In "On the Classification of Topological Field Theories" in Example 1.4.1, Lurie introduces the B-model with target an (even dimensional) Calabi-Yau variety $X$: The Hochschild cohomology $\operatorname{HH}^\*(X)$, together with the "canonical trace map" $\operatorname{HH}^\*(X) \rightarrow k$, form a graded commutative Frobenius algebra; and by a classical theorem that can, as he states, analogously applied to the graded setting, this defines a 2-dimensional Topological Quantum Field Theory $Z$ with values in complexes such that $Z(S^1)=\operatorname{HH}^\*(X)$.


While I can roughly follow his reasoning (although it would be helpful if someone could provide a short explanation on how this trace map occurs), as far as I knew the states of the B-model on $X$ were related to the (bounded) derived category $\operatorname{D}^b(\operatorname{Coh(X)})$, and I don't really see why Lurie's definition should describe the B-model as I know it, or why Hochschild cohomology occurs here. As Lurie only uses this statement as a motivation for introducing extended TQFTs with values in chain complexes, he doesn't give any further explanation or references, so I wondered if anyone here could. Thanks in advance and greetings,


Markus Zetto



B-model and Hochschild cohomology
The boundary conditions of the B-model, ie, the D-branes, are the objects in $\mathcal{D}^b(X)$. A little bit of playing with pictures gives that the space of closed string states must be in the center of the algebra of open strings for any given boundary condition. For a category $\mathcal{C}$, this gives a map to $\mathcal{Nat}(id\_\mathcal{C},id\_\mathcal{C})$. Extending this to the graded situation and using the description of natural transformations as bimodules, you get
$$
HH^\bullet(\mathcal{D}^b(X)) = \mathrm{Ext}^\bullet\_{X \times X}(\mathcal{O}\_\Delta,\mathcal{O}\_\Delta)
$$
where $\Delta$ is the diagonal. There's a version of the HKR theorem that relates this to cohomology of exterior powers of the tangent bundle, the usual space of states in the B-model.


A lot of this is explained in [Moore and Segal](https://arxiv.org/pdf/hep-th/0609042.pdf). I also explained some (with no claim to originality) in the beginning of my old paper [Deformations and D-branes](https://arxiv.org/pdf/hep-th/0609225.pdf).


