Givental-Dubrovin Analysis and Quantum Cohomology

In mathematics the art of proposing a question must be held of higher value than solving it.
Georg Cantor

In my last few posts, I studied quantum cohomology as well as the the Dubrovin meromorphic connection I analyzed here as well as the Givental-symplectic space here and finally derived via the orbifold Poincaré pairing embedding

    \[\begin{array}{c}{\widetilde {\not J}_\tau }:{\left( {{\pi _ * }\vartheta \left( F \right)} \right)_\tau }{ \to ^\dagger }{\widetilde {\rm H}_G}:{\widetilde {\not J}_\tau }\left( {\tau ,z} \right)\\: = {\widetilde {\not J}_\tau }\widehat 1 = L{\left( {\tau ,z} \right)^{ - 1}}\widehat 1\end{array}\]

and the Galois relation I derived here

    \[{\widetilde J_\tau }\left( {{{\widetilde G}^{{{\widetilde {\rm H}}_G}}}\left( \xi \right)\tau ,z} \right) = \widetilde G_\chi ^{\widetilde {\not H}}{\widetilde J_\tau }\left( {\tau ,z} \right)\]

the quantum cohomology central charge integral can be derived as

    \[\begin{array}{c}Z\left( V \right)\left( {\tau ,z} \right) = {\chi ^{{{\widetilde {\rm H}}_G}}} \cdot \\\left( {{{\widetilde {\not H}}_K}\left( {\tau ,{e^{\pi \widehat i}}z} \right) \otimes {V^ \vee }} \right) = \\\int\limits_{{I_\chi }} {{{\widetilde {\rm H}}_G}} \left( {\tau ,{e^{\pi \widehat i}}z} \right) \cup \widetilde {ch}\left( {{V^ \vee }} \right)\widetilde {Td}\left( {T\chi } \right)\end{array}\]

with \widetilde {ch} and \widetilde {Td} the Chern and Todd characteristic classes respectively I introduced here. I will continue my Givental-Dubrovin analysis in the context of differential geometry, and less from that of moduli spaces, to keep a more unificational faithfulness to Einstein’s as well as Witten’s intuitions.

Givental’s Symplectic Space and the Quantum Cohomology Central Integral

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas ~ G. H. Hardy

Last post, I derived the Orbifold Riemann–Roch Theorem via essential use of Todd-Chern Classes and the Lefschetz-grading operator by deducing the identity

    \[\frac{1}{{{{\left( {2\pi } \right)}^n}}}{\sum\limits_{v \in \,T} {\int\limits_{{\chi _v}} {\left( {{e^{\pi \widehat i\rho }}\widetilde \Gamma {{\left( {T\chi } \right)}_{inv\left( v \right)}}{{\left( {2\pi \widehat i} \right)}^{\frac{{\deg }}{2}}}\widetilde {ch}{{\left( {{V_1}} \right)}_v}} \right)} } _v}\]

    \[ = \]

    \[\begin{array}{c}\left( {{{\rm Z}_K}\left( {{V_1}} \right),{{\rm Z}_K}\left( {{V_2}} \right)} \right) = \\{\left( {{e^{\pi \widehat i\rho }}\Psi \left( {{V_1}} \right),{e^{\pi \widehat i\mu }}\Psi \left( {{V_2}} \right)} \right)_{ORB}}\end{array}\]

In this post, I will briefly analyze some crucial aspects of the Givental’s symplectic space of quantum cohomology and holomorphic theory