## 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-modeldefined 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 $\widehat {\not {\rm Z}}\left( V \right)$

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

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

- 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 ${\Gamma _\mathbb{R}}$

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

one gets

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

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

We hence can define:

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}$:

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

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

with

$d \in {H_2}\left( {X,\mathbb{Z}} \right)$ 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,

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

long-form,

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

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 $\tau '$

vertically,

with

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:

We shall get deeper next post.