Galois Action, Quantum D-Modules and Orbifold Gromov–Witten Theory

By George Shiber Sunday, January 17, 2016Monday, November 11, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory Calabi-Yau, Galois action

We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry ~ Ludwig Wittgenstein

In my last six posts, I finally showed that the quantum cohomological product ${ \bullet _\iota }$ is a family of commutative, associative products on $H_{orb}^ * \left( \chi \right) \otimes \mathbb{C}{\left[ {{\rm{Ef}}{{\rm{f}}_\chi }} \right]_f}$ parametrized by $\tau \in H_{orb}^ * \left( \chi \right)$ , which is defined by the formula

$\begin{array}{*{20}{c}}{\alpha { \bullet _\iota }\beta = \sum\limits_{d \in {\rm{Ef}}{{\rm{f}}_\chi }} {\sum\limits_{l{\kern 1pt} \ge 0} {\sum\limits_{k = 1}^N {\frac{1}{{l!}}} } } \cdot }\\{\left\langle {\alpha ,\beta ,\tau ,...,\tau ,{\phi _k}} \right\rangle _{o,l,d}^\chi {Q^d}{\phi ^k}}\end{array}$

long-form,

$\alpha { \bullet _\iota }\beta = \sum\limits_{d \in {\rm{Ef}}{{\rm{f}}_\chi }} {\sum\limits_{l{\kern 1pt} \ge 0} {\sum\limits_{k = 1}^N {\frac{1}{{l!}}} } } \left\langle {\alpha ,\beta ,\tau ,...,\tau ,{\phi _k}} \right\rangle _{o,l,d}^\chi {Q^d}{\phi ^k}$

with ${Q^d}$ the element of the group ring $H_{orb}^ * \left( \chi \right) \otimes \mathbb{C}{\left[ {{\rm{Ef}}{{\rm{f}}_\chi }} \right]_f}$ corresponding to $d \in {\rm{Ef}}{{\rm{f}}_\chi }$ . Now, decomposing $\tau \in H_{orb}^ * \left( \chi \right)$ as

$\left\{ {\begin{array}{*{20}{c}}{\tau = {\tau _{o,2}} + \tau '}\\{{\tau _{o,2}} \in {H^2}\left( \chi \right)}\\{\tau ' \in \underbrace {\widehat \oplus }_{k \ne 1}{H^{2k}}\left( \chi \right) \oplus \widehat \oplus {H^ * }\left( {{\chi _\nu }} \right)}\end{array}} \right.$

we finally get, by the Picard-divisor rank formula, the third proposition of the last post:

– proposition three: the quantum product can be viewed as a formal power series in ${e^{{\tau _{o,2}}}}Q$ and $\tau '$

$\alpha { \bullet _\iota }\beta = \sum\limits_{d \in {\rm{Ef}}{{\rm{f}}_\chi }} {\sum\limits_{l\, \ge 1pt \ge 0} {\sum\limits_{k = 1}^N {\frac{1}{{l!}}} } } \left\langle {\alpha ,\beta ,\tau ',...,\tau ',{\phi _k}} \right\rangle _{o,l + 3,d}^\chi {e^{\left\langle {{\tau _{o,2,d}}} \right\rangle }}{Q^d}{\phi ^k}$

vertically,

$\begin{array}{l}\alpha { \bullet _\iota }\beta = \sum\limits_{d \in {\rm{Ef}}{{\rm{f}}_\chi }} {\sum\limits_{l\, \ge 1pt \ge 0} {\sum\limits_{k = 1}^N {\frac{1}{{l!}}} } } \cdot \\\left\langle {\alpha ,\beta ,\tau ',...,\tau ',{\phi _k}} \right\rangle _{o,l + 3,d}^\chi {e^{\left\langle {{\tau _{o,2,d}}} \right\rangle }}{Q^d}{\phi ^k}\end{array}$

with

${e^{\left\langle {{\tau _{o,2,d}}} \right\rangle }}{Q^d}{\phi ^k}$

being the orbifold Poincaré ‘term’ and implies that the product ${o_\tau }$ defines an analytic family of commutative rings $\left( {{H_{orb}}\left( \chi \right),{o_\tau }} \right)$ over $I\chi$ , hence yielding the following deep (as we shall see) relation:

${o_\tau } \equiv { \bullet _\iota }\left| {_{Q = 1}} \right.$

with ${o_\tau }$ the orbifold quantum product.

