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What Chern-Simons theory assigns to a point.

André G Henriques1

  • 1Mathematical Institute, Oxford University, Oxford OX2 6GG, United Kingdom andre.henriques@maths.ox.ac.uk.

Proceedings of the National Academy of Sciences of the United States of America
|December 7, 2017
PubMed
Summary

Chern-Simons theory assigns representations of the based loop group to points. This study identifies these representations and proves they form bicommutant categories, a higher categorical analog of von Neumann algebras.

Keywords:
Chern–Simons theoryDrinfel’d centerconformal netsextended field theoryloop groups

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Area of Science:

  • Mathematical Physics
  • Quantum Field Theory
  • Category Theory

Background:

  • Chern-Simons theory is a topological quantum field theory.
  • Understanding the mathematical objects Chern-Simons theory assigns to points is crucial for its interpretation.
  • Loop groups are central to the structure of gauge theories.

Purpose of the Study:

  • To determine what mathematical objects Chern-Simons theory assigns to points.
  • To characterize the structure of these objects and their relationships within representation theory.

Main Methods:

  • Identification of specific projective unitary representations of the based loop group.
  • Definition and analysis of the fusion product of these representations.
  • Investigation of the Drinfel'd center of the representation category.
  • Characterization of bicommutant categories as higher categorical analogs of von Neumann algebras.

Main Results:

  • Chern-Simons theory assigns representations of the based loop group to points.
  • The Drinfel'd center of the representation category is shown to be equivalent to positive energy representations of the free loop group (modulo conjectures).
  • The category of representations of the based loop group is proven to be a bicommutant category (modulo conjectures).

Conclusions:

  • The study provides a precise answer to what Chern-Simons theory assigns to points: representations of the based loop group.
  • These representations form bicommutant categories, offering a new perspective on their algebraic structure.
  • The results hold for specific gauge groups and are contingent on certain mathematical conjectures.