Givental’s Symplectic Space and the Quantum Cohomology Central Integral

By George Shiber Monday, January 25, 2016Sunday, August 4, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory Givental’s Symplectic Space, Gromov–Witten Theory

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

In my 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 and derive the quantum cohomology central charge integral, which I will later show is crucial for world-brane cosmology. Keeping the following Mukai pairing formula in mind

$\begin{array}{c}\frac{1}{{{{\left( {2\pi } \right)}^n}}}{\sum\limits_{v \in \,T} {\left( {2\pi \widehat i} \right)} ^{\dim {\chi _v}}}\int\limits_{{\chi _v}} {\prod\limits_{f,i} \Gamma } \cdot \\\left( {1 - \overline f + \frac{{{\delta _{v,f,i}}}}{{2\pi \widehat i}}} \right)\Gamma \cdot \\\left( {1 - f - \frac{{{\delta _{v,f,i}}}}{{2\pi \widehat i}}} \right){e^{\frac{\rho }{2}}}\widetilde {ch}{\left( {{V_1}} \right)_v} \cdot \\{e^{\pi \widehat i\left( {{ \bullet _\iota } - \frac{n}{2} + \frac{{\deg }}{2}} \right)}}\widetilde {ch}{\left( {{V_2}} \right)_{inv\left( v \right)}}\end{array}$

we can proceed by letting ${A^ * }{\left( \chi \right)_\mathbb{C}}$ refer to the Chow ring of $\chi$ over $\mathbb{C}$ . Now set

${\widetilde {\not H}^ * }\left( {{\chi _v}} \right): = {\mathop{\rm Im}\nolimits} \left( {{A^ * }{{\left( {{\chi _v}} \right)}_\mathbb{C}} \to {{\widetilde {\not H}}^ * }\left( {{\chi _v}} \right)} \right)$

and define

$\widetilde {\not H}_{orb}^ * \left( \chi \right): = { \oplus _{v \in \,T}}{\widetilde {\not H}^ * }\left( {{\chi _v}} \right)$

Then the algebraic quantum D-module is definable as the holomorphic vector bundle

$\left\{ {\begin{array}{*{20}{c}}{\widetilde {\not H}_{orb}^ * \left( \chi \right) \times \left( {U' \times \mathbb{C}} \right) \to \left( {U' \times \mathbb{C}} \right)}\\{U' = U \cap \widetilde {\not H}_{orb}^ * \left( \chi \right)}\end{array}} \right.$

endowed with the restriction of the meromorphic Dubrovin connection $\nabla$ on

${\widetilde \pi ^ * }\overline T M \cong {H_X} \times \left( {M \times {{\widetilde A}^1}} \right)$

I defined here by

${\nabla _{\frac{{\not \partial }}{{\not \partial {t_i}}}}} = \frac{{\not \partial }}{{\not \partial {t_i}}} - \frac{1}{z}\left( {{\phi _i} * } \right)$

${\nabla _{z\frac{{\not \partial }}{{{{\not \partial }_z}}}}} = z\frac{{\not \partial }}{{{{\not \partial }_z}}} + \frac{1}{z}\left( {E * } \right) + \mu$

to $U'$ and the orbifold Poincaré pairing. Let me introduce the quantum cohomology central charge of $V \in K\left( \chi \right)$ associated to the $\widetilde \Gamma$ -class to be the function

$\begin{array}{c}Z\left( V \right)\left( {\tau ,z} \right): = c\left( z \right)\int\limits_\chi {\widetilde {{{\rm Z}_K}}} \left( {\tau ,z} \right) = \\c\left( z \right){\left( {\widehat 1,\widetilde {{{\rm Z}_K}}\left( V \right)\left( {\tau ,z} \right)} \right)_{orb}}\end{array}$

where

$c\left( z \right) = {\left( {2\pi z} \right)^{n/2}}/{\left( {2\pi \widehat i} \right)^n}$

is the $\chi$ –Calabi–Yau normalization factor given in terms of periods of the mirror and thus the Givental-symplectic space is the loop space on $\widetilde {\not H}_{orb}^ * \left( \chi \right)$ with a loop parameter $z$ that is identified with the space of sections of QDM– $\left( \chi \right)$ which are flat only in the $\tau$ -direction. Note, in the Givental space, QDM- $\left( \chi \right)$ can be realized as moving semi-infinite subspaces, which is a semi-infinite variation of Hodge structure $\frac{\infty }{2}$ -VHS.

