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Higher-form anomalies and state-operator correspondence beyond conformal invariance
by Stathis Vitouladitis
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Stathis Vitouladitis |
| Submission information | |
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| Preprint Link: | scipost_202507_00060v1 (pdf) |
| Date submitted: | July 22, 2025, 11:55 p.m. |
| Submitted by: | Stathis Vitouladitis |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We establish a state-operator correspondence for a class of non-conformal quantum field theories with continuous higher-form symmetries and a mixed anomaly. Such systems can always be realised as a relativistic superfluid. The symmetry structure induces an infinite tower of conserved charges, which we construct explicitly. These charges satisfy an abelian current algebra with a central extension, generalising the familiar Kac--Moody algebras to higher dimensions. States and operators are organised into representations of this algebra, enabling a direct correspondence. We demonstrate the correspondence explicitly in free examples by performing the Euclidean path integral on a $d$-dimensional ball, with local operators inserted in the origin, and matching to energy eigenstates on $S^{d-1}$ obtained by canonical quantisation. Interestingly, in the absence of conformal invariance, the empty path integral prepares a squeezed vacuum rather than the true ground state.
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
Strengths
1) Provides compelling evidence for state operator correspondence in non-conformal free compact boson theories.
2) Reformulates the (known) existence of infinitely many conserved charges in free boson theories as a Kac-Moody algebra, and shows that it persists when the free theory is deformed by higher-derivative corrections.
Weaknesses
1) Some derivation is quite technical and the reading is not very smooth. Moving some parts to appendices would make the whole paper much more enjoyable.
2) Sometimes the logic is not completely clear.
Report
In the free case the author argues that the algebra is sufficient to establish a state/operator correspondence on the sphere, even if the theory is not free. A very interesting finding is that the path integral without insertions does not prepare the vacuum, while the latter is prepared by the squeezing operators. This is likely to be a general fact in QFT. Moreover, even restricted to the class of theories considered, this is a particular case of a more general fact: the simple vertex operators do not correspond to energy eigenstates, but the former must be dressed with the same squeezing operator. Unfortunately no intuition about this interesting phenomenon is provided in the paper, but there is some hope that some intuition will arise in the future.
The author also has some speculative comments about some hidden integrable structure in the superfluid EFT, even for d>2, that would be very exciting.
In summary, the paper points out an interesting new fact in non-conformal theory, providing detailed derivation, and several interesting comments. Therefore the paper is suitable for publication in Scipost if the author can address a few points to improve/clarify the presentation.
Requested changes
1) Eq. 1.3 has a typo (missing tilde)
2) The logic behind the superfluidisation is not completely clear to me. From the presentation it seems that the author is simply stating that the superfluid EFT realizes the algebra. I don't see how is he arguing that any theory possessing that algebra is dual to a superfluid EFT with a a certain P-function. Moreover, how do we determine P?
3)Below 3.2. I think that g-->0 and g--> infty are both scale-invariant but non-conformal invariant theories. Notice, in fact, that g-->0 is a non-compact (d-2)-form gauge theory, and as such it behaves similarly to the non-compact scalar.
4) In eq. 3.7 I think it is important to emphasize that, differently from the d=2 case, the algebra depends on the space-time manifold through the \lambda_n. Does this means that the infinite symmetries are of higher-spin?
5)In 3.23, |\theta > is not a Goldstone state, but really a vacuum that breaks U(1)
6)Below 3.30. Why is this is a disorder operator? The latter are defined by imposing certain singularities for the fields in the path integral, while here an explicit expression in terms of fields is given.
7) It is not totally clear for me why the squeezing S is a local operator
8) Below 4.8. Why is J tilde conserved? To remove B_\mu the author turns on a background for the shift symmetry, and because of the mixed anomaly this should break the winding symmetry.
9) Eq. 5.4 requires \pi_1(G/H) to have infinite order (e.g. Z), not just being non-trivial.
Recommendation
Ask for minor revision
Strengths
- Explains a novel and generic method of extending the state-operator correspondence to non-conformal theories.
- Along the course, showed that the Kac-Moody algebra is present in such theories at low energies.
Weaknesses
- It was sometimes difficult (maybe just for me) to tell what is the regime of interest and the apprximations used.
Report
The manuscript concerns general quantum field theories in $d$ dimensions with a mixed anomaly between a $U(1)$ 0-form and a $U(1)$ $(d-2)$-form symmetry. Although the theory is in general non-conformal, the author argues that one has the state-operator correspondence, mapping a state on the sphere Cauchy slice in R^d to a local operator. The result is generic in the sense that there always exists a sector which is described by a superfluid EFT, at low energies. During the derivation, the author also pointed out that one can find charges which have the commutation relation of the Kac-Moody algebra.
