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...