Definition: Let $\vartheta \left( {{\mathbb{C}^ * }} \right)$ denote the space of holomorphic functions on ${\mathbb{C}^ * }$ with the coordinate $z$ . The Givental space ${\widetilde {\rm H}_G}$ is defined to be the free $\vartheta \left( {{\mathbb{C}^ * }} \right)$ –module

${\widetilde {\rm H}_G} = \widetilde {\not H}_{orb}^ * \left( \chi \right) \times \vartheta \left( {{\mathbb{C}^ * }} \right)$

endowed with the pairing

$\left( {.,.} \right):{\widetilde {\rm H}_G} \times {\widetilde {\rm H}_G} \to \vartheta \left( {{C^ * }} \right)$

${\left( {\alpha \left( z \right),\beta \left( z \right)} \right)_{{{\widetilde {\rm H}}_G}}}: = {\left( {\alpha \left( { - z} \right),\beta \left( z \right)} \right)_{orb}}$

and the symplectic form

$\begin{array}{c}\Omega \left( {\alpha \left( z \right),\beta \left( z \right)} \right) = {\rm{Re}}{{\rm{s}}_{z = 0}} \cdot \\dz{\left( {\alpha \left( z \right),\beta \left( z \right)} \right)_{orb}}\end{array}$

Using the fundamental solution $L\left( {\tau ,z} \right)$ I discussed here, in tandem with

$\begin{array}{c}\widehat {\not {\rm Z}}\left( {TX} \right)\left( {\tau ,z} \right): = {\left( {2\pi } \right)^{ - n/2}}L\left( {\tau ,z} \right)\\ \cdot \,{z^{ - \mu /2}}{z^{{c_1}\left( X \right)}} \cdot \\\left( {\widehat \Gamma \left( {TX} \right) \cup {{\left( {2\pi \widetilde i} \right)}^{{\rm{deg}}\,{\rm{2}}}}{\rm{ch}}\left( V \right)} \right)\end{array}$

we can identify ${\widetilde {\rm H}_G}$ with the space of sections of QDM- $\left( \chi \right)$ which are flat in the $\tau$ -direction,

$\begin{array}{c}{\widetilde {\rm H}_G} \triangleleft \alpha { \to ^\dagger }L\left( {\tau ,z} \right)\alpha \in \\\widetilde \Gamma \left( {U' \times {\mathbb{C}^ * },\vartheta \left( F \right)} \right)\end{array}$

Realize that under this identification, ${\left( {.,.} \right)_{{{\widetilde {\rm H}}_G}}}$ corresponds to ${\left( {.,.} \right)_F}$ and so the Galois action on flat sections induces a map ${\widetilde G^{{{\widetilde {\rm H}}_G}}}\left( \xi \right):{\widetilde {\rm H}_G} \to {\widetilde {\rm H}_G}$

$\begin{array}{c}{\widetilde G^{{{\widetilde {\rm H}}_G}}}\left( \xi \right)\left( {{\tau _0} \oplus \underbrace \otimes _{v \in \,T'}{\tau _v}} \right) = {e^{ - 2\pi \widehat i{\xi _0}/z}}\\ \cdot {\tau _0} \oplus \underbrace \otimes _{v \in \,T'}{e^{ - 2\pi \widehat i{\xi _0}{f_v}\left( \xi \right)}}{\tau _v}\end{array}$

with the following decomposition holding

${\widetilde {\rm H}_G}^\chi = { \oplus _{v \in \,T}}\widetilde {\not H}_{orb}^ * \left( {{\chi _v}} \right) \otimes \vartheta \left( {{\mathbb{C}^ * }} \right)$

Let me introduce the $\frac{\infty }{2}$ -VHS corresponding to quantum cohomology. Let $\pi :U \times \mathbb{C} \to U$ be the natural projection. The fiber ${\left( {{\pi _ * }\vartheta \left( F \right)} \right)_\tau }$ at $\tau \in U$ is then identified with the semi-infinite subspace ${\widetilde {\not F}_\tau }$ of ${\widetilde {\rm H}_G}$

Then the semi-infinite Hodge structure is ${\widetilde {\not F}_\tau }$ satisfying the following properties

$X{F_\tau } \subset {z^{ - 1}}{F_\tau }$

for a tangent vector $X \in {T_\tau }U$

${\widetilde F_\tau }$ isotropic with respect to $\Omega$

and

$\left( {2E + {\nabla _{z{{\not \partial }_z}}}} \right){\widetilde F_\tau } \subset {\widetilde F_\tau }$

Using the fact that $L{\left( {\tau ,z} \right)^{ - 1}}$ is the adjoint of $L\left( {\tau , - z} \right)$ with respect to the orbifold Poincaré pairing, one can calculate the embedding

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

explicitly as follows

$\begin{array}{c}{\widetilde {\not J}_\tau }\alpha = {e^{{\tau _0}2/z}}\left( {\alpha + \sum\limits_{d \in {\rm{Ef}}{{\rm{f}}_\chi }}^{\left( {d,l} \right) \ne \left( {0,0} \right)} {\sum\limits_{i = 1}^N {\frac{1}{{l!}}} } } \right. \cdot \\\left\langle {\alpha ,\tau ',...,\tau ',\frac{{{\phi _i}}}{{z - \psi }}} \right\rangle _{0,l + 2,d}^\chi \\{e^{\left\langle {{\tau _0},2,d} \right\rangle }}\left. {{\phi ^i}} \right)\end{array}$

Definition: The ${\widetilde {\not J}_\tau }$ -function is the image of the unit $\widehat 1$ section under the embedding

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

Because the unit section $\widehat 1$ is invariant under the Galois action, we have

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

Thus arriving at the result of this post: the quantum cohomology central charge integral can be derived as

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