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

Georg Cantor

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

and the Galois relation I derived here

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 one: The 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.…

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