Higher-Rank Tensor Non-Abelian Field Theory: Higher-Moment or Subdimensional Polynomial Global Symmetry, Algebraic Variety, Noether's Theorem, and Gauging

Abstract

With a view toward a fracton theory in condensed matter, we introduce a higher-moment polynomial degree-p global symmetry, acting on complex scalar/vector/tensor fields (e.g., ordinary or vector global symmetry for p$=0$ and p$=1$ respectively). We relate this higher-moment global symmetry of $n$-dimensional space, to a lower degree (either ordinary or higher-moment, e.g., degree-(p-$\ell$)) subdimensional or subsystem global symmetry on layers of $(n-\ell )$-submanifolds. These submanifolds are algebraic affine varieties (i.e., solutions of polynomials). The structure of layers of submanifolds as subvarieties can be studied via mathematical tools of embedding, foliation, and algebraic geometry. We also generalize Noether's theorem for this higher-moment polynomial global symmetry. We can promote the higher-moment global symmetry to a local symmetry, and derive a new family of higher-rank-m symmetric tensor gauge theory by gauging, with m = p$+1$. By further gauging a discrete ${\mathbb{Z}}_2^C$ charge conjugation (particle-hole) symmetry, we derive a new general class of rank-m tensor non-abelian gauge field theory (the gauge structure is non-commutative thus non-abelian but not an ordinary group): a hybrid class of (symmetric or non-symmetric) higher-rank-m tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879], interplaying between gapless and gapped sectors.

23 pages. Sequel to arXiv:1909.13879, arXiv:1912.13485. Dedicated to 90 years of Gauge Principle since Hermann Weyl. v2: clarifications added

References in corpus (38)

• Zoo of quantum-topological phases of matter
• Local stabilizer codes in three dimensions without string logical operators
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Quantum Glassiness
• Fractons
• A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations
• Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids
• Fracton Phases of Matter
• A symmetry principle for Topological Quantum Order
• Generalized Electromagnetism of Subdimensional Particles: A Spin Liquid Story
• The Fracton Gauge Principle
• Subsystem symmetry protected topological order
• Fracton Models on General Three-Dimensional Manifolds
• Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
• Fractal symmetries: Ungauging the cubic code
• Foliated fracton order from gauging subsystem symmetries
• Towards classification of Fracton phases: the multipole algebra
• Field Theories With a Vector Global Symmetry
• Generalized Elitzur's Theorem and Dimensional Reduction
• Higher-Spin Witten Effect and Two-Dimensional Fracton Phases
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
• Classification of subsystem symmetry-protected topological phases
• Twisted Fracton Models in Three Dimensions
• Cage-Net Fracton Models
• Symmetric Fracton Matter: Twisted and Enriched
• Symmetric Tensor Gauge Theories on Curved Spaces
• Scalar Field Theories with Polynomial Shift Symmetries
• Geometrical String and Spin Systems
• Twisted foliated fracton phases
• Gauging fractons: immobile non-Abelian quasiparticles, fractals, and position-dependent degeneracies
• Strong planar subsystem symmetry-protected topological phases and their dual fracton orders
• Self-Avoiding Gonihedric Srting and Spin Systems
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• Gauging permutation symmetries as a route to non-Abelian fractons
• Quantum Statistics and Spacetime Surgery
• Non-Abelian Gauged Fracton Matter Field Theory: New Sigma Models, Superfluids and Vortices
• Quantum Statistics and Spacetime Topology: Quantum Surgery Formulas
• Higher-Rank Tensor Field Theory of Non-Abelian Fracton and Embeddon

Cited by in corpus (23)

• Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry
• Non-Abelian Hybrid Fracton Orders
• Infinite families of fracton fluids with momentum conservation
• Fractonic superfluids. (II). Condensing subdimensional particles
• Boson-fermion duality with subsystem symmetry
• Boundary theory of the X-cube model in the continuum
• A symmetry principle for gauge theories with fractons
• Gapped boundaries of (3+1)d topological orders
• Fracton Self-Statistics
• Lattice Clifford fractons and their Chern-Simons-like theory
• Renormalization group analysis on emergence of higher rank symmetry and higher moment conservation
• Noncommutative Field Theory of the Tkachenko Mode: Symmetries and Decay Rate
• Fractionalizing Global Symmetry on Looplike Topological Excitations
• Boundary Modes in the Chamon Model
• Asymptotic symmetries and soft charges of fractons
• Bridging three-dimensional coupled-wire models and cellular topological states: Solvable models for topological and fracton orders
• Field Theories for type-II fractons
• Non-Abelian fracton order from gauging a mixture of subsystem and global symmetries
• Shift symmetries and duality web in gauge theories
• Fractonic superfluids. III. Hybridizing higher moments
• Electric-Magnetic Duality for Symmetric Tensor Gauge Theories and Immobile $p$-branes
• Infinite-component $BF$ field theory: Connection of fracton order, Toeplitz braiding, and non-Hermitian amplification
• Preparing Code States via Seed-Entangler-Enriched Sequential Quantum Circuits: Application to Tetra-Digit Topological Error-Correcting Codes