superdimension

Superdimension is a concept used in supergeometry and related areas of mathematics and theoretical physics to describe the size and structure of a superspace or a supervector space. A finite-dimensional supervector space V over a field (typically the real numbers or complex numbers) decomposes into an even part V0 and an odd part V1, written as V = V0 ⊕ V1. The most common way to denote its size is by the pair (dim V0 | dim V1), often abbreviated as (m | n), where m = dim V0 and n = dim V1. In some contexts, the superdimension is referred to as the difference m − n, reflecting the parity-based grading underlying the construction.

The pair (m | n) expresses the fundamental distinction between even (bosonic) and odd (fermionic) directions in a superspace. This grading influences algebraic, geometric, and analytical operations, including the construction of supermanifolds, morphisms, and integration theory. For a supermanifold, the local model is a space of dimension (m | n), with m even coordinates and n odd coordinates. The tangent space to such a manifold also carries a (m | n) superdimension.

A standard example is the flat superspace R^{m|n}, consisting of m real even coordinates and n real-anticommuting odd coordinates. Integration over this space uses the Berezin integral, which, over the odd coordinates, behaves similarly to taking a coefficient of the product of all odd variables, while integration over the even coordinates is conventional.

In practice, the superdimension guides the formulation of supersymmetric models, influencing the counting of independent degrees of freedom and the structure of superalgebras and supermanifolds.