Mathematics is the music of reason ~ 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 $J$-functions coincide, thus, the 'Picard-Lefschetz' Witten relation, for ${u_i} \in {\Omega ^0}\left( {{\wp _i}} \right)$ are

and

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 $\wp$ from part 1 refer $X$, the Calabi-Yau smooth projective variety corresponding to the Witten-potential, and ${H_X}$ the even part of ${H^ \bullet }\left( {X,\mathbb{Q}} \right)$, and ${X_{g,n,d}}$ the metaplectic moduli space of  $n$-pointed genus-$g$ stable maps to X of degree $d \in {H_2}\left( {X,\mathbb{Z}} \right)$. The following Kähler-Witten integral of $X$ is central to this post

with ${a_{1,...,}}{a_n} \in {H_X}$ and ${\rm{e}}{{\rm{v}}_k}:{X_{g,n,d}} \to X$ being the evaluation map at the $k$-th marked point and ${\psi _1},...,{\psi _n} \in {H^2}\left( {{X_{g,n,d}};\mathbb{Q}} \right)$ being the universal cotangent line classes. Now we can derive

Let me fix bases ${\phi _0},...,{\phi _N}$${\phi ^0},...,{\phi ^N}$ for ${H_X}$ satisfying

${\phi _0}$ being the identity element of ${H_X}$

${\phi _1},...,{\phi _r}$ is a nef-basis for ${H^2}\left( {X;\mathbb{Z}} \right) \subset {H_X}$

- each ${\phi _i}$ is Kähler-homogeneous

$\left( {{\phi _i}} \right)\,_{i = 0}^{i = N}$ and $\left( {{\phi ^j}} \right)\,_{j = 0}^{j = N}$ are Poincaré-pairing dual

with $r$ the rank of ${H_2}\left( X \right)$. To get to our quantum cohomology analysis, let the Novikov ring $\Lambda = \mathbb{Q}{\left\{ {{Q_1},...,{Q_r}} \right\}^\dagger }$ and for $d \in {H_2}\left( {X;\mathbb{Z}} \right)$, we write ${Q^d} = Q\,_1^{{d_1}}...Q\,_r^{{d_r}}$ with ${d_i} \equiv d \cdot {\phi _i}$ and now we can get to quantum cohomology. Letting ${t^0},...,{t^N}$ be the ${H_X}$ coordinates defined by the basis ${\phi _0},...,{\phi _N}$ such that $t \in {H_X}$ satisfies $t = {t^0}{\phi _0} + ... + {t^N}{\phi _N}$, we hence get the genus-zero Gromov-Witten potential

via

with  the first sum is over the set $NE\left( X \right)$ of degrees of effective curves in $X$

So the quantum product $*$ can thus be only defined in terms of the third partial derivatives of $F\,_X^0$ as

where $*$ is bilinear over $\Lambda$ therefore defining a formal family of algebras on ${H_X} \otimes \Lambda$ parametrized by ${t^0},...,{t^N}$ and that is the quantum super-cohomology of $X$ and has a Hodge–Tate type${H^{p,q}}\left( X \right) = 0$ for $p \ne q$. Since ${H_X} \otimes \Lambda$ is a Gromov-Witten scheme over $\Lambda$, letting $M$ be a topological neighbourhood of the origin in $X$, then the Euler vector field $E$ on $M$ is

Note now that the grading operator $\mu :{H_X} \to {H_X}$ is definable via

with $\pi :M \times {\widetilde A^1} \to M$ projection to the first factor. Hence, the Dubrovin connection is a meromorphic flat connection $\nabla$ on

defined by

$0 \le i \le N$ and $z$ the coordinate on ${\widetilde A^1}$. Now, by the Poincaré pairing, the Dubrovin connection equips $M$ with a Frobenius manifold with extended structure connection, and thus the genus-zero Gromov–Witten potential $F\,_X^0$ 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 $X$ 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.