A relation between chiral central charge and ground state degeneracy in 2+1-dimensional topological orders

Abstract

A bosonic topological order on $d$-dimensional closed space ${\Sigma }^d$ may have degenerate ground states. The space ${\Sigma }^d$ with different shapes (different metrics) form a moduli space ${\mathcal{M}}_{{\Sigma }^d}$. Thus the degenerate ground states on every point in the moduli space ${\mathcal{M}}_{{\Sigma }^d}$ form a complex vector bundle over ${\mathcal{M}}_{{\Sigma }^d}$. It was suggested that the collection of such vector bundles for $d$-dimensional closed spaces of all topologies completely characterizes the topological order. Using such a point of view, we propose a direct relation between two seemingly unrelated properties of 2+1-dimensional topological orders: (1) the chiral central charge $c$ that describes the many-body density of states for edge excitations (or more precisely the thermal Hall conductance of the edge), (2) the ground state degeneracy $D_g$ on closed genus $g$ surface. We show that $c{D}_g/2\in \mathbb{Z},g\ge 3$ for bosonic topological orders. We explicitly checked the validity of this relation for over 140 simple topological orders. For fermionic topological orders, let $D_{g,\sigma }^e$ ($D_{g,\sigma }^o$) be the degeneracy with even (odd) number of fermions for genus-$g$ surface with spin structure $\sigma$. Then we have $2c{D}_{g,\sigma }^e\in \mathbb{Z}$ and $2c{D}_{g,\sigma }^o\in \mathbb{Z}$ for $g\ge 3$.

arXiv comment8 pages. This paper supersedes Section XIV of an unpublished work arXiv:1405.5858. We add new results on fermionic topological orders and some numerical checks

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Cited by in corpus (2)

• M-theoretic Genesis of Topological Phases
• Ordering the topological order in the fractional quantum Hall effect