5,278 research outputs found
Cohomological Donaldson-Thomas theory
This review gives an introduction to cohomological Donaldson-Thomas theory:
the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau
threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their
numerical DT invariant. Local and global aspects of the theory are both
covered, including representations of quivers with potential. We will discuss
the construction of the DT sheaf, a nontrivial topological coefficient system
on such a moduli space, along with some cohomology computations. The
Cohomological Hall Algebra, an algebra structure on cohomological DT spaces,
will also be introduced. The review closes with some recent appearances, and
extensions, of the cohomological DT story in the theory of knot invariants, of
cluster algebras, and elsewhere.Comment: 33 pages, some references adde
Pfaffian quartic surfaces and representations of Clifford algebras
Given a nondegenerate ternary form of degree 4 over an
algebraically closed field of characteristic zero, we use the geometry of K3
surfaces and van den Bergh's correspondence between representations of the
generalized Clifford algebra associated to and Ulrich bundles on the
surface to construct a
positive-dimensional family of irreducible representations of
The main part of our construction, which is of independent interest, uses
recent work of Aprodu-Farkas on Green's Conjecture together with a result of
Basili on complete intersection curves in to produce simple
Ulrich bundles of rank 2 on a smooth quartic surface
with determinant This implies that every smooth quartic
surface in is the zerolocus of a linear Pfaffian, strengthening
a result of Beauville-Schreyer on general quartic surfaces.Comment: This paper contains a proof of the main result claimed in the
erroneous preprint arXiv:1103.0529. We also extend this result to all smooth
quartic surface
Logic and operator algebras
The most recent wave of applications of logic to operator algebras is a young
and rapidly developing field. This is a snapshot of the current state of the
art.Comment: A minor chang
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
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