The Fate of Discrete 1-Form Symmetries in 6d

We thank the Referees for their constructive criticism and comments. We hope to have addressed the issues raised in their reports in the new version of the manuscript in a satisfactory manner.

To Referee 1:

1.)
We certainly agree with the referee that generically, massless hypermultiplets will break parts of the 1-form symmetry explicitly. However, as our analysis shows, there are cases where the 1-form symmetry is strictly smaller than the center subgroup that is unbroken by massless hypers. Therefore, the obstruction we found poses an additional constraint that goes beyond gauge anomaly cancellation. Concrete examples are (3.15), and also (3.18), whose analysis we have extended, as requested by the Referee.

2.)
We are not aware of an alternative argument for why (correctly) gauging the flavor symmetry of conformal matter models does not break the center symmetries of the 0-form gauge groups. As the Referee suggests, a potential way could be to study the higher-form symmetries in the circle reduction, which however is outside the scope of our work. In connection to the previous point, we do want to point out that the gauging which preserves the 1-form symmetry must not be such that there are additional massless hypers (required by gauge anomaly cancellation) which are charged under the now-gauged flavor symmetries. For example, in the gauging of the flavor symmetries of (E6, E6) conformal matter, discussed in Section 5.3, the non-trivial global structure of the flavor symmetry (which requires also a non-trivial 1-form symmetry background of the SU(3) factor on the (-3)-curve) is only retained if the E6 factors are gauged on (-6)-curves; otherwise, there would be fundamental hypers charged just under the E6's, which would not allow for a non-trivial 1-form background.

3.)
As the Referee indicates the gauging of an electric 1-form symmetry generically leads to the appearance of a magnetic (D-3)-form symmetry in D dimensions. We do not discuss these magnetic higher-form symmetries in the present paper, but once one includes gravity these symmetries are supposed to be absent. What happens is that as the electric higher-form symmetries the magnetic once are broken once extended charged operators become dynamical. In the gravitational setup, therefore, the dual magnetic symmetries are broken explicitly by these objects and are absent as required. In string theory construction this can often be seen explicitly by the construction of electric and magnetic operators via wrapped branes. These are dynamical if the wrapped cycle has a finite volume. Activating gravity, demands that the internal space is compact so all the cycles have a finite volume and the associated states break the higher-form symmetry. In order to stress this point we included a brief discussion in the Introduction.

4.)
The theories discussed in Section 5.2 only have adjoint hypermultiplets, due to necessary blow-ups that remove non-minimal singularities which also separate the gauge divisors. Physically, this is to say that instead of bifundamental hypermultiplets, there are E-strings charged under two different gauge factors. Based on the hypermultiplet spectrum, one would expect the full center to be admissible. But only a diagonal part, isomorphic to the Mordell--Weil group, is unobstructed and can be gauged. We have extended the discussion after (5.17) to hopefully clarify it.

To Referee 2:

As suggested by the Referee we went through the draft and tried to streamline our presentation in order to work out the conclusions clearly. This resulted in several minor changes, most prominently in the Abstract, as well as in the Introduction. The obstruction to turn on a non-trivial 1-form symmetry background is now formulated throughout the manuscript as an incompatibility of the induced fractional string charges with Dirac quantization.