A place for me to talk about some of my recent mathematical musings.

## [11-27-23] More general structure than expected?

As mentioned in my previous post, we can construct product structures via the use of Gromov-Witten invariants. However, note the following theorem, due to (at least in part, I believe) Gromov:

** Theorem. ** Symplectic manifolds always have almost-complex structures which are compatible with the symplectic structure. Further, there exists an infinite-dimensional space of compatible almost-complex structures. This space is contractible however, and leads to the fact that all of these almost-complex structures are in fact homotopy equivalent.

So, in the case of manifolds, up to homotopy, the crucial data which allows for the application of Gromov-Witten theory to deform cohomology products does not come from symplectic structure, rather the almost-complex structure on the manifold. This extends to Lagrangians: it is known that a Lagrangian submanifold will be totally-real with respect to the almost-complex structure on the manifold. This motivates the following:

** If we consider a symplectic orbifold ** \(\mathcal{X}\) ** and a Lagrangian suborbifold ** \(\mathcal{L}\subset\mathcal{X}\), ** then the data required to extract the exotic product described in the previous post on ** \(H^\bullet(I_{\mathcal{X}}\mathcal{L})\) ** is dependent on some almost-complex/totally-real structure instead of symplectic/Lagrangian structure. **

## [11-22-23] Chen-Ruan product analogue for “Lagrangian suborbifolds”

Consider a symplectic manifold \((X,\omega)\) whose cohomology \(H^\bullet(X)\) is equipped with the PoincarĂ© pairing \(\langle\cdot,\cdot\rangle\).
Gromov-Witten theory is a powerful tool in symplectic geometry which connects to mathematical physics, namely Type IIA string theory. The main point of study are the Gromov-Witten invariants, which "count" genus \(g\) pseudo-holomorphic curves with \(n\) marked points, mapping into \(X\). That is, we consider * stable maps * \(f: \mathbb{C} P^1\to X\). It turns out that the set of these functions forms a moduli space, which allows us to define the Gromov-Witten invariants:

** Definition. ** Consider the stable maps \(\mathbb{C} P^1\to X\) of genus \(g\), where \(\mathbb{C}P^1\) has \(n\) marked points. Then we can form a moduli space \(\mathcal{M}_{g,n}(X)\) of such curves. Note that for our use, we only consider genus 0 curves with 3 marked points; that is, \(\mathcal{M}_{0,3}(X)\). We define the * evaluation maps * on the moduli space, \(\text{ev}_i:\mathcal{M}_{0,3}(X)\to X\), by sending a stable map to the value of the stable map at a marked point, \([f:(\mathbb{C}P^1,x_1,x_2,x_3)\to X]\mapsto f(x_i)\).

This moduli space and its evaluation maps allow us to define the Gromov-Witten invariants. Consider three cohomology classes \(\alpha,\beta,\gamma\) on \(X\). Then the * Gromov-Witten invariants * are given as
$$
GW_{0,3}(X)=\int_{\mathcal{M}_{0,3}(X)} \text{ev}_1^*\alpha\wedge\text{ev}_2^*\beta\wedge\text{ev}_3^*\gamma.
$$

Interestingly, if we identify the PoincarĂ© pairing with the Gromov-Witten invariants of \(X\),
$$
\langle\alpha\star\beta,\gamma\rangle=GW_{0,3}(X),
$$
it turns out that the product \(\star\) which satisfies this equality is the * quantum product * on the quantum ring \(H^\bullet(X)[[q]]\). If we now restrict to just constant curves by setting \(q=0\), this quantum product reduces to the standard cup product on \(H^\bullet(X)\).
We can do the same construction for a Lagrangian submanifold \(L\subset X\), but we instead consider a "disk" with three marked points instead of a projective sphere \(\mathbb{C}P^1\) as the source for the pseudo-holomorphic curves.

I spent the majority of the Summer of 2023 working with orbifolds; an extension of the ideas mentioned above would be to a symplectic orbifold \(\mathcal{X}\). It turns out that when similar processes as above are performed on \(\mathcal{X}\), we require the use of the cohomology of the * inertia orbifold* \(I\mathcal{X}\). Amazingly, when we identify
$$
\langle\alpha\star\beta,\gamma\rangle=GW_{0,3}(\mathcal{X})
$$
and restrict to \(q=0\), we retrieve the * Chen-Ruan product * on \(H^\bullet(I\mathcal{X})[[q]]\)! This is a remarkable result, as this product structure is not immediately observable when one studies the cohomology of orbifolds.

We also would like to study a Lagrangian suborbifold \(\mathcal{L}\subset\mathcal{X}\) as modeled in the manifold case. However, the notion of a "Lagrangian suborbifold" is not well defined. However, in a 2023 preprint of Chen, Ono, and Wang, the idea of the * dihedral twisted sector * \(I_{\mathcal{X}}\mathcal{L}\) is introduced -- as the notation may suggest, this acts as a sort of Lagrangian "relative" to \(\mathcal{X}\).

** So, we conjecture that another exotic cohomology product could be constructed on ** \(I_{\mathcal{X}}\mathcal{L}\) ** by restricting ** \(H^\bullet(I_\mathcal{X}\mathcal{L})[[q]]\) ** to ** \(q=0\).

In an alternative approach towards such a result, techniques from this paper suggest the need to construct a * double * inertia orbifold \(II_{\mathcal{X}}\mathcal{L}\) and an obstruction class in its \(K\)-theory. This would provide an explicit description of this conjectured product structure on the cohomology of the dihedral twisted sector.