Duality-Symmetry Enhancement in Maxwell Theory

Referee 1 changes and clarifications

1. We have corrected all the typos spotted and added a sentence before Corollary A.4, clarifying that we use the diffeomorphism between (S2xS2)^{#8} and M_{E8}#\bar{M}_{E8}.

2. The question of how the story changes when the metric is perturbed slightly away from its extremal value is an interesting one. It lies beyond the scope of the present paper and hence we have only added a short paragraph about it in the Discussion Section. However, we would like to point out that, on the one hand, the partial symmetries we have introduced are not inherited from dualities: By perturbing away either the metric from its extremal values or the coupling from the orbifold points, the partial symmetries are expected to be broken. On the other hand, as stressed in Section 2.2.3, analogously to what happens for SL(2,Z) at the orbifold points, specific subsets of the mapping class group of the space-time become space-time symmetries at the extremal values of the metric.

3. Non-invertible symmetries exist at smooth points of the moduli space. They are an intrinsic property of the theory corresponding to a given smooth value of the coupling and are not necessarily related to ordinary duality or triviality symmetries. However, it is possible to construct specific Z_2 or Z_3 subgroups of the stabilizer group of a given smooth point by taking paths in the moduli space which pass by the duality or the triality points respectively. The only caveat is that passing through the triality point gives rise to a well-defined (non-invertible) symmetry of the theory only when the partition function does not vanish at the triality point. But regardless of this, the non-invertible symmetry of the theory associated to the given smooth point remains well defined. We have tried to clarify all these subtle points by slightly modifying the discussion in the two paragraphs above formula (49) and around formula (65). We hope that this helps to improve the overall flow of Section 3.

4. Unfortunately, we do not have a way to find or guess the anomalous phase for non-invertible duality-symmetries. To detect their anomalies we can only rely on the vanishing of the partition function. We have estimated the approximate location of the zero of the generalized theta function of the Leech lattice. We have added formula (69), where we kept eight decimal digits. We have also added a short concluding paragraph, stressing that it would be nice to verify the presence of the claimed anomaly through a direct computation of the anomalous phase.

Referee 2 changes and clarifications

1. For the sake of clarity, we have summarized the main claim of the paper in lines 64-67.

2. We have simply eliminated the undefined phrase “infinite array of gaugings”, because unimportant for our purposes. We have done that in line 73, 625 and footnote 24.

3. We have added a sentence in line 39-40 to introduce the notion of mixed ’t Hooft anomaly in terms of obstruction to gauging.

4. The answer is affirmative. For all metrics leading to a factorization of the partition function, we conjecture that any of its zeros originates from a global symmetry (possibly non-invertible) plagued by a mixed anomaly with gravity. This is obviously the case, for instance, for all metrics sharing the same conformal class, because the theory only depends on the conformal class of metrics.

5. We have modified footnotes 2 and 14, substituting the wedge with the cup product and the field strength F with the first Chern class c_1 of the Maxwell line bundle.

6. We have modified the sentence in lines 116-118, to stress that it’s the de Rham representative of the first Chern class of the Maxwell line bundle to be locally equal to dA.

7. The gauge potential A is always defined only locally. We have added a sentence in line 137 to specify what we mean by the symbol “dA” in the action (4). We prefer to write it in terms of A, because the path integral is done on A.

8. We have modified the sentence in lines 193-194, in order to make it clear why formulas (14) and (13) signal a mixed 't Hooft anomaly between duality-symmetry and gravity.

9. We have also added the adjective “normalized” to the noun “partition function”, every time we refer to the quantity appearing in formulas (17), (18) and (20).

10. We have modified the sentence in line 280 to clarify that a metric is conjectured to exist, whose Hodge star is diagonal in the basis in which the intersection form has the form [+E_8] \oplus [-E_8].

11. In formula (22) there is no non-trivial monodromy phase associated with the ordinary duality-symmetry transformations. However, a symmetry transformation of the “partial” type, which we introduce, does produce a non-trivial phase in Eq. (22), and causes the partition function to vanish at the triality point. We have modified the sentence in lines 285-286 to clarify that indeed monodromies are in 1:1 correspondence with anomalies.

12. We have added the symbol $g$ in line 351 to stress that the quantity $G$, which depends on the Hodge star, is derived from the metric $g$.

13. We have corrected the typo in line 511.

14. The quantity (48) “defines” a projector. We have modified the sentence below accordingly.

15. The omitted determinant factors spoil the factorization in formula (51). We have clarified this point in footnote 26.

16. If we (even infinitesimally) perturb an extremal metric, we break the “partial” symmetry that we have defined. We have added a comment about that in lines 403-404.

17. While all of these are certainly interesting hints for further investigations, they lie beyond the scope of the present paper. We limited ourselves to shortly mentioning these suggestions in the Discussion Section.

18. We have slightly modified the discussion in the two paragraphs above formula (49) and around formula (65), in particular stressing the role of the triality point in defining non-invertible symmetries, as well as the fact that we rely on the vanishing of the partition function as a criterion to assess the presence of anomalous phases associated to non-invertible duality-symmetries. We hope that this helps improving the overall logical flow of Section 3.