nLab fat diagonal

Definition

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

$$\mathbf{\Delta}_X^n \hookrightarrow X^n$$

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

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

Properties

Relation to configuration spaces

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

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

Similarly, the blowup of the fat diagonal in $X^n$ 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