Which part of physical B model is not rigorous? the physical theory of B model,if it is not mathematical rigorous just because the Feynman integral,but it looks like for me the space is finite dimensional, what the problem is causing that it still not rigorous?

$\begingroup$ I see the articles constructing the B model mathematically,but I don't know why they do this,or do that ,because there is no relating to original physics. $\endgroup$ – HYYY Mar 6 '10 at 3:55

1$\begingroup$ What do you mean by the physical Bmodel? There's the IIB string and the topological Bmodel, but I don't know of a physical Bmodel. $\endgroup$ – Aaron Bergman Mar 6 '10 at 4:53

$\begingroup$ O,I mean topological B model, so, for example, why is it not rigorous constructed by physicist,the problem is in path integral(but it seems it is finite dimensional) or anything else?Thanks! $\endgroup$ – HYYY Mar 6 '10 at 13:37

$\begingroup$ By the way, what do we really mean by mathematical defintion of B model, and what is the relation to the definition by physicists?(I said no problem in feynman integral part) $\endgroup$ – HYYY Mar 7 '10 at 1:51
To define (as Kevin Lin does above) the Bmodel purely as the derived category of coherent sheaves is fine and rigorous, but it ignores the highergenus aspects of mirror symmetry  which was the original question. As I wrote above, Kevin Costello gives a rigorous description of the highergenus amplitudes, but it is still conjectural whether this agrees with the physics. The issue is that highergenus string amplitudes depend on an integration over the moduli space of Riemann surfaces (or a space of maps from them, depending on the model), and this demands compactification. The full, nontopological theory is of course an ordinary twodimensional quantum field theory, with all the usual difficulties in making the path integral rigorous.

$\begingroup$ Hi Eric, maybe this is a distinction between the B TFT and B topological strings  the derived category I think knows everything about the former in all genus via Costello/KontsevichSoibelman/Lurie reconstructions of closed strings from open, right? but of course as you say it knows nothing about integrating over moduli of curves/extension to nodal curves etc. $\endgroup$ – David BenZvi Mar 9 '10 at 12:56

$\begingroup$ Hi Costello's construction of the GW partition function (0509264) does involve the moduli space of Riemann surfaces. He works with topological conformal field theories in that paper. If his result differs from the physical Bmodel, it's due to a difference in the compactifications of the moduli spaces used. But the reconstruction results of Lurie et al (of course I don't myself understand how the cobordism hypothesis is proven in this case) do produce invariants at all genera that may have nothing to do with GW theory, as you state, due to use of TFT's vs. TCFT's. z $\endgroup$ – Eric Zaslow Mar 12 '10 at 2:58

$\begingroup$ Hi I was thinking of his 0412149, a precursor of the cobordism hypothesis where all genus topological Bmodel (eg) is recovered from the derived category. I didn't think there was any difference between TCFT and (differential graded) TFT, only a question of whether you pass from TFT to topological strings, calculating a topological string partition function by integrating over moduli spaces (as 0509264 does)? $\endgroup$ – David BenZvi Mar 17 '10 at 19:19
Kevin Costello's mathematical definition of the Bmodel (math/0509264) is rigorous. It's an open problem whether this definition agrees with the BCOV construction, as far as I know.

$\begingroup$ BCOV is closedstring Bmodel, correct? The Costello paper you reference does not give all of the closedstring Bmodel, only a part of it. $\endgroup$ – Kevin H. Lin Mar 6 '10 at 0:11
Openstring Bmodel seems to be ok: derived category of coherent sheaves in the CalabiYau case, matrix factorizations in the LandauGinzburg case.
The following are closedstring Bmodel, but genus 0 only:
See BarannikovKontsevich for the CalabiYau case: http://arxiv.org/abs/alggeom/9710032
For the isolated singularity LandauGinzburg case, see the work of Kyoji Saito (for example, take a look at the book of Hertling on Frobenius manifolds).
For the higher genus closedstring Bmodel, see this question.

$\begingroup$ Thank you for you two the answers. but I mean, the physical theory of B model,if it is not mathematical rigorous just because the Feynmann integral,but it looks like for me the space is finite dimensional, what the problem is causing it is still not rigorous? (Perhaps I still need to look at the physical theory of B model to see what is going on) $\endgroup$ – HYYY Mar 6 '10 at 1:05

$\begingroup$ I assume the space you are referring to here is, in the CalabiYau case, the cohomology (of polyvector fields) of the manifold, or the Jacobian ring in the isolated singularity LandauGinzburg case. Indeed these are finite dimensional. I do not know any physics and I do not understand the BCOV paper, but I think that they consider not just cohomology of polyvector fields $H^\ast(\Lambda^\ast T_X)$ but also $\Gamma(\Lambda^\ast T_X \otimes \Omega^{0,\ast}_X)$, which is infinite dimensional. Perhaps this is a source of problems, but I don't know  this is just a naive guess. $\endgroup$ – Kevin H. Lin Mar 6 '10 at 1:59

$\begingroup$ Thanks!Here space refers to the space of field configuration that the Feynman integral performs on. $\endgroup$ – HYYY Mar 6 '10 at 3:11