Lefschetz Categorical Monodromy Analysis of Galois Action and Gromov–Witten Theory

By George Shiber Wednesday, January 20, 2016Saturday, August 17, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory Calabi-Yau, Galois action

How wonderful that we have met with a paradox. Now we have some hope of making progress ~ Niels Bohr

Continuing my analysis of Gromov–Witten Theory, in this post, I will derive a fifth proposition and show that Galois actions are deeply connected to Lefschetz-monodromy transformations as well as proving the fourth proposition I derived in my last post, in the context of the Dubrovin connection. Recal that I showed such a 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. Identifying ${\phi _i}$ with the vector field $\not \partial /\not \partial {t^i}$ allows us to 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 ${c_1}\left( \chi \right) \equiv \sum\nolimits_{k = 1}^N {{r_k}{\phi _k}}$ . Furthermore, noting that the Euler-grading vector field satisfies the property

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

I then derived proposition four of last post thusly. 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. I showed that 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 ${\chi _\nu }$ .

– 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

– Proof of proposition four: for ${\alpha _1},...,{\alpha _l} \in H_{orb}^ * \left( \chi \right)$ , I can assume, without loss of generality, that

$\begin{array}{c}{\left\langle {{\alpha _1},...,{\alpha _l}} \right\rangle _{0,l,d}} = {e^{ - 2\pi \widehat i\left\langle {{\xi _0},d} \right\rangle }} \cdot \\{\left\langle {dG\left( \xi \right){\alpha _1},dG\left( \xi \right){\alpha _2},...,dG\left( \xi \right){\alpha _l}} \right\rangle _{0,l,d}}\end{array}$

and since there exists an orbifold stable map $f:\left( {C,{x_1},...,{x_l}} \right) \to \chi$ of degree $d$ , we have an orbifold line bundle ${f^ * }{L_\xi }$ on $C$ such that the monodromy at ${x_k}$ equals

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

where $e{v_k}\left( f \right) \in {\chi _{{v_k}}}$ . Then it follows that

$\deg {f^ * }{L_\xi } - \sum\limits_{k = 1}^l {{f_{{v_k}}}} \in \mathbb{Z}$

that is:

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

Given such an equality and

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

with

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

being the orbifold Poincaré ‘term’, we conclude the proof of proposition four. Let us proceed. Note that $U$ is invariant under the Galois action. Thus the quantum D-module Picard-descends to the quotient

$F/{H^2}\left( {\chi ,\mathbb{Z}} \right) \to \left( {U/{H^2}\left( {\chi ,\mathbb{Z}} \right)} \right) \times C$

This flat connection over $U/{H^2}\left( {\chi ,\mathbb{Z}} \right)$ is the quantum D-module. Now, the equation ${\nabla _s} = 0$ for a section $s$ of $F$ is the quantum differential equation. A fundamental solution $L\left( {\tau ,z} \right)$ to the quantum differential equation can be given by gravitational Picard-descendants. Let $pr:{I_\chi } \to \chi$ be the natural projection. We define the action of a class ${\tau _0} \in {H^ * }\left( \chi \right)$ on $H_{orb}^ * \left( \chi \right)$ by

$\left\{ {\begin{array}{*{20}{c}}{{\tau _0} \cdot \alpha = p{r^ * }\left( {{\tau _0}} \right) \cup \alpha }\\{\alpha \in H_{orb}^ * \left( \chi \right)}\end{array}} \right.$

with the right-hand side being the quantum-cup-product on ${I_\chi }$ . Now define

$\begin{array}{c}L\left( {\tau ,z} \right)\alpha : = \exp \left( { - {\tau _0},2/z} \right)\alpha - \\\sum\limits_{d \in {\rm{ef}}{{\rm{f}}_\chi },1 \le k \le N}^{\left( {d,l} \right) \ne \left( {0,0} \right)} {\frac{{{\phi ^k}}}{{l!}}} \cdot \\\left\langle {{\phi _k},\tau ',...,\tau ',\frac{{{e^{ - {\tau _0},2/{z_\alpha }}}}}{{z + \psi }}} \right\rangle _{0,l + 2,d}^\chi \cdot \\{e^{\left\langle {{\tau _0},2,d} \right\rangle }}\end{array}$

where $\tau = {\tau _{0,2}} + \tau '$ is the decomposition in

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

and

$1/\left( {z{\rm{ }} + {\rm{ }}\psi } \right)$

in the Galois-correlator expands in the series

$\infty {\rm{ }}k = 0\left( { - 1} \right)kz - k - 1\psi k$

Since the following holds for all manifolds, then

$L\left( {\tau ,z} \right)$

satisfies the following differential equations:

${\nabla _k}\;L\left( {\tau ,z} \right)\alpha = 0$

$\begin{array}{c}{\nabla _{z{{\not \partial }_z}}}L\left( {\tau ,z} \right)\alpha = L\left( {\tau ,z} \right) \cdot \\\left( {\mu \alpha - \frac{\rho }{z}\alpha } \right)\end{array}$

where $\rho : = {c_1}\left( {T{I_\chi }} \right) \in {H^2}\left( \chi \right)$ and $\mu$ is the grading operator. Now, the flat section $L\left( {\tau ,z} \right)\alpha$ in the $\tau$ -direction is characterized by the asymptotic initial condition: $L\left( {\tau ,z} \right)\alpha \sim {e^{ - {\tau _{0,2}}/z}}\alpha$ in the large radius limit, with $\tau ' = 0$ . Hence, setting

