Landau–Ginzburg pair and distinct categories

A Landau–Ginzburg pair is a specified smooth complex space $Y$ (or smooth algebraic variety over the named coefficient field) and a holomorphic (respectively regular) function $W:Y\to \mathbb{C}$ (respectively $W\in \mathrm{\Gamma }(Y,{\mathcal{O}}_Y)$), together with the coefficient and parameter conventions in Specified mathematical mirror datum. The critical locus is $\mathrm{Crit}(W)=\{y\in Y:d{W}_y=0\}$; no finiteness or nondegeneracy of it is implicit.

Two constructions must be specified separately. Fukaya–Seidel data consists of an exact or otherwise explicitly chosen symplectic structure on $Y$, coefficient ring, brane and grading conventions, critical values, admissible vanishing paths or stops, behavior at noncompact ends, and the selected A-infinity category or pre-category of vanishing-cycle branes. It is a category only after the relevant morphism complexes, polygon operations, regularization, compactness, gluing, and units have been supplied. Matrix-factorization data consists of objects and morphisms as in General Landau–Ginzburg matrix-factorization data, with the Hom differential of The matrix-factorization Hom differential is a dg differential.

The pair alone supplies neither category construction. No equivalence between the two, or between either category and a coherent-sheaf category, is part of this definition. In particular, the fixed-potential matrix-factorization interface imported for a different application does not replace the general $(Y,W)$ data above.

Specified mathematical mirror datum

Blocked construction interface for multi-object Fukaya categories and polygon morphisms

General Landau–Ginzburg matrix-factorization data

The matrix-factorization Hom differential is a dg differential