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 , isomorphic to the B-model of variations of Hodge structures associated to deformation of complex structures of the mirror . With the Euler constant and the Riemann zeta function and , I defined the Witten -flat connection

with and hence, defines the quantum cohomological -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 and a Laurent polynomial . The Landau–Ginzburg model then defines a -model -module which is decidable by an integral local system generated by Lefschetz thimbles of . By mirror symmetry, it follows that the quantum -module of a toric orbifold is isomorphic to the -model -module and derived the following two propositions

- Proposition one: Let be a weak Picard-Fano projective toric orbifold defined by the initial data satisfying , then, in light of mirror-symmetry, we get the -integral structure on the quantum -module, and it corresponds to the natural integral local system of the -model -module under the mirror isomorphism

with the quantum cohomology central charge of given by

- Proposition two: Under the same assumptions as proposition one, the quantum cohomology central charge of the structure sheaf is given by the Picard-integral over the real Lefschetz thimble

In this post, let me study, in an algebraic setting, orbifold quantum cohomology, in the context of the integral structure associated to the -group and the -class and derive a third proposition about the quantum product and how it is related to a power series . Let be the smooth Deligne–Mumford meta-stack over , with the inertia stack of over the fiber product of the diagonal morphisms . Now, points on are given by pairs for and with the pair-stabilizer. Let T be the Witten-index set of components of and the distinguished Euclidean-element corresponding to the trivial stabilizer: setting

one gets

with . Now pair a rational number to each connected component of and we get the degree shifting pair number of . Now define

as the eigenspace decomposition of relative to the stabilizer action, with acting on via

We hence can define:

and is independent of the choice of a point . I am now in a position to construct the orbifold cohomology group as the sum of the even degree cohomology of , :

with the degree of the orbifold cohomology a fractional number in general and factors in the right-hand side being the same as the cohomology group of qua topological space. Note now, there is an involution characterized by inducing an involution . So, let's then define the orbifold Poincaré pairing

with , the -components of , . The crucial properties of the orbifold Poincaré pairing is that it is symmetric, non-degenerate over and of degree , where . Since we can assume without loss of generality that the Poincaré-coarse moduli space of is projective, it follows that the genus zero Gromov– Witten invariants are integrals of the form

with

and a non-negative integer. Hence, is the virtual Yau-fundamental class of the Poincaré moduli stack of genus zero, -pointed meta-stable maps to of super-degree and is the symplectic evaluation map at the -th marked point, is the first Chern class of the line bundle over whose fiber at a meta-stable map is the cotangent space of the coarse curve at the -th marked point. Now Let and be bases of which are dual with respect to the orbifold Poincaré pairing, that is,

Then it follows that the orbifold quantum product is a family of commutative, associative products on parametrized by , which is defined by the formula

long-form,

with the element of the group ring corresponding to . Now, decomposing 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 and

vertically,

with

being the orbifold Poincaré 'term' and implies that the product defines an analytic family of commutative rings over , hence yielding the following deep (as we shall see) relation:

We shall get deeper next post.

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