Quantum realization and mathematical classification of the non-Abelian higher-rank tensor gauge theories obtained by gauging higher-moment polynomial symmetries

Wang, Xu, and Yau [Phys. Rev. Research 3, 013185 (2021)] constructed a class of continuum non-Abelian higher-rank tensor gauge theories: gauging a degree-$p$ higher-moment polynomial global symmetry yields an Abelian rank-$(p+1)$ tensor gauge theory, and further gauging the charge-conjugation symmetry $\mathbb{Z}_2^C$ produces a hybrid of a symmetric higher-rank tensor gauge theory and an antisymmetric-tensor topological sector whose gauge structure is non-commutative but not a group. The construction is, however, classical and its quantum-mechanical and mathematical identity is unknown. The open problem is to give these theories a well-defined quantum realization (continuum quantization or lattice Hamiltonian regularization, or a proof of obstruction), to classify the non-group gauge structure and its anomalies, to establish the precise correspondence with the polynomial-shift-symmetry and multipole-algebra frameworks of Griffin et al. and Gromov, and to resolve the gauging ambiguity by connecting the continuum constructions to lattice non-Abelian fracton models.

Background

Fractons are quasiparticles of certain three-dimensional gapped quantum phases that are individually immobile and can move only in neutral combinations; the X-cube model (an exactly solvable lattice model whose elementary point excitations cannot move individually, although pairs of them translate along coordinate axes) and the Chamon code (an exactly solvable lattice model whose point excitations are strictly immobile) are canonical examples. They lie outside the standard theory of topological order built on mobile anyons. A continuum route to fracton physics is the higher-rank tensor gauge theory: instead of an ordinary vector gauge field $A_\mu$ with redundancy $A_\mu \sim A_\mu + \partial_\mu \lambda$, one uses a spatial symmetric tensor $A_{ij}$ with a higher-derivative redundancy such as $A_{ij} \sim A_{ij} + \partial_i\partial_j\lambda$. Its Gauss law then implies conservation of both total charge and dipole moment, so isolated charges cannot move. The symmetry counterpart is the higher-moment (multipole) polynomial global symmetry: a complex field $\Phi$ transforms as $\Phi \to e^{iQ(x)}\Phi$ with $Q(x)$ a polynomial of degree $p$ in the spatial coordinates, where degree 0 is an ordinary global $U(1)$ symmetry and degree 1 enforces dipole conservation.

Wang, Xu, and Yau systematized this structure. They related degree-$p$ polynomial symmetries in $n$ dimensions to lower-degree subdimensional symmetries supported on layers of $(n-\ell)$-dimensional submanifolds that are algebraic affine varieties (solution sets of polynomial equations, studied through embedding and foliation), generalized Noether's theorem to these symmetries, and gauged the polynomial symmetry to derive a family of Abelian rank-$(p+1)$ tensor gauge theories. Crucially, they then further gauged the discrete charge-conjugation (particle-hole) symmetry $\mathbb{Z}_2^C$, which acts on the tensor gauge field as $A \to -A$ and does not commute with the polynomial symmetry. The result is a new class of non-Abelian tensor gauge theories: a hybrid of a (symmetric or non-symmetric) higher-rank tensor gauge theory and an antisymmetric-tensor topological field theory of BF type (a topological sector built from dual gauge fields $B$ and $C$, in the sense of Kalb-Ramond/BF topological gauge theory), whose gauge structure is non-commutative, hence called non-Abelian, but is not an ordinary group and therefore not a Yang-Mills theory. The construction generalizes the companion work of Wang and Xu, which introduced the $\mathbb{Z}_2^C$-gauging trick and the embeddon (the proposal that the foliation layers and spacetime embeddings of the theory become quantized excitations), and the companion work of Wang and Yau on non-Abelian gauged fracton sigma models, superfluids, and vortices.

All of these works are, however, essentially classical continuum constructions. The source paper itself leaves the following open. First, the continuum gauging procedures are not connected to the gauging procedures implemented on lattice models (orbifold operations, meaning lattice gauging by summing over symmetry-defect configurations, summing over background fields, condensation, or explicit lattice gauge constructions), so no mapping to discrete non-Abelian fracton models exists. Second, the precise correspondence with the polynomial shift symmetries studied by Griffin, Grosvenor, Hořava, and Yan and with the multipole algebra of Gromov is not worked out; the authors state they do not yet know the correspondence. Third, the full quantum treatment - quantization, operator algebra, spectrum, stability, and the quantum statistics of the fracton-like and embeddon excitations - is deferred; the companion paper states that field-theoretic quantization of even the matter sector is an open question, because the matter Lagrangian has no quadratic kinetic term, making the path integral non-Gaussian and leaving no free propagator, and that lattice Hamiltonian regularization is likewise open. Fourth, different gauging prescriptions applied to the same global symmetry may yield different theories, and classifying which prescriptions flow to which continuum or lattice universality classes is unresolved.

