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?
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.
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.