03.09.27 · modern-geometry / spin-geometry

Pure spinors and the spinor variety

shipped3 tiersLean: none

Anchor (Master): Lawson-Michelsohn §IV.9; Chevalley 1954; Cartan 1938 *Leçons sur la théorie des spineurs*; Harvey 1990 *Spinors and Calibrations* Ch. 12

Intuition Beginner

A spinor is a kind of square root of a vector — an object that the rotation group acts on, but in a way that only comes back to itself after turning twice around. Most spinors are complicated bookkeeping. A few are remarkably clean, and those are the pure spinors.

Here is the picture. Inside a complex vector space with a quadratic form, some subspaces are special: every vector in them has zero length, and they are as large as such a subspace can be. These are the maximal isotropic subspaces. A pure spinor is a spinor that "points at" exactly one of them. Each pure spinor singles out a maximal isotropic subspace, and each maximal isotropic subspace comes from one pure spinor (up to an overall scale). So pure spinors and these flat, maximal, length-zero planes are two names for the same thing.

Cartan called these the simple spinors, and they are the building blocks. In low dimensions every spinor is pure, so the clean case is the only case. As the dimension grows, most spinors stop being pure, and a set of equations — the purity equations — tells you which ones still are. The spinors that pass the test form a single beautiful surface called the spinor variety.

Why care? Because a maximal isotropic subspace is the same data as a way to turn the space into a complex space (an orthogonal complex structure). Pure spinors are therefore the algebra of complex structures, written in the language of spin. When a pure spinor can be carried around a curved space without changing, the space inherits a complex, Calabi-Yau-like shape.

Visual Beginner

Picture a half-spinor as an arrow living in its own 8-dimensional space. Most arrows are tangled. A pure spinor is an arrow that casts a clean shadow: a flat plane of length-zero vectors, sitting inside the original space, exactly as big as such planes are allowed to be.

The lower diagram shows the collection of all pure spinors, drawn up to scale, as a curved surface floating inside the projective space of every half-spinor direction. This surface is the spinor variety. The flat planes glide along it: as you move from point to point on the surface, the maximal isotropic plane each pure spinor points at rotates smoothly. The surface is the geometric home of every clean spinor at once.

Worked example Beginner

Take the space of dimension four — the first place where a pure spinor can fail to be pure — but start with the simpler dimension six to see purity for free.

In dimension six, the positive half-spinors fill a four-dimensional space. Every nonzero one of them is pure: each picks out a three-dimensional length-zero plane, and these planes are exactly the ones described by the points of a familiar surface, the quadric. So in dimension six there is nothing to check; purity is automatic, and the spinor variety is the whole projective half-spinor space.

Move up to dimension eight. Now the positive half-spinors fill an eight-dimensional space, and not every direction is pure. A single equation appears — one quadratic relation the spinor must satisfy. The spinors that satisfy it point at a four-dimensional length-zero plane; the ones that fail do not point at any clean plane at all. The surface of pure spinors is six-dimensional inside the seven-dimensional projective space, carved out by that one equation.

What this shows: purity is automatic in small dimensions and becomes a real condition exactly when the half-spinor space first outgrows the supply of length-zero planes. The number of equations grows with the dimension, and they always cut the pure spinors down to a single smooth surface.

Check your understanding Beginner

Formal definition Intermediate+

Let $V = C^{2 n}$ carry a nondegenerate symmetric bilinear form $b$ with associated quadratic form $q$ . Let $Cl (V)$ be its complex Clifford algebra and $S = S^{+} \oplus S^{-}$ the $2^{n}$ -dimensional spinor module, split into half-spinor modules $S^{\pm}$ of dimension $2^{n - 1}$ on which $Spin (2 n)$ acts irreducibly 03.09.13. Write $v \cdot ψ$ for Clifford multiplication of $v \in V$ on $ψ \in S$ ; it interchanges the two halves, $V \cdot S^{\pm} \subseteq S^{\mp}$ .

Definition (annihilator). For a nonzero $ψ \in S$ , its annihilator is $$ W_\psi := {, v \in V : v \cdot \psi = 0 ,} \subseteq V. $$

Since $v \cdot (v \cdot ψ) = q (v) ψ$ , any $v \in W_{ψ}$ has $q (v) = 0$ , so $W_{ψ}$ is isotropic: $b ∣_{W_{ψ}} = 0$ . A maximal isotropic subspace of $(V, q)$ has dimension exactly $n$ .

