We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered four-folds defined by Weierstrass models admitting a split $\tilde{A}_4$ singularity over a divisor of the discriminant locus. Such varieties are used to geometrically engineer SU(5) grand unified theories in F-theory. The desingularization is given by a small resolution of singularities. The $\tilde{A}_4$ fiber naturally appears after resolving the singularities in codimension-one in the base. The remaining higher codimension singularities are then beautifully described by a four-dimensional affine binomial variety which leads to six different small resolutions of the elliptically fibered four-fold. These six small resolutions define distinct four-folds connected to each other by a network of flop transitions forming a dihedral group. The location of these exotic fibers in the base is mapped to conifold points of the three-folds that defines the type IIB orientifold limit of the F-theory. The full resolution has interesting properties, specially for fibers in codimension-three: the rank of the singular fiber does not necessary increase and the fibers are not necessary in the list of Kodaira and some are not even (extended) Dynkin diagrams.

In this paper, we will outline computations of quantum sheaf cohomology for deformations of tangent bundles of toric varieties, for those deformations describable as deformations of toric Euler sequences. Quantum sheaf cohomology is a heterotic analogue of quantum cohomology, a quantum deformation of the classical product on sheaf cohomology groups, that computes nonperturbative corrections to analogues of $\overline{27}^3$ couplings in heterotic string compactifications. Previous computations have relied on either physics-based gauged linear sigma model (GLSM) techniques or computation-intensive brute-force Cech cohomology techniques. This paper describes methods for greatly simplifying mathematical computations, and derives more general results than previously obtainable with GLSM techniques. We will outline recent results (rigorous proofs will appear elsewhere).

We consider the moduli problem of stable maps from a Riemann surface into a supermanifold; in twistor-string theory, this is the instanton moduli space. By developing the algebraic geometry of supermanifolds to include a treatment of super-stacks we prove that such moduli problems, under suitable conditions, give rise to Deligne-Mumford superstacks (where all of these objects have natural definitions in terms of super-geometry). We make some observations about the properties of these moduli super-stacks, as well as some remarks about their application in physics and their associated Gromov-Witten theory.

An algebraic geometry method is used to calculate the moments of the electron density of states as a function of the energy for lattices in the tight binding approximation. Interpreting the moments as the Mellin transform of the density allows writing down a formula for the density as an inverse Mellin transform. The method is illustrated by working out the density function for the two-dimensional square and honeycomb lattices.

We apply the Arnowitt-Deser-Misner approach to obtain a Hamiltonian description of the Einstein-Hilbert action. In doing so we add four new ingredients: (i) we eliminate the diffeomorphism constraints, (ii) we replace the densities $\sqrt{g}$ by a function $\varphi(x, g_{ij})$ with the help of a fixed metric $\chi$ such that the Lagrangian and hence the Hamiltonian are functions, (iii) we consider the Lagrangian to be defined in a fiber bundle with base space $\mathcal{S}_0$ and fibers $F(x)$ which can be treated as Lorentzian manifolds equipped with the Wheeler-DeWitt metric. It turns out that the fibers are globally hyperbolic, and (iv) the Hamiltonian operator H is a normally hyperbolic operator in the bundle acting only in the fibers and the Wheeler-DeWitt equation $Hu = 0$ is a hyperbolic equation in the bundle. Since the corresponding Cauchy problem can be solved for arbitrary smooth data with compact support, we then apply the standard techniques of Algebraic Quantum Field Theory (QFT) which can be naturally modified to work in the bundle.

We explicitly compute the automorphism group of the large $N = 4$ conformal superalgebra and classify the twisted loop conformal superalgebras based on the large $N = 4$ conformal superalgebra. By considering the corresponding superconformal Lie algebras, we validate the existence of only, two (up to isomorphism) such algebras as described in the physics literature. Our approach is based on viewing the objects to be classified as "étale twisted forms" of objects over the Laurent polynomial ring $\mathbb{C}[t \pm 1]$. This allows methods from non-abelian cohomology (torsors) to enter into the picture. It is worth pointing out that the group of automorphisms of the large $N = 4$ conformal superalgebra is larger than the one described in the physics literature. Remarkably enough, both groups have the same étale cohomology over $\mathbb{C}[t \pm 1]$ which explains the agreement on the classification of the corresponding superconformal Lie algebras).

We prove the following: let $\Gamma\, \subset\, \text{SL}(2,{\mathbb C})$ be a cocompact lattice and let $\rho\,:\, \Gamma\, \longrightarrow\, \text{GL}(r,{\mathbb C})$ be an irreducible representation. Then the holomorphic vector bundle $E_\rho\, \longrightarrow\, \text{SL}(2,{\mathbb C})/ \Gamma$ associated to $\rho$ is polystable. The compact complex manifold $\text{SL}(2,{\mathbb C})/ \Gamma$ has natural Hermitian structures; the polystability of $E_\rho$ is with respect to these natural Hermitian structures. We show that the polystable vector bundle $E_\rho$ is not stable in general.

We prove the birationality of multiple Berglund-Hübsch-Krawitz (BHK) mirrors by using Shioda maps. We do this by creating a birational picture of the BHK correspondence in general. We give an explicit quotient of a Fermat variety to which the mirrors are birational.