## 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

the quantum cohomology central charge integral can be derived as

with  and  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  be an algebra of functions of the complex variables  with

and define  to be the algebra of differential operators generated by

with coefficients in  and  be a cyclic quantum -module, which is an algebraic version of the system of partial differential equations

where  belongs to a given function space , and the cyclic quantum -module independent of  and the vector space being the solution space of  with respect to the function space , which is isomorphic to the solution space  of the system. To a solution  there corresponds the -module homomorphism  given by . Note, the solution space is a complex vector space of dimension . Now let  be differential operators such that the equivalence classes  form a -module basis of  and define matrices

via

with

a 1-form with values in the space

of complex  matrices with  defining a connection in the trivial vector bundle , where  is the vector space spanned by , hence getting

and for any section

of that bundle, the following holds

Proposition oneThe connection  is flat. Let me prove that

Since

the curvature tensor of  is zero, hence flatness follows from

Hence

Proposition two: we have an isomorphism of vector spaces

with  being the dual connection to . One way to prove the isomorphism is to note that the left-hand side,  is regarded as the -module homomorphism , and on the right-hand side  is a solution of the system and the dual is defined by

where  is the dual basis to . So, a section

is covariant constant if the following expression is zero for all

To finalize the proof that it is an isomorphism, note that the kernel is zero due to , and that  by assumption. Next post, I will continue with a focus on the corresponding quantum cohomology -module.…