48 problems

Let $M$ be a compact pre-quantizable symplectic manifold admitting a Hamiltonian action of a compact connected Lie group $G$. Guillemin–Sternberg's conjecture. Quantization commute…

Let $\mathcal{P}$ be a polarization, let $\widetilde{U}$ be the relevant covering, and let $\xi$ be a Bohr–Sommerfeld point. Denote by…

Let $M$ be a $2n$-dimensional Poisson manifold with a real polarization $P$ and two integrable systems given by Lagrangian fibrations $\pi_i:M\to B_i$, with generically transverse…

GGT's normalizer conjecture. Every integrable irreducible representation of ${\cal B}$ can be extended to a quantization of $({\cal N}({\cal B}),{\cal B})$, and no nontrivial quant…

GGT's basic-set conjecture. For every basic set ${\cal B}\subset {\cal P}(M)$ whose generated Poisson algebra is finite-dimensional, the associated quantization should satisfy the…

Moment-mapping conjecture. If the geometric quantization is suitably normalized, then the restriction of the moment mapping to the intersection of the unit sphere with the space of…

Let $\pi:{\mathfrak X}\to S[[h]]$ and the quantization sheaf $\operatorname{\cal O}_h$ satisfy the assumptions of Theorem. Consider the global sections functor … from sheaves of fi…

Let $N$ be a possibly noncompact $G$-manifold, let ${\cal E}$ be a $G$-equivariant vector bundle on $N$ with $G$-invariant connection, and assume a $G$-invariant complex structure.…

Let $\Gamma$ be a quasi-symplectic groupoid, let $X$ be a Hamiltonian $\Gamma$-space with momentum map $J$, and let ${\cal O}\subset P$ be an orbit of the groupoid. The reduced spa…

Existence conjecture. There exist infinite-dimensional Kähler manifolds satisfying this condition and different from the projective spaces.

Let $M^{ss}_g$ denote the moduli space considered in the paper, and let its quantization be the corresponding quantized object. Thaddeus's conjecture. The quantization of…

Let $K$ be a compact Lie group and let $(M,omega,Phi)$ be a Hamiltonian $K$-manifold with proper moment map, prequantized by a line bundle $L$. Let…

Universality conjecture. As $\hbar\rightarrow 0$, the asymptotic behaviour of $\operatorname{dim}{\mathcal H}_{poly}$ is universal, as in the compact Kähler case.

Let $M$ be the symplectic manifold under consideration, let $U\subset TM$ be the domain of the map $F$, and fix $m\in M$. Set … where $\Psi(v,m_2)=(m_{12},m_{23})$, $dv$ is the fib…

Poisson-normalizer extension conjecture. Every integrable irreducible representation of $\wp({\cal B})$ can be extended to a quantization of

Dense basic-set existence conjecture. There exists a nontrivial quantization of

Groenewold–Van Hove conjecture. There is no nontrivial strong quantization of

Let $(N,\omega)$ be a compact symplectic manifold, and let $\hbar>0$. The Dirac quantization condition is … A contactification of $(N,\omega)$ is a contact-geometric lift as define…

Quantization conjecture. The Hilbert space $\mathcal H_B$, equipped with its $A$ and $A^{\mathrm{op}}$ actions, is a natural quantization of $\mathcal L_{\mathbb R}[B]$.

Let $\mathcal{H}_{\mathcal{P}}^\lambda$ and $\mathcal{H}_{\mathcal{P}_{\mu_{\mathrm{inv}}}}^\lambda$ be the $\lambda$-isotypical quantum spaces for the initial polarization…

Let $M$ carry the infinite-dimensional family of $G$-invariant Kähler polarizations, and let $\mathcal{P}_{\mu_{\mathrm{inv}}}$ be a polarization arising at infinite geodesic time…

Let ${\bf M}$ be a complete Kähler manifold satisfying the condition of Proposition as a Riemannian manifold, and let its Kähler form be the symplectic form. Assume that ${\bf M}$…

Let $M$ be a compact integrable system with singularities whose leaf-wise flat-section sheaf is used to define geometric quantization, and let $k_h$ denote the multiplicity of the…

The Poisson Sigma Model is a two-dimensional topological field theory whose AKSZ formulation provides a field-theoretic construction, and Hawkins' procedure associates to a symplec…

Let $(M,\omega)$ be a closed symplectic manifold, and let $\{T_k:C^\infty(M)\to\mathcal{L}(H_k)\}_{k\in\mathbb{N}}$ be a geometric quantization satisfying the trace axiom and star…