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Johan Martens

Publications and source records attributed to Johan Martens.

16 recordsLinked to original sources
arXiv2025

Abelianization of the $\operatorname{SL}_2$ Hitchin connection at level four

We prove that the Hitchin connection for $\operatorname{SL}_2$ at level four can be understood in terms of the Mumford-Welters connections on bundles of abelian theta functions for Prym torsors of all unramified double covers, and use this to show that its monodromy is finite. This builds on earlier works, for individual curves, of the last named author with Oxbury and Ramanan. The key ingredients in making this work on the level of connections are equivariant conformal embeddings, and anti-invariant level-rank duality.

math.AG↗
arXiv2025

Intersection cohomology of Popov-Vinberg varieties

The Popov-Vinberg variety of a simply connected, split, semisimple algebraic group $G$ is a singular affine variety that contains the basic affine space $G/U$ as a Zariski open subset. It is defined as the spectrum of the ring of functions on $G/U$, and can also be identified with the universal symplectic implosion for the maximal compact subgroup of $G$. We provide a recursive procedure for computing the intersection cohomology of this variety, with an emphasis on the case where $G = \operatorname{SL}_n$.

math.AG↗
arXiv2024

The Prym-Hitchin Connection and Anti-Invariant Level-Rank Duality

We construct a "Hitchin-type" connection on bundles of non-abelian theta functions on higher-rank Prym varieties, for unramified double covers of curves. We formulate a version of level-rank duality in this Prym setting (building on work of Zelaci), show it holds for level one, and establish that the duality respects the flat connections at all levels.

math.AG↗
arXiv2024

Complex Symplectic Contractions and 3d Mirrors

We propose magnetic quivers for the complex-symplectic contraction spaces, which are related to implosions and have a natural interpretation in terms of the Moore-Tachikawa category. We use 3-d mirrors to provide computational checks.

hep-th↗
arXiv2024

Implosion, Contraction and Moore-Tachikawa

We give a survey of the implosion construction, extending some of its aspects relating to hypertoric geometry from type $A$ to a general reductive group, and interpret it in the context of the Moore-Tachikawa category. We use these ideas to discuss how the contraction construction in symplectic geometry can be generalised to the hyperk\"ahler or complex symplectic situation.

math.SG↗
arXiv2023

SI methane hydrate confined in C8-grafted SBA-15: A highly efficient storage system enabling ultrafast methane loading and unloading

Confinement of water and methane in mesopores of hydrophobized SBA-15 is demonstrated to promote methane hydrate formation. In comparison to as-synthesized SBA-15, hydrophobization by C8 grafting accelerates the kinetics of methane storage in and delivery from the hydrate. C8 grafting density was determined at 0.5 groups nm-2 based on TGA and quantitative NMR spectroscopy. Multinuclear 1H-1H DQSQ and 1H-1H RFDR NMR provided spectroscopic evidence for the occurrence of C8 chains inside the mesopores of SBA-15, by showcasing close spatial proximity between the grafted C8 chains and pore-intruded water species. X-ray diffraction demonstrates formation of Structure I hydrate on SBA-15 C8. At 7.0 MPa and 248 K, the water-to-hydrate conversion on hydrophobized SBA-15 C8 reaches 96 pct. as compared to only 71 pct. on a pristine SBA-15 sample with comparable pore size, pore volume and surface area. The clathrate loading amounted to 14.8 g g-1. 2D correlation NMR spectroscopy (1H-13C CP-HETCOR, 1H-1H RFDR) reveals hydrate formation occurs within pores of SBA-15 C8 as well as in interparticle volumes. Following the initial crystallization of SBA-15 C8-supported methane hydrate taking several hours, a pressure swing process at 248 K allows to desorb and re-adsorb methane from the structure within minutes and without thawing the frozen water structure. Fast loading and unloading of methane was achieved in 19 subsequent cycles without losses in kinetics. The ability to harvest the gas and regenerate the structure without the need to re-freeze the water represents a 50 pct. energy gain with respect to melting and subsequently recrystallizing the hydrate at 298 K and 248 K, respectively. After methane desorption, a small amount of residual methane hydrate in combination with an amorphous yet locally ordered ice phase is observed using 13C and 2H NMR spectroscopy.

