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Canonical Charges in Flatland

Max Riegler, Céline Zwikel

TL;DR

This work analyzes how gravitational gauge symmetries at infinity and near horizons are encoded as canonical charges in 2+1 dimensions, emphasizing both metric and Chern-Simons formulations. It demonstrates the emergence of the bms_3 algebra with a central extension in flat space and explores broad boundary conditions that shape the asymptotic symmetry structure. Through near-horizon analyses, it identifies soft hair as zero-energy excitations generated by affine u(1) factors, linking boundary degrees of freedom to holography and information-theoretic challenges. Altogether, the results illustrate how boundary data in 3D gravity captures bulk physics, with implications for flat-space holography and black hole information paradox scenarios.

Abstract

In this series of lectures we give an introduction to the concept of asymptotic symmetry analysis with a focus on asymptotically flat spacetimes in 2+1 dimensions. We explain general ideas of quantizing gauge theories and then apply these ideas to gravity both in the metric as well as the Chern-Simons formulations. This enables one to compute the asymptotic symmetries of given gravitational configurations that in turn act as the basic underlying symmetries of a possible dual quantum field theory in the context of holography. We also briefly elaborate on the concept of "soft hair" excitations of black holes in this context.

Canonical Charges in Flatland

TL;DR

This work analyzes how gravitational gauge symmetries at infinity and near horizons are encoded as canonical charges in 2+1 dimensions, emphasizing both metric and Chern-Simons formulations. It demonstrates the emergence of the bms_3 algebra with a central extension in flat space and explores broad boundary conditions that shape the asymptotic symmetry structure. Through near-horizon analyses, it identifies soft hair as zero-energy excitations generated by affine u(1) factors, linking boundary degrees of freedom to holography and information-theoretic challenges. Altogether, the results illustrate how boundary data in 3D gravity captures bulk physics, with implications for flat-space holography and black hole information paradox scenarios.

Abstract

In this series of lectures we give an introduction to the concept of asymptotic symmetry analysis with a focus on asymptotically flat spacetimes in 2+1 dimensions. We explain general ideas of quantizing gauge theories and then apply these ideas to gravity both in the metric as well as the Chern-Simons formulations. This enables one to compute the asymptotic symmetries of given gravitational configurations that in turn act as the basic underlying symmetries of a possible dual quantum field theory in the context of holography. We also briefly elaborate on the concept of "soft hair" excitations of black holes in this context.

Paper Structure

This paper contains 31 sections, 2 theorems, 186 equations.

Table of Contents

  1. Introduction and Motivation
  2. Proper gauge transformations:
  3. Improper gauge transformations:
  4. Preliminaries
  5. Gravity in 3D
  6. Gravity as a Chern-Simons Theory
  7. Anti-de Sitter Spacetimes:
  8. Canonical Analysis and Asymptotic Symmetries
  9. General Ideas of Quantizing Gauge Theories
  10. The Lagrangian and Primary Constraints:
  11. The Canonical Hamiltonian:
  12. Secondary Constraints and the Total Hamiltonian:
  13. First Class and Second Class Constraints:
  14. 3D Chern-Simons Theory as an Explicit Example
  15. Computing Charges in the Metric Formulation
  16. ...and 16 more sections

Key Result

Theorem 3.1

Ordinary Noether theorem: Continuous global symmetries (defined up to gauge transformations) of a Lagrangian are in one-to-one correspondence with equivalence classes of conserved currents.

Theorems & Definitions (2)

  • Theorem 3.1
  • Theorem 3.2