Definition (pure spinor). A nonzero spinor $ψ \in S^{+}$ (or $S^{-}$ ) is pure if its annihilator $W_{ψ}$ is a maximal isotropic subspace, $dim_{C} W_{ψ} = n$ . Purity is scale-invariant, so it descends to a condition on $[ψ] \in P (S^{\pm})$ .

A maximal isotropic subspace $W$ is the same datum as an orthogonal complex structure: the decomposition $V = W \oplus \overline{W}$ (for a chosen conjugation, or $V = W \oplus W^{'}$ with $W^{'}$ a complementary maximal isotropic) defines a complex structure compatible with $q$ . The maximal isotropic subspaces of one family form the isotropic Grassmannian $O G (n, 2 n)$ , which has two connected components $O G^{\pm} (n, 2 n)$ distinguished by $dim (W \cap W_{0}) mod 2$ against a fixed reference $W_{0}$ ; the two components correspond to $S^{+}$ and $S^{-}$ .

The purity (Cartan) equations. Purity is cut out by quadratic equations in $ψ$ . The half-spinor module decomposes, via the contraction $S^{+} \otimes S^{+} \to Λ^{∙} V$ , into bilinear covariants $β_{k} (ψ) \in Λ^{n - 2 k} V$ . The top covariant $β_{0} (ψ) \in Λ^{n} V$ is the $n$ -form representing $W_{ψ}$ when $ψ$ is pure; $ψ$ is pure exactly when all the lower bilinear covariants vanish, $β_{k} (ψ) = 0$ for all $k \geq 1$ . These vanishing conditions are the purity equations; each is quadratic in $ψ$ .

Definition (spinor variety). The spinor variety is the image $$ \mathbb{S}^{\pm} := {, [\psi] \in \mathbb{P}(S^{\pm}) : \psi \text{ pure} ,} \subset \mathbb{P}(S^{\pm}), $$ the projectivisation of the pure spinors in one half. It is the highest-weight orbit $Spin (2 n) \cdot [ψ_{0}]$ of the lowest fundamental (half-spin) representation, the unique closed orbit. Pure spinors are exactly the highest-weight vectors and their $Spin (2 n)$ -translates; this identification is the foundational reason the spinor variety is the minimal orbit in $P (S^{\pm})$ ^{[Chevalley 1954]}.

Key theorem with proof Intermediate+

Theorem (Cartan; Lawson-Michelsohn IV.9.1). The assignment $ψ \mapsto W_{ψ}$ induces a bijection between projective pure spinors $[ψ] \in P (S^{+})$ and maximal isotropic subspaces $W \in O G^{+} (n, 2 n)$ of one family. Each pure spinor is determined up to scale by its annihilator, and every maximal isotropic subspace arises.

Proof. Well-defined and isotropic. For pure $ψ$ , $W_{ψ}$ is isotropic of dimension $n$ as shown above; it lies in a definite family because the $Spin (2 n)$ -action on $S^{+}$ matches the action on the component $O G^{+} (n, 2 n)$ .

Surjective. Fix a maximal isotropic $W = ⟨ w_{1}, \dots, w_{n} ⟩$ with complementary maximal isotropic $W^{'}$ pairing $b (w_{i}, w_{j}^{'}) = δ_{ij}$ . Realise the spinor module as the exterior algebra $S ≅ Λ^{∙} W^{'}$ , with $w_{i}^{'}$ acting by exterior product $w_{i}^{'} \land -$ and $w_{i}$ acting by the contraction $ι_{w_{i}}$ (interior product). The element $ψ_{W} := 1 \in Λ^{0} W^{'} \subset Λ^{∙} W^{'}$ is annihilated by every $w_{i}$ (a contraction kills a $0$ -form), so $W \subseteq W_{ψ_{W}}$ . Since $W$ is already maximal isotropic and $W_{ψ_{W}}$ is isotropic, $W = W_{ψ_{W}}$ , and $ψ_{W}$ is pure with annihilator $W$ . The parity of $1 \in Λ^{0}$ places $ψ_{W} \in S^{+}$ . So every $W \in O G^{+} (n, 2 n)$ is realised.

Injective (up to scale). Suppose $W_{ψ} = W_{ψ^{'}} = W$ . In the model $S ≅ Λ^{∙} W^{'}$ , the subspace annihilated by all of $W$ is exactly the line $Λ^{0} W^{'} = C \cdot 1$ , because contraction by a basis of $W$ has joint kernel of dimension one. Hence $ψ$ and $ψ^{'}$ are scalar multiples, and $[ψ] = [ψ^{'}]$ in $P (S^{+})$ .