cond-mat.mtrl-sci↗
arXiv2022

The Hitchin connection in arbitrary characteristic

We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan-Atiyah-Bott Kähler form that realizes the Chern class of the determinant-of-cohomology line bundle on the moduli space of bundles on a curve. As replacement we use an explicit realisation of the Atiyah class of this line bundle, based on the theory of the trace complex due to Beilinson-Schechtman and Bloch-Esnault.

math.AG↗
arXiv2017

Group Compactifications and Moduli Spaces

We give a summary of joint work with Michael Thaddeus that realizes toroidal compactifcations of split reductive groups as moduli spaces of framed bundles on chains of rational curves. We include an extension of this work that covers Artin stacks with good moduli spaces. We discuss, for complex groups, the symplectic counterpart of these compactifications, and conclude with some open problems about the moduli problem concerned.

math.AG↗
arXiv2016

The Witten-Reshetikhin-Turaev invariant for links in finite order mapping tori I

We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten-Reshetikhin-Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno.

math.GT↗
arXiv2016

Contraction of Hamiltonian $K$-spaces

In the spirit of recent work of Harada-Kaveh and Nishinou-Nohara-Ueda, we study the symplectic geometry of Popov's horospherical degenerations of complex algebraic varieties with the action of a complex linearly reductive group. We formulate an intrinsic symplectic contraction of a Hamiltonian space, which is a surjective, continuous map onto a new Hamiltonian space that is a symplectomorphism on an explicitly defined dense open subspace. This map is given by a precise formula, using techniques from the theory of symplectic reduction and symplectic implosion. We then show, using the Vinberg monoid, that the gradient-Hamiltonian flow for a horospherical degeneration of an algebraic variety gives rise to this contraction from a general fiber to the special fiber. We apply this construction to branching problems in representation theory, and finally we show how the Gel'fand-Tsetlin integrable system can be understood to arise this way.

math.SG↗
arXiv2015

Variations on a theme of Grothendieck

Grothendieck and Harder proved that every principal bundle over the projective line with split reductive structure group (and trivial over the generic point) can be reduced to a maximal torus. Furthermore, this reduction is unique modulo automorphisms and the Weyl group. In a series of six variations on this theme, we prove corresponding results for principal bundles over the following schemes and stacks: (1) a line modulo the group of nth roots of unity; (2) a football, that is, an orbifold of genus zero with two marked points; (3) a gerbe over a football whose structure group is the nth roots of unity; (4) a chain of lines meeting in nodes; (5) a line modulo an action of a split torus; and (6) a chain modulo an action of a split torus. We also prove that the automorphism groups of such bundles are smooth, affine, and connected.

math.AG↗
arXiv2015

Compactifications of reductive groups as moduli stacks of bundles

Let G be a split reductive group. We introduce the moduli problem of "bundle chains" parametrizing framed principal G-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of G. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple G, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev-Manin's spaces of weighted pointed curves and with Kausz's compactification of GL(n).

math.AG↗
arXiv2012

On Non-Abelian Symplectic Cutting

We discuss symplectic cutting for Hamiltonian actions of non-Abelian compact groups. By using a degeneration based on the Vinberg monoid we give, in good cases, a global quotient description of a surgery construction introduced by Woodward and Meinrenken, and show it can be interpreted in algebro-geometric terms. A key ingredient is the `universal cut' of the cotangent bundle of the group itself, which is identified with a moduli space of framed bundles on chains of projective lines recently introduced by the authors.

math.SG↗
arXiv2010

Moduli of Parabolic Higgs Bundles and Atiyah Algebroids

In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah algebroid over the moduli space of parabolic vector bundles. By considering the case of full flags, we get a Grothendieck-Springer resolution for all other flag types, in particular for the moduli spaces of twisted Higgs bundles, as studied by Markman and Bottacin and used in the recent work of Laumon-Ngô. We discuss the Hitchin system, and demonstrate that all these moduli spaces are integrable systems in the Poisson sense.

math.AG↗
arXiv2007

Equivariant volumes of non-compact quotients and instanton counting

Motivated by Nekrasov's instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-Kähler geometry by means of the Jeffrey-Kirwan residue-formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on $\R^{4}$ and rederive the formulas for the equivariant volumes obtained earlier by Nekrasov-Shadchin, expressing these volumes as iterated residues of a single rational function.

math.SG↗