${z^{ - \mu }}{z^\rho }: = \exp \left( { - \mu \log z} \right)\exp \left( {\rho \log z} \right)$

we thus have

$\;{\nabla _k}\left( {L\left( {\tau ,z} \right){z^{ - \mu }}{z^\rho }\alpha } \right) = 0$

${\nabla _{z{{\not \partial }_z}}}\left( {L\left( {\tau ,z} \right){z^{ - \mu }}{z^\rho }\alpha } \right) = 0$

$\begin{array}{c}{\left( {L\left( {\tau , - z} \right)\alpha ,L\left( {\tau ,z} \right)\rho } \right)_{orb}} = \\\left( {\alpha ,\beta } \right)_{orb}^\dagger \end{array}$

$\begin{array}{c}dG\left( \xi \right)L\left( {G{{\left( \xi \right)}^{ - 1}}\tau ,z} \right)\alpha = L\left( {\tau ,z} \right)\\ \cdot \exp \left( { - 2\pi \widehat i{\xi _0}/z} \right)\exp {\left( {2\pi \widehat i{f_v}\left( \xi \right)} \right)_\alpha }\end{array}$

with

$\alpha \in {H^ * }{\left( {{\chi _v}} \right)_{{f_{{v_k}}}}}{\left( \xi \right)^\dagger }$

Now, the space $S\left( \chi \right)$ of multi-valued $\nabla$ -flat sections of the quantum D-module $\left( {F,\nabla ,{{\left( {.,.} \right)}_F}} \right)$ is defined to be

$\widehat S\left( \chi \right): = \left\{ {s \in \Gamma \left( {U \times \widetilde {{C^ * }},\vartheta \left( F \right)} \right)} \right\}$

for ${\nabla _s} = 0$ , where $\widetilde {{C^ * }}$ is the universal cover of ${C^ * }$ . This is a finite-dimensional $C$ -vector space with $\dim \widehat S\left( \chi \right) = \dim H_{orb}^ * \left( \chi \right)$ and the pairing ${\left( {.,.} \right)_{\widehat S}}$ on $\widehat S\left( \chi \right)$ is given by

${\left( {{s_1},{s_2}} \right)_{\widehat S}}: = {\left( {{s_1}\left( {\tau ,{e^{\pi \widehat i}}z} \right),{s_2}\left( {\tau ,z} \right)} \right)_{orb}} \in \mathbb{C}$

where ${s_1}\left( {\tau ,{e^{\pi \widehat i}}z} \right)$ is the parallel translate of ${s_1}\left( {\tau ,z} \right)$ along the counter-clockwise path

$\left[ {0,1} \right] \triangleleft \theta \to '{e^{\widehat i\pi \theta }}z$

and since the right-hand side is a complex number which does not depend on $\left( {\tau ,z} \right)$ , it follows that the Galois action defines an automorphism of $\widehat S\left( \chi \right)$ for $\xi \in {H^2}\left( {\chi ,\mathbb{Z}} \right)$

${G^{\widehat S}}\left( \xi \right):\widehat S\left( \chi \right)$

$s\left( {\tau ,z} \right) \to 'dG\left( \xi \right)s\left( {G{{\left( \xi \right)}^{ - 1}}\tau ,z} \right)$

Define now the cohomology framing ${\widehat Z_{COH}}:H_{orb}^ * \left( \chi \right) \to \widehat S\left( \chi \right)$ by

${\widehat Z_{COH}}\left( \alpha \right): = L\left( {\tau ,z} \right){z^{ - \mu }}{z^\rho }\alpha$

Hence arriving at the proposition of this post: the pairing and the Galois action on $\widehat S\left( \chi \right)$ are uniquely determined by the cohomology framing

$\begin{array}{c}\left( {{{\widehat Z}_{COH}}\left( \alpha \right),{{\widehat Z}_{COH}}\left( \beta \right)} \right) = \\{\left( {{e^{\pi \widehat i\rho }}\alpha ,{e^{\pi \widehat i\mu }}\beta } \right)_{orb}}\end{array}$

$\begin{array}{c}{G^{\widehat S}}\left( \xi \right)\left( {{{\widehat Z}_{COH}}\left( \alpha \right)} \right) = \\{\widehat Z_{COH}}\left( {\left( {\underbrace \oplus _{v \in T}{e^{ - 2\pi \widehat i{\xi _0}}}{e^{2\pi \widehat i{f_v}\left( \xi \right)}}} \right)\alpha } \right)\end{array}$

and hence the Galois actions on $\widehat S\left( \chi \right)$ can be viewed as the monodromy transformations of the flat bundle $F/{H^2}\left( {\chi ,\mathbb{Z}} \right) \to \left( {U/{H^2}\left( {\chi ,\mathbb{Z}} \right)} \right) \times {C^ * }$ in the $\tau$ -direction. The monodromy with respect to $z$ is hence given by

$\begin{array}{c}{\left[ {{{\widehat Z}_{COH}}\left( \alpha \right)} \right]_{{ \bullet _\iota }{ \to ^{\dagger {e^{2\pi \widehat i}}z}}}} = {\widehat Z_{COH}} \cdot \\\left( {{e^{ - 2\pi \left| \mu \right.}}{e^{2\pi \widehat i\rho }}\alpha } \right)\end{array}$

which coincides with the Galois action

${\left( { - 1} \right)^n}{G^{\widehat S}}\left( {\left[ {{K_\chi }} \right]} \right)$

as well as the Serre functor of the Lefschetz category $D\chi$ with $\left[ {{K_\chi }} \right]$ the class of the canonical line bundle and $\chi$ is Calabi–Yau.