Problem Statement

The open problem is to give the non-Abelian higher-rank tensor gauge theories constructed by Wang, Xu, and Yau - obtained by gauging the degree-$p$ higher-moment polynomial global symmetry and then further gauging the charge-conjugation symmetry $\mathbb{Z}_2^C$ - a well-defined quantum-mechanical and mathematical identity. The common objective is to determine whether this continuum construction corresponds to a genuine quantum theory and quantum phase of matter, and to place it within the known landscape of fracton orders. Concretely, the open problem is settled by resolving the following four jointly supporting sub-questions for this class of theories, each by a definite outcome that may be affirmative or negative: (1) Quantum realization: provide a controlled quantization of the continuum theory (operator algebra, spectrum, stability of the hybrid gapless tensor and gapped topological sectors, quantum statistics of the fracton-like and embeddon excitations) or an explicit lattice Hamiltonian regularization whose Gauss laws and low-energy structure reproduce the continuum gauge structure, or prove an obstruction (for example an anomaly) showing that no such quantum theory exists. (2) Mathematical classification: identify and classify the non-commutative, non-group gauge structure produced by the $\mathbb{Z}_2^C$ gauging, including any anomalies it carries, within an appropriate mathematical framework. (3) Correspondence: establish the precise relation between this higher-moment polynomial symmetry framework and the polynomial shift symmetries of Griffin, Grosvenor, Hořava, and Yan and the multipole algebra of Gromov. (4) Gauging ambiguity and lattice mapping: determine which gauging prescriptions (promoting the symmetry to a local redundancy, summing over background fields, or condensation) applied to the same global symmetry yield equivalent or distinct theories, and identify which lattice non-Abelian fracton models, if any, the continuum theories flow to or describe.

References

1. Juven Wang, Kai Xu, Shing-Tung Yau, "Higher-rank tensor non-Abelian field theory: Higher-moment or subdimensional polynomial global symmetry, algebraic variety, Noether's theorem, and gauging", Phys. Rev. Research 3, 013185 (2021). DOI: 10.1103/PhysRevResearch.3.013185; arXiv:1911.01804; https://doi.org/10.1103/PhysRevResearch.3.013185
2. Juven Wang, Kai Xu, "Higher-rank tensor field theory of non-Abelian fracton and embeddon", Annals of Physics 424, 168370 (2021). DOI: 10.1016/j.aop.2020.168370; arXiv:1909.13879; https://doi.org/10.1016/j.aop.2020.168370
3. Juven Wang, Shing-Tung Yau, "Non-Abelian gauged fracton matter field theory: Sigma models, superfluids, and vortices", Phys. Rev. Research 2, 043219 (2020). DOI: 10.1103/PhysRevResearch.2.043219; arXiv:1912.13485; https://doi.org/10.1103/PhysRevResearch.2.043219
4. Tom Griffin, Kevin T. Grosvenor, Petr Hořava, Ziqi Yan, "Scalar Field Theories with Polynomial Shift Symmetries", Communications in Mathematical Physics 340, 985-1048 (2015). DOI: 10.1007/s00220-015-2461-2; arXiv:1412.1046; https://arxiv.org/abs/1412.1046
5. Andrey Gromov, "Towards classification of Fracton phases: the multipole algebra", Phys. Rev. X 9, 031035 (2019). DOI: 10.1103/PhysRevX.9.031035; arXiv:1812.05104; https://arxiv.org/abs/1812.05104
6. Nathan Seiberg, "Field Theories With a Vector Global Symmetry", SciPost Phys. 8, 050 (2020). DOI: 10.21468/SciPostPhys.8.4.050; arXiv:1909.10544; https://arxiv.org/abs/1909.10544
7. Yi-Hsien Du, Umang Mehta, Dung Xuan Nguyen, Dam Thanh Son, "Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry", SciPost Phys. 12, 050 (2022). DOI: 10.21468/SciPostPhys.12.2.050; arXiv:2103.09826; https://arxiv.org/abs/2103.09826
8. Yi-Ting Tu, Po-Yao Chang, "Non-Abelian fracton order from gauging a mixture of subsystem and global symmetries", Phys. Rev. Research 3, 043084 (2021). DOI: 10.1103/PhysRevResearch.3.043084; arXiv:2103.08603; https://arxiv.org/abs/2103.08603
9. Nathanan Tantivasadakarn, Wenjie Ji, Sagar Vijay, "Non-Abelian hybrid fracton orders", Phys. Rev. B 104, 115117 (2021). DOI: 10.1103/PhysRevB.104.115117; arXiv:2106.03842; https://arxiv.org/abs/2106.03842
10. Rabin Banerjee, Anwesha Chakraborty, "Shift symmetries and duality web in gauge theories", Nuclear Physics B 996, 116354 (2023). DOI: 10.1016/j.nuclphysb.2023.116354; arXiv:2210.12349; https://arxiv.org/abs/2210.12349
11. Daniel Bulmash, Oliver Hart, Rahul Nandkishore, "Multipole groups and fracton phenomena on arbitrary crystalline lattices", SciPost Phys. 15, 235 (2023). DOI: 10.21468/SciPostPhys.15.6.235; arXiv:2301.10782; https://arxiv.org/abs/2301.10782
12. Yuji Hirono, Minyoung You, Stephen Angus, Gil Young Cho, "A symmetry principle for gauge theories with fractons", SciPost Phys. 16, 050 (2024). DOI: 10.21468/SciPostPhys.16.2.050; arXiv:2207.00854; https://arxiv.org/abs/2207.00854
13. Davide Rovere, "Anomalies in Covariant Fracton Theories", Phys. Rev. D 110, 085012 (2024). DOI: 10.1103/PhysRevD.110.085012; arXiv:2406.06686; https://arxiv.org/abs/2406.06686
14. Ryuki Makino, Shin Sasaki, Kenta Shiozawa, "Electric-Magnetic Duality for Symmetric Tensor Gauge Theories and Immobile p-branes", JHEP 2025, 021 (2025). DOI: 10.1007/JHEP11(2025)021; arXiv:2505.11174; https://arxiv.org/abs/2505.11174
15. Bo-Xi Li, Peng Ye, "Infinite-component $BF$ topological field theory: Connection of fracton order, Toeplitz braiding, and non-Hermitian amplification", Phys. Rev. B 113, 125141 (2026). DOI: 10.1103/3dh7-ktmd; arXiv:2511.09301; https://arxiv.org/abs/2511.09301