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Abstract 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 non-zero 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 $$\frac{1}{n+1}x^{n+1}$$.
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Computing a categorical Gromov–Witten invariant
We compute the $g=1$ , $n=1$ B-model 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 non-trivial computation of a positive-genus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov–Witten invariants of a symplectic 2-torus computed by Dijkgraaf.
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- Award ID(s):
- 1811925
- PAR ID:
- 10293242
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 156
- Issue:
- 7
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 1275 to 1309
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X20007174
