On topology of the moduli space of gapped Hamiltonians for topological phases

A genus is a pair of an anyon model $B$ with a non-negative rational number c such that $∑iθidi2D=e2πic/8$⁠. The word genus comes from the case of abelian anyon models and lattice conformal field theories. In this case, c is the chiral central charge of the lattice CFT, and the abelian anyon model of the lattice CFT is given by the genus of the lattice.

More generally, one can vary the parameter over an auxiliary bulk such that there is no bulk dependence; this produces the integer class Berry phase,^15,17 which is the analog of the Wess–Zumino term in the nonlinear sigma model.

By Hamiltonian schema, we mean a well-defined procedure to write down Hamiltonians from some celluations of a space manifold, potentially with some extra background information such as orientation, branching, and framing.

The equalities hold under suitable choices of topology, using $BU(1)=CP∞=K(Z,2)$ and BG = K(G, 1) for G = U(1) with the discrete topology.

More generally, we can couple the system to particles in $ZN$ gauge theory that can have non-trivial statistics.

For given n₁, ω₂ there can be no solution for n₂. For instance, if we take M = S¹ × S², and n₁ is the integral volume form on S¹ modulo 2, and ω₂ is the integral volume form on S² modulo 2, then n₁ ∪ ω₂ + c₋ω₂ ∪₁ω₂ is the integral volume form on S¹ × S² modulo 2, which is closed but not exact in the cohomology with the $Z2$ coefficient, and thus the solution for n₂ does not exist.

Here it is an untwisted $ZN$ quantum double, corresponding to “minimally” gauging the $ZN$ symmetry without stacking extra symmetry protected topological phases.

One can also compute the higher Berry curvature from the formula in Ref. 15, which we will leave for future work.

It would also be interesting if the bulk-boundary correspondence could rule out isolated gapless theories without relevant deformations.

For N = 0 mod 4, the center of Spin(N) has three $Z2$ subgroups, and the three quotients $Spin(N)/Z2$ for these three $Z2$ subgroups give SO(N), Sc(N) and Ss(N).