Quantization of Chern-Simons topological invariants for H-type and L-type quantum systems

Abstract

In 2+1-dimensions (2+1D), a gapped quantum phase with no symmetry (i.e. a topological order) can have a thermal Hall conductance κxy = c π 2 k 2 B 3h T , where the dimensionless c is called chiral central charge. If there is a U1 symmetry, a gapped quantum phase can also have a Hall conductance σxy = ν e 2 h , where the dimensionless ν is called filling fraction. In this paper, we derive some quantization conditions of c and ν, via a cobordism approach to define Chern-Simons topological invariants which are associated with c and ν. In particular, we obtain quantization conditions that depend on the ground state degeneracies on Riemannian surfaces, and quantization conditions that depend on the type of spacetime manifolds where the topological partition function is non-zero. CONTENTSI. Introduction 1 A. Notations and conventions 3 II. The vector bundle on the moduli space for H-type bosonic systems with gapped liquid ground states 3 III. Chern-Simons invariants in 2+1D H-type bosonic U1-SET and U1-SPT orders 3 A. Bosonic gapped liquids with U1 symmetry in 2-dimensional space 3 B. Examples 5 1. Bosonic K-matrix Abelian topological orders 5 2. Bosonic non-Abelian topological orders described by SU (2) k Chern-Simons Theory 6 C. A general point of view 6 IV. Topological invariants in 2+1D H-type fermionic enriched topological orders with U f 1 symmetry 6 A. Symmetry twist of fermion system 6 B. U f 1 symmetry and spin C structure 7 C. Fermionic gapped liquids with U f 1 symmetry in 2-dimensional space 8 D. H-type invertible fermionic U f 1 -enriched topological orders 9 V. The topological invariants for L-type topological orders 9 A. Topological partition function for L-type topological order 9 B. The quantization of c for 2+1D bosonic topological orders with non-zero partition function 10 C. The quantization of c for 2+1D bosonic topological orders with vanishing partition functions only on non-spin manifolds and for 2+1D fermionic invertible topological orders 10 D. Framing anomaly 11 VI. The topological invariants for L-type topological orders with U1 symmetry 11 A. Quantization of c and ν for bosonic U1-SET orders with non-zero partition functions 11 B. Quantization of c and ν for fermionic U f 1 -enriched topological orders with non-zero partition functions 12 VII. Summary 12 A. Characteristic numbers of 4-manifolds 12 B. Characteristic numbers of surface bundles over surfaces 13 1. Definitions and relations 13 2. Realising characteristic numbers 14 C. Checking (29), (30), (31) 14 1. Spread polynomials Sn(x) 14 2. Symmetric polynomials e l and p l 14 3. Checking (29) 15 4. Checking (30) 15 5. Checking (31) 15 D. Group extension and trivialization 15 References 16