly exists, but instead there is the anomaly in the space of $\theta$-angles.
- Around Ref. [12], there are also other early works on higher-group symmetries in (topological) quantum field theories, such as the one by Kapustin and Thorngren https://arxiv.org/pdf/1309.4721.
- In the discussion around Eq. (2.8), it is not so clear why the sign ambiguity should be matched with Witten’s $SU(2)$ anomaly. Perhaps this point could be elaborated more.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
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 2-group symmetry that enforces the presence of a continuous 1-form symmetry, which in turn prohibits strongly-coupled non-abelian 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 well-written, 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 no-go theorem imply that the flavor symmetry must be broken at some scale?
2. On Page 5: "(Co)homological [22] (or (co)bordism-based [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 $U(1)$ one-form symmetry of $S^2$-target sigma model.
Recommendation
Ask for minor revision