Duality-Symmetry Enhancement in Maxwell Theory

This interesting work highlights subtle features of four-dimensional U(1) Maxwell theory on smooth, compact, oriented four-manifolds, and uses this setup as a laboratory to study duality symmetries. A key role is played by a so-called holomorphic factorization of the suitably normalized Maxwell partition function, which happens for special background field configurations. The authors contend that such a factorization can signal new discrete symmetries that are not associated with conventional dualities, but may have anomalies. They ask whether the zeros of the Maxwell partition function signal the presence of an anomalous duality symmetry.

1. Perhaps the authors' claim should be revised to say that the zeros of the Maxwell partition function signal the presence of a 't Hooft anomaly (potentially mixed) in a ``partial symmetry'' for a background that leads to holomorphic factorization.

2. The phrase ``infinite arrays of gaugings'' in line 68 is undefined, and perhaps could be slightly elaborated upon, for completeness.

3. It would be useful to define the phrase ``mixed 't Hooft anomaly with gravity.'' Sometimes, 't Hooft anomalies are defined as obstructions to gauging. In the present context, it would be useful to reconcile it with this definition.

4. In lines 82-85, the authors envision that the map

{ mixed anomalies between gravity and non-invertible dualities } ---> { zeros of the Maxwell partition function}

is surjective, when the background is so chosen to admit a holomorphic factorization of the partition function. There could in general be several configurations of background fields for which the partition function is holomorphically factorizaed. Do the authors expect surjectivity of the above map to hold for all such configurations?

1. Regarding footnotes 2 and 14: The wedge product should be replaced by a cup product, since the second Stiefel-Whitney class does not admit a de Rham representative.

2. In line 112, it is better to write [F]_{dR} =(loc) dA, because F is being used interchangeably for the de Rham representative as well as an integral cohomology class. Of course, under the assumptions that the authors make, the map H^{2}(M; 2\pi Z) \to H^{2}(M; R) is injective.

3. If an action includes an explicit gauge field A, then it makes sense only if A is globally defined as a 1-form. In particular, the path integral over \tilde{F} will yield \mathcal{F} = dA globally if equation (4) is taken literally. I believe here the authors have in mind the usual setting where the connection is only locally defined as a 1-form. While the conclusions of this section are consistent, it is a bit confusing to write "loc" when the action presupposes global definitions. Why must \tilde{A} exist globally?

4. In lines 186-187, the authors say, "Consequently, non-trivial monodromies of the form (14) signal the presence of a mixed ’t Hooft anomaly between the duality-symmetry and gravity." Here, I think it would be quite useful to motivate exactly why the structure of equations (14) and (13) signals a ``mixed 't Hooft anomaly between duality-symmetry and gravity,'' especially in view of any metric dependence that is suppressed on account of the omitted terms of footnote (10).

I also suggest that the normalized quantities (17), (18), and (20) not be called the partition function of Maxwell theory. This is correctly addressed in later sections.

1. Line 272, which reads "Moreover, a metric whose Hodge star simultaneously diagonalize is conjectured to exist," is confusing. (Minor typo: diagonalize --> diagonalizes). Simultaneous with what?

2. Although monodromies of Z have been suggested as signals of anomalies, the discussion around equation (22) seems to suggest that monodromies of the partition function are not in 1-1 correspondence with anomalies, but zeros of the partition function are still believed to signal some anomaly. This point could be emphasized for pedagogical clarity.

3. Line 343: The phrase "metric associated with G" seems redundant.

4. Typo in line 502: the criterion on gcd(N_e, N_m) is missing. I believe it should read gcd(N_e, N_m) = 1.

5. Below line (48), do the authors mean that C_{(a b)^T} is a projector or that it defines a projector? This should be clarified.

6. In equations (50) and (51), how do the omitted determinant factors affect the factorization, if at all? It may be useful to clarify this.

7. The discussion of the mixed anomaly involving gravity and non-invertible symmetries could be improved, and the section on extremal metrics could comment on how mild perturbations to the extremal metric would affect the result.

8. It would be quite interesting to compare with other 4d theories, e.g., changing the gauge group, adding matter fields, etc. While I recognize that a detailed investigation is beyond the scope of the present paper, which is focused on Maxwell theory, I wonder if the authors considered the interplay between perturbed metrics (away from extremal metrics) and matter fields, and whether judicious choices of additional background fields could help with the factorization of the partition function.

9. The logical flow of section 3 could be improved.

With minimal changes and clarifications, I suggest the publication of this paper.

1. Footnotes 2 and 14 should be corrected.
2. Address the above-mentioned issues with lines 272, 343, 502, and the remark around equation (48).
3. Minor clarifications of some of the issues/questions posed above.
4. The logical flow of section 3 could be improved.

