Symmetry Topological Field Theory (SymTFT)

• SymTFT is a framework that assigns a (d+1)-dimensional topological field theory to a d-dimensional QFT, capturing global symmetry and anomaly data.
• It employs dimensional reduction from string/M-theory and refined differential cohomology to systematically incorporate both continuous and discrete torsion contributions.
• Applications range from 7d Super-Yang-Mills to 5d SCFTs, with dual derivations confirming its universal role in describing anomaly inflow and higher-form symmetries.

Symmetry Topological Field Theory (SymTFT) is a construction that associates to any $d$-dimensional quantum field theory (QFT) a $(d+1)$-dimensional topological field theory. The SymTFT systematically encodes the generalized global symmetries—including higher-form and possibly non-invertible symmetries—and their 't Hooft anomalies of the QFT in a way that is robust under deformations of the local dynamics. In string and M-theory settings, the SymTFT arises from the reduction of the topological sector of the higher-dimensional supergravity action on the boundary of a non-compact internal space, capturing both the continuous and discrete, including torsional, symmetry data via refined topological techniques such as differential cohomology.

1. Construction from Higher-Dimensional Supergravity

The core construction of SymTFT arises from the compactification of string/M-theory backgrounds of the form $$\mathcal{M}_{11} = \mathcal{W}_{d+1} \times X^{10-d}$$, with $X$ a non-compact space whose asymptotic boundary $\partial X$ serves as a link (typically a closed manifold). The topological (typically Chern–Simons–type) term in the higher-dimensional action—for M-theory, the $G_4 \wedge G_4 \wedge G_4$ coupling—when dimensionally reduced along the link, gives rise to a $(d+1)$-dimensional effective topological field theory.

A crucial technical ingredient is the expansion of the $p$-form fluxes (for example, $G_4$ in M-theory) in terms of both harmonic representatives and torsion (finite-order) classes of $H^*(\partial X, \mathbb{Z})$. This Kaluza–Klein–like ansatz takes the form: $$G_4 = \sum_{p, \alpha_p} F^{\alpha_p}_{4-p} \cdot v_{p}(\alpha_p) + \sum_{p, i_p} B^{i_p}_{4-p} \cdot t_{p}(i_p) + (\text{trivial terms}),$$ where $v_p$ and $t_p$ are the generators of the free and torsion subgroups of the relevant cohomology groups. The resulting reduced topological action, after integrating over the link $\partial X$, yields the SymTFT, typically of BF-type, encoding the background field structure, symmetry classes, and 't Hooft anomalies for the engineered $d$-dimensional QFT (Apruzzi et al., 2021).

2. Differential Cohomology and Torsion Data

Standard approaches relying only on harmonic forms do not capture torsion contributions, which are crucial for discrete and higher-form symmetries. SymTFTs employ the framework of differential cohomology, where a class $a \in \hat{H}^p(M)$ refines an ordinary cohomology class $I(a) \in H^p(M,\mathbb{Z})$ by also including curvature representative $R(a)$ and holonomy data. The product structure in differential cohomology (denoted “$\star$”) is crucial, obeying

$$I(a \star b) = I(a) \cup I(b), \quad R(a \star b) = R(a) \wedge R(b).$$

Torsion elements (with $R(t_p) = 0$) generate nontrivial contributions through the product, for example through nonzero triple products like $\int_L t_2 \star t_2 \star t_2 \neq 0$. This enables the rigorous derivation of discrete BF-type terms in the SymTFT that encode higher-form background fields and anomalies associated to the torsion structure (Apruzzi et al., 2021).

3. Applications: Explicit Examples from String/M-Theory

A. 7d Super-Yang-Mills via M-theory on $\mathbb{C}^2/\Gamma_{ADE}$

For $X=\mathbb{C}^2/\Gamma_{ADE}$ with asymptotic link $L=S^3/\Gamma$, the reduction yields a SymTFT in 8d that encodes the discrete 1-form symmetry (or its magnetic dual 4-form symmetry) of the engineered SYM. The discrete data are captured by the relative homology quotient $$\Gamma^{(1)} = H_2(X, \partial X; \mathbb{Z})/H_2(X; \mathbb{Z})$$. The resulting SymTFT includes couplings such as

$$S_{8d} \supset \frac{1}{2} \int_{\mathcal{W}_8} \gamma_4 \star B_2 \star B_2,$$

with anomaly coefficients determined by Chern–Simons invariants computed through refined intersection theory (Gordon–Litherland formula, self-intersection of divisors $Z$). The anomalies are encoded as

$$\operatorname{CS}[S^3/\Gamma, t_2] = \left[\frac{Z \cdot Z}{2n^2}\right]_{\,\mathrm{mod}\,1},$$

matching precisely the field theory expectations for the quadratic anomaly and the global form of the gauge group (Apruzzi et al., 2021).

