We construct Symmetry Topological Field Theories (SymTFTs) for continuous subsystem symmetries, which are inherently non-Lorentz-invariant. Our framework produces dual bulk descriptions—gapped foliated and exotic SymTFTs—that generate gapless boundary theories with spontaneous subsystem symmetry breaking via interval compactification. In analogy with the sandwich construction of SymTFT, we call this Mille-feuille. This is done by specifying gapped and symmetry-breaking boundary conditions. In this way we obtain the foliated dual realizations of a topological theory enjoying subsystem symmetries as well, i.e. a foliated BF-type theory for various models, including the XY plaquette, XYZ cube, and \phi ϕ , \hat{\phi} ϕ ̂ theories. This provides a systematic method for generating free theories that non-linearly realize subsystem symmetries.
A bstract Symmetry topological field theory (SymTFT), or topological holography, posits a correspondence between symmetries in a d -dimensional theory and topological order in a ( d + 1)-dimensional theory. In this work, we extend this framework to subsystem symmetries and develop subsystem SymTFT as a systematic tool to characterize and classify subsystem symmetry-protected topological (SSPT) phases. For (2 + 1)D gapped phases, we introduce a 2-foliated (3 + 1)D exotic tensor gauge theory (which is equivalent to 2-foliated (3 + 1)D BF theory via exotic duality) as the subsystem SymTFT and systematically analyze its topological boundary conditions and linearly rigid subsystem symmetries. Taking subsystem symmetry groups G = ℤ N and G = ℤ N × ℤ M as examples, we demonstrate how to recover the classification scheme
C[G] C G = H 2 ( G ×2 , U(1))/ H 2 ( G , U(1))) 3 , which was previously derived by examining topological invariant under linear subsystem-symmetric local unitary transformations in the lattice Hamiltonian formalism. To illustrate the correspondence between field-theoretic and lattice descriptions, we further analyze ℤ 2 × ℤ 2 and ℤ N × ℤ M cluster state models as concrete examples.
A bstract We generalize the idea of symmetry topological field theory (SymTFT) for subsystem symmetry. We propose the 2-foliated BF theory with level N in (3 + 1)d as subsystem SymTFT for subsystem ℤ N symmetry in (2 + 1)d. Focusing on N = 2, we investigate various topological boundaries. The subsystem Kramers-Wannier and Jordan-Wigner dualities can be viewed as boundary transformations of the subsystem SymTFT and are included in a larger duality web from the subsystem SL(2, ℤ 2 ) symmetry of the bulk foliated BF theory. Finally, we construct the condensation defects and twist defects of S -transformation in the subsystem SL(2, ℤ 2 ), from which the fusion rule of subsystem non-invertible operators can be recovered.
We provide a generalization of the Symmetry Topological Field Theory (SymTFT) framework to characterize phase transitions and gapless phases with categorical symmetries. The central tool is the club sandwich, which extends the SymTFT setup to include an interface between two topological orders: there is a symmetry boundary, which is gapped, and a physical boundary that may be gapless, but in addition, there is also a gapped interface in the middle. The club sandwich generalizes so-called Kennedy-Tasaki (KT) transformations. Building on the results in [L. Bhardwaj et al., Phys. Rev. Lett. 133, 161601 (2024); L. Bhardwaj et al., SciPost Phys. 18, 032 (2025)] on gapped phases with categorical symmetries, we construct gapless theories describing phase transitions with non-invertible symmetries by applying suitable KT transformations on known phase transitions provided by the critical Ising model and the 3-state Potts model. We also describe in detail the order parameters in these gapless theories characterizing the phase transitions, which are generally mixtures of conventional and string-type order parameters mixed together by the action of categorical symmetries. Additionally, removing the physical boundary from the club sandwiches results in club quiches, which characterize all possible gapped boundary phases with (possibly non-invertible) symmetries that can arise on the boundary of a bulk gapped phase. We also provide a mathematical characterization of gapped boundary phases with symmetries as pivotal tensor functors whose targets are pivotal multi-fusion categories.
