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A (smooth) dg algebra is called (left) Calabi-Yau if (see for example here) $$ A^! = A[-n]$$ Here we use the inverse dualizing complex $A^!=\mathbf{R}\operatorname{Hom}_{(A^e)^{op}}(A,A^e)$. In topological string theory (e.g. in the context of matrix factorizations, see here or here), it is interesting to regard $A$ as a dg-category and form a sort of Yondeda-embedding $\hat{A} := \operatorname{Int}(A-{\operatorname{Mod}})$ defined in Toen of it, the full subcategory of fibrant-cofibrant objects in the category of $A$-dg-modules. 9 My questions are the following:

  • Is the dg-category $\hat{A}$ (where the dg-structure is inherited from $A-{\operatorname{Mod}}$ automatically a Calabi-Yau $A_\infty$-category when regarded as an $A_\infty$-category? So, are both of these definitions of Calabi-Yau compatible? Due to the work of Costello, that would be required for the above construction to actually define a TCFT, and it is often stated it actually does - however I have never actually seen a discussion of this subtle point.
  • If this is not always the case, is the homotopy category of $\hat{A}$ at least a Calabi-Yau category?
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