The Witten Equation, Quantum Cohomology and the Dubrovin Connection

By George Shiber Saturday, January 2, 2016Friday, October 26, 2018 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory, Supersymmetry Calabi-Yau, 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

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 $*$