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arXiv:2603.11681

Foliated-Exotic Duality and Anomaly Inflow in Fracton Quantum Field Theories

Shutaro Shimamura

Published Mar 12, 2026• 0 citations
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Abstract

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 $\mathbb{Z}_N \times \mathbb{Z}_N$ subsystem symmetry. This anomaly is captured by a subsystem symmetry-protected topological (SSPT) phase for $\mathbb{Z}_N \times \mathbb{Z}_N$ 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 $U(1) \times U(1)$ subsystem symmetry. The anomaly is captured by an SSPT phase for $U(1) \times U(1)$ 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.

DOI
10.48550/arxiv.2603.11681

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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 BFBFBF theories in 2+1 dimensions, which exhibit the mixed 't Hooft anomaly of ZN×ZN\mathbb{Z}_N \times \mathbb{Z}_NZN​×ZN​ subsystem symmetry. This anomaly is captured by a subsystem symmetry-protected topological (SSPT) phase for ZN×ZN\mathbb{Z}_N \times \mathbb{Z}_NZN​×ZN​ subsystem symmetry in one dimension higher. By extending the foliated-exotic duality in the fractonic BFBFBF 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 ϕϕϕ-theory in 2+1 dimensions -- a fractonic gapless scalar field theory, which has the 't Hooft anomaly of U(1)×U(1)U(1) \times U(1)U(1)×U(1) subsystem symmetry. The anomaly is captured by an SSPT phase for U(1)×U(1)U(1) \times U(1)U(1)×U(1) subsystem symmetry in 3+1 dimensions via the anomaly inflow mechanism. Extending the foliated-exotic duality to the ϕϕϕ-theory, we establish field correspondences in the ϕϕϕ-theory and construct the foliated ϕϕϕ-theory that is equivalent to the exotic ϕϕϕ-theory. This provides the first example of the foliated-exotic duality in gapless theories.
We study 't Hooft anomalies and the related anomaly inflow for subsystem global symmetries. These symmetries and anomalies arise in a number of exotic systems, including models with fracton order such as the XXX-cube model. As is the case for ordinary global symmetries, anomalies for subsystem symmetries can be canceled by anomaly inflow from a bulk theory in one higher dimension; the corresponding bulk is therefore a nontrivial subsystem symmetry protected topological (SSPT) phase. We demonstrate these phenomena in several examples with continuous and discrete subsystem global symmetries, as well as time-reversal symmetry. For each example we describe the boundary anomaly, and present classical continuum actions for the corresponding bulk SSPT phases, which describe the response of background gauge fields associated with the subsystem symmetries. Interestingly, we show that the anomaly does not uniquely specify the bulk SSPT phase. In general, the latter may also depend on how the symmetry and the associated foliation structure on the boundary are extended into the bulk.
There has been proposed two continuum descriptions of fracton systems: foliated quantum field theories (FQFTs) and exotic quantum field theories. Certain fracton systems are believed to admit descriptions by both, and hence a duality is expected between such a class of FQFTs and exotic QFTs. In this paper we study this duality in detail for concrete examples in 2+1 2 + 1 and 3+1 3 + 1 dimensions. In the examples, both sides of the continuum theories are of BF B F -type, and we find the explicit correspondences of gauge-invariant operators, gauge fields, parameters, and allowed singularities and discontinuities. This deepens the understanding of dualities in fractonic quantum field theories.
We investigate the properties of foliated gauge fields and construct several foliated field theories in 3+1d that describe foliated fracton orders both with and without matter, including the recent hybrid fracton models. These field theories describe Abelian or non-Abelian gauge theories coupled to foliated gauge fields, and they fall into two classes of models that we call the electric models and the magnetic models. We show that these two classes of foliated field theories enjoy a duality. We also construct a model (using foliated gauge fields and an exactly solvable lattice Hamiltonian model) for a subsystem-symmetry protected topological (SSPT) phase, which is analogous to a one-form symmetry protected topological phase, with the subsystem symmetry acting on codimension-two subregions. We construct the corresponding gauged SSPT phase as a foliated two-form gauge theory. Some instances of the gauged SSPT phase are a variant of the X-cube model with the same ground state degeneracy and the same fusion, but different particle statistics.
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] \mathcal{C}\left[G\right] 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.
Fracton phases of matter are gapped phases of matter that, by dint of their sensitivity to UV data, demand non-standard quantum field theories to describe them in the IR. Two such approaches are foliated quantum theory and exotic field theory. In this paper, we explicitly construct a map from one to the other and work out several examples. In particular, we recover the equivalence between the foliated and exotic fractonic BF theories recently demonstrated at the level of operator correspondence. We also demonstrate the equivalence of toric code layers and the anisotropic model with lineons and planons to the foliated BF theory with one and two foliations, respectively. Finally, we derive new exotic field theories that provide simple descriptions of hybrid fracton phases from foliated field theries known to do so. Our results both provide new examples of exotic field theories and pave the way toward their systematic construction from foliated field theories.
Abstract Fracton phases of matter are gapped phases of matter that, by dint of their sensitivity to UV data, demand non-standard quantum field theories to describe them in the IR. Two such approaches are foliated quantum theory and exotic field theory. In this paper, we explicitly construct a map from one to the other and work out several examples. In particular, we recover the equivalence between the foliated and exotic fractonic BF theories recently demonstrated at the level of operator correspondence. We also demonstrate the equivalence of toric code layers and the anisotropic model with lineons and planons to the foliated BF theory with one and two foliations, respectively. Finally, we derive new exotic field theories that provide simple descriptions of hybrid fracton phases from foliated field theries known to do so. Our results both provide new examples of exotic field theories and pave the way toward their systematic construction from foliated field theories.