skip to main content

Title: 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).
Award ID(s):
Publication Date:
Journal Name:
The Quarterly Journal of Mathematics
Sponsoring Org:
National Science Foundation
More Like this
  1. A bstract We revisit the proposal that the ensemble average over free boson CFTs in two dimensions — parameterized by Narain’s moduli space — is dual to an exotic theory of gravity in three dimensions dubbed U(1) gravity. We consider flavored partition functions, where the usual genus g partition function is weighted by Wilson lines coupled to the conserved U(1) currents of these theories. These flavored partition functions obey a heat equation which relates deformations of the Riemann surface moduli to those of the chemical potentials which measure these U(1) charges. This allows us to derive a Siegel-Weil formula which computes the average of these flavored partition functions. The result takes the form of a “sum over geometries”, albeit with modifications relative to the unflavored case.
  2. We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 2 curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role.
  3. Abstract

    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 ofgr= 0.97 ± 0.02,ri= 0.43 ± 0.02, andrz= 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 »is brighter than expected. This is consistent with outgassing from CO or CO2. Using surface brightness profiles, we estimate an upper limit of ∼0.01 kg s−1for micron-sized dust that can be produced without us detecting it for the inactive portion of the light curve, and upper limits of ∼1 kg s−1for CO and ∼1.5 kg s−1for CO2between 20 and 14.7 au.

    « less
  4. In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the q-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of equivariant K-theoretic mirrors.
  5. One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each Teichmüller space of a closed surface of genus g≥2. The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential.