Writings:

4.) In this preprint I record the Calabi-Yau version of Kontsevich's formality morphism from deformation quantization. 

3.) My thesis combines papers 1. and 2. below. In addition, we propose ideas towards a categorical formulation of twisted holography at the level of partition functions, given as input a Calabi-Yau category and one of its objects.

2.) A crucial ingredient in the theory of Categorical Enumerative Invariants is a splitting of the non-commutative Hodge filtration (related to a trivialization of the circle action on Hochschild chains of a Calabi-Yau category), from which a formality morphism is constructed. In my second paper I explain how to construct a formality morphism in the open-closed theory of Categorical Enumerative Invariants, again depending on a splitting. Physically, an instance of such a open-closed theory is  BCOV theory coupled to holomorphic Chern-Simons, as developed by Costello-Li.

1.) In my first paper I explain how various versions of the Loday-Quillen-Tsygan map, respectively a generalization that applies to (cyclic) A-infinity categories, various graph complexes, additionally equiped with shifted Poisson or BV-algebra structure, and Kontsevich's cocycle construction are linked together. This is related to open String Field Theory (SFT). Its cousin is closed SFT, as described by Sen-Zwiebach, Costello, Căldăraru-Tu, which led to the definition of Categorical Enumerative Invariants.