Mathematical Description of Bulk Boundary Correspondances

by Tommy on 5/02/2017

This is some great stuff if you are into this kind of mathematics as I am.

Boundary-bulk relation in topological orders, Liang Kong, Xiao-Gang Wen, Hao Zheng (2 February 2017)

In this paper, we study the relation between an anomaly-free n+1D topological order, which are often called n+1D topological order in physics literature, and its nD gapped boundary phases. We argue that the n+1D bulk anomaly-free topological order for a given nD gapped boundary phase is unique. This uniqueness defines a notion of the “bulk” for a given gapped boundary phase. In this paper, we show that the n+1D “bulk” phase is given by the “center” of the nD boundary phase. In other words, the geometric notion of the “bulk” corresponds precisely to the algebraic notion of the “center”. We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of “bulk” satisfying the same universal property as that of the “center” of an algebra in mathematics, i.e. “bulk = center”. The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions.

There are some very interesting physical ramifications here too.

Happy Groundhog’s Day! Again.

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