SymTFTs for Continuous non-Abelian Symmetries

Topological defects and operators give a far-reaching generalization of symmetries of quantum fields. An auxiliary topological field theory in one dimension higher than the QFT of interest, known as the SymTFT, provides a natural way for capturing such operators. This gives a new perspective on several applications of symmetries, but fails to capture continuous non-Abelian symmetries. The main aim of this work is to fill this gap. Guided by geometric engineering and holography, we recover various known features of representation theory of the non-Abelian symmetry from a SymTFT viewpoint. Central to our approach is a duality between (flat) free Yang-Mills and non-Abelian BF theories. Our results extend naturally to models without supersymmetry.

💡 Research Summary

This paper presents a significant advancement in the Symmetry Topological Field Theory (SymTFT) program by extending its framework to encompass continuous non-Abelian symmetries. The SymTFT is a powerful tool that encodes the generalized symmetries of a d-dimensional quantum field theory (QFT) within an auxiliary (d+1)-dimensional topological field theory coupled to a topological boundary condition. While highly successful for finite (discrete) symmetries, capturing continuous non-Abelian symmetries within this framework had remained an open challenge.

The core proposal of this work is that the SymTFT for a QFT with a continuous non-Abelian symmetry group G contains a sector described by a free Yang-Mills theory—i.e., Yang-Mills theory in the limit of zero gauge coupling (g → 0). In this limit, only flat gauge connections contribute, and the Wilson lines become topological operators, satisfying a key requirement for a SymTFT. The authors establish a crucial duality showing that this free, flat Yang-Mills sector is equivalent to the non-Abelian BF theory proposed by Horowitz. By introducing a Lagrange multiplier (d-1)-form field h, the Yang-Mills action can be rewritten, and taking the g→0 limit yields the topological BF action S_BF = ∫ tr(h ∧ f). This BF theory possesses gauge symmetries for both the connection A and the field h.

The paper provides substantial evidence for this proposal from two primary physical contexts: geometric engineering in string theory and holography (AdS/CFT). In geometric engineering (e.g., M-theory on Calabi-Yau singularities), non-Abelian flavor symmetries often originate from non-compact loci of singularities. The “link” of the singularity at infinity contains compact cycles associated with these symmetries, but their infinite volume in the decoupling limit corresponds to taking the gauge coupling to zero. The authors illustrate this explicitly with the example of 5D T_N SCFTs, where the SU(N)^3 global symmetry leads to a sum of three BF terms in the 6D SymTFT.

In holography, a global symmetry in the boundary CFT is dual to a gauge symmetry in the AdS bulk. The analysis shows that near the conformal boundary of AdS_(d+1), the effective gauge coupling of the bulk Yang-Mills theory vanishes due to the metric warp factor (or IR freedom in high dimensions). Neglecting the subleading kinetic term leaves precisely the non-Abelian BF theory action, effectively providing the SymTFT description. The 6D (1,0) E-string theory with its E_8 global symmetry, dual to AdS_7 × S^4/Z_2, is discussed as a key example.

An important application discussed is how this SymTFT framework naturally encodes the representation theory of the non-Abelian group. The topological operators in the bulk—Wilson lines and Gukov-Witten operators—are related to the Cartan subalgebra and Weyl group action, recovering features like charge quantization and the structure of multiplets from a bulk-boundary perspective. The Gukov-Witten operators, in particular, are noted to form a non-invertible subalgebra.

The authors emphasize that their construction, while motivated by supersymmetric examples from geometric engineering and holography, extends naturally to non-supersymmetric models. They also clarify a key difference from the finite symmetry case: while the SymTFT for finite symmetries can describe gauging via topological interfaces, gauging a continuous symmetry introduces new dynamical degrees of freedom and cannot be captured by purely topological manipulations. Nevertheless, the SymTFT perspective remains invaluable for understanding the action of global continuous symmetries, their anomalies, and their realization across different phases of a QFT. This work fills a major gap and paves the way for exploring more general non-finite symmetry structures in quantum field theory.