Equivariance. For $g \in Spin (2 n)$ acting on $S^{+}$ with image $\overset{g}{ˉ} \in SO (2 n)$ on $V$ , Clifford multiplication transforms as $g (v \cdot ψ) = (\overset{g}{ˉ} v) \cdot (g ψ)$ , so $W_{g ψ} = \overset{g}{ˉ} W_{ψ}$ . The bijection intertwines the $Spin (2 n)$ -actions, and the orbit of $[ψ_{W}]$ is all of $S^{+}$ . $□$

Corollary (dimension). The spinor variety $S^{\pm}$ has complex dimension $(2 n)$ . Indeed $dim_{C} O G^{+} (n, 2 n) = (2 n)$ (the dimension of one component of the isotropic Grassmannian), and the bijection is an isomorphism of projective varieties.

Bridge. The pure-spinor / maximal-isotropic dictionary builds toward the geometry of complex and special-holonomy structures, where it appears again in the guise of a parallel spinor: the foundational reason a parallel pure spinor forces a complex structure is exactly this annihilator construction, run fibrewise on a spinor bundle 03.09.05. This is dual to the calibration story 03.09.19, where a parallel spinor squares to a calibrating form; here the same spinor instead points at a maximal isotropic plane, and the central insight is that the two descriptions are the spinor read two ways. Putting these together, the algebra of this unit generalises the low-dimensional accident that every spinor is pure into the variety that organises complex structures on a fixed quadratic space.

Exercises Intermediate+

Exercise 2 (easy, short-answer).

State the dimension of the spinor variety $S^{\pm}$ for $n = 2, 3, 4, 5$ , and identify the values of $n$ for which it fills the whole projective half-spinor space.

Hint

The variety has dimension $(2 n)$ ; the ambient projective space has dimension $2^{n - 1} - 1$ .

Answer

The spinor variety has dimension $(2 n)$ , while $P (S^{\pm})$ has dimension $2^{n - 1} - 1$ .

$n$	$dim S^{\pm} = (2 n)$	$dim P (S^{\pm}) = 2^{n - 1} - 1$
2	1	1
3	3	3
4	6	7
5	10	15

For $n = 2$ and $n = 3$ the two dimensions agree, so the variety fills the whole projective space and every spinor is pure. From $n = 4$ onward the variety is a proper subvariety, first of codimension one (the single purity equation at $n = 4$ ).

Exercise 3 (medium, proof).

In the exterior-algebra model $S ≅ Λ^{∙} W^{'}$ , with $W$ acting by contraction, show that the joint kernel of all of $W$ is the line $Λ^{0} W^{'}$ .

Hint

Pick a basis $w_{1}, \dots, w_{n}$ of $W$ acting as $ι_{w_{1}}, \dots, ι_{w_{n}}$ on $Λ^{∙} W^{'}$ , with $b (w_{i}, w_{j}^{'}) = δ_{ij}$ . A form is jointly annihilated when every contraction kills it.

Answer

Let $w_{1}^{'}, \dots, w_{n}^{'}$ be the basis of $W^{'}$ dual to $w_{1}, \dots, w_{n}$ under $b$ , so $ι_{w_{i}} (w_{j}^{'}) = δ_{ij}$ . A general element of $Λ^{∙} W^{'}$ is a sum $\sum_{I} c_{I} w_{I}^{'}$ over multi-indices $I$ , where $w_{I}^{'} = w_{i_{1}}^{'} \land \dots \land w_{i_{k}}^{'}$ .

The contraction $ι_{w_{i}}$ deletes the factor $w_{i}^{'}$ from each monomial containing it (with sign) and kills monomials not containing it. If a monomial $w_{I}^{'}$ with $∣ I ∣ \geq 1$ has $i \in I$ , then $ι_{w_{i}} (w_{I}^{'}) \neq = 0$ . So for $\sum_{I} c_{I} w_{I}^{'}$ to be annihilated by every $ι_{w_{i}}$ , no monomial of positive degree may appear with nonzero coefficient: applying $ι_{w_{i}}$ for any $i \in I$ produces a surviving term that cannot cancel against terms of different multidegree.

Concretely, the lowest-degree monomial of positive degree, say $w_{J}^{'}$ with $∣ J ∣ = k \geq 1$ minimal among nonzero $c_{I}$ , survives under $ι_{w_{j}}$ for $j \in J$ as a degree- $(k - 1)$ term not produced by any other monomial of degree $> k$ . Hence $c_{J} = 0$ , a contradiction. So only $Λ^{0} W^{'} = C \cdot 1$ survives, a one-dimensional joint kernel. $□$

Exercise 4 (medium, symbolic).

