52 problems

Type IIB supergravity has an $SL(2,\mathbb R)$ duality group, with the axio-dilaton transforming under this symmetry. Type IIB S-duality conjecture. The subgroup $SL(2,\mathbb Z)$…

The new type IIB vacua under discussion are described by elliptically fibered Calabi–Yau manifolds, with the dilaton varying over an elliptic curve. On the heterotic side, consider…

Let $B_H$ be the heterotic base, let ${\cal N}$ be the line bundle appearing in the spectral-cover description of the heterotic vector bundle, and let ${\cal M}$ be the line bundle…

Let $B'$ be a $\mathbf{P}^1$-bundle over $B$, with $B'=\mathbf{P}({\mathcal O}\oplus {\mathcal T})$, and let $X\to B'\to B$ be the corresponding elliptically fibered fourfold. Let…

Let $W$ be the fourfold appearing in the degeneration $X=W_1\cup_Z W_2$, and let $J_W$ denote its intermediate Jacobian. Consider $E_8$ bundles whose restriction to each fiber is f…

Loop-index conjecture. The effective action of F-theory compactified on $S^1$ could be computed by the same index-theoretical method as the effective action of M-theory, with loop-…

Orbifold conjecture. An ideal F-theory could be obtained by an orbifolding analogous to the construction in homotopy theory that produces $TMF$ or its connected form $tmf$.

F-theory conjecture. This theory should be related to F-theory compactified on the elliptic curve $E$.

Let $k$ be the degree of a multisection in a genus-one-fibered Calabi–Yau threefold. Toric hypersurfaces realize $k=2,3$, complete intersections realize a $4$-section, and more sop…

Let $k$ be the degree of a multisection in a genus-one-fibered Calabi–Yau threefold. Toric hypersurface constructions realize only $k=2,3$, complete intersections realize a $4$-sec…

Let $B'$ be the base obtained by blowing up $B$ at five points, so that $h^{1,1}(B')=10$, and let the nonabelian gauge group be $SU(15)\times SU(10)^3\times SU(5)^3$. The nonabelia…

Let a Weierstrass model be defined over $\mathbb{P}^2$ with an $SU(2)$ gauge symmetry tuned over a degree-nine curve. The associated threefold has a Mordell–Weil group whose additi…

Let $B$ be the base of a genus-one fibration, let $N$ be the degree of its multisection, and let $N_k$, for $k=1,\ldots,N-1$, denote the numbers of fibral curves intersecting the…

Let $Z$ be the compact surface serving as the base of an elliptic Calabi–Yau 3-fold, and let $\rho(Z)$ denote its Picard number. Morrison's maximal Picard rank conjecture. The maxi…

Consider a six-dimensional F-theory compactification with a frozen $7$-brane localized along a divisor $D$ and residual discriminant $\widetilde{\Delta}$. Neutral-hypermultiplet co…

Let $X$ be a smooth Calabi–Yau fourfold, and let the vertical subgroup be the subgroup of the middle cohomology generated by products of divisor classes. Its intersection pairing c…

Let $X$ be an elliptically fibered Calabi–Yau variety. F-theory/M-theory duality conjecture. F-theory compactified on $X\times S^1$ is dual to M-theory compactified on $X$. This ex…

Rank-$n$ $E_8$ compact duality conjecture. The following theories are equivalent:

Compact TCS $G_2$ duality conjecture. The following theories are equivalent:

M-theory–F-theory duality conjecture. The following physical theories are equivalent:

Harmonic intersection-cohomology conjecture. The vector space of $\mathbb{V}$-valued $i$-forms that are harmonic and $L^2$ with respect to the Kähler metric is isomorphic to

Intersection-cohomology conjecture. For the physical cases of interest,

Normalizability conjecture. Closed normalizable one-forms correspond to abelian gauge fields in the reduced 8D or 6D theory, whereas non-normalizable one-forms correspond to gauge…

Bulk–brane decomposition conjecture. This decomposition matches the separation between bulk and brane degrees of freedom expected from string theory.

Bulk–brane correspondence. Forms that are $L^2$ correspond to fields associated with bulk supergravity degrees of freedom, while forms that are not normalizable correspond to degre…