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.

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.

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 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).

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.