Gravitational greybody factors are analytically computed for static, spherically symmetric black holes in $d$-dimensions, including black holes with charge and in the presence of a cosmological constant (where a proper definition of greybody factors for both asymptotically de Sitter and anti-de Sitter (Ads) spacetimes is provided). This calculation includes both the low-energy case — where the frequency of the scattered wave is small and real — and the asymptotic case — where the frequency of the scattered wave is very large along the imaginary axis — addressing gravitational perturbations as described by the Ishibashi–Kodama master equations, and yielding full transmission and reflection scattering coefficients for all considered spacetime geometries. At low frequencies a general method is developed, which can be employed for all three types of spacetime asymptotics, and which is independent of the details of the black hole. For asymptotically de Sitter black holes the greybody factor is different for even or odd spacetime dimension, and proportional to the ratio of the areas of the event and cosmological horizons. For asymptotically Ads black holes the greybody factor has a rich structure in which there are several critical frequencies where it equals either one (pure transmission) or zero (pure reflection, with these frequencies corresponding to the normal modes of pure Ads spacetime). At asymptotic frequencies the computation of the greybody factor uses a technique inspired by monodromy matching, and some universality is hidden in the transmission and reflection coefficients. For either charged or asymptotically de Sitter black holes the greybody factors are given by non-trivial functions, while for asymptotically Ads black holes the greybody factor precisely equals one (corresponding to pure blackbody emission).

We study the $q$-deformed Knizhnik–Zamolodchikov ($qKZ$) equation in path representations of the Temperley–Lieb algebras. We consider two types of open boundary conditions, and in both cases we derive factorized expressions for the solutions of the $qKZ$ equation in terms of Baxterized Demazurre–Lusztig operators. These expressions are alternative to known integral solutions for tensor product representations. The factorized expressions reveal the algebraic structure within the $qKZ$ equation, and effectively reduce it to a set of truncation conditions on a single scalar function. The factorized expressions allow for an efficient computation of the full solution once this single scalar function is known. We further study particular polynomial solutions for which certain additional factorized expressions give weighted sums over components of the solution. In the homogeneous limit, we formulate positivity conjectures in the spirit of Di Francesco and Zinn-Justin. We further conjecture relations between weighted sums and individual components of the solutions for larger system sizes.

We construct a surprisingly large class of new Calabi–Yau 3-folds $X$ with small Picard numbers and propose a construction of their mirrors $X^∗$ using smoothings of toric hypersurfaces with conifold singularities. These new examples are related to the previously known ones via conifold transitions. Our results generalize the mirror construction for Calabi– Yau complete intersections in Grassmannians and flag manifolds via toric degenerations. There exist exactly 198849 reflexive four-polytopes whose two-faces are only triangles or parallelograms of minimal volume. Every such polytope gives rise to a family of Calabi–Yau hypersurfaces with at worst conifold singularities. Using a criterion of Namikawa we found 30241 reflexive four-polytopes such that the corresponding Calabi–Yau hypersurfaces are smoothable by a flat deformation. In particular, we found 210 reflexive four-polytopes defining 68 topologically different Calabi–Yau 3-folds with $h_11 = 1$. We explain the mirror construction and compute several new Picard–Fuchs operators for the respective oneparameter families of mirror Calabi–Yau 3-folds.

We calculate the ground state current densities for $(2 + 1)$-dimensional free fermion theories with local, translationally invariant boundary states. Deformations of the bulk wave functions close to the edge and boundary states both may cause edge current divergencies, which have to cancel in realistic systems. This yields restrictions on the parameters of quantum field theories which can arise as low-energy limits of solid-state systems. Some degree of Lorentz invariance for boosts parallel to the boundary can be recovered, when the cutoff is removed.

The moduli space of multiply connected Calabi–Yau threefolds is shown to contain codimension-one loci on which the corresponding variety develops a certain type of hyperquotient singularity. These have local descriptions as discrete quotients of the conifold, and are referred to here as hyperconifolds. In many (or possibly all) cases such a singularity can be resolved to yield a distinct compact Calabi–Yau manifold. These considerations therefore provide a method for embedding an interesting class of singularities in compact Calabi–Yau varieties, and for constructing new Calabi–Yau manifolds. It is unclear whether the transitions described can be realized in the string theory.

Simplicial homology manifolds are proposed as an interesting class of geometric objects, more general than topological manifolds but still quite tractable, in which questions about the microstructure of spacetime can be naturally formulated. Their string orientations are classified by $H^3$ with coefficients in an extension of the usual group of $D$-brane charges, by cobordism classes of homology three-spheres with trivial Rokhlin invariant.