Nikita Markarian’s mathblog

On quantum cohomology

Posted in Mathematics by nikitamarkarian on May 5, 2010

Nowadays the theory of quantum cohomology is treated as a part of symplectic geometry. But it is not clear (for me) how much information about the symplectic structure does quantum cohomology contain. Perhaps, the situation is analagous to the Morse theory, where the definition is given in differential geometric terms (integral curves and so on) but the result is purely topological.

If one take as a starting point the paper of Witten and two papers of Givental (this and this), one may develop a theory of a different flavour sketched below. Conjecturally it coincides with the usual one, at least in some cases.

Let M be a Kahler (it may be relaxed) compact manifold with h^{2,0}=0 and h^{1,1}=d. Let \tilde M be the fiber bundle over M, that is the fiber product of a set of line bundles that form basis of \mathop{Jac}(M). This space is acted by {\mathbb{C^*}}^d and the factor is M. Let L\tilde M be the space of free loops (in the usual sence) of \tilde M . This space is acted by L \mathbb{C^*}^d. The latter group contains (S^1)^d\times \mathbb{Z}^d.

Consider the (S^1)^d-equivariant twisted K-theory K^\zeta_{(S^1)^d}(L\tilde M) (M. Atiyah, G. Segal, “Twisted $K$-theory”) of L\tilde M. The twisting \zeta is given by the semi-infinite determinant of the tangent bundle. For a loop that bounds a holomorphic curve this is the determinant of the subbundle of the tangent bundle that may be continued on the whole curve. Different bounding curves give subspaces that differs by a finite-dimensional vector spaces. Thus the semi-infinite determinant of the tangent bundle gives a \mathbb C^*-gerbe. I would like to have a sharper definition of this twisting gerbe, which is the only non-topological ingredient of the construction.

Firstly, the twisted equivariant K-theory is acted by \mathbb Z^d, because the group acts on the loop space and the gerbe is invariant under it. So \mathbb C\otimes K^\mu_{(S^1)^d}(L\tilde M) is a module over the group algebra \mathbb C [q_1^{\pm 1}, \dots , q_d^{\pm 1}] of \mathbb Z^d. Then, due to the seminal paper of Chas and Sullivan, the twisted K-theory is a BV algebra. In fact a lot of work should be done here: one should replace cohomology in Chas-Sullivan theory with K-theory and show that the gerbe is multiplicative in a proper sense (UPD: this point is very weak, it is not clear for me how the gerbe behaves under compositions of loops; it seems much more plausible that there is an action of the non-twisted K-theory on the twisted one, see the end of the next post). Then the this BV algebra is a module over \mathbb C [q_1^{\pm 1}, \dots , q_d^{\pm 1}] and moreover it is an algebra over this ring. Thus we have a family of BV algebras. Finally, one should show that this family is curvy trivial and trivialized in the sense of this post.

Now we may apply methods of the mentioned post. Spherical homology of BV algebras form a D-module over \mathop{Spec} \mathbb C [q_1^{\pm 1}, \dots , q_d^{\pm 1}], and this is exactly D-module that play the major role in the mentioned papers of A. Givental: “Homological geometry and mirror symmetry” and “Homological geometry I. Projective hypersurfaces” . Moreover, it seems that my definition is (a part of) what Givental means, but written in a slightly different language. It is natural to propose that S-homology is responsible for higher genera. Note that the spherical homology if our BV algebra is not naturally isomorphic to cohomology of M, it is not even clear that dimensions are equal.

In the paper E. Witten, “The Verlinde Algebra And The Cohomology Of The Grassmannian” it is shown that  in some cases quantum cohomology of a Grassmanian equals to some Verlinde algebra; by quantum cohomlogy Witten means the fiber of the Givental D-module at q_1=\dots=q_d=1. In the light of our definition it is not so suprising. By D. S. Freed, M. J. Hopkins, C. Teleman, “Loop groups and twisted K-theory I”, the Verlinde algebra of a compact Lie group G equals to the twisted K-theory of free loop space of the classifying space of G with some special twisting (level). The quantum cohomology of a Grassmanian at q_1=\dots=q_d=1 in our definition is twisted K-theory of free loop space of the Grassmanian. It is natural to propose that under the natural embedding of our Grassmanian into the classifying space, \zeta is induced from some level and the inverse image gives the isomorphism stated by Witten.

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