# nLab pseudoscalar

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

Given a symmetry group $G$ equipped with a homomorphism $G \overset{\phi}{\longrightarrow} O(s,t)$ to an orthogonal group (for instance a Pin group), a pseudoscalar is an element of the 1-dimensional linear representation (over a given ground field $k$)

$\mathbf{1}_{{}_{sgn}} \;\in\; Rep_k(G)$

that is given by forming the determinant (sign representation):

(1)$\mathbf{1}_{{}_{sgn}} \;\;\coloneqq\;\; \left( \array{ G \times k &\longrightarrow& k \\ (g, c ) &\mapsto& det\big( \phi(g) \big) } \right) \,.$

More generally, given a function with values in $\mathbf{1}_{{}_{sgn}}$, or yet more generally a section of a fiber bundle with typical fiber $\mathbf{1}_{{}_{sgn}}$, this whole function/section is often called a pseudoscalar; more precisely: a pseudoscalar field. This as opposed to scalar fields, which take values in the 1-dimensional trivial representation $\mathbf{1}$.

If the fiber bundle in question is a “canonical bundle”/determinant line bundle of a Riemannian manifold of dimension $n$, hence the top exterior power of a tangent bundle (i.e. top degree (Kähler) differential form-bundle) then such “pseudoscalar fields” of physics are what are called densities in mathematics.