'**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

**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 a quasi-homogeneous 'super-potential' polynomial and a section of the corresponding orbifold line bundle on a Riemann surface . 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 , degree with the following relation

On the Calabi-Yau 'side', one analyses the intersection 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 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

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

with

central for Calabi-Yau n-foldings, and with

holding, and runs over a basis for . On the Landau-Ginzburg side, the polynomials are interpretable as the equations for singularities in . Hence, since the state space 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 , we get the -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 -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 . Note now by explicit change of variables

for -valued functions and , and thus, the -function matches , hence

For the Landau-Ginzburg side, we have

for the cubic

The key fact about -functions is the fact that the family lives on the Lagrangian cone on which the -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 ,

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

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