---
tag: Z0SB9HQBAJ
kind: concept
name: Supermanifold
msc: [58A50]
sources:
  - url: https://en.wikipedia.org/wiki/Supermanifold
same_as:
  wikidata: Q3978062
  wikipedia: Supermanifold
---

## Summary

A supermanifold is a space whose functions depend on ordinary commuting coordinates and also on anticommuting coordinates.
Formally, it is a topological space with a [[sheaf|TB6TM6R4MD]] of supercommutative [[superalgebras|6JZW4JC68Z]] that looks locally like smooth functions with values in an [[exterior algebra|0MDMZ965VE]].
Supermanifolds are the geometric setting of supersymmetry, where bosons are described by even and fermions by odd coordinates.
Every smooth supermanifold is built, up to isomorphism, from an ordinary [[vector bundle|P4DZM975BV]] over an ordinary manifold.

## Explanations

### L1 human

On an ordinary manifold, functions are built from coordinates $x_1, \dots, x_p$ that commute.
A supermanifold adds coordinates $\theta_1, \dots, \theta_q$ that anticommute: $\theta_i \theta_j = -\theta_j \theta_i$, so in particular $\theta_i^2 = 0$.
A function in one even and one odd variable is then always of the form $f(x) + g(x)\, \theta$, since higher powers of $\theta$ vanish.
These odd coordinates have no points of their own; they live only in the algebra of functions.