For $n = 4$ , the half-spinor space $S^{+}$ is $8$ -dimensional and there is a single purity equation. Argue that it is a quadratic form on $S^{+}$ , and that the spinor variety is the quadric it defines in $P (S^{+}) = P^{7}$ .

Hint

The lone lower covariant lives in $Λ^{0} V ≅ C$ , so it is a single scalar — a quadratic in $ψ$ . Triality relates $S^{+}$ to the vector representation.

Answer

For $n = 4$ , the bilinear covariants $β_{k} (ψ) \in Λ^{4 - 2 k} V$ occur at $k = 0$ (a $4$ -form, the top covariant) and $k = 2$ (a $0$ -form, i.e. a scalar). Purity demands the vanishing of every covariant below the top, which here is the single scalar $β_{2} (ψ) \in Λ^{0} V ≅ C$ . As a function of $ψ$ this scalar is quadratic, since each covariant is built from $S^{+} \otimes S^{+} \to Λ^{∙} V$ .

So the purity locus is ${[ψ] : β_{2} (ψ) = 0}$ , the zero set of one quadratic form $Q$ on the $8$ -dimensional $S^{+}$ , i.e. a smooth quadric hypersurface in $P^{7}$ . By triality on $Spin (8)$ 03.09.13, $S^{+}$ is isomorphic to the vector representation $V ≅ C^{8}$ , and $Q$ is the invariant quadratic form; the spinor variety $S^{+} ≅ O G^{+} (4, 8)$ is exactly this six-dimensional quadric, matching $(2 4) = 6$ . The other family $O G^{-} (4, 8)$ is the quadric in $P (S^{-})$ , and the three quadrics ${V, S^{+}, S^{-}}$ are permuted by triality.

Exercise 5 (medium, short-answer).

Why does the isotropic Grassmannian $O G (n, 2 n)$ have two connected components, and how do they match $S^{+}$ and $S^{-}$ ?

Hint

Compare a maximal isotropic $W$ with a fixed reference $W_{0}$ via $dim (W \cap W_{0}) mod 2$ , and recall that $Cl (V)$ acts with even/odd parity.

Answer

Fix a reference maximal isotropic $W_{0}$ . For any maximal isotropic $W$ , the parity of $dim (W \cap W_{0})$ is a deformation invariant: a continuous family of maximal isotropic subspaces cannot change $dim (W \cap W_{0}) mod 2$ , because that intersection dimension jumps by even amounts within a family. This splits $O G (n, 2 n)$ into two pieces $O G^{\pm} (n, 2 n)$ , each connected and homogeneous under $SO (2 n)$ (the full orthogonal group $O (2 n)$ swaps them).

Under the pure-spinor bijection, the parity of $dim (W_{ψ} \cap W_{0})$ matches the parity of the spinor: pure spinors in $S^{+}$ correspond to one component and those in $S^{-}$ to the other. In the exterior model, $ψ_{W} = w_{I}^{'}$ for the multi-index $I$ with $W = ⟨ w_{i} : i \in / I ⟩$ -type description sits in even or odd exterior degree according to that intersection parity, which is exactly the $S^{+} / S^{-}$ grading.

Exercise 6 (hard, proof).

Show that two pure spinors $ψ, ψ^{'}$ satisfy $b_{S} (ψ, ψ^{'}) \neq = 0$ (the spinor inner product) precisely when their annihilators are complementary, $W_{ψ} \oplus W_{ψ^{'}} = V$ , and use this to describe the spinor variety's secant structure.

Hint

Work in the exterior model with $ψ_{W} = 1 \in Λ^{0} W^{'}$ . The pairing of two pure spinors detects the top component $Λ^{n}$ of a product, which is nonzero exactly when no isotropic direction is shared.

Answer

Model $S ≅ Λ^{∙} W^{'}$ with $ψ = ψ_{W} = 1$ . A second pure spinor $ψ^{'}$ with annihilator $W^{''} \in O G (n, 2 n)$ is, up to scale, represented by the generator $Λ^{top} (W^{'} \cap (complement of W^{''})$ ; equivalently $ψ^{'} = ω$ where $ω \in Λ^{k} W^{'}$ encodes $W^{''} = (ker on ψ^{'})$ . The spinor inner product $b_{S}$ pairs $Λ^{j} W^{'}$ with $Λ^{n - j} W^{'}$ into $Λ^{n} W^{'} ≅ C$ .

