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This paper studies the action of the Fourier-Mukai transform on moduli spaces of vertical torsion sheaves on elliptic Calabi-Yau threefolds in Weierstrass form. Moduli stacks of semistable one dimensional sheaves on such threefolds are…
We review a proof of the well know result stating that moduli spaces of stable sheaves with fixed Chern character on a polarized $K3$ surface are deformations of a hyperk\"ahler variety of Type $K3^{[n]}$ (if a suitable numerical hypothesis…
We revisit type II$_0$ limits in the vector multiplet moduli space of compactifications of Type IIB string theory on Calabi-Yau threefolds. These limits are special because they cannot be described using mirror symmetry. While we showed in…
This paper contains some applications of Bridgeland-Douglas stability conditions on triangulated categories, and Joyce's work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of…
We investigate certain fixed points in the boundary conformal field theory representation of type IIA D-branes on Gepner points of K3. They correspond geometrically to degenerate brane configurations, and physically lead to enhanced gauge…
The disk partition function of the open topological string computes the spacetime superpotential for D-branes wrapping cycles of a compact Calabi-Yau threefold. We use string duality to show that when appropriately formulated, the problem…
We study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and the subgroup of the N\'eron-Severi group generated…
We study the duality between four-dimensional N=2 compactifications of heterotic and type IIA string theories. Via adiabatic fibration of the duality in six dimensions, type IIA string theory compactified on a K3-fibred Calabi-Yau threefold…
For a K3 surface S, we study motivic invariants of stable pairs moduli spaces associated to 3-fold thickenings of S. We conjecture suitable deformation and divisibility invariances for the Betti realization. Our conjectures, together with…
In the framework of type IIB string compactifications on Calabi-Yau orientifolds we describe how to construct consistent global embeddings of models with fractional D3-branes and connected `flavour' D7-branes at del Pezzo singularities with…
We discuss gauge-fixing, propagators and effective potentials for topological A-brane composites in Calabi-Yau compactifications. This allows for the construction of a holomorphic potential describing the low-energy dynamics of such…
We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich-Polishchuk, Kuznetsov, Lieblich, and…
We prove the modularity of Maschke's octic and two Calabi-Yau threefolds derived from it as double octic and quotient thereof by the Heisenberg group, as conjectured by Bini and van Geemen. The proofs rely on automorphisms of the varieties…
This article gives a complete account of the modular properties and Verlinde formula for conformal field theories based on the affine Kac-Moody algebra sl(2) at an arbitrary admissible level k. Starting from spectral flow and the structure…
In this paper, we show the moduli spaces of stable sheaves on K3 surfaces are irreducible symplectic manifolds, if the associated Mukai vectors are primitive. More precisely, we show that they are related to the Hilbert scheme of points. We…
We present the most complete list of mirror pairs of Calabi-Yau complete intersections in toric ambient varieties and develop the methods to solve the topological string and to calculate higher genus amplitudes on these compact Calabi-Yau…
We systematically analyze a broad class of dual heterotic and F-theory models that give four-dimensional supergravity theories, and compare the geometric constraints on the two sides of the duality. Specifically, we give a complete…
In this Part II, D(10.2), of D(10), we take D(10.1) (arXiv:1302.2054 [math.AG]) as the foundation to define the notion of $Z$-semistable morphisms from general Azumaya nodal curves, of genus $\ge 2$, with a fundamental module to a…
We prove that the weight-two Hodge structure of moduli spaces of torsion-free sheaves on a K3 surface is as described by Mukai (the rank is arbitrary but we assume the first Chern class is primitive). We prove the moduli space is an…
We study the duality between M-theory on compact holonomy G2-manifolds and the heterotic string on Calabi-Yau three-folds. The duality is studied for K3-fibered G2-manifolds, called twisted connected sums, which lend themselves to an…