Matrix factorization - phenomenon and contexts

By “matrix factorization”, we’ll refer to the general phenomenon that

• Descent data along $0:pt\to {\mathbb{A}}^1$ is some circle action

That is, starting from data over the line, the stalk at the origin carries some additional symmetry, encoding infinitesimal data near the origin.

Remark: Observe that this immediately needs to be using an exceptional pushforward along the proper map 0: if we were to use the ordinary *-pushforward, the descent data would be trivial, as the functor constructing skyscrapers is fully faithful.

There are various contexts where this can be unpacked as explicit theorems:

• $QCoh(-)$, this is firmly in the realm of ordinary higher algebra
• categorifications $nQCoh(-)$
  • intermediate cases: e.g. $Al{g}^{(1)}(QCoh(-))$ sits between category level 1 and 2 (as the pointed connective 2-categories).
• $CAlg(QCoh(-))$, here the exceptional left adjoint computes derived de rham/crystalline cohomology
• ${\mathbb{G}}_m$ equivariant version of all of the above.
  • For example, in the case of $2QCoh(-)$, this is the content of Waldhaus.pdf, see Spectra as periodic 2-deformations. For example, this says that topological Fukaya categories (Dyckerhoff’s flavor) arise this way.
  • Carefully unpacking the algebra case here $CAlg(QCoh(-/{\mathbb{G}}_m))$ should provide a setting for $E_{\infty }$-rings in synthetic spectra and motivic filtrations, recollection - synthetic spectra, Even Cyclotomic Formal Geometry