D-branes and Discrete Torsion II

We derive D-brane gauge theories for C^3/Z_n x Z_n orbifolds with discrete torsion and study the moduli space of a D-brane at a point. We show that, as suggested in previous work, closed string moduli do not fully resolve the singularity, but the resulting space -- containing n-1 conifold singularities -- is somewhat surprising. Fractional branes also have unusual properties. We also define an index which is the CFT analog of the intersection form in geometric compactification, and use this to show that the elementary D6-brane wrapped about T^6/Z_n x Z_n must have U(n) world-volume gauge symmetry.Comment: harvmac, 25 p

The Breakdown of Topology at Small Scales

We discuss how a topology (the Zariski topology) on a space can appear to break down at small distances due to D-brane decay. The mechanism proposed coincides perfectly with the phase picture of Calabi-Yau moduli spaces. The topology breaks down as one approaches non-geometric phases. This picture is not without its limitations, which are also discussed.Comment: 12 pages, 2 figure

Solitons in Seiberg-Witten Theory and D-branes in the Derived Category

We analyze the "geometric engineering" limit of a type II string on a suitable Calabi-Yau threefold to obtain an N=2 pure SU(2) gauge theory. The derived category picture together with Pi-stability of B-branes beautifully reproduces the known spectrum of BPS solitons in this case in a very explicit way. Much of the analysis is particularly easy since it can be reduced to questions about the derived category of CP1.Comment: 20 pages, LaTex2

A Point's Point of View of Stringy Geometry

The notion of a "point" is essential to describe the topology of spacetime. Despite this, a point probably does not play a particularly distinguished role in any intrinsic formulation of string theory. We discuss one way to try to determine the notion of a point from a worldsheet point of view. The derived category description of D-branes is the key tool. The case of a flop is analyzed and Pi-stability in this context is tied in to some ideas of Bridgeland. Monodromy associated to the flop is also computed via Pi-stability and shown to be consistent with previous conjectures.Comment: 15 pages, 3 figures, ref adde

Strings, Junctions and Stability

Identification of string junction states of pure SU(2) Seiberg-Witten theory as B-branes wrapped on a Calabi-Yau manifold in the geometric engineering limit is discussed. The wrapped branes are known to correspond to objects in the bounded derived category of coherent sheaves on the projective line \cp{1} in this limit. We identify the pronged strings with triangles in the underlying triangulated category using Pi-stability. The spiral strings in the weak coupling region are interpreted as certain projective resolutions of the invertible sheaves. We discuss transitions between the spiral strings and junctions using the grade introduced for Pi-stability through the central charges of the corresponding objects.Comment: 15 pages, LaTeX; references added. typos correcte

C^2/Z_n Fractional branes and Monodromy

We construct geometric representatives for the C^2/Z_n fractional branes in terms of branes wrapping certain exceptional cycles of the resolution. In the process we use large radius and conifold-type monodromies, and also check some of the orbifold quantum symmetries. We find the explicit Seiberg-duality which connects our fractional branes to the ones given by the McKay correspondence. We also comment on the Harvey-Moore BPS algebras.Comment: 34 pages, v1 identical to v2, v3: typos fixed, discussion of Harvey-Moore BPS algebras update

Distributions of flux vacua

We give results for the distribution and number of flux vacua of various types, supersymmetric and nonsupersymmetric, in IIb string theory compactified on Calabi-Yau manifolds. We compare this with related problems such as counting attractor points.Comment: 43 pages, 7 figures. v2: improved discussion of finding vacua with discrete flux, references adde

Quantum symmetries and exceptional collections

We study the interplay between discrete quantum symmetries at certain points in the moduli space of Calabi-Yau compactifications, and the associated identities that the geometric realization of D-brane monodromies must satisfy. We show that in a wide class of examples, both local and compact, the monodromy identities in question always follow from a single mathematical statement. One of the simplest examples is the Z_5 symmetry at the Gepner point of the quintic, and the associated D-brane monodromy identity

Quivers from Matrix Factorizations

We discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity. This method also yields explicit geometrical interpretations of D-branes (i.e., quiver representations) on a resolution given in terms of Grassmannians. As an example we analyze some non-toric singularities which are resolved by a single CP1 but have "length" greater than one. These examples have a much richer structure than conifolds. A picture is proposed that relates matrix factorizations in Landau-Ginzburg theories to the way that matrix factorizations are used in this paper to perform noncommutative resolutions.Comment: 33 pages, (minor changes

On D0-branes in Gepner models

We show why and when D0-branes at the Gepner point of Calabi-Yau manifolds given as Fermat hypersurfaces exist.Comment: 22 pages, substantial improvements in sections 2 and 3, references added, version to be publishe