nLab
fat diagonal

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Given a topological space XX and a natural number nn, writing X nX^n for the nn-fold product topological space of XX with itself, its fat diagonal

Δ X nX n \mathbf{\Delta}_X^n \hookrightarrow X^n

is the topological subspace of n-tuples of points for which at least one pair of components coincide:

Δ X n{(x i)X n|x i=x jfor someij}. \mathbf{\Delta}_X^n \;\coloneqq\; \left\{ (x_i) \in X^n \;\vert\; x_i = x_j \, \text{for some}\, i \neq j \right\} \,.

Properties

Relation to configuration spaces

The complement X nΔ X nX^n \setminus \mathbf{\Delta}_X^n of the nn-fold Cartesian product by its fat diagonal may be understood as the configuration space of nn distinguishable points in XX. The quotient space of that by the action of the symmetric group S nS_n given by permutation of points in XX yields the configuration space of nn indistinguishable points in XX:

Conf n(X)(X nΔ X n)/S n. Conf_n(X) \;\coloneqq\; \left( X^n \setminus \mathbf{\Delta}_X^n\right)/S_n \,.

Similarly, the blowup of the fat diagonal in X nX^n yields the Fulton-MacPherson compactification of configuration spaces of points.

Relation to renormalization

Closely related to the role of the fat diagonal in configuration spaces is its role in renormalization of perturbative quantum field theories, which may be formulated as the choice of extensions of distributions (of time-ordered products) from the complement of a fat diagonal to the fat diagonal (the locus of coinciding “virtual particles” where interactions take place).

For more on this see at geometry of physics – perturbative quantum field theory the chapter Interacting quantum fields

Last revised on November 4, 2018 at 13:01:23. See the history of this page for a list of all contributions to it.