Then $b_{S} (ψ, ψ^{'}) = b_{S} (1, ω)$ picks out the $Λ^{n} W^{'}$ -component of $1 \land ω$ , which is nonzero iff $ω$ has a full top-degree part, i.e. iff $W^{''}$ meets $W = W_{ψ}$ only in $0$ . Since both are maximal isotropic of dimension $n$ in $V = C^{2 n}$ , $W \cap W^{''} = 0$ is equivalent to $W \oplus W^{''} = V$ . Hence $b_{S} (ψ, ψ^{'}) \neq = 0$ iff $W_{ψ} \oplus W_{ψ^{'}} = V$ .

Geometrically: two points of the spinor variety span a secant line transverse to the variety exactly when their isotropic subspaces are complementary; when they share isotropic directions the pairing degenerates and the points lie on a common linear subspace of $S^{\pm}$ . This is the spinorial shadow of the fact that the spinor variety is cut out by quadrics and its secant geometry encodes incidence of maximal isotropic subspaces. $□$

Exercise 7 (hard, proof).

Prove that a parallel pure spinor on a Riemannian spin manifold $M^{2 n}$ induces an integrable orthogonal complex structure, so that $M$ is Kähler (and, when the spinor is also harmonic of the right type, Calabi-Yau).

Hint

Run the annihilator construction fibrewise. A parallel spinor gives, at each point, a maximal isotropic $W_{x} \subset T_{x}^{C} M$ varying parallel-ly; parallelism of $W$ gives both an almost complex structure $J$ and $\nabla J = 0$ .

Answer

Let $ψ$ be a parallel pure spinor, $\nabla ψ = 0$ , on the spinor bundle 03.09.05 of $M$ . At each $x$ , purity gives a maximal isotropic subspace $W_{x} = {v \in T_{x}^{C} M : v \cdot ψ = 0}$ of $T_{x}^{C} M = C^{2 n}$ . The decomposition $T_{x}^{C} M = W_{x} \oplus \overline{W_{x}}$ defines an orthogonal almost complex structure $J_{x}$ with $W_{x}$ its $(+ i)$ -eigenspace.

Because $\nabla ψ = 0$ and Clifford multiplication is parallel, $W$ is a parallel subbundle: if $v \cdot ψ = 0$ then $\nabla_{X} (v \cdot ψ) = (\nabla_{X} v) \cdot ψ + v \cdot \nabla_{X} ψ = (\nabla_{X} v) \cdot ψ$ , so $\nabla_{X} v$ stays in $W$ whenever $v$ does. Hence $\nabla J = 0$ : $J$ is parallel for the Levi-Civita connection.

A parallel almost complex structure is automatically integrable (its Nijenhuis tensor is built from $\nabla J$ and so vanishes) and the metric is Kähler, since $J$ parallel and orthogonal makes the Kähler form $ω = g (J \cdot, \cdot)$ closed. The holonomy is therefore reduced to $U (n)$ . When the parallel pure spinor lies in the appropriate half and is moreover the square root realising a parallel holomorphic volume form, the holonomy reduces further to $SU (n)$ , the Calabi-Yau condition — recovering the special-holonomy ambient of 03.09.19. $□$

Advanced results Master

The minimal orbit and the highest-weight vector. Inside $P (S^{+})$ , the half-spin representation of $Spin (2 n)$ , the pure spinors are precisely the highest-weight vectors and their orbit. The highest-weight orbit of any irreducible representation is the unique closed orbit and the minimal-dimensional one; here it is $S^{+} = Spin (2 n) \cdot [ψ_{0}]$ , of dimension $(2 n)$ . This is the spinor variety, a homogeneous projective variety $Spin (2 n) / P$ for a maximal parabolic $P$ , and it is one of the minuscule varieties of the $D_{n}$ Dynkin diagram. The half-spin node of $D_{n}$ is minuscule, which is the structural reason the purity equations are quadratic and the variety is cut out cleanly by the second symmetric power.

The Cartan relations as the Plücker analogue. Just as the ordinary Grassmannian sits in projective space via the Plücker embedding cut out by the Plücker quadrics, the spinor variety $O G (n, 2 n) ↪ P (S^{\pm})$ is cut out by the Cartan purity relations — the vanishing of all lower bilinear covariants. For $n \leq 3$ there are no relations; for $n = 4$ one quadric (triality); for $n = 5$ the variety is the $10$ -dimensional spinor tenfold $S_{10} \subset P^{15}$ , a celebrated Fano variety cut out by ten quadrics; for general $n$ the number of independent purity quadrics is the dimension of the relevant covariant spaces.

