Gromov–Witten Theory, Quantum Cohomology and Mirror Symmetry

"It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances" ~ Hannah Arendt

In this post, I will begin the first part of a study of the cohomological-Dubrovin connection in the context of mirror-symmetry for Calabi-Yau Gromov–Witten theory. In particular, I will derive two propositions about quantum cohomology. In my last three posts, I finally derived, after studying the Witten Equation as well as the Landau-Ginzburg/Calabi-Yau correspondence, the Witten super-Dubrovin compactification Chern-Simons formula

\begin{array}{*{20}{l}}{2\left[ {{{\hat c}_1}\left( {{{\dot \nabla }^{\dagger *}}} \right) - {c_1}\left( {{{\dot \nabla }^{\dagger - 1}}} \right)} \right] = }\\{2\left[ {{{\hat c}_1}\left( {{{\dot \nabla }^{\dagger *}}} \right) - {{\hat c}_1}\left( {{{\rm{A}}^\dagger }\left( {{{\dot \nabla }^{\dagger *}}} \right)} \right)} \right]}\\{ = \frac{{ - 1}}{{\pi i}}\int_{{S_X}^1} {{\rm{tr}}{\mkern 1mu} \left( {\frac{{1 - i\beta }}{2}{{\rm{A}}^{\dagger - 1}}{{\dot \nabla }^\dagger }_{\not \partial /{{\not \partial }_u}}} \right)} {\mkern 1mu} du}\end{array}


{\dot \nabla ^\dagger } - {{\rm A}^\dagger }\left( {{{\dot \nabla }^\dagger } - 1} \right) = \, - \frac{{1 - i\beta }}{2}{{\rm A}^\dagger }^{ - 1}{\dot \nabla ^\dagger }{{\rm A}^\dagger }


{\dot \nabla ^\dagger }_{\not \partial /{{\not \partial }_u}} = {\not \partial _u} + \frac{1}{2}\alpha

the super-Dubrovin relation. Realizing that mirror symmetry for Calabi–Yau manifolds is interpretable as an isomorphism of variations of Hodge structures, with the A-model, defined by the genus zero Gromov–Witten theory of Xisomorphic to the B-model of variations of Hodge structures associated to deformation of complex structures of the mirror Y. Now, while the B-model of the variations of Hodge structures has integral local system {H^n}\left( {Y,\mathbb{Z}} \right), the A-model does not. So: what is the integral local system in the A-model mirrored from the B-model?! I showed here that analysis on compact toric orbifolds entail that the K-group of X should give the integral local system in the A-model. So, the genus zero Gromov-Witten theory induces a family of super-commutative algebras

\left( {{H^ * }\left( X \right),{o_\tau }} \right)

on the cohomology group metaplectically parametrized by

\tau \in {H^ * }\left( X \right)

which is the Witten quantum cohomology whose D-module is defined by a flat connection ^W\nabla on the bundle {H^ * }\left( X \right) \times {H^ * }\left( X \right) \to {H^ * }\left( X \right), with a parameter z \in {\mathbb{C}^ * }, and is the Dubrovin connection studied by me here, and is quantum-cohomologically given by

^W{\nabla _X} = {d_X} + \frac{1}{z}X{o_\tau }

with X \in {H^ * }\left( X \right), where \tau denotes a point on the base and {d_X} is the directional derivative. One can then extend this connection in the direction of the parameter z yielding a flat {H^ * }\left( X \right)-bundle over {H^ * }\left( X \right) \times {\mathbb{C}^ * }

Now a symplectic solution to the differential equation

^W{\nabla _s}\left( {\tau ,z} \right) = 0

has general Picard-form

s\left( {\tau ,z} \right) = L\left( {\tau ,z} \right){z^{ - \mu }}{c_1}\left( X \right)

where \mu is the Hodge grading operator

\mu \left( {{\phi _k}} \right): = \left( {\frac{1}{2}{\rm{deg}}\,{\phi _k} - \frac{n}{2}} \right){\phi _k}

with L\left( {\tau ,z} \right) is the master solution

L\left( {\tau ,z} \right)\alpha : = {e^{ - {\tau _{0,2/z}}}}\alpha - \sum\limits_{\left( {d,l} \right) \ne \left( {0,0} \right)}^{d \in {\rm{Ef}}{{\rm{f}}_\chi },1 \le k \le N} {\frac{{{\phi ^k}}}{{1!}}} \left\langle {{\phi _k},\tau ',...,\tau ',\frac{{{e^{ - {\tau _{0,2/z}}}}}}{{z + \psi }}} \right\rangle _{0,l + z,d}^\chi {e^{\left\langle {{\tau _0},2,d} \right\rangle }}

Let me separate the terms

L\left( {\tau ,z} \right)\alpha : = {e^{ - {\tau _{0,2/z}}}}\alpha

\sum\limits_{\left( {d,l} \right) \ne \left( {0,0} \right)}^{d \in {\rm{Ef}}{{\rm{f}}_\chi },1 \le k \le N} {\frac{{{\phi ^k}}}{{1!}}}

\left\langle {{\phi _k},\tau ',...,\tau ',\frac{{{e^{ - {\tau _{0,2/z}}}}}}{{z + \psi }}} \right\rangle _{0,l + z,d}^\chi {e^{\left\langle {{\tau _0},2,d} \right\rangle }}

which is asymptotic to {e^{ - \tau /z}} in the large radius limit

\left\{ {\begin{array}{*{20}{c}}{\Re \left\langle {{\tau _{0,2}},d} \right\rangle \to - \infty }\\{\forall d \in {\rm{Ef}}{{\rm{f}}_\chi }\backslash \left\{ 0 \right\}}\\{\tau ' \to 0}\end{array}} \right.

