### 'Edward Witten is often likened to Einstein; one colleague reached even further back for a comparison, suggesting that Witten possessed the greatest mathematical mind since Newton' ~ John Horgan

This is a series of posts that hopefully will culminate in a book I am working on. The aim is to show that the Witten equation in the context of Picard-Lefschetz Theory, leads to the non-forking, super-stability and hyper-categoricity of M-theory, thus making it the 'only-game-in-town' indeed. First, I will analyse the Witten nonlinear elliptic system of PDEs associated with a quasi-homogeneous polynomial-super-potential by showing its depth via the Landau-Ginzburg/Calabi-Yau correspondence. The Witten equation is deceiving in its simplicity:

However, as you will see at the end of this post, it gets tricky:

with $W$ a quasi-homogeneous 'super-potential' polynomial and ${u_i}$ a section of the corresponding orbifold line bundle on a Riemann surface $\not C$. To appreciate the depth of the Witten equation, one must understand the Landau-Ginzburg/Calabi-Yau correspondence: a connection between the geometry of Calabi-Yau hyper-complete intersections in projective space and the Landau-Ginzburg model, where the polynomials defining the intersections are interpreted as singularities, and the Calabi-Yau side is the Gromov-Witten theory of the hyper-complete intersection. To give a taste of such relation(s), take a nondegenerate collection of quasihomogeneous polynomials

with weights ${c_1},...,{c_N}$, degree $d$ with the following relation

On the Calabi-Yau 'side', one analyses the intersection $X$ in weighted projective space cut out by the polynomials: the cohomology of this intersection is quasimorphic to the state-space from which insertions to Gromov-Witten invariants of $X$ are selected. Now, for any choice of

and any

we have a Gromov-Witten invariant

which is the hyper-intersection number on the moduli space of stable maps to $X$

Hence, the genus-zero invariants are encoded by a $J$-function

with

central for Calabi-Yau n-foldings, and with

holding, and ${\varphi _\alpha }$ runs over a basis for ${H_{GW}}$. On the Landau-Ginzburg side,  the polynomials ${W_i}$ are interpretable as the equations for singularities in ${\mathbb{C}^N}$. Hence, since the state space ${H_{hyb}}$ and its members can be used as the insertions to hybrid invariants

and are metaplectic hyper-intersection numbers on a moduli space parameterizing stable maps to projective space together with a collection of line bundles on the source curve whose tensor powers satisfy equations determined by the polynomials ${W_i}$, we get the $J$-functional genus-encoding relation

and this time,

is key for to solving the equations for singularities. The genus-zero Landau-Ginzburg/Calabi-Yau correspondence is the assertion that there is a degree-preserving isomorphism between the two state spaces and that after certain identifications, the small $J$-function coincide. So one gets, along the way to show the depth of the Witten equation, a derivation of the Picard-Fuchs equation

and

where ${\not D_q} = q\frac{{\not \partial }}{{\not \partial q}}$. Note now by explicit change of variables

for $\mathbb{C}$-valued functions ${g_{GW}}$ and ${f_{GW}}$, and thus, the $J$-function ${J_{GW}}$ matches ${I_{GW}}$, hence

For the Landau-Ginzburg side, we have

for the cubic

The key fact about $I$-functions is the fact that the family ${I_{hyb}}\left( {t, - z} \right)$ lives on the Lagrangian cone $\widetilde {{{\not L}_{hyb}}}$ on which the $J$-function is a slice, as will be shown. Hence, we get

with

key to Calabi-Yau n-folding, hence getting the 'Picard-Lefschetz' Witten relation, for ${u_i} \in {\Omega ^0}\left( {{\wp _i}} \right)$,

In the next post, I will make contact with quantum cohomology.