Low-dimensional exceptional identifications. The pure-spinor dictionary specialises beautifully: for $n = 1$ the variety is a point; $n = 2$ gives $S^{\pm} ≅ P^{1}$ (the two rulings of a quadric surface, $Spin (4) = SU (2) \times SU (2)$ ); $n = 3$ gives $S^{\pm} ≅ P^{3}$ with $Spin (6) = SU (4)$ and $O G (3, 6)$ a $P^{3}$ ; $n = 4$ gives the quadric sixfold tied to triality; $n = 5$ gives the spinor tenfold $S_{10}$ , whose homogeneous coordinate ring is generated by the $16$ pure-spinor coordinates with the ten Cartan quadrics as relations.

Pure spinors in physics and twistor theory. Pure spinors organise solutions of the massless field equations and supply Penrose's twistor correspondence in dimension four (where every spinor is pure). In ten dimensions ( $n = 5$ ), the spinor tenfold $S_{10}$ is the configuration space of the Berkovits pure-spinor superstring, where the ghost field is constrained to be a pure spinor; the cohomology of the pure-spinor constraint computes the superstring spectrum. The purity equations are then physical constraints, and the $(2 5) = 10$ -dimensional spinor variety is the target of the ghost sector.

Relation to calibrations and special holonomy. A parallel pure spinor on a Riemannian $2 n$ -manifold reduces the holonomy: it forces a parallel orthogonal complex structure, hence a Kähler structure, and with the appropriate reality the full Calabi-Yau $SU (n)$ holonomy 03.09.19. This is dual to the calibration construction: there the parallel spinor squares to a calibrating form; here the same spinor instead names a maximal isotropic plane field whose real-and-imaginary form decomposition recovers the Kähler and holomorphic-volume calibrations. The two are the spinor read covariantly versus read through its annihilator.

Synthesis. Pure spinors are the highest-weight vectors of the half-spin representation, and the spinor variety is their projective orbit, the minimal closed orbit of dimension $(2 n)$ . This is exactly the isotropic Grassmannian $O G (n, 2 n)$ of maximal isotropic subspaces, so the central insight is that a pure spinor and the length-zero plane it annihilates are one object in two languages — the bridge is the annihilator map, and the purity equations are the Cartan-quadric analogue of the Plücker relations. The construction generalises the low-dimensional accident that every spinor is pure (true for $n \leq 3$ ) into the variety governing complex structures on a quadratic space, and it is dual to calibrated geometry: the foundational reason a parallel pure spinor builds a Calabi-Yau structure is the same annihilator construction run fibrewise, putting these together so that complex geometry, special holonomy, and the pure-spinor superstring all read off the one spinor variety.

Full proof set Master

Proposition (purity is equivalent to the maximality of an isotropic annihilator, with the line recovered). Let $ψ \in S^{+} ∖ {0}$ . Then $dim_{C} W_{ψ} \leq n$ , with equality iff $ψ$ is, up to scale, the spinor $ψ_{W}$ of a maximal isotropic $W$ ; and in the equality case $ψ$ is recovered from $W$ as the generator of the one-dimensional joint kernel of $W$ acting by Clifford multiplication.

Proof. The annihilator $W_{ψ}$ is isotropic (Exercise 1), so $dim W_{ψ} \leq n$ since maximal isotropic subspaces of $(C^{2 n}, q)$ have dimension $n$ . Suppose $dim W_{ψ} = n$ , so $W := W_{ψ}$ is maximal isotropic. Choose a complementary maximal isotropic $W^{'}$ with $b (W, W^{'})$ a perfect pairing, and present $S ≅ Λ^{∙} W^{'}$ as in the Key theorem, with $W$ acting by contraction. By Exercise 3 the joint kernel of $W$ on $Λ^{∙} W^{'}$ is the line $Λ^{0} W^{'} = C \cdot 1$ . Since $ψ$ is annihilated by all of $W$ , $ψ$ lies in this line: $ψ = c \cdot 1 = c ψ_{W}$ for a scalar $c \neq = 0$ . So $ψ$ is, up to scale, the pure spinor $ψ_{W}$ , and it is recovered as the generator of the joint kernel.

Conversely if $ψ = c ψ_{W}$ for maximal isotropic $W$ , then $W_{ψ} = W_{ψ_{W}} \supseteq W$ and by isotropy $dim W_{ψ} \leq n = dim W$ , forcing $W_{ψ} = W$ of dimension $n$ . So $dim W_{ψ} = n$ iff $ψ$ is pure. $□$

Proposition (equivariance and orbit). The map $[ψ] \mapsto W_{ψ}$ is $Spin (2 n)$ -equivariant via the covering $Spin (2 n) \to SO (2 n)$ , and the pure spinors of one half form a single orbit, so $S^{+} = Spin (2 n) \cdot [ψ_{0}] ≅ Spin (2 n) / P$ for a maximal parabolic $P$ .

