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Steven Rayan

Publications and source records attributed to Steven Rayan.

At least 55 records · Page 3Linked to original sources
arXiv2020

Moduli spaces of generalized hyperpolygons

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 explictly. 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).

math.AG↗
arXiv2019

Higgs bundles without geometry

Higgs bundles appeared a few decades ago as solutions to certain equations from physics and have attracted much attention in geometry as well as other areas of mathematics and physics. Here, we take a very informal stroll through some aspects of linear algebra that anticipate the deeper structure in the moduli space of Higgs bundles. (This note was produced for the MFO Snapshots of Modern Mathematics series, which is "designed to promote the understanding and appreciation of modern mathematics and mathematical research in the interested public world-wide.")

math.AG↗
arXiv2019

Twisted cyclic quiver varieties on curves

We study the algebraic geometry of twisted Higgs bundles of cyclic type along complex curves. These objects, which generalize ordinary cyclic Higgs bundles, can be identified with representations of a cyclic quiver in a twisted category of coherent sheaves. Referring to the Hitchin fibration, we produce a fibre-wise geometric description of the locus of such representations within the ambient twisted Higgs moduli space. When the genus is 0, we produce a concrete geometric identification of the moduli space as a vector bundle over an associated (twisted) A-type quiver variety; we count the number of points at which the cyclic moduli space intersects a Hitchin fibre; and we describe explicitly certain $\mathbb{C}^\times$-flows into the nilpotent cone. We also extend this description to moduli of certain twisted cyclic quivers whose rank vector has components larger than 1. We show that, for certain choices of underlying bundle, such moduli spaces decompose as a product of cyclic quiver varieties in which each node is a line bundle.

math.AG↗
arXiv2019

The Calogero-Françoise integrable system: algebraic geometry, Higgs fields, and the inverse problem

We review the Calogero-Françoise integrable system, which is a generalization of the Camassa-Holm system. We express solutions as (twisted) Higgs bundles, in the sense of Hitchin, over the projective line. We use this point of view to (a) establish a general answer to the question of linearization of isospectral flow and (b) demonstrate, in the case of two particles, the dynamical meaning of the theta divisor of the spectral curve in terms of mechanical collisions. Lastly, we outline the solution to the inverse problem for CF flows using Stieltjes' continued fractions.

math-ph↗
arXiv2018

Nonabelian Hodge theory and vector valued modular forms

We examine the relationship between nonabelian Hodge theory for Riemann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequality on the weights of free bases of vector valued modular forms associated to complex, finite dimensional, irreducible representations of the modular group. This conjecture is known for irreducible unitary representations and for all irreducible representations of dimension at most 12. We prove new instances of the three-term inequality for certain nonunitary representations, corresponding to a class of maximally-decomposed variations of Hodge structure, by considering the same inequality with respect to a new type of modular form, called a "Higgs form", that arises naturally on the Dolbeault side of nonabelian Hodge theory. The paper concludes with a discussion of a strategy for reducing the general case of nilpotent Higgs bundles to the case under consideration in our main theorem.

math.NT↗
arXiv2018

Aspects of the topology and combinatorics of Higgs bundle moduli spaces

This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinatorial nature.

math.AG↗
arXiv2018

Principal co-Higgs bundles on $\mathbb{P}^1$

For complex connected, reductive, affine, algebraic groups $G$, we give a Lie-theoretic characterization of the semistability of principal $G$-co-Higgs bundles on the complex projective line $\mathbb{P}^1$ in terms of the simple roots of a Borel subgroup of $G$. We describe a stratification of the moduli space in terms of the Harder-Narasimhan type of the underlying bundle.

math.AG↗
arXiv2018

Geodesics on a Kerr-Newman-(anti-)de Sitter instanton

We study geodesics along a noncompact Kerr-Newman instanton, where the asymptotic geometry is either de Sitter or anti-de Sitter. We use first integrals for the Hamilton-Jacobi equation to characterize trajectories both near and away from horizons. We study the interaction of geodesics with special features of the metric, particularly regions of angular degeneracy or "theta horizons" in the de Sitter case. Finally, we characterize a number of stable equilibrium orbits.

