Recall that two posts ago, I mentioned that Landsberg et al. observed that the matchgate identities are merely the defining equations of the spinor varieties. In this post, I will make explore what this really means in the hopes of describing to the reader their vision of the beautiful geometric reasons why holographic algorithms work. The big picture is that holographic algorithms are all about the efficient computation of some custom-made Holant that captures the answer to the combinatorial problem at hand. We come up with this exponentially large vector pairing (i.e. tensor contraction) and magically manage to compute it in polynomial time basically by computing a determinant. When do such efficiently computable pairings occur in geometry? Well if you take vectors in the cones of and (informally, vectors close enough to a -plane that they can be regarded as decomposable in or respectively) they turn out to be vectors consisting of determinants of minors of some matrix, and their pairing is merely the determinant of a related matrix. As it will turn out, if you take points in the variety of pure of spinors, they can likewise be characterized as vectors of “sub-Pfaffians” of some matrix, and a pairing between two such vectors is merely the Pfaffian of a related matrix. These are the kinds of efficiently computable pairings holographic algorithms take advantage of.

But it’s not enough to know that there exist special vectors whose pairings are easy to compute. After all, as a stupid example, the pairing between two -vectors supported in only dimensions is also computable in time. What distinguishes the vectors that feature in holographic algorithms from these vectors is the **surprising availability of the former in complexity theory problems**. Whereas there are way too many vectors in not supported on only polynomially many dimensions, the codimensions of for of low dimension and the spinor variety over low dimensions are quite small, and just as we’ve seen with matchgates, the vector spaces involved can often be decomposed into small enough “local” components (this local decomposition is key, because the abovementioned codimensions blow up very quickly).

In what follows, I hope to give a rigorous treatment of these ideas. My main references Chevalley’s *The Algebraic Theory of Spinors and Clifford Algebras* and the fifth chapter of Varadarajan’s lecture notes on supersymmetry for the development of the basic theory of Clifford algebras and spinors, and Landsberg’s *Tensors: Geometry and Applications* and Manivel’s *On Spinor Varieties and Their Secants* for establishing the connection between the matchgate identities and the variety of pure spinors. As always, if there are any mistakes in presentation below, they are entirely my fault.