Orbifold Quantum Cohomology, Gromov–Witten Theory and the Quantum-Product

By George Shiber Thursday, January 14, 2016Friday, October 26, 2018 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Physics, Super String Theory Calabi-Yau, Gromov–Witten

In physics, you don’t have to go around making trouble for yourself – nature does it for you ~ Frank Wilczek … Why I always say: ‘lucky we have mathematicians’

In my last four posts, I derived two propositions about quantum cohomology, with mirror-symmetry for Calabi-Yau manifolds being an isomorphism of variations of Hodge structures, with the A-model, defined by the genus zero Gromov–Witten theory of $X$ , isomorphic to the B-model of variations of Hodge structures associated to deformation of complex structures of the mirror $Y$ . With $\gamma$ the Euler constant and $\xi (s)$ the Riemann zeta function and $V \in K\left( X \right)$ , I defined the Witten $^W\nabla$ -flat connection $\widehat {\not {\rm Z}}\left( V \right)$

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

with $n = \dim X$ and hence, $\widehat {\not {\rm Z}}\left( V \right)$ defines the quantum cohomological $\widehat \Gamma$ -integral structure. The mirror-symmetric image of a compact toric orbifold is then given by a Landau–Ginzburg model, which is a pair of a torus ${Y_q} = {\left( {{\mathbb{C}^ * }} \right)^n}$ and a Laurent polynomial ${W_q}:{Y_q} \to \mathbb{C}$ . The Landau–Ginzburg model then defines a $B$ -model $D$ -module which is decidable by an integral local system generated by Lefschetz thimbles of ${W_q}$ . By mirror symmetry, it follows that the quantum $D$ -module of a toric orbifold is isomorphic to the $B$ -model $D$ -module and derived the following two propositions

– Proposition one: Let $\chi$ be a weak Picard-Fano projective toric orbifold defined by the initial data satisfying $\widehat \rho \in {\widetilde C_\chi }$ , then, in light of mirror-symmetry, we get the $\Gamma$ -integral structure on the quantum $D$ -module, and it corresponds to the natural integral local system of the $B$ -model $D$ -module under the mirror isomorphism

$\begin{array}{l}{\rm{Mirr}}:\left( {{{\widehat {R'}}^{\left( 0 \right)}}{,^W}\nabla ,{{\left( {.,.} \right)}_{{{\widehat {R'}}^{\left( 0 \right)}}}}} \right)\left| {_{{V_\varepsilon }}} \right. \cong \\{\left( {\tau \times {\rm{id}}} \right)^ * }\left( {\left( {F{,^W}\nabla ,{{\left( {.,.} \right)}_F}} \right)/{H^2}\left( {\chi ,\mathbb{Z}} \right)} \right)\end{array}$

with the quantum cohomology central charge of $V \in K\left( X \right)$ given by

$\widehat {\not Z}\left( V \right)\left( {\tau ,z} \right): = \left( {\frac{{{{\left( {2\pi z} \right)}^{n/2}}}}{{2\pi {{\widetilde i}^n}}}} \right)\int_X {\widehat {\not Z}} \left( {V\left( {\tau ,z} \right)} \right)$

– Proposition two: Under the same assumptions as proposition one, the quantum cohomology central charge of the structure sheaf ${\vartheta _\chi }$ is given by the Picard-integral over the real Lefschetz thimble ${\Gamma _\mathbb{R}}$

$\begin{array}{l}\widehat {\not Z}\left( {{\vartheta _\chi }} \right)\left( {\tau (q),z} \right) = \frac{1}{{{{\left( {2\pi \widehat i} \right)}^n}}} \cdot \\\int_{{\Gamma _\mathbb{R}}} {\exp \left( { - {W_q}(y)/z} \right)} \,{\omega _q}\end{array}$

In this post, let me study, in an algebraic setting, orbifold quantum cohomology, in the context of the integral structure associated to the $K$ -group and the $\Gamma$ -class and derive a third proposition about the quantum product and how it is related to a power series . Let $\chi$ be the smooth Deligne–Mumford meta-stack over $\mathbb{C}$ , with $I\chi$ the inertia stack of $\chi$ over the fiber product $\chi { \times _{\chi \times \chi }} \times \chi$ of the diagonal morphisms $\Delta :\chi \times \chi$ . Now, points on $I\chi$ are given by pairs $\left( {x,g} \right)$ for $x \in \chi$ and $g \in {\rm{Aut}}\left( x \right)$ with $g$ the pair-stabilizer. Let T be the Witten-index set of components of $I\chi$ and $o \in {\rm T}$ the distinguished Euclidean-element corresponding to the trivial stabilizer: setting

${\rm T}' = {\rm T}\backslash \left\{ 0 \right\}$

one gets

$I\chi = \coprod\limits_{\nu \in {\rm{T}}} {{\chi _\nu }} = {\chi _o} \cup \coprod\limits_{\nu \in {{\rm{T}}^\prime }} {{\chi _\nu }}$

with ${\chi _o} = \chi$ . Now pair a rational number ${\iota _\nu }$ to each connected component ${\chi _\nu }$ of $I\chi$ and we get the degree shifting pair number of $I\chi$ . Now define