B. 5d Superconformal Field Theories from Toric Calabi–Yau Threefolds

For 5d SCFTs engineered from toric Calabi–Yau cones, the link is a 5-manifold $L$ (e.g., Sasaki–Einstein manifolds). Here, $H_1(L, \mathbb{Z})$ classifies 2-form symmetries; $H_3(L, \mathbb{Z})_{\text{free}}$ corresponds to continuous instanton symmetries. The reduced SymTFT contains cubic and mixed anomaly terms, for instance: $$\mathcal{A}_{B^3} = \frac{q p(p-1)(p-2)}{6\,\gcd(p,q)^3} B_2^3,$$ with the cubic B-field term $B_2^3$ determining the 1-form symmetry anomaly in accordance with field theory calculations for IR $SU(p)_q$ gauge theories and extending to non-Lagrangian SCFTs (Apruzzi et al., 2021).

4. IIB 5-Brane Webs: Dual Derivation and Consistency

SymTFTs in 5d can also be constructed from the perspective of type IIB (p,q) 5-brane webs realizing 5d SCFTs. The topological sector from the IIB coupling $\int F_5 \wedge H_3 \wedge F_3$ is expanded over the asymptotic S$^4$ with punctures (from semi-infinite 5-branes), and harmonic forms encoding the flux sectors. The reduction produces BF-type couplings in 6d mirroring the M-theory derivation. Linking pairings and noncommuting flux operators (e.g., operators obeying

$$\Phi(\mathcal{T}_3) \Phi(\mathcal{T}_6) = e^{2\pi i\, \ell^{-1}} \Phi(\mathcal{T}_6) \Phi(\mathcal{T}_3)$$

with linking pairing $\ell$) manifest as anomaly coefficients in the 6d SymTFT. This dual construction demonstrates that different string theory realizations lead to consistent, physically equivalent SymTFTs, confirming the universality of the topological structures encoding global symmetries and anomalies (Apruzzi et al., 2021).

5. Generalizations: Lagrangian, Non-Lagrangian, and Higher-Dimensional Theories

The method of deriving SymTFTs via reduction on the link $\partial X$ is broadly applicable:

• Both Lagrangian theories (with conventional field theory Lagrangians) and non-Lagrangian, strongly coupled SCFTs can be analyzed through their engineered singularity.
• The framework is robust to singularities and nontrivial global topology in $X$, since all the relevant data are captured by the closed manifold $\partial X$.
• The use of differential cohomology is essential for capturing discrete, higher-form, and non-invertible global symmetry structures and their 't Hooft anomalies, including contributions from torsion classes.
• The approach generalizes across dimensions: M-theory on singular Calabi–Yau $n$-folds gives rise to effective QFTs in various dimensions (for example, 7d, 5d, 3d) with their SymTFTs constructed by similar reduction (Apruzzi et al., 2021).

6. Mathematical Structure and Physical Interpretation

The mathematical backbone of SymTFTs in this context is the use of refined cohomological tools—differential cohomology and torsion pairings—to encode all higher-form symmetries. The origin of BF-type couplings in the boundary SymTFT connects to linking numbers and self- and mutual-intersections in the homology of the link. In a physical picture, these topological couplings capture the fusion and non-commutativity of symmetry backgrounds; their integrals count global features such as Wilson line and surface operator commutations and encode the full anomaly inflow structure. This mathematical structure also enables the systematic application of SymTFTs in classifying global symmetry data, distinguishing quantum field theories with the same local operator content but different anomaly and symmetry backgrounds.

7. Implications and Consistency Across Constructions

The construction of SymTFTs via reduction of the supergravity topological sector (using differential cohomology for precise torsion data) yields a universal, rigorous encoding of higher-form symmetries, backgrounds, and anomalies for theories engineered in string theory. The equivalence between derivations from M-theory and from 5-brane webs in IIB supports the notion that SymTFTs encapsulate the intrinsic global symmetry and anomaly content independent of the detailed microscopic realization. The approach applies equally to Lagrangian and non-Lagrangian theories, and naturally generalizes to compactifications leading to QFTs in lower and higher dimensions (Apruzzi et al., 2021).

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In summary, the Symmetry Topological Field Theory (SymTFT) framework provides, via dimensional reduction and differential cohomology, a comprehensive and universal mechanism for encoding higher-form symmetries and 't Hooft anomalies of QFTs arising in string/M-theory engineering. Its validity across Lagrangian and non-Lagrangian examples, equivalence across different string-theoretic constructions, and robustness in the presence of singularities and torsion cohomology underpin its centrality in the modern understanding of quantum field theoretic global symmetry structure.