Odd Knörrer periodicity as a double cover

Technical Summary: Odd Knörrer Periodicity as a Double Cover

Problem Statement
The paper addresses the global behavior of Knörrer periodicity, a fundamental equivalence in the theory of matrix factorizations. Classically, Knörrer periodicity establishes that adding a quadratic term $x^2$ to a superpotential $w$ (a "stabilization") preserves the derived category of matrix factorizations if the rank of the quadratic form is even, while odd-rank stabilizations can be resolved by passing to an equivariant category. While global versions of even-rank periodicity are well-understood (relating matrix factorizations on a vector bundle to the derived category of the base), a global statement for rank-one quadratic forms has been elusive. Specifically, the pointwise equivalence $MF_{|x|=1}(\mathbb{A}^1, x^2) \simeq D^b(2 \text{ pts})$ suggests a relationship between matrix factorizations and branched double covers, but existing frameworks for graded matrix factorizations are insufficient to handle the necessary "odd" grading data.

The core difficulty lies in the fact that standard definitions of matrix factorizations rely on the splitting of objects into even and odd parts ($E_0 \leftrightarrows E_1$), a phenomenon unique to evenly-graded rings. When the linear fiber coordinate has odd cohomological degree, the structure sheaf is no longer super-commutative, the differential on morphism sheaves fails to be linear, and the standard definition of $RHom$ breaks down. Consequently, the derived category of matrix factorizations for such "oddly-graded" Landau–Ginzburg (LG) models cannot be constructed using traditional methods.

Methodology
The author develops a framework for odd matrix factorizations where the linear fiber coordinate $q$ has odd cohomological degree ($|q|=1$).

1. Reformulation of Definitions: Instead of relying on the $E_0 \leftrightarrows E_1$ splitting or $RHom$ on the base space $X$, the paper adopts the object-level definition from Segal and constructs the derived category as a Verdier quotient of the homotopy category of matrix factorizations by acyclic objects (following Orlov and Preygel).
2. Local-to-Global via Quotient Stacks: To recover a notion of internal homs and perform local-to-global computations, the author observes that while morphisms do not form sheaves over the oddly-graded space $X$, they do form sheaves over the quotient stack $[X/\mathbb{Z}_2]$. The induced grading on $[X/\mathbb{Z}_2]$ is even, allowing the application of standard theory for curved algebras over an evenly-graded base.
3. Fourier–Mukai Transform: The main theorem is proven by constructing an explicit Fourier–Mukai kernel. The equivalence is realized via a curved bimodule over the total space of the line bundle and the branched double cover. The proof carefully accounts for Koszul signs arising from the non-commutative nature of the tensor product of oddly-graded algebras.

Key Contributions and Results

• Theorem A (Main Equivalence): The paper establishes an equivalence between the derived category of a branched double cover $\tilde{X} = \{y^2 = f\} \subset \text{Tot}(L)$ and the category of matrix factorizations on the total space of the dual line bundle $\text{Tot}(L^{-1})$ with a fiberwise quadratic potential.
  $$MF(\text{Tot}(L^{-1}), fq^2) \simeq D^b(\tilde{X})$$
  Here, the fiber coordinate $q$ is assigned odd degree, making the potential $fq^2$ have total degree 2. This result globalizes the pointwise equivalence $MF_{|q|=1}(\mathbb{A}^1, q^2) \simeq D^b(2 \text{ pts})$.

• Corollary B (Orbifold Version): An analogous equivalence is proven for the associated root stack (stacky quotient):
  $$MF([\text{Tot}(L^{-1})/\mathbb{Z}_2], fq^2) \simeq D^b(\sqrt{f}/X)$$
  This stays within the realm of evenly-graded matrix factorizations, as the $\mathbb{Z}_2$ action on the quotient stack trivializes the odd grading.

• Resolution of the "Paradox" of Iteration: The paper clarifies why iterating the double cover construction does not trivially lead to a contradiction with even Knörrer periodicity. When iterating the construction on an oddly-graded base, the resulting algebra becomes non-commutative (specifically, a Clifford algebra). These non-commutative algebras are Morita trivial, resolving the apparent paradox that an iterated double cover (non-trivial) should be equivalent to an even stabilization (trivial).

• Applications to Mirror Symmetry and Resolutions:

  • Mirror Symmetry: The framework naturally accommodates 1-dimensional mirror symmetry examples involving orbifold surfaces and wrapped Fukaya categories, where odd gradings arise from line fields on orbifold points.
  • Orlov-type Equivalences: The result provides a dimensional reduction of Orlov's theorem for double covers, relating the derived category of a Fano branched double cover to matrix factorizations on an affine orbifold.
  • Categorical Resolutions: The paper applies the theorem to construct and analyze categorical resolutions of singularities (e.g., $A_2$ and $A_3$ curve singularities) via partial compactifications of LG models. It demonstrates how kernel subcategories of these resolutions can be computed using the established equivalences, matching results from other constructions (e.g., Kuznetsov, Kalck).
  • Exoflops: The theory is used to reinterpret "exoflops" (phenomena where resolutions of singularities correspond to partial compactifications of LG models) in the context of odd periodicity, showing how the flop can be realized via GIT variation without orbifold structures.

Significance and Claims
The author claims that the primary significance of this work is providing a rigorous global statement for rank-one quadratic forms that necessitates odd grading data. The paper argues that existing approaches to graded matrix factorizations are inadequate for this purpose because they assume super-commutativity and even splitting. By developing the theory of oddly-graded matrix factorizations and proving the equivalence to branched double covers, the paper:

1. Unifies the pointwise equivalence (1.1) with global geometric constructions.
2. Offers a new tool for understanding matrix factorizations with superpotentials of the form $fq^2$.
3. Provides a geometric interpretation for homological projective duality in the context of diagonal quadric families and singular double covers.
4. Clarifies the relationship between iterated double covers and Clifford algebras, resolving potential confusion regarding the iteration of Knörrer periodicity.

The paper remains modest regarding future implications, noting that while the results suggest connections to homological mirror symmetry for singular double covers and minimal categorical resolutions, these are areas for further investigation rather than established conclusions of the current work. The work is presented as a foundational step in formalizing the "odd" side of Knörrer periodicity.