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Categorical Enumerative Invariants (CEI) are invariants associated with unital cyclic $A_\infty$-categories that are smooth, proper and satisfy the Hodge-to-de-Rham degeneration property. In this paper, we formulate and prove their Morita invariance. In particular, when applied to derived categories of coherent sheaves, this yields new birational invariants of smooth and proper Calabi-Yau $3$-folds., Comments welcome!

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.

We study categorical primitive forms for Calabi--Yau $A_\infty$ categories with semi-simple Hochschild cohomology. We classify these primitive forms in terms of certain grading operators on the Hochschild homology. We use this result to prove that, if the Fukaya category ${{\sf Fuk}}(M)$ of a symplectic manifold $M$ has semi-simple Hochschild cohomology, then its genus zero Gromov--Witten invariants may be recovered from the $A_\infty$-category ${{\sf Fuk}}(M)$ together with the closed-open map. An immediate corollary of this is that in the semi-simple case, homological mirror symmetry implies enumerative mirror symmetry., Comment: Comments welcome. v2: sign corrections and other improvements. v3: some improvements

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 $\frac{1}{n+1}x^{n+1}$.

In this paper we prove an inverse function theorem in derived differential geometry. More concretely, we show that a morphism of curved $L_\infty$ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. This theorem simultaneously generalizes the inverse function theorem for smooth manifolds and the Whitehead theorem for $L_\infty$ algebras. The main ingredients are the obstruction theory for $L_\infty$ homomorphisms (in the curved setting) and the homotopy transfer theorem for curved $L_\infty$ algebras. Both techniques work in the $A_\infty$ case as well., Comment: Comments welcome! Title change in the second version. v4: references updated

Given two cyclic A ∞ A_\infty -algebras A A and B B , in this paper we prove that there exists a cyclic A ∞ A_\infty -algebra structure on their tensor product A ⊗ B A\otimes B which is unique up to a cyclic A ∞ A_\infty -quasi-isomorphism. Furthermore, the Kontsevich class of A ⊗ B A\otimes B is equal to the cup product of the Kontsevich classes of A A and B B on the moduli space of curves.

In this paper, we present an explicit cyclic minimal $A_\infty$ model for the category of matrix factorizations $\MF(W)$ of an isolated hypersurface singularity. The key observation is to use Kontsevich's deformation quantization technique. Pushing this idea further, we use the Tsygan formality map to obtain a comparison theorem that the categorical Variation of Semi-infinite Hodge Structure of $\MF(W)$ is isomorphic to Saito's original geometric construction in primitive form theory. An immediate corollary of this comparison result is that the analogue of Caldararu's conjecture holds for the category $\MF(W)$., Comment: 31 pages

Let $W\in \mathbb{C}[x_1,\cdots,x_N]$ be an invertible polynomial with an isolated singularity at origin, and let $G\subset {{\sf SL}}_N\cap (\mathbb{C}^*)^N$ be a finite diagonal and special linear symmetry group of $W$. In this paper, we use the category ${{\sf MF}}_G(W)$ of $G$-equivariant matrix factorizations and its associated VSHS to construct a $G$-equivariant version of Saito's theory of primitive forms. We prove there exists a canonical categorical primitive form of ${{\sf MF}}_G(W)$ characterized by $G_W^{{\sf max}}$-equivariance. Conjecturally, this $G$-equivariant Saito theory is equivalent to the genus zero part of the FJRW theory under LG/LG mirror symmetry. In the marginal deformation direction, we verify this for the FJRW theory of $\big(\frac{1}{5}(x_1^5+\cdots+x_5^5),\mathbb{Z}/5\mathbb{Z}\big)$ with its mirror dual B-model Landau-Ginzburg orbifold $\big(\frac{1}{5}(x_1^5+\cdots+x_5^5), (\mathbb{Z}/5\mathbb{Z})^4\big)$. In the case of the Quintic family $\mathcal{W}=\frac{1}{5}(x_1^5+\cdots+x_5^5)-\psi x_1x_2x_3x_4x_5$, we also prove a comparison result of B-model VSHS's conjectured by Ganatra-Perutz-Sheridan., Comment: 30 pages, comments welcome! Minor modification in V2

We give a construction of NC-smooth thickenings [a notion defined by Kapranov (J Reine Angew Math 505:73–118, 1998)] of a smooth variety equipped with a torsion free connection. We show that a twisted version of this construction realizes all NC-smooth thickenings as $$0$$ th cohomology of a differential graded sheaf of algebras, similarly to Fedosov’s construction in (J Differ Geom 40:213–238, 1994). We use this dg resolution to construct and study sheaves on NC-smooth thickenings. In particular, we construct an NC version of the Fourier–Mukai transform from coherent sheaves on a (commutative) curve to perfect complexes on the canonical NC-smooth thickening of its Jacobian. We also define and study analytic NC-manifolds. We prove NC-versions of some of GAGA theorems, and give a $$C^\infty $$ -construction of analytic NC-thickenings that can be used in particular for Kahler manifolds with constant holomorphic sectional curvature. Finally, we describe an analytic NC-thickening of the Poincare line bundle for the Jacobian of a curve, and the corresponding Fourier–Mukai functor, in terms of $$A_\infty $$ -structures.

We interpret symplectic geometry as certain sheaf theory by constructing a sheaf of curved A_\infty algebras which in some sense plays the role of a "structure sheaf" for symplectic manifolds. An interesting feature of this "structure sheaf" is that the symplectic form itself is part of its curvature term. Using this interpretation homological mirror symmetry can be understood by well-known duality theories in mathematics: Koszul duality or Fourier- Mukai transform. In this paper we perform the above constructions over a small open subset inside the smooth locus of a Lagrangian torus fibration. In a subsequent work we shall use the language of derived geometry to obtain a global theory over the whole smooth locus. However we do not know how to extend this construction to the singular locus. As an application of the local theory we prove a version of homological mirror symmetry between a toric symplectic manifold and its Landau-Ginzburg mirror.

Let \pi: M \ra B be a Lagrangian torus fibration with singularities such that the fibers are of Maslov index zero, and unobstructed. The paper constructs a rigid analytic space M_0^\chk over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus B_0 of \pi. Transition functions of M_0^\chk are obtained via A-\infty homomorphisms which capture the wall-crossing phenomenon of moduli spaces of holomorphic disks., Comment: Final version, minor changes. To appear in Advances in Mathematics

Let h \subset g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module U(g)/U(g)h whose associated graded h-module is isomorphic to S(n). We give a necessary and sufficient condition for the existence of a splitting of this filtration. In turn such a splitting yields an isomorphism between the h-modules U(g)/U(g)h and S(n). For the diagonal embedding h \subset h \oplus h the condition is automatically satisfied and we recover the classical Poincae-Birkhoff-Witt theorem. The main theorem and its proof are direct translations of results in algebraic geometry, obtained using an ad hoc dictionary. This suggests the existence of a unified framework allowing the simultaneous study of Lie algebras and of algebraic varieties, and a closely related work in this direction is on the way., Major revision, proofs of several results rewritten. Added a section explaining the case of a general representation, as opposed to the trivial one. 20 pages, LaTeX

In this paper we prove a version of curved Koszul duality for Z/2Z-graded curved coalgebras and their coBar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations MF(R,W). We show how Dyckerhoff's generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel-Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category MF(R,W). Similar results are also obtained in the orbifold case and in the graded case., Comment: Latex 34 pages, rewritten introduction, deleted an appendix, minor modification on proofs, final version