Which part of physical B model is not rigorous? 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?
 A: To define (as Kevin Lin does above) the B-model purely as the derived category of coherent sheaves is fine and rigorous, but it ignores the higher-genus aspects of mirror symmetry -- which was the original question.  As I wrote above, Kevin Costello gives a rigorous description of the higher-genus amplitudes, but it is still conjectural whether this agrees with the physics.  The issue is that higher-genus 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, non-topological theory is of course an ordinary two-dimensional quantum field theory, with all the usual difficulties in making the path integral rigorous.
A: Kevin Costello's mathematical definition of the B-model (math/0509264) is rigorous.
It's an open problem whether this definition agrees with the BCOV
construction, as far as I know.
A: Open-string B-model seems to be ok: derived category of coherent sheaves in the Calabi-Yau case, matrix factorizations in the Landau-Ginzburg case.
The following are closed-string B-model, but genus 0 only:


*

*See Barannikov-Kontsevich for the Calabi-Yau case: http://arxiv.org/abs/alg-geom/9710032

*For the isolated singularity Landau-Ginzburg case, see the work of Kyoji Saito (for example, take a look at the book of Hertling on Frobenius manifolds).
For the higher genus closed-string B-model, see this question.
