Moduli Spaces of Generalized Hyperpolygons
Abstract We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-g Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of Gelfand–Tsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explicitly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of Kapustin–Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).
Authors:
;
Award ID(s):
Publication Date:
NSF-PAR ID:
10232976
Journal Name:
The Quarterly Journal of Mathematics
ISSN:
0033-5606
Manx comets are objects on long-period comet orbits that are inactive as they approach perihelion. They are of particular interest because they may help constrain solar system formation models. 2013 LU28 was discovered as an inactive asteroidal object on 2013 June 8 at a heliocentric distance of 21.8 au. Images and photometric data were obtained of 2013 LU28 from multiple telescopes from pre-discovery data in 2010 until the present. Its spectral reflectivity is consistent with typical organic-rich comet surfaces with colors of$g′−r′$= 0.97 ± 0.02,$r′−i′$= 0.43 ± 0.02, and$r′−z′$= 0.65 ± 0.03, corresponding to a spectral reflectivity slope of 30 ± 3%/100 nm. There is no obvious indication of dust coma in deep stacked images. We estimate the nucleus radius to be ∼55.7 ± 0.3 km assuming an albedo of 4%. This is much smaller than the 1σupper limits on the nucleus size of 79.9 km from the NEOWISE survey assuming the same albedo, since the NEOWISE survey is not very sensitive to objects this small at this distance. The heliocentric light curve suggests possible activity betweenr∼ 17 and 13 au where 2013 LU28more »
5. Abstract We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F , what is c 1 ( n , F ), the least integer d such that if G is an n -vertex 3-graph with minimum vertex-degree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G ? We asymptotically determine c 1 ( n , F ) when F is the generalized triangle $K_4^{(3)-}$ , and we give close to optimal bounds in the case where F is the tetrahedron $K_4^{(3)}$ (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n -vertex graph G with $m> n^2/4$ edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.