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.

In this post, part one, I will derive two propositions, one about the flatness of the connection and another about an isomorphism related to quantum vector spaces. Let K be an algebra of functions of the complex variables {q_{1,...,}}{q_r} with

    \[{\not \partial _i} \equiv \frac{{\not \partial }}{{{{\not \partial }_{{t_i}}}}} = {q_i}\frac{{\not \partial }}{{{{\not \partial }_{{q_i}}}}}\]

and define D to be the algebra of differential operators generated by

    \[\frac{{\not \partial }}{{{{\not \partial }_{{q_1}}}}},...,\frac{{\not \partial }}{{{{\not \partial }_{{q_r}}}}}\]

with coefficients in K and M = D/\left( {{D_{1,}}...,{D_u}} \right) be a cyclic quantum D-module, which is an algebraic version of the system of partial differential equations

    \[{D_1}f = ... = {D_u}f = 0\]

where f belongs to a given function space F, and the cyclic quantum D-module independent of F and the vector space Ho{m_D}\left( {M,F} \right) being the solution space of M with respect to the function space F, which is isomorphic to the solution space \left\{ {f \in F\left| {{D_1}f = ...{D_u} = 0} \right.} \right\} of the system. To a solution f there corresponds the D-module homomorphism M \to F given by P{ \to ^ * }Pf. Note, the solution space is a complex vector space of dimension s + 1. Now let {P_0},...,{P_s} be differential operators such that the equivalence classes \left[ {{P_0}} \right],...,\left[ {{P_s}} \right] form a K-module basis of M and define matrices

    \[{\Omega _i} = {\left( {\Omega _{kj}^i} \right)_{0 \le k,j \le s}}\]

via

    \[\left[ {{{\not \partial }_i}{P_j}} \right] = \sum\limits_{k = 0}^s {{{\left( {\Omega _{kj}^i} \right)}_{0 \le k,j \le s}}} \]

with

    \[\Omega \equiv \sum\nolimits_{i = 1}^r {{\Omega _i}} d{t_i}\]

a 1-form with values in the space

    \[{\rm{End}}\left( {{\mathbb{C}^{s + 1}}} \right)\]

of complex \left( {s + 1} \right) \times \left( {s + 1} \right) matrices with \nabla = d + \Omega defining a connection in the trivial vector bundle {\mathbb{C}^r} \times {\mathbb{C}^{s + 1}} \to {\mathbb{C}^r}, where {\mathbb{C}^{s + 1}} is the vector space spanned by \left[ {{P_0}} \right],...,\left[ {{P_s}} \right], hence getting

    \[{\nabla _{{{\not \partial }_i}}}\left[ {{P_j}} \right] = \sum\nolimits_{k = 0}^s {\Omega _{kj}^i} \left[ {{P_k}} \right]\]

and for any section

    \[\sum\nolimits_{j = 0}^s {{y_i}} \left[ {{P_j}} \right]\]

of that bundle, the following holds

    \[\begin{array}{c}{\nabla _{{{\not \partial }_i}}}\left( {\sum\nolimits_{i = 0}^s {{y_j}\left[ {{P_j}} \right]} } \right) = \\\sum\nolimits_{j = 0}^s {{{\not \partial }_i}} {y_i}\left[ {{P_i}} \right] + \\\sum\nolimits_{j = 0}^s {{y_i}} {\nabla _{{{\not \partial }_i}}}\left[ {{P_j}} \right]\end{array}\]

Proposition oneThe connection \nabla is flat. Let me prove that

Since

    \[{\nabla _{{{\not \partial }_i}}}{\nabla _{{{\not \partial }_j}}} = {\nabla _{{{\not \partial }_j}}}{\nabla _{{{\not \partial }_i}}}\]

the curvature tensor of \nabla is zero, hence flatness follows from

    \[d\Omega + \Omega \wedge \Omega = 0\]

Hence

Proposition two: we have an isomorphism of vector spaces

    \[Ho{m_D}\left( {M,F} \right) \to \left\{ {{\rm{covariant constant sections of}}\,{\nabla ^ * }} \right\}\]

    \[f{ \to ^ * }\left( {\begin{array}{*{20}{c}}{{P_o}f}\\{{P_s}f}\end{array}} \right)\]

with {\nabla ^*} being the dual connection to \nabla. One way to prove the isomorphism is to note that the left-hand side, f is regarded as the D-module homomorphism P{ \to ^ * }Pf, and on the right-hand side f is a solution of the system {D_1}f = ...{D_u}f = 0 and the dual is defined by

    \[\begin{array}{c}\left( {\nabla _{{{\not \partial }_i}}^ * {{\left[ {{P_j}} \right]}^ * }} \right)\left[ {{P_k}} \right] = - {\left[ {{P_j}} \right]^ * } \cdot \\\left( {{\nabla _{{{\not \partial }_i}}}\left[ {{P_k}} \right]} \right)\end{array}\]

where {\left[ {{P_o}} \right]^ * },...,{\left[ {{P_s}} \right]^ * } is the dual basis to \left[ {{P_o}} \right],...,\left[ {{P_s}} \right]. So, a section

    \[\sum\nolimits_{j = 0}^s {{y_j}} {\left[ {{P_j}} \right]^ * }\]

is covariant constant if the following expression is zero for all k

    \[\left( {\nabla _{{{\not \partial }_i}}^ * {{\sum\limits_{j = 0}^s {{y_i}\left[ {{P_j}} \right]} }^ * }} \right)\left[ {{P_k}} \right]\]

    \[ = \]

    \[\begin{array}{c}\left( {\sum\limits_{j = 0}^s {{{\not \partial }_i}{{\left[ {{P_j}} \right]}^ * } + \sum\limits_{j = 0}^s {{y_i}\nabla _{{{\not \partial }_i}}^ * {{\left[ {{P_j}} \right]}^ * }} } } \right)\\ \cdot \left[ {{P_k}} \right]\end{array}\]

    \[ = \]

    \[{\not \partial _i}{y_k} - \sum\limits_{j = 0}^s {{y_j}} \left( {{{\left[ {{P_j}} \right]}^ * }\sum\limits_{l = 0}^s {\Omega _{lk}^i\left[ {{P_k}} \right]} } \right)\]

    \[{\not \partial _i}{y_k} - \sum\limits_{j = 0}^s {{y_j}\,} \Omega _{lk}^i\]

To finalize the proof that it is an isomorphism, note that the kernel is zero due to {P_0}f = f, and that {\rm{Dim}}\,Ho{m_D}\left( {M,F} \right) = s + 1 by assumption. Next post, I will continue with a focus on the corresponding quantum cohomology D-module.…