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

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)

\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:

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

We shall get deeper next post.

Science is the belief in the ignorance of experts ~ Richard Feynman, again ... as I say, 'lucky we have mathematicians'

No Comments Yet.

Leave a Reply

Your email address will not be published. Required fields are marked *