M-Theory, the Witten Equation and Picard-Lefschetz Theory

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:

    \[\overline {\not \partial \,} {u_i} + \frac{{\overline {\not \partial W} }}{{\not \partial {u_i}}} = 0\]

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

    \[\left\{ {\begin{array}{*{20}{c}}{\overline {\not \partial } \,{u_i} \in {\Omega ^{0,1}}\left( {{\wp _i}} \right)}\\{\frac{{\overline {\not \partial W} }}{{\not \partial {u_i}}} \in \Omega _{{\rm{log}}}^{o,1}\left( {\overline \wp _i^1} \right)}\end{array}} \right.\]

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

    \[{W_1},...,{W_r} \in \mathbb{C}\left[ {{x_1},...,{x_N}} \right]\]

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

    \[dr = \sum\limits_{j = 1}^N {{c_j}} \]

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

    \[{\varphi _1},...,{\varphi _n} \in {H_{GW}} = {H^ * }\left( X \right)\]

and any

    \[{a_1},...,{a_n} \in {\mathbb{Z}^{ \le 0}}\]

we have a Gromov-Witten invariant

    \[\left\langle {{\tau _{a1}}\left( {{\varphi _1}} \right),...,{\tau _{an}}\left( {{\varphi _n}} \right)} \right\rangle _{g,n,\beta }^{GW}\]

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

    \[{J_{GW}}\left( {\not t,z} \right) = z + \not t + \sum\limits_{n,\beta } {\frac{1}{{n!}}} \left\langle {\not t\left( \psi \right),...\not t\left( \psi \right),\frac{{{\varphi _\alpha }}}{{z - \psi }}} \right\rangle _{0,n + 1,\beta }^{GW}\]

with

    \[\left\langle {\not t\left( \psi \right),...\not t\left( \psi \right),\frac{{{\varphi _\alpha }}}{{z - \psi }}} \right\rangle _{0,n + 1,\beta }^{GW}\]

central for Calabi-Yau n-foldings, and with

    \[\not t\left( z \right) = {t_0} + {t_1}z + {t_2}{z^2} \in {H_{GW}}\left[ z \right]\]

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

    \[\left\langle {{\tau _{a1}}\left( {{\phi _1}} \right),...,{\tau _{an}}\left( {{\phi _n}} \right)} \right\rangle _{g,n.\beta }^{hyb}\]

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

    \[{J_{hyb}}\left( {\not t,z} \right) = z + \not t + \sum\limits_{n,\beta } {\frac{1}{{n!}}} \left\langle {\not t\left( {\overline \psi } \right),...\not t\left( {\overline \psi } \right),\frac{{{\phi _\alpha }}}{{z - \overline \psi }}} \right\rangle _{o,n + 1,\beta }^{hyb}{\varphi ^\alpha }\]

and this time,

    \[\left\langle {\not t\left( {\overline \psi } \right),...\not t\left( {\overline \psi } \right),\frac{{{\phi _\alpha }}}{{z - \overline \psi }}} \right\rangle _{o,n + 1,\beta }^{hyb}{\varphi ^\alpha }\]

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

    \[\left[ {\not D_q^4 - {3^6}q\left( {{{\not D}_q} + \frac{1}{3}} \right){{\left( {{{\not D}_q} + \frac{2}{3}} \right)}^2}} \right]{I_{GW}} = 0\]

and

    \[\left[ {\not D_q^4 - {2^8}q{{\left( {{{\not D}_q} + \frac{1}{2}} \right)}^4}} \right]{I_{GW}} = 0\]

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

    \[q' = \frac{{{g_{GW}}\left( q \right)}}{{{f_{GW}}\left( q \right)}}\]

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

    \[\frac{{{I_{GW}}\left( {q,z} \right)}}{{{f_{GW}}\left( q \right)}} = {J_{GW}}\left( {q',z} \right)\]

For the Landau-Ginzburg side, we have

    \[\begin{array}{c}{I_{hyb}}\left( {t,z} \right) = \sum\limits_{d\, \ge 0}^{d \ne 1,{\rm{mod}}3} {\frac{{z{e^{\left( {d + 1 + \frac{{{H^{\left( {d + 1} \right)}}}}{2}} \right)}}}}{{{3^6}\left[ {\frac{d}{3}} \right]}}} \\ \cdot \frac{{\prod\limits_{1 \le b \le d}^{b \buildrel \wedge \over = d + 1,{\rm{mod}}3} {{{\left( {{H^{\left( {d + 1} \right)}} + bz} \right)}^4}} }}{{\prod\limits_{1 \le b \le d}^{b \ne d + 1,{\rm{mod}}3} {{{\left( {{H^{\left( {d + 1} \right)}} + bz} \right)}^2}} }}\end{array}\]

for the cubic

    \[\begin{array}{c}{I_{hyb}}\left( {t,z} \right) = \sum\limits_{d\, \ge 0}^{d \ne 1,{\rm{mod}}2} {\frac{{z{e^{\left( {d + 1 + \frac{{{H^{\left( {d + 1} \right)}}}}{2}} \right)t}}}}{{{2^8}\left[ {\frac{d}{2}} \right]}}} \\ \cdot \frac{{\prod\limits_{1 \le b \le d}^{b \buildrel \wedge \over = d + 1,{\rm{mod}}2} {{{\left( {{H^{\left( {d + 1} \right)}} + bz} \right)}^4}} }}{{\prod\limits_{1 \le b \le d}^{b \ne d + 1,{\rm{mod}}2} {{{\left( {{H^{\left( {d + 1} \right)}} + bz} \right)}^4}} }}\end{array}\]

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

    \[{I_{hyb}}\left( {t,z} \right) = \omega _1^{hyb}\left( t \right) \cdot {1^{\left( 1 \right)}} \cdot z + \omega _2^{hyb} + {\rm O}\left( {{z^{ - 1}}} \right)\]

with

    \[\omega _2^{hyb} + {\rm O}\left( {{z^{ - 1}}} \right)\]

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

    \[\left\{ {\begin{array}{*{20}{c}}{\overline {\not \partial } \,{u_i} \in {\Omega ^{0,1}}\left( {{\wp _i}} \right)}\\{\frac{{\overline {\not \partial W} }}{{\not \partial {u_i}}} \in \Omega _{{\rm{log}}}^{o,1}\left( {\overline \wp _i^1} \right)}\end{array}} \right.\]

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