Symmetry TFTs from String Theory

Abstract

We determine the d + 1 dimensional topological field theory, which encodes the higher-form symmetries and their 't Hooft anomalies for d-dimensional QFTs obtained by compactifying M-theory on a non-compact space X. The resulting theory, which we call the Symmetry TFT, or SymTFT for short, is derived by reducing the topological sector of 11d supergravity on the boundary ∂X of the space X. Central to this endeavour is a reformulation of supergravity in terms of differential cohomology, which allows the inclusion of torsion in cohomology of the space ∂X, which in turn gives rise to the background fields for discrete (in particular higher-form) symmetries. We apply this framework to 7d super-Yang Mills, where X = C 2 /Γ ADE , as well as the Sasaki-Einstein links of Calabi-Yau three-fold cones that give rise to 5d superconformal field theories. This M-theory analysis is complemented with a IIB 5-brane web approach, where we derive the SymTFTs from the asymptotics of the 5-brane webs. Our methods apply to both Lagrangian and non-Lagrangian theories, and allow for many generalisations.

0-Form, 1-Form and 2-Group Symmetries via Cutting and Gluing of Orbifolds

“…The quiver (4.54) engineers 5D so(8 + 2m) gauge theory with half hypermultiplets in the bifundamental representation of so 8+2m × sp 2m . Choosing a purely electric polarization, the gauge group is Spin(8 + 2m) and we match the results presented in [52]. In section 5 we turn to a closely related example of this sort in the context of 4D N = 1 theories engineered on local G 2 spaces.…”

“…Given this, it would seem important to extract further details, as captured by topologically robust quantities such as anomalies. Perhaps this can be calculated along the lines of reference [52] (see also [41]).…”

Higher Symmetries of 5d Orbifold SCFTs

“…In particular, it would be desirable to extract the Postnikov class β directly from the geometry of a string compactification. We think that in order to clarify this interplay it will be very fruitful to look at the symmetry TQFT [41] for these orbifold singularities, which arises from the reduction of the topological Chern-Simons term of M-theory on the horizon S 5 /Γ. Further compactification of these 5d SCFTs will give rise to a rich class of lower-dimensional systems.…”

“…As shown in [109], the base B is always of the form C 2 /Γ U (2) for Γ U (2) a particular set of finite subgroups of U (2), and in all these cases, ∂B = S 3 /Γ. In this case, the corresponding defect group is associated with a two-form symmetry, as specified by string-like defects of the 6d SCFT [13] (see also [16,22,23,30,41,44]). While we leave a more complete analysis for future work, in this section we observe that in situations where the geometry faithfully reproduces the 0-form symmetry of the system, Ab[Γ] is closely correlated with the 2-group symmetry of the 5d SCFT.…”

Comments on Global Symmetries and Anomalies of $5d$ SCFTs

“…As described in [19], the instantonic and center symmetries have a mixed 't Hooft anomaly, whose derivation is analogous to the three-dimensional anomaly (2.6) (a recent derivation using string theory methods is given in [49]). Again, we turn on a background gauge field for U(1)…”