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

The Universe Was Not Created From Q-'Nothingness': A Lefschetz-Hodge-Fourier Refutation

Beethoven tells you what it’s like to be Beethoven and Mozart tells you what it’s like to be human. Bach tells you what it’s like to be the universe. ― Douglas Adams … And I will tell you why it did not come from (Quantum) nothingness!

An equation means nothing to me unless it expresses a thought of God.
S. Ramanujan

Keep this:

\frac{d}{{dt}}{\left( {{\Delta ^C} = \underbrace {\int_0^{\delta fK} {\frac{{\frac{d}{{d{t^ \circ }}}{\Psi _{{{\not Q}_V}}}}}{{\alpha ({t^ \circ })}}\not D{{\not Z}_V}} }_{{\rm{creation}}} + \underbrace {\int_{\delta f}^{fK} {\frac{{\frac{d}{{d{t^ \circ }}}{\Psi _{U({t^ \circ })}}}}{{\alpha ({t^ \circ })}}d\Omega {{({\phi _{\exp t}})}^{2\pi i\xi t}}dt} }_{{\rm{quantum gravity}}}} \right)_{t = 0}}

and the Hodge 'creation' identity':