Now, with {\delta _{1,}}...,{\delta _n} the Chern roots of the tangent bundle TX, one defines a Picard-transcendental characteristic class \widehat \Gamma \left( {T\chi } \right) via

\begin{array}{l}\widehat \Gamma \left( {TX} \right): = \prod\limits_{i = 1}^n \Gamma \left( {1 + {\delta _i}} \right) = \\e\left( { - \gamma {c_1}\left( X \right) + \sum\limits_{k \ge 2} {{{\left( { - 1} \right)}^k}\left( {k - 1} \right)!\xi \left( k \right){\rm{c}}{{\rm{h}}_k}\left( {TX} \right)} } \right)\end{array}


e\left( { - \gamma {c_1}\left( X \right) + \sum\limits_{k \ge 2} {{{\left( { - 1} \right)}^k}\left( {k - 1} \right)!\xi \left( k \right){\rm{c}}{{\rm{h}}_k}\left( {TX} \right)} } \right)

key for the Landau-Ginzburg/Calabi-Yau correspondence, with \gamma  the Euler constant and \xi (s) the Riemann zeta function. For V \in K\left( X \right), define a ^W\nabla -flat connection \widehat {\not {\rm Z}}\left( V \right)

\begin{array}{c}\widehat {\not {\rm Z}}\left( {TX} \right)\left( {\tau ,z} \right): = {\left( {2\pi } \right)^{ - n/2}}L\left( {\tau ,z} \right)\\ \cdot \,{z^{ - \mu /2}}{z^{{c_1}\left( X \right)}} \cdot \\\left( {\widehat \Gamma \left( {TX} \right) \cup {{\left( {2\pi \widetilde i} \right)}^{{\rm{deg}}\,{\rm{2}}}}{\rm{ch}}\left( V \right)} \right)\end{array}

with n = \dim X and hence, \widehat {\not {\rm Z}}\left( V \right) define the quantum cohomological \widehat \Gamma -integral structure. The mirror-symmetric image of a compact toric orbifold is then given by a Landau–Ginzburg model, which is a pair of a torus {Y_q} = {\left( {{\mathbb{C}^ * }} \right)^n} and a Laurent polynomial {W_q}:{Y_q} \to \mathbb{C}. The Landau–Ginzburg model then defines a B-model D-module which is decidable by an integral local system generated by Lefschetz thimbles of {W_q}. By mirror symmetry, it follows that the quantum D-module of a toric orbifold is isomorphic to the B-model D-module: hence the two propositions of this post can be stated

- Proposition one: Let \chi  be a weak Picard-Fano projective toric orbifold defined by the initial data satisfying \widehat \rho \in {\widetilde C_\chi }, then, in light of mirror-symmetry, we get the \Gamma -integral structure on the quantum D-module, and it corresponds to the natural integral local system of the B-model D-module under the mirror isomorphism

\begin{array}{l}{\rm{Mirr}}:\left( {{{\widehat {R'}}^{\left( 0 \right)}}{,^W}\nabla ,{{\left( {.,.} \right)}_{{{\widehat {R'}}^{\left( 0 \right)}}}}} \right)\left| {_{{V_\varepsilon }}} \right. \cong \\{\left( {\tau \times {\rm{id}}} \right)^ * }\left( {\left( {F{,^W}\nabla ,{{\left( {.,.} \right)}_F}} \right)/{H^2}\left( {\chi ,\mathbb{Z}} \right)} \right)\end{array}

with the quantum cohomology central charge of V \in K\left( X \right) given by

\widehat {\not Z}\left( V \right)\left( {\tau ,z} \right): = \left( {\frac{{{{\left( {2\pi z} \right)}^{n/2}}}}{{2\pi {{\widetilde i}^n}}}} \right)\int_X {\widehat {\not Z}} \left( {V\left( {\tau ,z} \right)} \right)

- Proposition two: Under the same assumptions as proposition one, the quantum cohomology central charge of the structure sheaf {\vartheta _\chi } is given by the Picard-integral over the real Lefschetz thimble {\Gamma _\mathbb{R}}

\begin{array}{l}\widehat {\not Z}\left( {{\vartheta _\chi }} \right)\left( {\tau (q),z} \right) = \frac{1}{{{{\left( {2\pi \widehat i} \right)}^n}}} \cdot \\\int_{{\Gamma _\mathbb{R}}} {\exp \left( { - {W_q}(y)/z} \right)} \,{\omega _q}\end{array}

In the nest post, I will go further and study Orbifold quantum cohomology.

Physics is becoming too difficult for the physicists ~ David Hilbert

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