We introduce a categorical analogue of Saito’s notion of primitive forms. For the category $\textsf{MF}(\frac{1}{n+1}x^{n+1})$ of matrix factorizations of $\frac{1}{n+1}x^{n+1}$, we prove that there exists a unique, up to nonzero constant, categorical primitive form. The corresponding genus zero categorical Gromov–Witten invariants of $\textsf{MF}(\frac{1}{n+1}x^{n+1})$ are shown to match with the invariants defined through unfolding of singularities of $\frac{1}{n+1}x^{n+1}$.
Computing a categorical Gromov–Witten invariant
We compute the $g=1$ , $n=1$ Bmodel Gromov–Witten invariant of an elliptic curve $E$ directly from the derived category $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ . More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic $\mathscr{A}_{\infty }$ model of $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ described by Polishchuk. This is the first nontrivial computation of a positivegenus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (noncategorical) Gromov–Witten invariants of a symplectic 2torus computed by Dijkgraaf.
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 Award ID(s):
 1811925
 NSFPAR ID:
 10293242
 Date Published:
 Journal Name:
 Compositio Mathematica
 Volume:
 156
 Issue:
 7
 ISSN:
 0010437X
 Page Range / eLocation ID:
 1275 to 1309
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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