9 problems

Geometric engineering conjecture. Partition functions in $4$-dimensional $\mathcal N=2$ supersymmetric gauge theories are equal to limits of generating functions of local Gromov–Wi…

In the geometric description, surgery circles and matter circles carry length parameters, while the associated physical parameters are the Fayet–Iliopoulos parameter $xi$ and the m…

Consider a matter node coupled to gauge nodes $ullet_{k_i}$, with charge $q_i$ under each gauge group $U(1)_{k_i}$. The charge $q_i$ is interpreted as the winding number between th…

The plumbing description uses three-manifolds with surgery circles and matter nodes; the charge of a chiral multiplet is interpreted as a winding number. Geometric loop conjecture.…

Shapere–Vafa conjecture. These are necessary and sufficient conditions for the isolated hypersurface singularity to define a four-dimensional $\mathcal{N}=2$ superconformal field t…

Let $X_N$ be the geometric-engineering Calabi–Yau threefold, let $N_j=\mathcal O_{D_j}(D_j)[1]$ be the noncompact D4-brane for a fixed $1\leq j\leq N-1$, and let $(P_i,Q_i)$ be the…

Let $X$ be a toric Calabi–Yau threefold with local-mirror family $\mathcal Z$, let $\mathcal Z_0\to\mathcal B$ be its field-theory-limit family, and let $\mathcal Y\to\mathcal B$ b…

Let $X$ be the toric Calabi–Yau threefold associated with the geometric-engineering construction, let $D^b(X)$ be its derived category, and let $\\{\mathcal L_\alpha\\}$ be an exce…

Let $X_N$ be the geometric-engineering Calabi–Yau threefold, and let $\rho_{\rm LCS}$ and $\rho_{\rm FT}$ denote the large-complex-structure and field-theory scaling regions in its…