Semisimple quantum cohomology

Question

Dubrovin's conjecture (or Bayer's modified version, if prefer) establishes a condition for the semisimplicity of the quantum cohomology of a manifold X, but, Why is important to know that the quantum cohomology of some manifold X is semisimple?

Answer 1

One reason is Givental's conjecture, which says that in the semisimple case, genus 0 GW invariants determine higher genus GW invariants. See this paper of Teleman, in which the conjecture is proved.

The theory of Frobenius manifolds in general is quite complicated. I guess semisimple Frobenius manifolds form a relatively tractable set of examples. Here are some basic references for the theory of semisimple Frobenius manifolds:

• Manin: Frobenius manifolds, quantum cohomology, and moduli spaces

• Dubrovin: Geometry of 2d topological field theories

• Lee, Pandharipande: Frobenius manifolds, Gromov-Witten theory, and Virasoro constraints (so far unpublished and incomplete; available at Pandharipande's webpage)

Semisimple Frobenius manifolds also arise in singularity theory, when studying for instance isolated hypersurface singularities (see Hertling's book Frobenius manifolds and moduli spaces for singularities; the three references above probably also talk about this), or in the physics terminology "Landau-Ginzburg (B-)models". I don't know whether Frobenius manifolds (in particular non-semisimple ones) arise more generally in singularity theory...? In any case, these Frobenius manifolds coming from singularity theory are supposed to be related to those coming from Gromov-Witten theory via mirror symmetry.*

Another comment: Quantum cohomology of, for example, $\mathbb{P}^n$ is semisimple. Then perhaps this makes quantum cohomology and GW theory of projective varieties more tractable, because of quantum Lefschetz ... but I don't really know anything about this. But very roughly speaking, I think this is the strategy of Givental in his proof of the "mirror conjecture" of Candelas et. al. regarding the genus 0 GW theory of the quintic 3-fold, though I might be wrong.

*Edit: For example, this paper of Etienne Mann seems to prove a mirror theorem relating the quantum cohomology Frobenius manifolds of (weighted) projective spaces and the Frobenius manifolds associated to the mirror Landau-Ginzburg B-models. As Arend mentions, germs of semisimple Frobenius manifolds are specified by a finite set of data, and I think the strategy of Mann's paper is to show that these data coincide for the two Frobenius manifolds.

Answer 2

• One of the first interesting questions in GW theory asks how much information is needed to determine all GW invariants. The germ of a semisimple Frobenius is determined by a finite amount of data - this follows either from Dubrovin's or Manin's classification data via their structure connections. This implies that all genus-zero GW invariants are determined by this data; combined with Givental's conjecture/Teleman's theorem, this shows that all GW invariants are determined by the same finite amount of data. It also means that you could prove a mirror theorem (an isomorphism of Frobenius manifolds, in this case) by comparing finite amount of data on each side.

• The mirror partner of $P^n$ (or any toric variety) is, as Kevin mentioned, constructed from the versal deformation of a function with isolated singularities (essentially this was proved by Givental as the first step of his mirror theorem). The Frobenius manifold associated to such a versal deformation is always generically semisimple. It seems reasonable to ask which other manifolds could have a function with isolated singularities as mirror partner. Clearly, having semisimple quantum cohomology is necessary (and one could motivate Dubrovin's conjecture by suggesting that it is a sufficient condition as well).

While I post here I should as well point out that "Bayer's modified version" was a little to optimistic. A counterexample is given by minimal surfaces of general type which have an exceptional vector bundle. At least one set of examples are the surfaces constructed by Yongnam Lee, Jongil Park in arXiv:math/0609072, on which a construction by Paul Hacking (which is included in arXiv:0808.1550) applies to produce exceptional vector bundles. On the other hand, all genus zero invariants vanish on minimal surfaces of general type by dimension reasons.