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4
votes
2answers
741 views

complexified kahler form

In mirror symmetry one usually considers a complexified kahler form $B+iw$ instead of kahler form $w$ itself.(Or their moduli) Here is the question: What does $B$ correspond to? what kind of ...
2
votes
1answer
263 views

balanced curves in Calabi-Yau 3-folds

A balanced smooth rational curve in a calabi-Yau X is a smooth rational curve whose normal bundle is $O(-1)\oplus O(-1)$. We usually like these curves because of their rigidity. But, Is there any ...
4
votes
2answers
315 views

Shrinking Fano surfaces to a point in Calabi-Yau 3-folds

Let X be a Calabi-Yau 3-fold, and $D\subset X$, smooth,Fano divisor. Since $K_X=0$ we have $N_D^X=K_D$. I have seen the following fact in many papers: By deforming X within Kahler moduli, we can ...
1
vote
1answer
327 views

About topological B model

I was heard (by an expert) that, in mirror symmetry, we have constructed a Quantum Master Equation associated to topological B model, and a solution to it. But I can't find any material about this. Is ...
7
votes
1answer
1k views

“Fourier-Mukai” functors for Fukaya categories?

I just skimmed a bit of this fresh-off-the-press paper on homological mirror symmetry for general type varieties. One thing that intrigued me was statement (ii) of Conjecture 3.3. It suggests that, ...
4
votes
4answers
981 views

Mirror of local Calabi-Yau

What is the mirror manifold of the total space of the bundle $O(-1)\oplus O(-1)$ over $P^1$? I have tried to find the answer on the web but failed. Is there a good reference for this? Thanks.
4
votes
3answers
537 views

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 ...
15
votes
5answers
2k views

Mirror symmetry mod p?! … Physics mod p?!

In his answer to this question, Scott Carnahan mentions "mirror symmetry mod p". What is that? (Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than ...
9
votes
2answers
1k views

Higher genus closed string B-model

The closed string A-model is mathematically described by Gromov-Witten invariants of a compact symplectic manifold $X$. The genus 0 GW invariants give the structure of quantum cohomology of $X$, which ...
2
votes
1answer
488 views

Understanding formula in Frenkel-Witten

I'm not the person to understand everything in Geometric Endoscopy and Mirror Symmetry, but some parts of it are reasonably clear to me. In particular, one of the main objects, mathematically ...
6
votes
2answers
2k views

Picard-Fuchs equations

If I have the periods $$\pi_1(\lambda)=\int_0^1\frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$$ and $\pi_2(\lambda)$ similarly defined of the cubic curve $$y^2z=z(x-z)(x-\lambda z)$$ Such functions will be ...
14
votes
4answers
3k views

Is the Fukaya category “defined”?

Sometimes people say that the Fukaya category is "not yet defined" in general. What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact ...
4
votes
1answer
413 views

Singularity theory references

I am looking for some good references on singularity theory. I'm interested in singularity theory in the context of mirror symmetry, so this means I'm interested in things like Picard-Lefschetz ...
5
votes
1answer
1k views

Mirror symmetry for noncompact Calabi-Yau manifolds

In analogy with the Hodge diagram for ordinary de Rham cohomology, we should have some kind of diagram for Alexander-Spanier cohomology. Doing all the relevant duality stuff and assuming that now our ...