Keep this identity in mind, it will be key to this post’s proposition

${e^{ - 2\pi \widehat i\left\langle {{\xi _o},d} \right\rangle }}\prod\limits_{i = 1}^l {{e^{2\pi \widehat i{f_{{\nu _1}}}\left( \xi \right)}}} = 1$

Now let me study ${o_\tau } \equiv { \bullet _\iota }\left| {_{Q = 1}} \right.$ further in this post and derive one more (of many) propositions. Now, given that the orbifold quantum product ${o_\tau }$ is convergent over an open set ${I_\chi } \subset H_{orb}^ * \left( \chi \right)$ of the form

${I_\chi } = \left\{ {\begin{array}{*{20}{c}}{\tau \in H_{orb}^ * \left( \chi \right)}\\{\Re \left\langle {{\tau _{0,2,d}}} \right\rangle \le - M}\\{\forall d \in {\rm{Ef}}{{\rm{f}}_\chi }\backslash \left\{ 0 \right\}}\\{\left\| {\tau '} \right\| \le {e^{ - M}}}\end{array}} \right.$

for a large $M > 0$ , where $\tau = {\tau _{0,2}} + \tau '$ is the decomposition in

and $\left\| \cdot \right\|$ is the metaplectic norm on $H_{orb}^ * \left( \chi \right)$ , the domain ${I_\chi }$ then has the following large radius limit direction conditions

$\left\{ {\begin{array}{*{20}{c}}{\Re \left\langle {{\tau _{0,2,d}}} \right\rangle }\\{\forall d \in {\rm{Ef}}{{\rm{f}}_\chi }\backslash \left\{ 0 \right\}}\\{\tau ' \to 0}\end{array}} \right.$

Under such large radius limit, ${o_\tau }$ goes to the orbifold cup product ${ \cup _{ORB}}$ . Now associate a meromorphic quantum D-module to the orbifold quantum cohomology and introduce quantum D-module-automorphic Galois action and take a homogeneous basis $\left\{ {{\phi _k}} \right\}_{k = 1}^N$ of $H_{orb}^ * \left( \chi \right)$ and let $\left\{ {t'} \right\}_{k = 1}^N$ be the linear co-ordinate system on $H_{orb}^ * \left( \chi \right)$ dual to $\left\{ {{\phi _k}} \right\}_{k = 1}^N$ and let

$\tau = \sum\nolimits_{k = 1}^N {t'} {\phi _k}$

be a general point on ${I_\chi } \subset H_{orb}^ * \left( \chi \right)$ with $\left( {\tau ,z} \right)$ be a general point on ${I_\chi } \times \mathbb{C}$ and $\left( - \right):{I_\chi } \times \mathbb{C} \to {I_\chi } \times \mathbb{C}$ be the map sending $\left( {\tau ,z} \right)$ to $\left( {\tau , - z} \right)$ .

So, let the quantum D-module be the tuple $\left( {F,\nabla ,{{\left( {.,.} \right)}_F}} \right)$ consisting of the holomorphic vector bundle $F: = {H^ * }\left( \chi \right) \times {I_\chi } \times \mathbb{C} \to {I_\chi } \times \mathbb{C}$ , the meromorphic flat connection $\nabla$

$\left\{ {\begin{array}{*{20}{c}}{{\nabla _k} = {\nabla _{\frac{{\not \partial }}{{\not \partial {t^k}}}}} = \frac{{\not \partial }}{{\not \partial {t^k}}} + \frac{1}{z}{\phi _k}{o_\tau }}\\{{\nabla _{z{{\not \partial }_z}}} = z\frac{{\not \partial }}{{\not \partial z}} - \frac{1}{z}E{o_\tau } + \mu }\end{array}} \right.$

with the $\nabla$ -flat pairing ${\left( {.,.} \right)_F}$ being

${\left( {.,.} \right)_F}:{\left( - \right)^*}\vartheta \left( F \right) \times \vartheta \left( F \right) \to {\vartheta _{{I_\chi } \times \mathbb{C}}}$

which is induced from the orbifold Poincaré pairing

${F_{\left( {\tau , - z} \right)}} \times {F_{\left( {\tau ,z} \right)}} = H_{orb}^ * \left( \chi \right) \times H_{orb}^ * \left( \chi \right) \to \mathbb{C}$

with $E \in \vartheta \left( F \right)$ being the Euler vector field

$\begin{array}{c}E: = {c_1}\left( {T\chi } \right) + \sum\limits_{k = 1}^N {\left( {1 - \frac{1}{2}{\rm{deg}}{\varphi _k}} \right)} \\ \cdot t'{\phi _k}\end{array}$

and $\mu \in {\rm{End}}\left( {H_{orb}^ * \left( \chi \right)} \right)$ being the Hodge grading operator

$\mu \left( {{\phi _k}} \right): = \left( {\frac{1}{2}{\rm{deg}}{\phi _k} - \frac{n}{2}} \right){\phi _k}$

