Xiao-Gang Wen

Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond

The challenge of identifying symmetry-protected topological states (SPTs) is due to their lack of symmetry-breaking order parameters and intrinsic topological orders. For this reason, it is impossible to formulate SPTs under Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and topology-dependent ground state degeneracy. However, the partition functions from path integrals with various symmetry twists are universal SPT invariants, fully characterizing SPTs. In this work, we use gauge fields to represent those symmetry twists in closed spacetimes of any dimensionality and arbitrary topology. This allows us to express the SPT invariants in terms of continuum field theory. We show that SPT invariants of pure gauge actions describe the SPTs predicted by group cohomology, while the mixed gauge-gravity actions describe the beyond-group-cohomology SPTs. We find new examples of mixed gauge-gravity actions for U(1) SPTs in (4+1)D via the gravitational Chern-Simons term. Field theory representations of SPT invariants not only serve as tools for classifying SPTs, but also guide us in designing physical probes for them. In addition, our field theory representations are independently powerful for studying group cohomology within the mathematical context.

Non-Abelian string and particle braiding in topological order: Modular SL(3,Z) representation and (3 + 1)-dimensional twisted gauge theory

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group G and a 4-cocycle twist ω 4 of G s cohomology group H 4 ( G , R / Z ) in three-dimensional space and one-dimensional time ( 3 + 1 D ) . We establish the topological spin and the spin-statistics relation for the closed strings and their multistring braiding statistics. The 3 + 1 D twisted gauge theory can be characterized by a representation of a modular transformation group, SL ( 3 , Z ) . We express the SL ( 3 , Z ) generators S x y z and T x y in terms of the gauge group G and the 4-cocycle ω 4 . As we compactify one of the spatial directions z into a compact circle with a gauge flux b inserted, we can use the generators S x y and T x y of an SL ( 2 , Z ) subgroup to study the dimensional reduction of the 3D topological order C 3 D to a direct sum of degenerate states of 2D topological orders C b 2 D in different flux b sectors: C 3 D = ⊕ b C b 2 D . The 2D topological orders C b 2 D are described by 2D gauge theories of the group G twisted by the 3-cocycle ω 3 ( b ) , dimensionally reduced from the 4-cocycle ω 4 . We show that the SL ( 2 , Z ) generators, S x y and T x y , fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into a three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.

Gapped Domain Walls, Gapped Boundaries, and Topological Degeneracy

Gapped domain walls, as topological line defects between (2+1)D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which (2+1)D topological orders must have gapless edge modes, namely, which (1+1)D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix W, whose entries are the fusion-space dimensions W(ia), to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.

Boundary degeneracy of topological order

We introduce the concept of boundary degeneracy of topologically ordered states on a compact orientable spatial manifold with boundaries, and emphasize that the boundary degeneracy provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of the fully gapped edge modes depends on boundary gapping conditions. By associating different types of boundary gapping conditions as different ways of particle or quasiparticle condensations on the boundary, we develop an analytic theory of gapped boundaries. By Chern-Simons theory, this allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which encodes more than the fusion algebra of fractionalized quasiparticles. We apply our theory to Kitaevs toric code and Levin-Wen string-net models. We predict that the

Bosonic anomalies, induced fractional quantum numbers, and degenerate zero modes: The anomalous edge physics of symmetry-protected topological states

Z_2

Multikink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF field theories

toric code and

A Lattice Non-Perturbative Hamiltonian Construction of 1+1D Anomaly-Free Chiral Fermions and Bosons - on the equivalence of the anomaly matching conditions and the boundary fully gapping rules

Z_2

Quantum Statistics and Spacetime Surgery

double-semion model (more generally, the

A Lattice Non-Perturbative Definition of 1+1D Anomaly-Free Chiral Fermions and Bosons

Z_k

Field theory representation of mixed gauge-gravity symmetry-protected topological invariants, group cohomology and beyond

gauge theory and the