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I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer theory", http://arxiv.org/abs/1105.5123), the homology ring being computed is given in terms of a Superpotential function. I understand that the relations in the homology ring are somehow encoded in its derivatives, and that it's critical point somehow correspond to Toric fibers with non vanishing homology, but nothing more than that and I have no clue about the details.

What I am looking for is some exposition on the subject, some lecture notes, or somewhere where I can see a simple example being calculated completely from scratch.

Can anyone please refer me so such sources?

Thank you

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2 Answers 2

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For compact toric manifolds, Floer cohomology of non-displaceable Lagrangians can be detected by their superpotentials. This is in some sense the $\mathfrak{m}_0$ term in the $A_\infty$ structure which represents the obstruction to defining Floer cohomology. This construction can be found in the paper of FOOO: http://arxiv.org/abs/1009.1648. For more elementary expositions, see the paper of Cho: http://arxiv.org/abs/math/0412414.

In fact, the superpotantial encodes informations about the non-triviality of the $A_\infty$ structures of non-displaceable Lagrangians. What you want can be deduced by passing to the cohomology level. In fact, it is conjectured by Kontsevich in the compact case that the Hochschild cohomology of the Fukaya category is equal to the quantum cohomology. On the other hand, homological mirror symmetry predicts that the quantum cohomology ring should be identified with the Jacobi ring of the superpotential. The latter thing is just the Hochschild cohomology of the triangulated category of singularities in the sense of Orlov. That's why the informations of quantum cohomology are encoded in the derivatives of the superpotential. A large part of this conjecture in the toric case is proved by FOOO in the above paper. In the compact toric case, it is the work in preparation of Abouzaid-FOOO that the Fukaya category is generated by certain torus fibers of the toric fibration on these manifolds. But as toric fibers, their disc enumeration and therefore the superpotential can be determined explicitly using the work of Cho-Oh: http://arxiv.org/abs/math/0308225. The work of Cho-Oh only concerns the monotone case. For indefinite case like Hirzebruch surfaces, you need to deal with non-trivial issues coming from Kuranish structures, which involves techniques of FOOO developed in their book. There are non-trivial sphere bubblings and the discs involved are not regular.

The same construction also generalizes to symplectic manifolds with local toric charts, and the superpotential and Floer cohomology can be deduced in a similar way as Cho did, see for example, the work of Abouzaid-Auroux-Katzarkov: http://arxiv.org/abs/1205.0053.

Many interesting explicit examples can be found in the paper of Ritter-Smith: http://arxiv.org/abs/1201.5880. They give computations in both compact and non-compact cases. In the non-compact setting, you need to replace $QH^\ast(X)$ by the symplectic cohomology $SH^\ast(X)$.

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  • $\begingroup$ Great answer!! It is worthwhile to note that the first appearance of a description in Landau-Ginzburg form appeared first in section 9 of Batyrev's work (93) while Orlov's machinery of singular derived categories was introduced later (04) $\endgroup$ Aug 7, 2017 at 12:49
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You can find some very accessible discussion on quantum cohomology of toric (Fano) manifolds in On the quantum homology algebra of toric Fano manifolds by Ostrover & Tyomkin (and references therein). Also check out Batyrev's original Quantum Cohomology Rings of Toric Manifolds - before going to FOOO.

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