In the domain of mathematics known as representation theory, pure spinors (or simple spinors) are spinors that are annihilated, under the Clifford algebra representation, by a maximal isotropic subspace of a vector space ${\displaystyle V}$ with respect to a scalar product ${\displaystyle Q}$. They were introduced by Élie Cartan in the 1930s and further developed by Claude Chevalley.

They are a key ingredient in the study of spin structures and higher dimensional generalizations of twistor theory, introduced by Roger Penrose in the 1960s. They have been applied to the study of supersymmetric Yang-Mills theory in 10D, superstrings, generalized complex structures and parametrizing solutions of integrable hierarchies.

Consider a complex vector space ${\displaystyle V}$, with either even dimension ${\displaystyle 2n}$ or odd dimension ${\displaystyle 2n+1}$, and a nondegenerate complex scalar product ${\displaystyle Q}$, with values ${\displaystyle Q(u,v)}$ on pairs of vectors ${\displaystyle (u,v)}$. The Clifford algebra ${\displaystyle Cl(V,Q)}$ is the quotient of the full tensor algebra on ${\displaystyle V}$ by the ideal generated by the relations

Spinors are modules of the Clifford algebra, and so in particular there is an action of the elements of ${\displaystyle V}$ on the space of spinors. The complex subspace ${\displaystyle V_{\psi }^{0}\subset V}$ that annihilates a given nonzero spinor ${\displaystyle \psi }$ has dimension ${\displaystyle m\leq n}$. If ${\displaystyle m=n}$ then ${\displaystyle \psi }$ is said to be a pure spinor. In terms of stratification of spinor modules by orbits of the spin group ${\displaystyle Spin(V,Q)}$, pure spinors correspond to the smallest orbits, which are the Shilov boundary of the stratification by the orbit types of the spinor representation on the irreducible spinor (or half-spinor) modules.

Pure spinors, defined up to projectivization, are called projective pure spinors. For ${\displaystyle \,V\,}$ of even dimension ${\displaystyle 2n}$, the space of projective pure spinors is the homogeneous space ${\displaystyle SO(2n)/U(n)}$; for ${\displaystyle \,V\,}$ of odd dimension ${\displaystyle 2n+1}$, it is ${\displaystyle SO(2n+1)/U(n)}$.

Following Cartan and Chevalley, we may view ${\displaystyle V}$ as a direct sum

where ${\displaystyle V_{n}\subset V}$ is a totally isotropic subspace of dimension ${\displaystyle n}$, and ${\displaystyle V_{n}^{*}}$ is its dual space, with scalar product defined as

or