Note that the flat connection $\nabla$ is the Dubrovin connection. Hence, the connection $\nabla$ defines a map

$\begin{array}{c}\nabla :\vartheta \left( F \right) \to \vartheta \left( F \right)\left( {{I_\chi } \times \mathbb{C}} \right)\\{ \otimes _{U \times \mathbb{C}}}\left( {{{\widehat \pi }^ * }\Omega _U^1 \oplus {\vartheta _{U \times \mathbb{C}}}\frac{{dz}}{z}} \right)\end{array}$

with $U \equiv {T^\dagger }_{{I_\chi }}$ and $\widehat \pi :U \times \mathbb{C} \to U$ the Picard-projection. Identify ${\phi _i}$ with the vector field $\not \partial /\not \partial {t^i}$ : thus, one can view $E$ as the vector field over $U$

$\begin{array}{c}E = \sum\limits_{k = 1}^N {{r_k}} \frac{{\not \partial }}{{\not \partial {t^k}}} + \sum\limits_{k = 1}^N {\left( {1 - \frac{1}{2}{\rm{deg}}{\phi _k}} \right)} \\ \cdot \,t'\frac{{\not \partial }}{{\not \partial {t^k}}}\end{array}$

where ${c_1}\left( \chi \right) \equiv \sum\nolimits_{k = 1}^N {{r_k}{\phi _k}}$ . Realizing the crucial point: that the Euler-grading vector field satisfies the property

$Gr: = 2\left( {{{\widehat \nabla }_{z{{\not \partial }_z}}} + {{\widehat \nabla }_E} + \frac{n}{2}} \right)$

we are ready to state the proposition of this post. Let ${H^2}\left( {\chi ,\mathbb{Z}} \right)$ refer to the cohomology of the constant sheaf $\mathbb{Z}$ on the topological stack $X$ but not on the corresponding topological space. This group defines the set of isomorphism classes of topological orbifold line bundles on $X$ . Letting ${L_\xi } \to \chi$ be the orbifold line bundle corresponding to $\xi \in {H^2}\left( {\chi ,\mathbb{Z}} \right)$ and $0 \le {f_\nu }\left( \xi \right) < 1$ be the rational number such that the stabilizer of ${\chi _\nu }\quad {\rm{,}}\quad \nu \in {\rm{T}}$ acts on

${L_\xi }\left| {_{{\chi _\nu }}} \right.$

via a complex number

$\exp \left( {2\pi \widehat i{f_\nu }\left( \xi \right)} \right)$

with ${f_\nu }\left( \xi \right)$ the symplectic-age of ${L_\xi }$ along ${\chi _\nu }$ .

– So now, we are in a position to state proposition four:

For $\xi \in {H^2}\left( {\chi ,\mathbb{Z}} \right)$ , the bundle isomorphism of $F$ defined by

$H_{orb}^ * \left( \chi \right) \times \left( {U \times \mathbb{C}} \right) \to H_{orb}^ * \left( \chi \right) \times \left( {U \times \mathbb{C}} \right)$

$\left( {\alpha ,\tau ,z} \right) \to '\left( {dG\left( \xi \right)\alpha ,G\left( \xi \right)\left( {\tau ,z} \right)} \right)$

gives an automorphism of the quantum D-module that preserves the flat connection $\nabla$ and the pairing ${\left( {.,.} \right)_F}$ , with $G\left( \xi \right)$ ,

$dG\left( \xi \right):H_{orb}^ * \left( \chi \right) \to H_{orb}^ * \left( \chi \right)$

are defined by

$\begin{array}{c}G\left( \xi \right)\left( {{\tau _0} \oplus \underbrace {\widehat \otimes }_{\nu \in T'}{\tau _0}} \right) = \\\left( {{\tau _0} - 2\pi \widehat i{\xi _0}} \right) \oplus \underbrace {\widehat \otimes }_{\nu \in T'}{e^{2\pi \widehat i{f_\nu }\left( \xi \right)}}{\tau _0}\end{array}$

$\begin{array}{c}dG\left( \xi \right)\left( {{\tau _0} \oplus \underbrace {\widehat \otimes }_{\nu \in T'}{\tau _0}} \right) = \\{\tau _0} \oplus \underbrace {\widehat \otimes }_{\nu \in T'}{e^{2\pi \widehat i{f_\nu }\left( \xi \right)}}{\tau _0}\end{array}$

where ${\tau _\nu } \in {H^ * }\left( {{\chi _\nu }} \right)$ and ${\xi _0}$ is the image of $\xi$ in the $\chi$ –quantum D-module: and this is the Galois action of ${H^2}\left( {\chi ,\mathbb{Z}} \right)$ on $\chi$ –quantum D-module.