Proof. For $g \in Spin (2 n)$ with image $\overset{g}{ˉ} \in SO (2 n)$ , Clifford multiplication satisfies $g (v \cdot ψ) = (\overset{g}{ˉ} v) \cdot (g ψ)$ , so $v \cdot ψ = 0$ iff $(\overset{g}{ˉ} v) \cdot (g ψ) = 0$ , giving $W_{g ψ} = \overset{g}{ˉ} W_{ψ}$ . The action of $SO (2 n)$ on one family $O G^{+} (n, 2 n)$ of maximal isotropic subspaces is transitive, hence so is the $Spin (2 n)$ -action on $S^{+}$ . The stabiliser of $[ψ_{0}]$ is the preimage of the stabiliser of $W_{0}$ , a parabolic subgroup $P$ corresponding to the half-spin node of $D_{n}$ ; thus $S^{+} ≅ Spin (2 n) / P$ , a homogeneous projective variety of dimension $dim Spin (2 n) - dim P = (2 n)$ . $□$

Proposition (the purity equations are quadratic and define $S^{\pm}$ ). The pure spinors in $S^{+}$ are exactly the common zeros of the lower bilinear covariants $β_{k}$ , $k \geq 1$ ; each $β_{k}$ is a quadratic map $S^{+} \to Λ^{n - 2 k} V$ , and their joint vanishing cuts out $S^{+}$ scheme-theoretically.

Proof. The contraction $S^{+} \otimes S^{+} \to Λ^{∙} V$ is bilinear, so each component $β_{k} (ψ) = μ_{k} (ψ \otimes ψ)$ is quadratic in $ψ$ . For pure $ψ = ψ_{W}$ in the exterior model, $ψ_{W} \otimes ψ_{W}$ contracts to the top form representing $W$ and nothing below: $β_{0} (ψ_{W}) \neq = 0$ and $β_{k} (ψ_{W}) = 0$ for $k \geq 1$ . Conversely, the highest-weight orbit is exactly the locus where all but the top covariant vanish — this is the standard description of a minuscule highest-weight orbit as the common zeros of the lower components of the symmetric square, since $Sym^{2} (S^{+})$ decomposes with the lower covariants as the non-top summands and the orbit closure is their common zero scheme. By minuscule-orbit theory the scheme is reduced, so the purity quadrics define $S^{+}$ scheme-theoretically. $□$

Connections Master

Half-spinor representations / triality 03.09.13 — pure spinors are the highest-weight vectors of the half-spin representation $S^{\pm}$ of $Spin (2 n)$ ; the spinor variety is the minimal $Spin (2 n)$ -orbit in $P (S^{\pm})$ . At $n = 4$ the single purity quadric is permuted with the vector quadric by triality, so the three quadrics ${V, S^{+}, S^{-}}$ are interchanged — the purity equation is the triality-invariant quadratic form. Builds-on: spinor variety as highest-weight orbit of the half-spin representation [conn:441.half-spinor-rep-pure-spinor].
Spinor bundle 03.09.05 — running the annihilator construction fibrewise on a spinor bundle turns a field of pure spinors into a field of maximal isotropic subspaces, hence an almost complex structure; a parallel pure-spinor section gives an integrable one. This is the mechanism by which spin geometry produces complex geometry on a manifold. Builds-on: pure-spinor annihilator fibred over the spinor bundle [conn:442.spinor-bundle-pure-spinor].
Calibrated geometries / parallel spinors 03.09.19 — a parallel pure spinor is dual to a calibration: the spinor that squares to a calibrating form is the same spinor that annihilates a maximal isotropic plane. A parallel pure spinor forces a Calabi-Yau structure, the special-holonomy ambient on which the Special Lagrangian calibration lives. Lateral: parallel pure spinor yields the special-holonomy structure underlying calibrations [conn:443.pure-spinor-parallel-calibration, anchor: parallel pure spinor ⇒ complex/Calabi-Yau structure dual to the calibrating form].
Isotropic Grassmannian / flag varieties 07.04.01 — the projectivised pure spinors are the isotropic Grassmannian $O G (n, 2 n)$ , a homogeneous space $Spin (2 n) / P$ for a maximal parabolic; the Cartan purity quadrics are the spinorial analogue of the Plücker relations cutting out an ordinary Grassmannian. The minuscule half-spin node of the $D_{n}$ Dynkin diagram is the reason the variety is quadratically cut.

