Modular extension of topological orders from congruence representations

Abstract

We present an efficient method to compute the modular extension of both fermionic topological orders and ${\mathbb{Z}}_2$-symmetric bosonic topological orders in two spatial dimensions, basing on congruence representations of ${\mathrm{SL}}_2(\mathbb{Z})$ and its subgroups. To demonstrate the validity of our approach, we provide explicit calculations for topological orders with rank up to 10 for the fermionic cases and up to 6 for the bosonic cases. Along the way, we clarify the relation between fermionic rational conformal field theories, which live on the boundary of the corresponding fermionic topological orders, and modular extensions. In particular, we show that the ${\mathrm{SL}}_2(\mathbb{Z})$ representation of the R-R sector can be determined from the NS-NS sector using the modular extensions.

96 pages, no figure

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• Non-Abelian Anyons and Topological Quantum Computation
• Twisted gauge theories in 3D Walker-Wang models
• Fermionic CFTs from classical codes over finite fields
• On Classification of Fermionic Rational Conformal Field Theories
• Reconstruction of modular data from $S{L}_2(\mathbb{Z})$ representations
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Cited by in corpus (2)

• Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders
• Gapped boundaries of fermionic topological orders and higher central charges