A Landau-Ginzburg language for Spectral Sequences

The Story in Broad Strokes

By broad strokes, I mean: I’ll tell the story first without specifying a specific geometric context—that is, without saying what stacks mean or what commutativity means. A birds-eye view of the construction.

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Cast of Geometric Objects

• The Multiplicative Group Scheme
  A commutative group stack ${\mathbb{G}}_m$.

• The Cartier Stack
  A pointed stack over $B{\mathbb{G}}_m$ with:

  • underlying ${\mathbb{G}}_m$–equivariant stack $\mathbb{A}/{\mathbb{G}}_m\to B{\mathbb{G}}_m$
  • basepoint $0:B{\mathbb{G}}_m\to \mathbb{A}$

• A Quantized Stack
  A stack $X$ equipped with a map

  $$f:X\to \mathbb{A}/{\mathbb{G}}_m.$$

  There are two physical perspectives on how to read this:

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Physical Interpretation of a Quantized Stack

1. Deformation Quantization Perspective
   $\mathbb{A}=\mathrm{Spec}(\mathbb{k}[\hslash ])/(\text{scaling }\hslash )$.
   The map $f$ exhibits $X$ as an equivariant deformation quantization of the central fiber

   $$X_0:=X{\times }_{\mathbb{A}/{\mathbb{G}}_m}B{\mathbb{G}}_m.$$

   The $\hslash$-scaling equivariance means we’re constructing a theory where we mostly care about what happens at $\hslash =1$ and $\hslash =0$—a theory that’s allowed to forget detailed information about what happens, say, from $\hslash =0.1$ to $\hslash =0.15$.

2. Potential / Phase Space Perspective
   $f$ can also be thought of as a “potential” $V=d\mathcal{S}$: a derivative of an action functional on a phase space of states $X$.
   The central fiber $X_0$ again has a physical interpretation: it’s the classical locus—the space of states carved out by the Euler–Lagrange equations.

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Staring at the Diagram Defining the Central Fiber

We perform a formal completion of the entire picture:

By switching from the intersection $f\cap 0$ to the formally completed intersection $\hat{f}\cap \hat{0}$, we move from the world of “all geometry” to the world of formal geometry (all of this over $B{\mathbb{G}}_m$).

Once in the world of formal geometry, we can use the formal moduli problem form of Koszul duality to turn questions about geometry on $\hat{X}$ into questions about algebra on $X_0$.

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Descriptions of the Category of Matrix Factorizations

We can now give several ways to describe the category

$$MF(X,f)\text{or more precisely }MF(\hat{X},\hat{f})$$

of matrix factorizations on $X$ for the potential $f$.

1. Quantized Complete Sheaves

• $\hat{X}$ gives rise to the category $\mathrm{Coh}(\hat{X})$.
• $\hat{f}$ equips it with an action by the category $\mathrm{Coh}(\widehat{\mathbb{A}/{\mathbb{G}}_m})$.

Remember, in this introduction we’re not going into technical details in a specific geometric setting. So for now, $\mathrm{Coh}$ is some notion of formally complete sheaves on formal moduli stacks.

Matrix factorizations are the data of this complete filtered category.
Fixing the total space $\hat{X}$ and changing the potential $f$ changes the filtration.

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2. Representations of Automorphisms

A characteristic property of formal moduli problems under $X_0$ is that we have equivalences

$$\hat{X}\simeq {\beta }_{X_0}\mathfrak{aut}(X_0/X),$$

where $\mathfrak{aut}(X_0/-)$ is the formal group of symmetries of the base point of $\hat{X}$, and $\beta$ is its inverse functor (this is the defining property of formal moduli problems).

Aside. We can use this to get a different description of the object in the above section: the completion along the upstairs arrow gives us

$$\mathrm{Coh}(\hat{X})\simeq \mathrm{Rep}(\mathfrak{aut}(X_0/X);\mathrm{Coh}(X_0)).$$

If we remember the map $X_0\to B{\mathbb{G}}_m$, then this category acquires a grading.

The main construction here actually uses Koszul duality on the downstairs arrow, which tells us there’s a categorical action by the formal group

$$\mathfrak{aut}(B{\mathbb{G}}_m/(\mathbb{A}/{\mathbb{G}}_m))$$

on the category $\mathrm{Coh}(X_0)$.

This categorical representation

$$\mathrm{Coh}(X_0)\in \mathrm{Rep}(\mathfrak{aut}(0/\text{Cartier Stack});2\mathrm{QCoh}(B{\mathbb{G}}_m))$$

is also said to be the matrix factorization category of $(\hat{X},\hat{f})$.

These two types of 2-categorical objects are both “matrix factorizations,” and that makes sense because there’s an equivalence between the theories of the gadgets on each side:

$$\mathrm{Rep}(\mathfrak{aut}(0/\text{Cartier Stack});2\mathrm{QCoh}(B{\mathbb{G}}_m))\simeq 2\mathrm{Coh}(\widehat{\mathbb{A}/{\mathbb{G}}_m}).$$

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Next Orders of Business

• Example / Application: the Koszul duality diagram in Hodge theory.
• Koszul-dual description of Synthetic Spectra.
• Physical interpretation.