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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

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2 Answers 2

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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. I also explained some (with no claim to originality) in the beginning of my old paper Deformations and D-branes.

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  • $\begingroup$ Thanks, these both seem like very good references! I'll still have a bit of reading to do to figure out what exactly is going on here and how exactly this factors into Lurie's definition (I figure the fact that $Z(S^1)=\operatorname{HH}^*(X)$ just tells me exactly what you are saying, i.e. what the space of states for a closed string is), but this makes it very clear why Hochschild cohomology apears. $\endgroup$ Apr 30, 2020 at 13:19
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    $\begingroup$ You might also want to check out Kevin Costello's work, which focuses only on the 2D case so might be easier to understand: arxiv.org/pdf/math/0412149.pdf $\endgroup$ Apr 30, 2020 at 14:10
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For correct attribution, one should at least mention the paper which the preprint of Moore and Segal itself quotes as the source for the particular case of the algebraic description of open-closed TFTs which they give without proof. The following paper was published about 5 years before the preprint of Moore and Segal:

C. I. Lazaroiu, On the structure of open-closed topological field theory in two dimensions, Nucl.Phys.B603 (2001) 497-530, https://arxiv.org/abs/hep-th/0010269

Notice that the spaces of boundary states in the B-model with CY manifold target X are Z-graded, since the physically correct category of B-type topological D-branes is not D^b(X) but rather its shift completion. The Z/2Z grading in Lazaroiu's axioms is the mod 2 reduction of that natural Z-grading -- and this Z/2Z grading is ignored by Moore and Segal. Shift completions were explained in:

C. I. Lazaroiu, Graded D-branes and skew-categories, JHEP 0708 (2007) 088, https://arxiv.org/abs/hep-th/0612041

(which also explains the twist-completion that occurs, for example, in B-type orbifold LG models).

One should also note that the argument using HH(D^b(X)) and the holomorphic HKR isomorphism is not due to Lurie but to Kontsevich. Furthermore this argument (while very nice and natural) is not needed for the B-model, since the closed string state space of the B-model is known independently from localization of the path integral in the bulk sector, which produces the space of polyvector fields endowed with its (Dolbeault) differential and Serre trace. This goes back to Witten and to the paper of BCOV in the early 1990's. The advantage of the polyvector approach is that it provides an off-shell model which also appears in the BCOV theory and has a dual description using a BV bracket.

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  • $\begingroup$ Without digging too deep into this particular issue, I will note that the Moore and Segal results had been circulating and were available on Greg’s webpage many years before the release of the preprint. $\endgroup$ Jun 28, 2020 at 0:46
  • $\begingroup$ @Aaron What I have heard is that the only thing available (as per Greg Moore himself) were transparencies of a talk that had been unknown to Lazaroiu and which did not contain Lazaroiu's results but only a preliminary attempt at the problem. Lazaroiu (who was a first year post-doc at the time) mentions this in an added note and quotes those transparencies in the second version of his preprint, since he received a very strongly worded email from Greg Moore attaching those transparencies. It is dangerous to make claims about priority which are not supported by documentary evidence. $\endgroup$ Jun 28, 2020 at 14:04

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