Crepant resolutions and open strings. (English) Zbl 1428.14087
Consider a toric Calabi-Yau orbifold \(\mathfrak X\) with semi-projective moduli space, and let \(Y\to X\) be a crepant resolution of its coarse moduli space. Choose a Lagrangian boundary (Aganagic-Vafa brane)
condition \(L\) on \(\mathfrak X\) and denote by \(L'\)
its transform in \(Y\). Fix a \(\mathbb C^*\)-action with zero-dimensional
fixed loci such that the resolution morphism is equivariant. Denote the equivariant parameter by \(\nu\). For \(\mathcal Z= \mathfrak X\) or \(Y\), let \(H(\mathcal Z)\) be the equivariant Chen-Ruan cohomology ring of \(\mathcal Z\), \(\mathcal H_\mathcal{Z}=H(\mathcal Z)((z^{-1}))\) be its Givental’s symplectic vector space of \(\mathcal Z\), and \(\Delta_\mathcal{Z}\) be the free module over \(\mathbb C[\nu]\) spanned by equivariant lifts of
orbifold cohomology classes with degree less than or equal to 2.
The paper under review defines a family of elements of Givental space \(\mathbb F^{\text{disk}}_{L,\mathfrak X}\colon H(\mathfrak X)\to \mathcal H_\mathfrak{X}\) called the winding neutral disk potential. It encodes disk invariants of \((\mathfrak X, L)\) at any winding \(d\). \(\mathbb F^{\text{disk}}_{L',Y}\) is defined similarly. The paper then proposes the following version of open crepant resolution conjecture (OCRC):
There exists a \(\mathbb C((z^{-1}))\)-linear map of Givental spaces \(\mathbb O \colon \mathcal H_{\mathfrak X}\to \mathcal H_Y\) and analytic functions \(\mathfrak H_\mathfrak{X}\colon \Delta_\mathfrak{X} \to\mathbb C\), \(\mathfrak H_Y\colon \Delta_Y \to\mathbb C\) such that \(\mathfrak H^{1/z}_Y\; \mathbb F^{\text{disk}}_{L',Y}|_{\Delta_Y}=\mathfrak H^{1/z}_\mathfrak{X}\; \mathbb O \; \mathbb F^{\text{disk}}_{L,\mathfrak X}|_{\Delta_\mathfrak{X}}\) after analytic continuation of quantum cohomology parameters. Moreover, both \(\mathbb O\) and \(\mathfrak H\) are completely determined by the classical toric geometry of \(\mathfrak X\) and \(Y\). Furthermore, when \(\mathfrak X\) is a Hard Lefschetz Calabi-Yau orbifold, the OCRC comparison extends to all of \(H(\mathcal Z)\) and hence also allows proposing a comparison for potentials encoding higher genus invariants with arbitrary boundary conditions.
The main result of the paper under review proves OCRC conjecture above for the \(A_n\) orbifold \(\mathfrak X=[\mathbb C^2/\mathbb Z_{n+1}]\times \mathbb C\) any choice of Aganagic-Vafa brane.
The paper under review defines a family of elements of Givental space \(\mathbb F^{\text{disk}}_{L,\mathfrak X}\colon H(\mathfrak X)\to \mathcal H_\mathfrak{X}\) called the winding neutral disk potential. It encodes disk invariants of \((\mathfrak X, L)\) at any winding \(d\). \(\mathbb F^{\text{disk}}_{L',Y}\) is defined similarly. The paper then proposes the following version of open crepant resolution conjecture (OCRC):
There exists a \(\mathbb C((z^{-1}))\)-linear map of Givental spaces \(\mathbb O \colon \mathcal H_{\mathfrak X}\to \mathcal H_Y\) and analytic functions \(\mathfrak H_\mathfrak{X}\colon \Delta_\mathfrak{X} \to\mathbb C\), \(\mathfrak H_Y\colon \Delta_Y \to\mathbb C\) such that \(\mathfrak H^{1/z}_Y\; \mathbb F^{\text{disk}}_{L',Y}|_{\Delta_Y}=\mathfrak H^{1/z}_\mathfrak{X}\; \mathbb O \; \mathbb F^{\text{disk}}_{L,\mathfrak X}|_{\Delta_\mathfrak{X}}\) after analytic continuation of quantum cohomology parameters. Moreover, both \(\mathbb O\) and \(\mathfrak H\) are completely determined by the classical toric geometry of \(\mathfrak X\) and \(Y\). Furthermore, when \(\mathfrak X\) is a Hard Lefschetz Calabi-Yau orbifold, the OCRC comparison extends to all of \(H(\mathcal Z)\) and hence also allows proposing a comparison for potentials encoding higher genus invariants with arbitrary boundary conditions.
The main result of the paper under review proves OCRC conjecture above for the \(A_n\) orbifold \(\mathfrak X=[\mathbb C^2/\mathbb Z_{n+1}]\times \mathbb C\) any choice of Aganagic-Vafa brane.
Reviewer: Amin Gholampour (College Park)
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
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