37 problems

Teleman's conjecture. A Hamiltonian $G$-action on $X$ should be mirror to a holomorphic fibration

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

Hyperkähler Kirwan surjectivity conjecture. The hyperkähler Kirwan map is surjective.

Let $M$ be compact, and let $X$ be a Kähler manifold with a ${\mathbb C}^{\times}$-action, prequantum line bundle $L$, symplectic cuts $X_+$ and $X_-$, and reduced space $X_0$, wit…

Guillemin–Sternberg's conjecture. The equivariant holomorphic Euler characteristics should agree:

For each $t\in\mathbb{R}$, $X\in\mathfrak{g}$, let $\Theta(M,t)(X):=\int_M e^{-itd_X\lambda^{M_{\mathcal O}}}\alpha(X)$, and let $\Theta_0^{\mathcal O}$ denote the generalized-func…

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…

Let $(F,calpha,cmu)$ be a regular SFP, and suppose that its symplectic quotient $F_cmu$ is compact and that both the standard Gromov–Witten invariants of $F_cmu$ and the gauge theo…

Invariant quotient conjecture. For special choices of $\alpha$ and $B$, the invariant $\Phi^{C,k}_{X,B}(\alpha)$ should coincide with a Gromov–Witten invariant of $\hat X$.

Let the original Dirac theory be the constrained Hamiltonian formulation obtained using the canonical Hamiltonian together with the primary constraints and the consistency algorith…

Let $M$ be a Kähler manifold with a Hamiltonian $S^1$-action, and let $M_+$ and $M_-$ be the associated symplectic cut spaces, with reduced space $M_0$ and corresponding holomorphi…

Let $X$ be a Hamiltonian $T$-manifold with moment map $mu$, and let $X//_tT=mu^{-1}(t)/T$ be the smooth symplectic reduction. Let $Pi_X^{\rm eq}$ denote the equivariant quantum vol…

Let $Y$ carry a Hamiltonian action of a torus $T$ with moment map $\mu_Y$, let $\lambda$ be a value of the moment map, and write $Y\sslash_{\lambda}T=\mu_Y^{-1}(\lambda)/T$ for the…

Let $X$ be a monotone almost complex manifold with a Hamiltonian action of a group $G$, moment map $\mu$, and symplectic quotient $X/\!/G=\mu^{-1}(0)/G$. Let $\mathcal{F}(X)$ denot…

Let $G$ be a Lie group and $H$ a subgroup, and consider the quotient $G/H$. When $H$ is closed, cotangent bundle reduction identifies the reduced space $(T^*G)/\!/H$ with…

Let $G$ be a Lie group with closed subgroup $H$, and let $X$ and $Y$ be Hamiltonian $G$- and $H$-spaces, respectively. Write … for the Frobenius reciprocity bijection, where the tw…

Quantum Kirwan conjecture. The sum $I$ is supported on $C_Y^\vee\cap N_1^T(X)$ as a $\widehat{S}$-power series and gives a point on the Givental cone of $Y$, defined over the exten…

Let $V$ and $W$ be simple vertex superalgebras of CFT type containing rank-one Heisenberg vertex algebras of opposite levels, with decompositions…

Let $X$ be a Liouville manifold equipped with a Hamiltonian $S^1$ action such that the fixed locus $X^{S^1}$ has codimension four and there are no finite non-trivial stabiliser gro…

Big-cell integrability conjecture. Degenerate integrability also holds on the Poisson manifold ${\mathcal M}_{d*}^{\mathrm{red}}$.

Sufficiency conjecture. The so far exhibited ${\rm U}(n)$-invariant constants of motion are sufficient for degenerate integrability of the reduced system on the Poisson manifold…

Donnelly–Freidel gluing conjecture. The phase space of the glued surface should be the symplectic reduction

Non-commuting reduction by stages conjecture. The reduction by the product action should satisfy

Generic symmetry-reduced integrability conjecture. A generic $G$-invariant Hamiltonian system on $P^N_M$ is integrable only when these geometric conditions are such that symplectic…

Equivariant reduction conjecture. The theory computes the $A$-model correlator of the reduced symplectic manifold $\mu^{-1}(0)/G$.