# nLab asymptotic isometry

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Considering a class of diffeomorphisms which are not isometries but change metrics a tiny bit, with some asymptotic boundary conditions.

This is useful to obtain conformal symmetry algebras in some physical theories of gravity, esp. in 3D.

• J. Brown, Marc Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Comm. Math. Phys. 104 (1986) 207 (key paper!!) euclid

• Coussaert, Marc Henneaux, P. van Driel, Class. Quant. Grav. 12 (1995) 2961

…Brown-Henneaux boundary condition

New developments

• Marc Henneaux, Soo-Jong Rey, Nonlinear W(infinity) algebra as asymptotic symmetry of 3-dimensional higher spin Anti-de Sitter gravity, arxiv/1008.4579