Mathematics is the music of reason ~
James Joseph Sylvester
James Joseph Sylvester
In my last post (part 1) in this series on the Witten-equation, I showed that the genus-zero Landau-Ginzburg/Calabi-Yau correspondence amounts to the assertion that there is a degree-preserving isomorphism between the two state spaces and that after certain identifications, the small -functions coincide, thus, the 'Picard-Lefschetz' Witten relation, for are
Keep the quantum product
in mind throughout the Witten Equation post-series.
In this post, part 2, I will connect the Witten equation
with the Gromov–Witten invariants and the Kähler-Witten integral in a deep way via the Dubrovin connection (see below) in the context of quantum cohomology, with the Dubrovin connection and the Gromov–Witten invariants playing a central role on the corresponding throat-bulk smooth projective variety. Let me change notation from my previous post: in this post, let from part 1 refer , the Calabi-Yau smooth projective variety corresponding to the Witten-potential, and the even part of , and the metaplectic moduli space of -pointed genus- stable maps to X of degree . The following Kähler-Witten integral of is central to this post
with and being the evaluation map at the -th marked point and being the universal cotangent line classes. Now we can derive
Let me fix bases , for satisfying
- being the identity element of
- is a nef-basis for
- each is Kähler-homogeneous
- and are Poincaré-pairing dual
with the rank of . To get to our quantum cohomology analysis, let the Novikov ring and for , we write with and now we can get to quantum cohomology. Letting be the coordinates defined by the basis such that satisfies , we hence get the genus-zero Gromov-Witten potential
with the first sum is over the set of degrees of effective curves in
So the quantum product can thus be only defined in terms of the third partial derivatives of as
where is bilinear over therefore defining a formal family of algebras on parametrized by and that is the quantum super-cohomology of and has a Hodge–Tate type: for . Since is a Gromov-Witten scheme over , letting be a topological neighbourhood of the origin in , then the Euler vector field on is
Note now that the grading operator is definable via
with projection to the first factor. Hence, the Dubrovin connection is a meromorphic flat connection on
and the coordinate on . Now, by the Poincaré pairing, the Dubrovin connection equips with a Frobenius manifold with extended structure connection, and thus the genus-zero Gromov–Witten potential converges to an analytic function, allowing a definition of a Fredholm-Calabi-Yau measure for the Kähler–Witten integral, due to the quantum product
and allows the Gromov–Witten invariants
to be metaplectic invariants on the homotopy group-manifold of again due to, and since it is, equipped with the quantum product .
As we shall see, these two relations are key to the unificational uniqueness, up to isomorphism, of M-theory and Branewold cosmology.