Givental-Dubrovin Analysis and Quantum Cohomology

By George Shiber Thursday, January 28, 2016Friday, October 26, 2018 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Philosophy Of Science, Physics, Super String Theory Dubrovin Connection, Givental

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 one: The 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.