# Applications of schemes to mathematical physics

Could anyone cite some applications or developments in mathematical physics or string theory that use schemes?

I find curious the fact that while things like derived algebraic geometry and stacks have certain applications to mathematical physics I cannot find such applications for the case of (underived) schemes.

Just to clarify: I know that for example you can have Calabi-Yau threefolds that are projective algebraic varieties and that in some sense algebraic varieties ⊂ schemes ⊂ stacks. However I am looking for specific (underived) scheme constructions like Fano schemes, Hilbert schemes, scheme-theoretic interesections etc

• Physics cares about analytic structures, not the Zariski topology, so schemes aren't going to be fundamental. But they can be a good way to get a handle on analytic spaces through GAGA theorems and the like. – Aaron Bergman Oct 2 '19 at 1:13

In string theory, gauge symmetries on D-branes can be studied using classical scheme theory.

This was introduced around the late 1990s and early 2000s, cf. this 1998 paper, this 2000 note, these 2003 notes, and this 2007 paper to start.

The Hilbert scheme of points on a K3 surface plays an important rôle in providing a strong coupling test of S-duality by Vafa and Witten. This is the original paper on what is known as Vafa-Witten theory.

• Thank you very much, that seems pretty interesting. I will check it in more detail. – Chequez Oct 2 '19 at 1:47
• Not however that this particular Hilbert scheme is a variety. Nonreduced schemes appear for example in the mathematical theory of BPS states @Chequez – SWS Oct 2 '19 at 21:51
• @Chequez Regarding BPS states, the discussion in this PhysicsOverflow question is really nice. – Théo Oct 3 '19 at 1:09

Given the dearth of answers to this question I'll offer up some of my work as it provides an answer to this question as follows:

1. Topological phases of matter are described by topological quantum field theories, and Hamiltonian models of them can be constructed using String-Nets.
2. The String-Net Hamiltonian takes as one of its inputs a spherical fusion category, and in particular the arithmetic information associated to this fusion category.
3. Said arithmetic information is obtained from solutions to polynomial equations (called pentagon, pivotal, and hexagon equations) which, of course, define an algebraic set and thus a scheme.

As to how this is used, consider the following:

One problem in studying both fusion categories and topological phases is their classification. In particular, if one has two sets of solutions to a set of pentagon equations, how do we know that they correspond to monoidally inequivalent categories? Well, it's pretty straightforward to show equivalent solutions correspond to orbits of the action of an algebraic group, namely the gauge group of the fusion category, on the algebraic set.

So how, then, do we distinguish orbits? Through some straightforward arguments these orbits are closed and there are finitely many of them, so it turns out that they all have the same dimension. Putting this all in the language of schemes, we have a sufficient set of conditions to invoke Theorems 1.1 and 1.3 from Mumford's "Geometric Invariant Theory" to show that there exist gauge invariant rational functions which distinguish each orbit from the other. This can be repeated by applying it to the automorphism group of the underlying Grothendieck ring so that one ends up with a 0-Dimensional algebraic set where each point corresponds to a monoidal class of fusion categories.

While the language of schemes is, in general, overkill with regards to obtaining this result it is true that fusion categories are classified by the points of their associated moduli spaces.