The Witten Equation, Quantum Cohomology and the Dubrovin Connection

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

\overline {\not \partial } \,{u_i} \in {\Omega ^{0,1}}\left( {{\wp _i}} \right)

and

\frac{{\overline {\not \partial W} }}{{\not \partial \,u}} \in \Omega _{{\rm{log}}}^{0,1}\left( {{{\overline \wp }_i}^1} \right)

Keep the quantum product

{\phi _\alpha } * {\phi _\beta } = \sum\limits_{\gamma = 0}^{\gamma = N} {\frac{{{{\not \partial }^3}F\,_X^0}}{{\not \partial {t^\alpha }\not \partial {t^\beta }\not \partial {t^\gamma }}}} \,{\phi ^\gamma }

in mind throughout the Witten Equation post-series.

In this post, part 2, I will connect the Witten equation

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

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

\int_{{{\left[ {{X_{g,n.d}}} \right]}^{{\rm{vir}}}}} {\prod\limits_{k = 1}^{k = n} {{\rm{ev}}\,_k^ * } } \left( {{a_k}} \right) \cup \psi \,_k^{ik}

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

\left\langle {{a_1}\psi _1^{{i_1}},...,{a_n}\psi _n^{{i_n}}} \right\rangle \,_{g,n,d}^X = \int_{{{\left[ {{X_{g,n.d}}} \right]}^{{\rm{vir}}}}} {\prod\limits_{k = 1}^{k = n} {{\rm{ev}}{\mkern 1mu} _k^*} } \left( {{a_k}} \right) \cup \psi {\mkern 1mu} _k^{ik}

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

 

F\,_X^0 \in \Lambda {\left\{ {{t^0},...,{t^N}} \right\}^\dagger }

via

F\,_X^0 = \sum\limits_{d \in NE\left( X \right)} {\sum\limits_{n = 0}^\infty {\frac{{{Q^d}}}{{n!}}} } \left\langle {t,...,t} \right\rangle \,_{0,n,d}^X

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

{\phi _\alpha } * {\phi _\beta } = \sum\limits_{\gamma = 0}^{\gamma = N} {\frac{{{{\not \partial }^3}F\,_X^0}}{{\not \partial {t^\alpha }\not \partial {t^\beta }\not \partial {t^\gamma }}}} \,{\phi ^\gamma }

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

E = {t^0}\frac{{\not \partial }}{{\not \partial {t^0}}} + \sum\limits_{i = 1}^r {{\rho ^i}} \frac{{\not \partial }}{{\not \partial {t^i}}} + \sum\limits_{i = r + 1}^N {\left( {1 + \frac{1}{2}{\rm{deg}}{\phi _i}} \right)} \,{t^i}\frac{{\not \partial }}{{\not \partial {t^i}}}

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

 

\mu \left( {{\phi _i}} \right) = \left( {\frac{1}{2}{\rm{deg}}{\phi _i} - \frac{1}{2}{\rm{di}}{{\rm{m}}_\mathbb{C}}X} \right)\,{\phi _i}

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

{\widetilde \pi ^ * }\overline T M \cong {H_X} \times \left( {M \times {{\widetilde A}^1}} \right)

defined by

{\nabla _{\frac{{\not \partial }}{{\not \partial {t_i}}}}} = \frac{{\not \partial }}{{\not \partial {t_i}}} - \frac{1}{z}\left( {{\phi _i} * } \right)

{\nabla _{z\frac{{\not \partial }}{{{{\not \partial }_z}}}}} = z\frac{{\not \partial }}{{{{\not \partial }_z}}} + \frac{1}{z}\left( {E * } \right) + \mu

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  *

\int_{{{\left[ {{X_{g,n.d}}} \right]}^{{\rm{vir}}}}} {\prod\limits_{k = 1}^{k = n} {{\rm{ev}}\,_k^ * } } \left( {{a_k}} \right) \cup \psi \,_k^{ik}

and allows the Gromov–Witten invariants

\left\langle {{a_1}\psi _1^{{i_1}},...,{a_n}\psi _n^{{i_n}}} \right\rangle \,_{g,n,d}^X

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.

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