Historical & philosophical context Master

The notion originates with Élie Cartan, who in his 1938 Leçons sur la théorie des spineurs ^{[Cartan 1938]} singled out what he called simple spinors — those associated, as he put it, to a single totally isotropic plane. Cartan's spinors were defined precisely so that the geometric data of a spinor could be read off as an isotropic subspace, and he understood that in low dimensions every spinor is simple while in higher dimensions a system of quadratic relations intervenes. His treatment was concrete and computational, built on explicit isotropic frames rather than the later representation-theoretic language.

Claude Chevalley's 1954 The Algebraic Theory of Spinors ^{[Chevalley 1954]} recast Cartan's simple spinors in the exterior-algebra model $S ≅ Λ^{∙} W^{'}$ , which is the model used throughout this unit. Chevalley's framing made the annihilator construction algebraically transparent and connected pure spinors to the structure theory of Clifford algebras; the word pure (in place of Cartan's simple) became standard through this line. The purity relations are sometimes called the Cartan relations and sometimes the Chevalley relations, reflecting this dual ancestry. Lawson and Michelsohn's Spin Geometry §IV.9 gives the modern textbook synthesis, stating the bijection (Theorem IV.9.1) and the variety structure (Theorem IV.9.2) in the form used here.

The philosophical thread is the recurring spin-geometry theme that a spinor is geometric data in disguise. Where Cartan's earlier triality (1925) showed that in dimension eight the vector and spinor pictures become interchangeable, the pure-spinor theory shows what a spinor sees: a maximal isotropic plane, equivalently a complex structure. Reese Harvey's 1990 Spinors and Calibrations ^{[Harvey 1990]} makes this the organising principle, deriving calibrations and complex structures alike from the pure-spinor dictionary. The modern reappearance in the Berkovits pure-spinor superstring closes a long arc: an object Cartan introduced to clarify the geometry of isotropic planes became, seventy years later, the constraint defining a quantisation of the superstring.

Bibliography Master

Cartan, É., Leçons sur la théorie des spineurs, Hermann, 1938. The original notion of a simple (pure) spinor associated to an isotropic plane.
Chevalley, C., The Algebraic Theory of Spinors, Columbia University Press, 1954. Ch. III — the exterior-algebra model and the quadratic purity relations.
Lawson, H. B. & Michelsohn, M.-L., Spin Geometry, Princeton University Press, 1989. §IV.9 (Theorems IV.9.1, IV.9.2).
Harvey, F. R., Spinors and Calibrations, Academic Press, 1990. Ch. 12 — pure spinors, isotropic subspaces, and the calibration link.
Penrose, R. & Rindler, W., Spinors and Space-Time, Vol. 2, Cambridge University Press, 1986. Pure spinors and twistor geometry in dimension four.
Berkovits, N., "Super-Poincaré covariant quantization of the superstring", JHEP 04 (2000), 018. The pure-spinor superstring; the spinor tenfold as ghost target.
Igusa, J., "A classification of spinors up to dimension twelve", American Journal of Mathematics 92 (1970), 997–1028. Orbit classification of spinors and the pure-spinor stratum.

Prerequisites

03.09.05
03.09.13
03.09.19

Tier anchors

beginner: 3Blue1Brown / Strogatz analogous level — a spinor that 'points at' a plane
intermediate: Lawson-Michelsohn §IV.9 (Theorems IV.9.1, IV.9.2); Chevalley *The Algebraic Theory of Spinors*
master: Lawson-Michelsohn §IV.9; Chevalley 1954; Cartan 1938 *Leçons sur la théorie des spineurs*; Harvey 1990 *Spinors and Calibrations* Ch. 12

References

Lawson-Michelsohn — Spin Geometry · §IV.9 (pure spinors, Theorems IV.9.1 and IV.9.2; the spinor variety)
Chevalley — The Algebraic Theory of Spinors · Columbia University Press, 1954 — Ch. III, the original treatment of pure spinors and their quadratic relations
Cartan — Leçons sur la théorie des spineurs · Hermann, 1938 — Cartan's original notion of a 'simple' (pure) spinor
Harvey — Spinors and Calibrations · Academic Press, 1990 — Ch. 12, pure spinors, isotropic subspaces, and the link to calibrations

Estimated time

beginner: 22m
intermediate: 62m
master: 108m