${{\rm{T}}_x}\chi \equiv \underbrace \oplus _{0{\kern 1pt} \le f < 1}{\left( {{{\rm{T}}_x}\chi } \right)_f}$

as the eigenspace decomposition of ${{\rm{T}}_x}I\chi {\rm{ }}$ relative to the stabilizer action, with $g$ acting on ${\left( {{{\rm T}_x}\chi } \right)_f}$ via

${e^{\,2\,\pi \,\hat i\,f}}$

We hence can define:

${\iota _\nu }: = \sum\limits_{0\, \le f < 1} {f\,{\rm{Di}}{{\rm{m}}_\mathbb{C}}} {\left( {{{\rm T}_x}\chi } \right)_f}$

and is independent of the choice of a point $\left( {x,g} \right) \in {\chi _\nu }$ . I am now in a position to construct the orbifold cohomology group $H_{orb}^ * \left( \chi \right)$ as the sum of the even degree cohomology of ${\chi _\nu }$ , $\nu \in {\rm T}$ :

$H_{orb}^k\left( \chi \right) = \underbrace \oplus _{\nu \in {\rm T};k - 2{\iota _\nu } \equiv o\left( 2 \right)}{H^{k - {2_{{\iota _\nu }}}}}\left( {{\chi _\nu },\mathbb{C}} \right)$

with the degree $k$ of the orbifold cohomology a fractional number in general and factors ${H^ * }\left( {{\chi _\nu },\mathbb{C}} \right)$ in the right-hand side being the same as the cohomology group of ${\chi _\nu }$ qua topological space. Note now, there is an involution $inv:I\chi \to I\chi$ characterized by $inv\left( {x,g} \right) = \left( {x,{g^{ - 1}}} \right)$ inducing an involution $inv:{\rm T} \to {\rm T}$ . So, let’s then define the orbifold Poincaré pairing

$\begin{array}{c}{\left( {\alpha ,\beta } \right)_{orb}}: = \sum\limits_{\nu \in {\rm T}} {\int_{{\chi _\nu }} {{\alpha _\nu } \cup {\beta _{inv\left( \nu \right)}}} } = \\\int_{I\chi } {\alpha \cup in{v^ * }} \left( \beta \right)\end{array}$

with ${\alpha _\nu }$ , ${\beta _\nu }$ the $\nu$ -components of $\alpha$ , $\beta$ . The crucial properties of the orbifold Poincaré pairing is that it is symmetric, non-degenerate over $\mathbb{C}$ and of degree $- 2n$ , where $n = {\rm{Di}}{{\rm{m}}_\mathbb{C}}\chi$ . Since we can assume without loss of generality that the Poincaré-coarse moduli space $\chi$ of $\chi$ is projective, it follows that the genus zero Gromov– Witten invariants are integrals of the form

$\begin{array}{c}\left\langle {{\alpha _1}{\psi ^{{k_1}}},...,{\alpha _l}{\psi ^{{k_l}}}} \right\rangle _{0,l,d}^\chi = \\\int\limits_{{{\left[ {{\chi _{o,l,d}}} \right]}^{{\rm{vir}}}}} {\prod\limits_{i = 1}^l {{\rm{ev}}_i^ * } } \left( {{\alpha _i}} \right)\psi _i^{{k_i}}\end{array}$

$\left\langle {{\alpha _1}{\psi ^{{k_1}}},...,{\alpha _l}{\psi ^{{k_l}}}} \right\rangle _{0,l,d}^\chi = \int\limits_{{{\left[ {{\chi _{o,l,d}}} \right]}^{{\rm{vir}}}}} {\prod\limits_{i = 1}^l {{\rm{ev}}_i^ * } } \left( {{\alpha _i}} \right)\psi _i^{{k_i}}$

with

${\alpha _i} \in H_{orb}^ * \left( \chi \right)$

$d \in {H_2}\left( {X,\mathbb{Z}} \right)$ and ${k_i}$ a non-negative integer. Hence, ${\left[ {{\chi _{o,l,d}}} \right]^{{\rm{vir}}}}$ is the virtual Yau-fundamental class of the Poincaré moduli stack ${\chi _{o,l,d}}$ of genus zero, $l$ -pointed meta-stable maps to $\chi$ of super-degree $d$ and ${\rm{e}}{{\rm{v}}_i}:{\chi _{o,l,d}} \to I\chi$ is the symplectic evaluation map at the $i$ -th marked point, ${\psi _i}$ is the first Chern class of the line bundle over ${\chi _{o,l,d}}$ whose fiber at a meta-stable map is the cotangent space of the coarse curve at the $i$ -th marked point. Now Let $\left\{ {{\phi _k}} \right\}_{k = 1}^N$ and $\left\{ {{\phi ^k}} \right\}_{k = 1}^N$ be bases of $H_{orb}^ * \left( \chi \right)$ which are dual with respect to the orbifold Poincaré pairing, that is,

${\left( {{\phi _i},{\phi ^j}} \right)_{orb}} = \delta _i^j$

Then it follows that the orbifold quantum 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 desired result of this 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: