Symmetry TFTs from String Theory

TL;DR

This work builds the Symmetry TFT (SymTFT) program for QFTs arising from string theory by reducing the topological sector of M-theory on the boundary of the internal space and using differential cohomology to faithfully incorporate torsion and discrete higher-form backgrounds. It provides a concrete, boundary-based KK reduction framework that yields eight-dimensional BF-type theories encoding the global structure choices and 't Hooft anomalies of 7d SYM and 5d SCFTs, with cross-checks from IIB $(p,q)$ 5-brane webs and little string theory holography. The authors derive explicit anomaly couplings, including $B^3$ and mixed $B^2 F$ terms, for ADE cases and toric non-Lagrangian models, and demonstrate that the resulting SymTFTs reproduce known field-theoretic results while extending to strongly coupled and non-geometric settings. The approach unifies geometric engineering, brane constructions, and holographic perspectives, providing a versatile tool to study discrete higher-form symmetries and their anomalies in a broad class of theories. It sets the stage for further applications to other compactifications (e.g., G2, CY4/ CY5) and for exploring higher-group structures within a differential-cohomology framework.

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 $\partial X$ 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 $\partial X$, 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 $X= \mathbb{C}^2/Γ_{ADE}$, 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.

Figures (4)

• Figure 1: The cone $C(L)$ over the link $L$, and the deformation, shown on the right, to a long cylinder where the singularity is at the far end.
• Figure 2: On the left: Under the assumption $H_{d-3}(X_d;\mathbb{Z}) =0$, any $(d-3)$-cycle $a \in Z_{d-3}(L_{d-1})$ in the link can be realised as boundary of a $(d-2)$-chain $\kappa \in C_{d-2}(X_d)$ in the bulk $X_d$. On the right: If $a$ represents a torsional homology class, $na = \partial u$ for some $(d-2)$-chain $u\in C_{d-2}(L_{d-1})$ in the link, which can also be naturally regarded as an element in $C_{d-2}(X_d)$. Combining the chains $u$ and $n\kappa$ we get a cycle, $\partial(n\kappa - u) = 0$. This cycle can now be smoothly retracted to the interior of $X_d$, and can therefore be thought of as a compact cycle. Its homology class $[n\kappa -u]$ represents $Z \in H_{d-2}(X_d;\mathbb{Z})$.
• Figure 3: The toric diagram for the 5d SCFT realization of $SU(p)_q$. The example shown is $p=6$, $q= 6-(k_x + k_y) =3$, i.e. $SU(6)_3$, which has $\mathbb{Z}_3$ 1-form symmetry.
• Figure 4: $B_N$ and $B_N^{(i)}$ non-Lagrangian toric diagrams for $N=4$.