Spin Representation

Question

I have been reading Cassels's book on "Rational Quadratic Forms". Most part of his book is written perfectly, but there is a chapter about the "Spin Representation" on his book, which I can not really understand, basically because I don't have enough motivation why such a thing would be important.

So I want to understand, the motivation behind the spin representation, and more importantly some application of the concept.

Presumably this notion came from the Geometry, and I guess this is why, for example, they prove spin group is a double cover of proper orthogonal group of a quadratic form. Which kind of result one might obtain by considering this geometric interpolation in the context of quadratic forms?

Thank you.

Answer 1

It was a longstanding problem to decide equivalence of indefinite forms. The showpiece of the spinor genus is that, for indefinite forms in at least three variables over the rational integers, the spinor genus and the equivalence class coincide.

The phenomena that are most directly explained occur in three variables and positive forms, first identified in Jones and Pall (1939). The first example is that $$ g(x,y,z) = 2 x^2 + 2 y^2 + 5 z^2 + 2 y z + 2 z x \neq m^2,$$ where all prime factors of $m$ are congruent to $1 \pmod 4.$ In comparison, and in the same genus, $$ f(x,y,z) = x^2 + y^2 + 16 z^2$$ does represent all squares, and primitively represents all $m^2.$ Other than the difference noted, the two forms represent the same numbers. $g$ and $f$ are in different spinor genera. $f$ is called regular, as it represents all numbers locally eligible, while $g$ is called spinor regular, as it represents everything eligible for its spinor genus.

Kaplansky, A. Schiemann, and I found all possible regular positive ternary forms in 1997. Later, in contact with Andrew Earnest, i found a total of 29 spinor regular forms that are not regular. This is probably the complete list. Earnest contacted me recently about completing the project by proving the 29 are all. I sent him some background on what is necessary to complete such a proof.

My own little toy, to appear next year, is infinite families of genera where membership in a spinor genus can be decide by a single number. If $N$ is squarefree and $N = u^2 + v^2$ in integers, then consider the genus of $$ h(x,y,z) = x^2 + y^2 + 16 N z^2. $$ It turns out that the genus of $h$ has exactly two spinor genera, there are spinor exceptional integers, which are all the $Nm^2.$ The theorem, quite unusual, is that a form $k(x,y,z)$ in the same genus as $h$ is in the same spinor genus as $h$ if and only if $k$ integrally represents $N.$ This is pretty rare. There is usually no assurance about the small numbers a form represents. All we really know is that a form resents all sufficiently large numbers that are primitively represented by some form in the same spinor genus; this is Duke and Schulze-Pillot (1990). I have about a dozen such infinite families and some related conjectures.

Probably enough for now.

Answer 2

I believe the historical motivation for considering the spin representation is actually mathematical physics. In his attempt to formulate a counterpart of the Schrodinger equation in quantum mechanics which is compatible with special relativity, Dirac decided that he needed to find a first order differential operator whose square is the Laplacian:

$$ D = \sum c_i \partial_i $$ $$ D^2 = -\sum \partial_i^2 $$

If you sit down and work out the relations that the $c_i$'s have to satisfy, you find that $c_i c_j + c_j c_i = 0$ if $i \neq j$ and $c_i^2 = -1$. Of course, there are no real or complex numbers which satisfy these relations; Dirac realized (not in this language) that the $c_i$'s are generators for the Clifford algebra $\mathbb{C}_n$. Thus $D$ is a vector valued operator which takes values in a vector space $S$ on which $\mathbb{C}_n$ acts by endomorphisms. Ideally we would like to take $S$ as small as possible, in the sense that $End(S) \cong \mathbb{C}_n$.

If this physicsy motivation isn't enough, Atiyah and Singer discovered a few decades later that Dirac's idea has profound applications in differential topology. One says that a Riemannian $n$-manifold $M$ is spin if there is a principal $Spin(n)$-bundle $P \to M$ equipped with a bundle map $P \to SO(TM)$ (where $SO(TM)$ is the principal $SO(n)$-bundle of orthonormal frames) which on fibers is the double cover $Spin(n) \to SO(n)$. If $P$ exists then there is a natural vector bundle $S \to M$ coming from the spin representation via the "associated bundle" construction, and $S$ has the property that its endomorphism bundle is isomorphic to the Clifford algebra bundle associated with $TM$. In this case there is a global "Dirac operator" $D$ whose square is the Laplacian, and this operator is central to the Atiyah-Singer index theorem. In fact, it serves as the fundamental class in real K-homology theory.