658 research outputs found
Brauer group of moduli spaces of pairs
We show that the Brauer group of any moduli space of stable pairs with fixed
determinant over a curve is zero.Comment: 12 pages. Final version, accepted in Communications in Algebr
The K\"ahler Potential of Abelian Higgs Vortices
We calculate the K\"ahler potential for the Samols metric on the moduli space
of Abelian Higgs vortices on \mathbbm{R}^{2}, in two different ways. The
first uses a scaling argument. The second is related to the Polyakov conjecture
in Liouville field theory. The K\"ahler potential on the moduli space of
vortices on \mathbbm{H}^{2} is also derived, and we are led to a geometrical
reinterpretation of these vortices. Finally, we attempt to find the K\"ahler
potential for vortices on \mathbbm{R}^{2} in a third way by relating the
vortices to SU(2) Yang-Mills instantons on \mathbbm{R}^{2}\times S^{2}. This
approach does not give the correct result, and we offer a possible explanation
for this.Comment: 25 page
Moduli of vortices and Grassmann manifolds
We use the framework of Quot schemes to give a novel description of the
moduli spaces of stable n-pairs, also interpreted as gauged vortices on a
closed Riemann surface with target Mat(r x n, C), where n >= r. We then show
that these moduli spaces embed canonically into certain Grassmann manifolds,
and thus obtain natural Kaehler metrics of Fubini-Study type; these spaces are
smooth at least in the local case r=n. For abelian local vortices we prove
that, if a certain "quantization" condition is satisfied, the embedding can be
chosen in such a way that the induced Fubini-Study structure realizes the
Kaehler class of the usual L^2 metric of gauged vortices.Comment: 22 pages, LaTeX. Final version: last section removed, typos
corrected, two references added; to appear in Commun. Math. Phy
On the geometry of moduli spaces of coherent systems on algebraic curves
Let be an algebraic curve of genus . A coherent system on consists
of a pair , where is an algebraic vector bundle over of rank
and degree and is a subspace of dimension of the space of sections
of . The stability of the coherent system depends on a parameter .
We study the geometry of the moduli space of coherent systems for different
values of when and the variation of the moduli spaces when
we vary . As a consequence, for sufficiently large , we compute
the Picard groups and the first and second homotopy groups of the moduli spaces
of coherent systems in almost all cases, describe the moduli space for the case
explicitly, and give the Poincar\'e polynomials for the case .Comment: 38 pages; v3. Appendix and new references added; v4. minor
corrections, two added references; v5. final version, one typo corrected and
one reference delete
Moduli spaces of coherent systems of small slope on algebraic curves
Let be an algebraic curve of genus . A coherent system on
consists of a pair , where is an algebraic vector bundle over of
rank and degree and is a subspace of dimension of the space of
sections of . The stability of the coherent system depends on a parameter
. We study the geometry of the moduli space of coherent systems for
. We show that these spaces are irreducible whenever they are
non-empty and obtain necessary and sufficient conditions for non-emptiness.Comment: 27 pages; minor presentational changes and typographical correction
Quaternionic Monopoles
We present the simplest non-abelian version of Seiberg-Witten theory:
Quaternionic monopoles. These monopoles are associated with
Spin^h(4)-structures on 4-manifolds and form finite-dimensional moduli spaces.
On a Kahler surface the quaternionic monopole equations decouple and lead to
the projective vortex equation for holomorphic pairs. This vortex equation
comes from a moment map and gives rise to a new complex-geometric stability
concept. The moduli spaces of quaternionic monopoles on Kahler surfaces have
two closed subspaces, both naturally isomorphic with moduli spaces of
canonically stable holomorphic pairs. These components intersect along
Donaldsons instanton space and can be compactified with Seiberg-Witten moduli
spaces. This should provide a link between the two corresponding theories.
Notes: To appear in CMP The revised version contains more details concerning
the Uhlenbeck compactfication of the moduli space of quaternionic monopoles,
and possible applications are discussed. Attention ! Due to an ununderstandable
mistake, the duke server had replaced all the symbols "=" by "=3D" in the
tex-file of the revised version we sent on February, the 2-nd. The command
"\def{\ad}" had also been damaged !Comment: LaTeX, 35 page
The dynamics of vortices on S^2 near the Bradlow limit
The explicit solutions of the Bogomolny equations for N vortices on a sphere
of radius R^2 > N are not known. In particular, this has prevented the use of
the geodesic approximation to describe the low energy vortex dynamics. In this
paper we introduce an approximate general solution of the equations, valid for
R^2 close to N, which has many properties of the true solutions, including the
same moduli space CP^N. Within the framework of the geodesic approximation, the
metric on the moduli space is then computed to be proportional to the Fubini-
Study metric, which leads to a complete description of the particle dynamics.Comment: 17 pages, 9 figure
The topology of moduli spaces of free group representations
For any complex affine reductive group G and a fixed choice of maximal
compact subgroup K, we show that the G-character variety of a free group
strongly deformation retracts to the corresponding K-character space, which is
a real semi-algebraic set. Combining this with constructive invariant theory
and classical topological methods, we show that the SL(3,C)-character variety
of a rank 2 free group is homotopic to an 8 sphere and the SL(2,C)-character
variety of a rank 3 free group is homotopic to a 6 sphere.Comment: 37 pages, 2 figures, version 2 corrects typos, generalizes context,
and adds a corollary, it is the accepted version to appear in Mathematische
Annale
An Exploratory Look at Supermarket Shopping Paths
We present analyses of an extraordinary new dataset that reveals the path taken by individual shoppers in an actual grocery store, as provided by RFID (radio frequency identification) tags located on their shopping carts. The analysis is performed using a multivariate clustering algorithm not yet seen in the marketing literature that is able to handle data sets with unique (and numerous) spatial constraints. This allows us to take into account physical impediments (such as the location of aisles and other inaccessible areas of the store) to ensure that we only deal with feasible paths. We also recognize that time spent in the store plays an important role, leading to different cluster configurations for short, medium, and long trips. The resulting three sets of clusters identify a total of 14 canonical path types that are typical of grocery store travel, and we carefully describe (and cross-validate) each set of clusters These results dispel certain myths about shopper travel behavior that common intuition perpetuates, including behavior related to aisles, end-cap displays, and the racetrack. We briefly relate these results to previous research (using much more limited datasets) covering travel behavior in retails stores and other related settings
- âŠ