We give an expository account of a conjecture, developed by Coates–Iritani–Tseng and Ruan, which relates the quantum cohomology of a Gorenstein orbifold $𝒳$ to the quantum cohomology of a crepant resolution $Y$ of $𝒳$. We explore some consequences of this conjecture, showing that it implies versions of both the Cohomological Crepant Resolution Conjecture and of the Crepant Resolution Conjectures of Ruan and Bryan–Graber. We also give a ‘quantized’ version of the conjecture, which determines higher-genus...

In a previous paper, the author introduced an integral structure in quantum cohomology defined by the $K$-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle to the previous results, we find an explicit relationship between solutions to the quantum differential equation of toric complete intersections and the periods (or oscillatory integrals) of their mirrors. We describe in detail the mirror isomorphism of...

We construct pairs of compact Kähler-Einstein manifolds $(M_i,g_i,{\omega }_i)(i=1,2)$ of complex dimension $n$ with the following properties: The canonical line bundle $L_i={\bigwedge }^n{T}^{*}{M}_i$ has Chern class $[{\omega }_i/2\pi ]$, and for each positive integer $k$ the tensor powers $L_1^{\otimes k}$ and $L_2^{\otimes k}$ are isospectral for the bundle Laplacian associated with the canonical connection, while $M_1$ and $M_2$ – and hence $T^{*}{M}_1$ and $T^{*}{M}_2$ – are not homeomorphic. In the context of geometric quantization, we interpret these examples as magnetic fields which are quantum equivalent but not classically equivalent....

We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the $r$-spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity $W$ of type $A$ our construction of the stack of $W$-curves is canonically isomorphic to the stack of $r$-spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an $r$-spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the...

The conformal infinity of a quaternionic-Kähler metric on a $4n$-manifold with boundary is a codimension $3$ distribution on the boundary called quaternionic contact. In dimensions $4n-1$ greater than $7$, a quaternionic contact structure is always the conformal infinity of a quaternionic-Kähler metric. On the contrary, in dimension $7$, we prove a criterion for quaternionic contact structures to be the conformal infinity of a quaternionic-Kähler metric. This allows us to find the quaternionic-contact structures...

The first two authors have recently defined Rabinowitz Floer homology groups $RF{H}_{*}(M,W)$ associated to a separating exact embedding of a contact manifold $(M,\xi )$ into a symplectic manifold $(W,\omega )$. These depend only on the bounded component $V$ of $W\setminus M$. We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$, which in turn maps to Rabinowitz Floer homology $RF{H}_{*}(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RF{H}_{*}(S{T}^{*}L,T^{*}L)$, where $S{T}^{*}L$ is the unit cosphere bundle of a closed manifold...

We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the...

We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\mathbb{Z}$-graded chain complex of vector spaces over ${\mathbb{Z}}_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a...