The result is of potential interest to various subfields of physics, and is presented clearly. I recommend this paper for publication.
That said, here are some points that I noticed could be imporved, even though the paper can be published as is.
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Tower of conserved currents, in my understanding, usually refer to tower of higher spin currents, starting from the stress tensor. However, in line 176, for example, it is used to refer to the mode expansion of the usual current itself (if I understand correctly). Maybe a rephrasing could clarify what the author means.
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In (2.5), the author decided to throw out all terms with higher derivatives; a clarification of why it is okay to do so could be beneficial. (As I understand, this is because the EFT as written will reproduce any $n$-point functions in the IR limit. Is this correct?)
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In (2.15), please define $d^\dagger (\propto d ?)$
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In line 248, it would be great to explain more why they are gauge redundant.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
I thank the referee for their detailed review and thoughful and constructive comments. I reply below to the points in order:
- In this context, the terminology refers to the fact that, unlike the generic situation in QFT, the modes of the current are themselves local conserved currents. Therefore, each such current gives rise to a tower in the sense that each mode corresponds to a distinct conserved quantity. With the referee's understanding, I would prefer to retain the original phrasing, which I believe is appropriate in this setting.
- This understanding is largely correct. I have expanded footnote 6, explaining when higher derivative terms are suppressed and can be consistently neglected within the EFT framework.
- The definition of $d^\dagger$ has now been added to the text.
- I have added an explanation clarifying the gauge redundancy associated with the dressing functions.
Author: Stathis Vitouladitis on 2025-11-21 [id 6058]
(in reply to Report 2 on 2025-11-07)I thank the referee for their careful reading, comments, and sharp and interesting questions. I address each of the points raised below:
While eq. (1.3) was meant to be schematic, I agree that the omission of the tilde can be misleading. I have added a definition of $\widetilde{Q}_n$ just after eq. (1.2), and updated eq. (1.3) to include the tilde for clarity.
One argument is as follows. In the spirit of ref. [30], one may dualise $\widetilde{J}$ to $K = \star \widetilde{J}$ which is now a closed form, and represent the conservation equation by taking $K = d\phi$ (this is a minimal choice which can always be done). This enables one to build a standard EFT of the form $L = \alpha K^2 + \beta K^4 + \cdots$, which reproduces the superfluid EFT. I hope the logic is clearer now. Regarding the function $P$: this is a good point. It seems to me that its precise form cannot be fixed solely from symmetry considerations and I have now made this point clearer in the text.
This is a subtle and important point.
The $g \to \infty$ limit corresponds to a non-compact scalar, which is generally accepted to be a conformal field theory in flat space. I suspect the referee's comment was aimed at the subtlety that in curved spacetime conformal invariance requires turning on the conformal coupling, $R \phi^2$, arguably modifying the theory. I have added a clarifying comment in the manuscript to acknowledge this. While one might see this as modifying the theory, the interpretation in the manuscript follows the standard one in the literature.
In contrast, note that the $g \to 0$ limit (which indeed is dual to corresponding to a non-compact $(d-2)$-form gauge theory) is not conformal even in flat space. A very clear discussion highlighting this contrast can be found in Sections 4.3 and 4.4 of [JHEP 10 (2015) 171], which I now cite in the revised manuscript.
With these clarifications, I would prefer to retain the rest of the discussion unchanged, unless the referee has further concerns.
I have added a remark to clarify that the structure of the algebra does indeed depend on the spatial slice through the $\lambda_n$. As for the interpretation in terms of higher-spin symmetries, I am somewhat hesitant. The conserved currents involved are still spin-1, so I would be inclined to view them as encoding higher moments or multipoles of the original symmetries, though this is not sharply defined either.
Indeed, $\left|\theta\right>$ is a symmetry breaking vacuum. I have corrected that in the manuscript.
I agree with the referee's comment and have removed the mention of disorder operators to avoid confusion.
The squeezing operator, $\mathcal{S}$, is built from $B_{\ell m}$ and their adjoints. While in general ladder operators are delocalised, in this case (as well as in the 2d case which is a CFT and much better understood), the construction in eq. (3.46) can be understood in the limit $r \to 0$, where it becomes effectively local, just like eq. (3.29). Since the $B_{\ell m}$ can be viewed as local operators (they function as such in correlation functions), so does $\mathcal{S}$. I hope this clarifies the question. I have added a comment on this point in the revised manuscript.
Good point. For a constant chemical potential, it follows that $B_\mu$ is also constant, and so the compensating background field is necessarily flat. In this case, the mixed anomaly does not turn on. I have added a note around line 669 clarifying this and emphasising that the discussion is restricted to constant chemical potentials.
The referee is correct that eq. (5.4) requires $\pi_1(G/H)$ to be of infinite order to get a conserved current. I have corrected this statement in the revised version.