We establish a relation, conjectured recently by E. Witten, between the hypermultiplet moduli space in compactifications of the heterotic string on an A-D-E singularities, and the moduli spaces of three dimensional pure gauge theories with the corresponding A-D-E gauge groups. It is possible to add a bounded number of heterotic fivebranes sitting in the singularity, while (in leading order in α...

The moduli space of curves has proven itself a central object in geometry. The past decade has seen substantial progress in understanding the moduli space of curves, involving ideas, for example, from geometry (algebraic, symplectic, and differential), physics, topology, and combinatorics. Many of the new ideas are related to the tautological ring of the moduli space, a subring of the cohomolog...

In the moduli space of quadratic differentials over complex structures on a surface, we construct a set of full Hausdorff dimension of points with bounded Teichmüller geodesic trajectories. The main tool is quantitative nondivergence of Teichmüller horocycles, due to Minsky and Weiss. This has an application to billiards in rational polygons.

We explore the moduli space of heterotic strings in two dimensions. In doing so, we introduce new lines of compactified theories with Spin(24) gauge symmetry and discuss compactifications with Wilson lines. The phase structure of d = 2 heterotic string theory is examined by classifying the hypersurfaces in moduli space which support massless quanta or discrete states. Finally, we compute the to...

We construct the moduli space of r−jets of Riemannian metrics at a point on a smooth manifold. The construction is closely related to the problem of classification of jet metrics via differential invariants. The moduli space is proved to be a differentiable space which admits a finite canonical stratification into smooth manifolds. A complete study on the stratifica-tion of moduli spaces is car...

We describe an algebro-geometric approach to Vakil-Zinger’s desingularization of the main component of the moduli of genus one stable maps to P [6, 7]. The new approach provides complete local structural results for this moduli space as well as for the desingularization of the entire moduli space and should fully extend to higher genera.

Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected quiver is Dynkin or extended Dynkin if and only if all moduli spaces of its representations are smooth.

We classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced in [10]. The full moduli space is found for ≤ 3 points, and a restricted moduli space for 4 points. The topological part of the moduli space is found for ≤ 9 points based on the known atlas of regular graphs. We also discuss asp...

We construct the moduli space of r−jets at a point of Riemannian metrics on a smooth manifold. The construction is closely related to the problem of classification of jet metrics via differential invariants. The moduli space is proved to be a differentiable space which admits a finite canonical stratification into smooth manifolds. A complete study on the stratifica-tion of moduli spaces is car...

In this section, we will give a sketch of the construction of the moduli space Mg of curves of genus g and the closely related moduli space Mg,n of n-pointed curves of genus g using two different approaches. Throughout this section we always assume that 2g − 2 + n > 0. In the first approach, we will embed curves in a projective space P using a sufficiently high power n of their dualizing sheaf....