We propose a general framework to characterize gapped infra-red (IR) phases of theories with non-invertible (or categorical) symmetries. In this paper we focus on (1+1)d gapped phases with fusion category symmetries. The approach that we propose uses the Symmetry Topological Field Theory (SymTFT) as a key input: associated to a field theory in d d spacetime dimensions, the SymTFT lives in one dimension higher and admits a gapped boundary, which realizes the categorical symmetries. It also admits a second, physical, boundary, which is generically not gapped. Upon interval compactification of the SymTFT by colliding the gapped and physical boundaries, we regain the original theory. In this paper, we realize gapped symmetric phases by choosing the physical boundary to be a gapped boundary condition as well. This set-up provides computational power to determine the number of vacua, the symmetry breaking pattern, and the action of the symmetry on the vacua. The SymTFT also manifestly encodes the order parameters for these gapped phases, thus providing a generalized, categorical Landau paradigm for (1+1)d gapped phases. We find that for non-invertible symmetries the order parameters involve multiplets containing both untwisted and twisted sector local operators, and hence can be interpreted as mixtures of conventional and string order parameters. We also observe that spontaneous breaking of non-invertible symmetries can lead to vacua that are physically distinguishable: unlike the standard symmetries described by groups, non-invertible symmetries can have different actions on different vacua of an irreducible gapped phase. This leads to the presence of relative Euler terms between physically distinct vacua. Along with the physical description of symmetric gapped phases, we also provide a mathematical one as pivotal 2-functors whose source 2-category is the delooping of the fusion category characterizing the symmetry and the target 2-category is the Euler completion of 2-vector spaces.
Fracton phases are new types of phases of matter characterized by subsystem global symmetry, which is a generalized global symmetry whose symmetry operator is partially topological. Their continuum low-energy effective descriptions admit two different formulations: an exotic quantum field theory (QFT) using exotic tensor gauge fields, and a foliated QFT constructed from a foliation structure and foliated gauge fields. For certain fracton QFTs, these two descriptions are equivalent, which is called the foliated-exotic duality. In this dissertation, we extend the foliated-exotic duality by combining it with the anomaly inflow mechanism for 't Hooft anomalies of subsystem symmetries. This dissertation has two main results. First, we discuss the exotic and foliated
BF theories in 2+1 dimensions, which exhibit the mixed 't Hooft anomaly of
ZN×ZN subsystem symmetry. This anomaly is captured by a subsystem symmetry-protected topological (SSPT) phase for
ZN×ZN subsystem symmetry in one dimension higher. By extending the foliated-exotic duality in the fractonic
BF theory to the SSPT phase, we establish the field correspondences in the SSPT phase and construct the foliated description of the SSPT phase. Second, we discuss the exotic
ϕ-theory in 2+1 dimensions -- a fractonic gapless scalar field theory, which has the 't Hooft anomaly of
U(1)×U(1) subsystem symmetry. The anomaly is captured by an SSPT phase for
U(1)×U(1) subsystem symmetry in 3+1 dimensions via the anomaly inflow mechanism. Extending the foliated-exotic duality to the
ϕ-theory, we establish field correspondences in the
ϕ-theory and construct the foliated
ϕ-theory that is equivalent to the exotic
ϕ-theory. This provides the first example of the foliated-exotic duality in gapless theories.
The study of gapless phases with categorical (or so-called non-invertible) symmetries is a formidable task, in particular in higher than two space-time dimensions. In this paper we build on previous works [L. Bhardwaj et al. SciPost Phys. 19, 056 (2025); L. Bhardwaj et al. arXiv:2502.20440] on gapped phases in (2+1)d and provide a systematic framework to study phase transitions with categorical symmetries. The Symmetry Topological Field Theory (SymTFT) is, as often in these matters, the central tool. Applied to gapless theories, we need to consider the extension of the SymTFT to interfaces between topological orders, so-called “club sandwiches”, which realize generalizations of so-called Kennedy-Tasaki (KT) transformations. This requires an input phase transition for a smaller symmetry, such as the Ising transition for \mathbb{Z}_2 ℤ 2 , and the SymTFT constructs a transformation to a gapless phase with a larger categorical symmetry. We carry this out for categorical symmetries whose SymTFT is a (3+1)d Dijkgraaf-Witten (DW) theory for a finite group G G with twist - so-called all bosonic fusion 2-categories. We classify such interfaces using a physically motivated picture of generalized gauging, as well as with a complementary analysis using (bi-)module 2-categories. This is exemplified in numerous Abelian and non-Abelian DW theories, giving rise to interesting gapless phases such as intrinsically gapless symmetry protected phases (igSPTs) and spontaneous symmetry breaking phases (igSSBs) from Abelian, S_3 S 3 , and D_8 D 8 DW theories.