990 research outputs found
Orlov's Equivalence and Maximal Cohen-Macaulay Modules over the Cone of an Elliptic Curve
We describe a method for doing computations with Orlov's equivalence between
the bounded derived category of certain hypersurfaces and the stable category
of graded matrix factorisations of the polynomials describing these
hypersurfaces. In the case of a smooth elliptic curve over an algebraically
closed field we describe the indecomposable graded matrix factorisations of
rank one. Since every indecomposable Maximal Cohen-Macaulay module over the
completion of a smooth cubic curve is gradable, we obtain explicit descriptions
of all indecomposable rank one matrix factorisations of such potentials.
Finally, we explain how to compute all indecomposable matrix factorisations of
higher rank with the help of a computer algebra system.Comment: 26 page
Vector Bundles on the Moduli Stack of Elliptic Curves
We study vector bundles on the moduli stack of elliptic curves over a local
ring R. If R is a field or a discrete valuation ring of (residue)
characteristic not 2 or 3, all these vector bundles are sums of line bundles.
For R the 3-local integers, we construct higher rank indecomposable vector
bundles and give a classification of vector bundles that are iterated
extensions of line bundles. For R the 2-local integers, we show that there are
even indecomposable vector bundles of arbitrary high rank.Comment: 26 page
Continuous cohomology of topological quandles
A continuous cohomology theory for topological quandles is introduced, and
compared to the algebraic theories. Extensions of topological quandles are
studied with respect to continuous 2-cocycles, and used to show the differences
in second cohomology groups for specific topological quandles. A method of
computing the cohomology groups of the inverse limit is applied to quandles.Comment: 17 page
Massless particles on supergroups and AdS3 x S3 supergravity
Firstly, we study the state space of a massless particle on a supergroup with
a reparameterization invariant action. After gauge fixing the
reparameterization invariance, we compute the physical state space through the
BRST cohomology and show that the quadratic Casimir Hamiltonian becomes
diagonalizable in cohomology. We illustrate the general mechanism in detail in
the example of a supergroup target GL(1|1). The space of physical states
remains an indecomposable infinite dimensional representation of the space-time
supersymmetry algebra. Secondly, we show how the full string BRST cohomology in
the particle limit of string theory on AdS3 x S3 renders the quadratic Casimir
diagonalizable, and reduces the Hilbert space to finite dimensional
representations of the space-time supersymmetry algebra (after analytic
continuation). Our analysis provides an efficient way to calculate the
Kaluza-Klein spectrum for supergravity on AdS3 x S3. It may also be a step
towards the identification of an interesting and simpler subsector of
logarithmic supergroup conformal field theories, relevant to string theory.Comment: 16 pages, 10 figure
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