Abstract
The 2+1D topological order can be characterized by the mapping-class-group representations for Riemann surfaces of genus-1, genus-2, etc. In this paper, we use those representations to determine the possible gapped boundaries of a 2+1D topological order, as well as the domain walls between two topological orders. We find that mapping-class-group representations for both genus-1 and genus-2 surfaces are needed to determine the gapped domain walls and boundaries. Our systematic theory is based on the fixed-point partition functions for the walls (or the boundaries), which completely characterize the gapped domain walls (or the boundaries). The mapping-class-group representations give rise to conditions that must be satisfied by the fixed-point partition functions, which leads to a systematic theory. Such conditions can be viewed as bulk topological order determining the (non-invertible) gravitational anomaly at the domain wall, and our theory can be viewed as finding all types of the gapped domain wall given a (non-invertible) gravitational anomaly. We also developed a systematic theory of gapped domain walls (boundaries) based on the structure coefficients of condensable algebras.
Highlights
Topological order is a new kind of order in gapped quantum states of matter beyond Landau symmetry breaking theory [1,2]
We develop a systematic approach to the gapped domain walls between two topological orders A and B. (If B is the trivial topological order, the domain wall becomes the boundary of topological order A.) Our systematic approach is based on the topological partition function WBIBAIA,g of the domain wall g, which is a Riemann surface of arbitrary genus g, which is a multicomponent partition function labeled by IA, IB
The multicomponent partition function WBIBAIA,g is expected since the domain wall has a noninvertible gravitational anomaly [35] as characterized by topological orders A and B
Summary
Topological order is a new kind of order in gapped quantum states of matter beyond Landau symmetry breaking theory [1,2]. To obtain a full description of 2 + 1D topological orders, the modular data for a genus-1 surface are not enough [12,13,14]; we must use the non-Abelian geometric phases (i.e., the mapping-class-group representations) for genus-2 surfaces [15]. We try to address the following question: Given a 2 + 1D topological order described by (S, T, c) and the data from higher genus, how do we describe and classify different gapped 1+1D boundary phases? If we find that there is no gapped 1+1D boundary for a type of 2 + 1D topological order, such a type of topological order must have a gapless boundary It can be stated in the following way: Given a type of gravitational anomaly, how do we describe different gapped 1+1D phases? Our approach can be directly generalized to higher dimensions, to systematically find gapped boundaries of a topological order from the data that describe mapping-class-group action in higher dimensions [35]
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