1,056 research outputs found
Quiver varieties and cluster algebras
Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452],
we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the
Grothendieck ring R of the category of representations of quantum loop algebras
U_q(Lg) of a symmetric Kac-Moody Lie algebra g, studied earlier by the author
via perverse sheaves on graded quiver varieties [arXiv:math/9912158]. Graded
quiver varieties controlling the image can be identified with varieties which
Lusztig used to define the canonical base. The cluster monomials form a subset
of the base given by the classes of simple modules in R, or Lusztig's dual
canonical base. The positivity and linearly independence (and probably many
other) conjectures of cluster monomials [arXiv:math/0104151] follow as
consequences, when there is a seed with a bipartite quiver. Simple modules
corresponding to cluster monomials factorize into tensor products of `prime'
simple ones according to the cluster expansion.Comment: 45 pages, v3: minor corrections, v4: Odd cohomology vanishing of
quiver Grassmann for an acyclic quiver is proved in an appendix, v5: minor
correction
The spectrum of BPS branes on a noncompact Calabi-Yau
We begin the study of the spectrum of BPS branes and its variation on lines
of marginal stability on O_P^2(-3), a Calabi-Yau ALE space asymptotic to
C^3/Z_3. We show how to get the complete spectrum near the large volume limit
and near the orbifold point, and find a striking similarity between the
descriptions of holomorphic bundles and BPS branes in these two limits. We use
these results to develop a general picture of the spectrum. We also suggest a
generalization of some of the ideas to the quintic Calabi-Yau.Comment: harvmac, 45 pp. (v2: added references
N=2 gauge theories, instanton moduli spaces and geometric representation theory
We survey some of the AGT relations between N=2 gauge theories in four
dimensions and geometric representations of symmetry algebras of
two-dimensional conformal field theory on the equivariant cohomology of their
instanton moduli spaces. We treat the cases of gauge theories on both flat
space and ALE spaces in some detail, and with emphasis on the implications
arising from embedding them into supersymmetric theories in six dimensions.
Along the way we construct new toric noncommutative ALE spaces using the
general theory of complex algebraic deformations of toric varieties, and
indicate how to generalise the construction of instanton moduli spaces. We also
compute the equivariant partition functions of topologically twisted
six-dimensional Yang-Mills theory with maximal supersymmetry in a general
Omega-background, and use the construction to obtain novel reductions to
theories in four dimensions.Comment: 55 pages; v2: typos corrected and reference added; Final version to
appear in the Special Issue "Instanton Counting: Moduli Spaces,
Representation Theory and Integrable Systems" of the Journal of Geometry and
Physics, eds. U. Bruzzo and F. Sal
Report on "Geometry and representation theory of tensors for computer science, statistics and other areas."
This is a technical report on the proceedings of the workshop held July 21 to
July 25, 2008 at the American Institute of Mathematics, Palo Alto, California,
organized by Joseph Landsberg, Lek-Heng Lim, Jason Morton, and Jerzy Weyman. We
include a list of open problems coming from applications in 4 different areas:
signal processing, the Mulmuley-Sohoni approach to P vs. NP, matchgates and
holographic algorithms, and entanglement and quantum information theory. We
emphasize the interactions between geometry and representation theory and these
applied areas
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