Ask for minor revision

This paper presents a very interesting extension of generalized symmetries, that appears when the partition function of a QFT factorizes in several sectors. The QFT that the authors study is Maxwell in 4d. Then, the core idea is to first notice that some dualities can have anomalies on some specific spacetimes (hence dubbed mixed duality-gravitational anomalies). If the dualities are promoted to symmetries for specific values of the Maxwell coupling, then the partition function has to vanish. They then ask what happens at a generic zero of the partition function, which duality symmetry is anomalous? It turns out they need to generalize to partial dualities, i.e. acting only on the (anti)self-dual fluxes that appear in one of the factors of the partition function. The generalization to non-invertible duality symmetries is particularly interesting since in that case it becomes a way to detect the presence of a (mixed) anomaly.

Section 2 starts with a nice review of how the partition function of Maxwell changes under a generic SL(2,Z) duality action, with a multiplicative factor that depends on the Euler characteristic and a phase that depends on the signature of the manifold. When dualities become symmetries, this phase is interpreted as a mixed duality-gravitational anomaly. The partition function is then written as a sum over fluxes, normalized to the one on $S^4$, and crucially, the conditions are discussed for it to factorize into two sums, one holomorphic and the other anti-holomorphic in the coupling. In fact, it is enough that a holomorphic or anti-holomorphic factor is present. Then its modular properties can imply that it vanishes at some point (typically the triality point). The symmetry that is anomalous is not a full triality, but one that acts only on the fluxes pertinent to that (anti)self-dual subsector.

A question is what happens to these symmetries if the metric is moved slightly away from its extremal value. It would seem the partition function no longer factorizes, hence one just cannot define the symmetry action. This is contrary to the requirement to be at the triality point. Even slightly away from it, one can still define the S(T) action, it will just change the value of $\tau$. This question could be addressed more explicitly.

There is an attempt to understand the action of such partial duality symmetry on lines, but it is eventually not completely satisfactory (to me at least, but it feels also to the authors), since the outcome is a hybrid (not a dyonic line), in the sense that it can be described by a background gauge field with a magnetic self-dual and electric anti-self-dual parts (for the simple starting point of a Wilson line). [There is a minor typo in Eq. (36), $F$ instead of $J$ in the last expression.]

In section 3, they turn to considering the possibility of having zeros of the (factorized) partition function at smooth points of the conformal manifold. They should be related to anomalies of non-invertible symmetries, that can in principle be defined there. [There is a typo in (45) (an extra = sign on the second line), and in the paragraph just below it (the gcd has to be equal to 1).]

However they argue that the interesting new partial symmetries at the triality point cannot be promoted to non-invertible symmetries precisely because the partition function vanishes at that point. This last point could be explained better, also in light of what they find afterwards, namely how is this argument circumvented by their later findings?

They then proceed to show that generically, the $\tau$ dependent part of the partition function factorizes when the manifold is a connected sum. One can then define a partial gauging of the electric and magnetic 1-form symmetries in a single factor. This allows to define partial non-invertible symmetries, by combining with partial dualities. As mentioned above, it is not really answered how this observation circumvents the problem with the vanishing of the partition function at the triality point.

The section ends rather abruptly in a sort of anti-climax. The authors first correctly comment that non-invertible symmetries relate points in the conformal manifold that are physically distinct. They then try to discuss where the mixed anomaly between gravity and such non-invertible symmetries might arise. They consider mixed anomalies between 1-form symmetries and gravity, only to dismiss them (here both the premise and the argument are obscure to me). Finally, they end with an example. They show that a particular connected sum is not only diffeomorphic to an extremal manifold for which the partition function vanishes at the triality point, but it is also diffeomorphic to another extremal manifold, related to the Leech lattice, for which the partition function has to vanish at a smooth point. Strangely, they do not determine such point (I would have expected at least a numerical hint of where it lies!), and do not discuss further the anomalous phase. It feels like this section is not finished.

In section 4 they discuss prospects, in particular related to the 6d origin of Maxwell theory.

There is a technical appendix on how to ensure there are indeed extremal geometries. [There is a typo in Theorem A.3 (a -1 should be a +1); in corollary A.4, is the relation discussed above Eq. (22) taken into account?]

In conclusion, I think this is a very interesting paper, it deserves to be published, but I would like the authors to (briefly) address the few clarifications that, I think, would improve the understanding (besides fixing the few typos that I spotted):
-discuss how the partial symmetries are broken when moving slightly away from the extremal geometries.
-discuss how the partial gauging relates to the problem with the vanishing of the partition function at the triality point—are the smooth points discussed later expected to be related to the triality point by partial gauging or not?
-add some closing remarks after the example that is supposed to have a partial symmetry at the smooth zero. Is there a hint where the zero lies? Is there a way to find or guess the anomalous phase?

Ask for minor revision