math.DG↗
arXiv2018

Twisted argyle quivers and Higgs bundles

Ordinarily, quiver varieties are constructed as moduli spaces of quiver representations in the category of vector spaces. It is also natural to consider quiver representations in a richer category, namely that of vector bundles on some complex variety equipped with a fixed sheaf that twists the morphisms. Representations of A-type quivers in this twisted category --- known in the literature as "holomorphic chains" --- have practical use in questions concerning the topology of the moduli space of Higgs bundles. In that problem, the variety is a Riemann surface of genus at least 2, and the twist is its canonical line bundle. We extend the treatment of twisted A-type quiver representations to any genus using the Hitchin stability condition induced by Higgs bundles and computing their deformation theory. We then focus in particular on so-called "argyle quivers", where the rank labelling alternates between 1 and integers $r_i\geq1$. We give explicit geometric identifications of moduli spaces of twisted representations of argyle quivers on $\mathbb{P}^1$ using invariant theory for a non-reductive action via Euclidean reduction on polynomials. This leads to a stratification of the moduli space by change of bundle type, which we identify with "collision manifolds" of invariant zeroes of polynomials. We also relate the present work to Bradlow-Daskalopoulos stability and Thaddeus' pullback maps to stable tuples. We apply our results to computing $\mathbb{Q}$-Betti numbers of low-rank twisted Higgs bundle moduli spaces on $\mathbb{P}^1$, where the Higgs fields take values in an arbitrary ample line bundle. Our results agree with conjectural Poincaré series arising from the ADHM recursion formula.

math.AG↗
arXiv2017

Spectral curves for the triple reduced product of coadjoint orbits for SU(3)

We give an identification of the triple reduced product of three coadjoint orbits in SU(3) with a space of Hitchin pairs over a genus 0 curve with three punctures, where the residues of the Higgs field at the punctures are constrained to lie in fixed coadjoint orbits. Using spectral curves for the corresponding Hitchin system, we identify the moment map for a Hamiltonian circle action on the reduced product. Finally, we make use of results of Adams, Harnad, and Hurtubise to find Darboux coordinates and a differential equation for the Hamiltonian.

math.AG↗
arXiv2017

Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices

We study holomorphic integrable systems on the hyperkähler manifold $G\times S_{\text{reg}}$, where $G$ is a complex semisimple Lie group and $S_{\text{reg}}$ is the Slodowy slice determined by a regular $\mathfrak{sl}_2(\mathbb{C})$-triple. Our main result is that this manifold carries a canonical \textit{abstract integrable system}, a foliation-theoretic notion recently introduced by Fernandes, Laurent-Gengoux, and Vanhaecke. We also construct traditional integrable systems on $G\times S_{\text{reg}}$, some of which are completely integrable and fundamentally based on Mishchenko and Fomenko's argument shift approach.

math.SG↗
arXiv2017

A vanishing theorem for co-Higgs bundles on the moduli space of bundles

We consider smooth moduli spaces of semistable vector bundles of fixed rank and determinant on a compact Riemann surface $X$ of genus at least $3$. The choice of a Poincaré bundle for such a moduli space $M$ induces an isomorphism between $X$ and a component of the moduli space of semistable sheaves over $M$. We prove that $h^0(M, \text{End}({\mathcal E})\otimes TM)= 1$ for a vector bundle $\mathcal E$ on $M$ coming from this component. Furthermore, there are no nonzero integrable co-Higgs fields on $\mathcal E$.

math.AG↗
arXiv2016

The quiver at the bottom of the twisted nilpotent cone on $\mathbb P^1$

For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimize the function are representations of $\mbox{A}_1$. For twisted Higgs bundles on the projective line, the quiver describing the bottom of the cone is more complicated. We determine it here. We show that the moduli space is topologically connected whenever the rank and degree are coprime, thereby verifying conjectural lowest Betti numbers coming from high-energy physics.

math.AG↗
arXiv2015

Some results on equivariant contact geometry for partial flag varieties

We study equivariant contact structures on complex projective varieties arising as partial flag varieties $G/P$, where $G$ is a connected, simply-connected complex simple group of type $ADE$ and $P$ is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. The result can be deduced from Boothby's classification of compact simply-connected complex contact manifolds with transitive action by contact automorphisms, but our proof is completely independent and relies on properties of $G$-equivariant vector bundles on $G/P$. A byproduct of our argument is a canonical, global description of the unique $SO_{2n}(\mathbb C)$-invariant contact structure on the isotropic Grassmannian of $2$-planes in $\mathbb C^{2n}$.

math.RT↗
arXiv2014

Hyperpolygons and Hitchin systems

We study the hyperkähler analogues of moduli spaces of semistable n-gons in complex projective space. We prove that the hyperkähler Kirwan map is surjective and produce a formula that recursively calculates the Betti numbers of these spaces for all ranks. Building on a natural analogy between hyperpolygons and parabolic Higgs bundles, we identify hyperpolygon spaces with certain degenerate Hitchin systems, and use this to establish their complete integrability, for ranks up to and including 3.

math.AG↗
arXiv2013

Co-Higgs bundles on P^1

Co-Higgs bundles are Higgs bundles in the sense of Simpson, but with Higgs fields that take values in the tangent bundle instead of the cotangent bundle. Given a vector bundle on P^1, we find necessary and sufficient conditions on its Grothendieck splitting for it to admit a stable Higgs field. We characterize the rank-2, odd-degree moduli space as a universal elliptic curve with a globally-defined equation. For ranks r=2,3,4, we explicitly verify the conjectural Betti numbers emerging from the recent work of Chuang, Diaconescu, Pan, and Mozgovoy on the ADHM formula. We state the result for r=5.

math.AG↗