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BIMSA TQFT and Higher Symmetries Seminar
Extracting symmetry fractionalisation data from microscopics
Extracting symmetry fractionalisation data from microscopics
Organizers
Speaker
Siddharth Vadnerkar
Time
Tuesday, October 29, 2024 9:40 AM - 11:40 AM
Venue
A3-2-301
Online
Zoom 468 248 1222
(BIMSA)
Abstract
Based on "An operator algebraic approach to symmetry defects and fractionalisation", (arxiv:2411.XXXX). Joint work with Kyle Kawagoe and Daniel Wallick.
Symmetry has always played an important role in physics, and in the last 2 decades, the notion of topological order has taken the centre stage in many sub-disciplines of physics. A Symmetry Enriched Topological (SET) order is when one marries the concept of topological order and symmetry and considers sysems that have a global symmetry, as well as topological order. It is well known that gapped topological order in 2+1 dimensions is mathematically described by Unitary Modular Tensor Categories, and houses long-range quasi-particle excitations called anyons. It is natural to study the consequences of imposing a symmetry on such a system. The landmark paper by BBCW (arxiv:1410.4540) worked out the algebraic theory of such a resulting system, with the conclusion that such a theory must be a $G$-crossed braided monoidal category. This approach was based on a UMTC as a starting point and completely independent of microscopics.
In this talk, we take an entirely different approach. Starting from the algebra of local observables of any lattice system in 2+1D, we propose a "selection criterion" to extract global data of such system. We then work out the resultant theory of such data and it turns out to be a $G$-crossed braided monoidal category as expected. We will illustrate this technique using lattice models and see that the resultant category is equivalent to the algebraic theory one gets using the procedure of BBCW.
Symmetry has always played an important role in physics, and in the last 2 decades, the notion of topological order has taken the centre stage in many sub-disciplines of physics. A Symmetry Enriched Topological (SET) order is when one marries the concept of topological order and symmetry and considers sysems that have a global symmetry, as well as topological order. It is well known that gapped topological order in 2+1 dimensions is mathematically described by Unitary Modular Tensor Categories, and houses long-range quasi-particle excitations called anyons. It is natural to study the consequences of imposing a symmetry on such a system. The landmark paper by BBCW (arxiv:1410.4540) worked out the algebraic theory of such a resulting system, with the conclusion that such a theory must be a $G$-crossed braided monoidal category. This approach was based on a UMTC as a starting point and completely independent of microscopics.
In this talk, we take an entirely different approach. Starting from the algebra of local observables of any lattice system in 2+1D, we propose a "selection criterion" to extract global data of such system. We then work out the resultant theory of such data and it turns out to be a $G$-crossed braided monoidal category as expected. We will illustrate this technique using lattice models and see that the resultant category is equivalent to the algebraic theory one gets using the procedure of BBCW.