arXiv subjects

Gabriele Rembado

Publications and source records attributed to Gabriele Rembado.

13 recordsLinked to original sources

arXiv2025

Wild genus-zero quantum de Rham spaces

The wild de Rham spaces parameterize isomorphism classes of (stable) meromorphic connections, defined on principal bundles over wild Riemann surfaces. Working on the Riemann sphere, we will deformation-quantize the standard open part of de Rham spaces, which corresponds to the moduli of linear ordinary differential equations with meromorphic coefficients. We treat the general untwisted/unramified case with nonresonant semisimple formal residue, for any polar divisor and reductive structure group. The main ingredients are: (i) constructing the quantum Hamiltonian reduction of a (tensor) product of quantized coadjoint orbits in dual truncated-current Lie algebras, involving the corresponding category-O Verma modules; and (ii) establishing sufficient conditions on the coadjoint orbits, so that generically all meromorphic connections are stable, and the (semiclassical) moment map for the gauge-group action is faithfully flat.

math.QA↗

arXiv2025

Twisted local G-wild mapping class groups

We consider the isomonodromic deformations of irregular-singular connections defined on principal bundles over complex curves: for any complex reductive structure group G, and any polar divisor; allowing for a twisted/ramified formal normal form at each pole, and for twists in the interior of the curve. (This covers the general case in 2-dimensional meromorphic gauge theory.) We focus on the irregular moduli of the connections, studying the fundamental groups of the spaces of admissible deformations of their irregular types/classes, i.e., the local wild mapping class groups in the title. To describe them, we first take the viewpoint of (nonsplit) reflections cosets in Springer/Lehrer--Springer theory, which yields in particular new modular interpretations of complex reflection groups -- and their braid groups. Then we introduce new `fission' trees to treat structure groups of any (simple) classical type, leading to a complete classification of the corresponding hyperplane arrangements, and singling out an infinite family of noncrystallographic examples in type D. Moreover, we reinterpret Bessis' lift of Springer theory as the study of `quasi-generic' deformations, corresponding to irregular singularities whose leading coefficient is regular semisimple upon pullback along a local cyclic covering of the base curve. Finally, we rephrase much of this material in terms of generalized root-valuation stratifications.

math.GT↗

arXiv2024

Moduli spaces of untwisted wild Riemann surfaces

We construct moduli stacks of wild Riemann surfaces in the (pure) untwisted case, for any complex reductive structure group, and we define the corresponding (pure) wild mapping class groups.

math.AG↗

arXiv2024

Wild orbits and generalised singularity modules: stratifications and quantisation

We study truncated gauge-orbits through principal parts of irregular-singular connection germs, in the untwisted/unramified setting: for any connected complex reductive structure group $G$, in the general multilevel case. In particular, we compute the stabilisers of the formal normal forms using filtrations of Levi root systems, showing that they are connected. When the residue is semisimple we then stratify the space of orbits by the conjugacy class of the stabilisers, i.e., by quotients of root-valuation strata; the dense stratum corresponds to the generic setting of isomonodromic deformations, \`a la Jimbo--Miwa--Ueno. Then we adapt a result of Alekseev--Lachowska to deformation-quantise nongeneric orbits. The $\ast$-product involves affine-Lie-algebra modules, extending: (i) the parabolic Verma modules (in the case of regular singularities); and (ii) the `singularity' modules of F.--R. (in the case of generic irregular singularities). They contain Whittaker vectors for the Gaiotto--Teschner/Bonelli--Maruyoshi--Tanzini Virasoro pairs in irregular Liouville conformal field theory, and they provide all the quotients obtained by leaving the aforementioned dense strata. We also construct Shapovalov forms for the corresponding representations of truncated-current Lie algebras, which enter into the category $\mathcal O$ of Chaffe--Topley; and we state a sharp irreducibility criterion. Finally, we use these representations to construct vector bundles of genus-zero vacua/covacua, equipped with flat connections \`a la Knizhnik--Zamolodchikov/Reshetikhin.

math.QA↗

arXiv2022

Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids

Following the completion of the algebraic construction of the Poisson wild character varieties (B.--Yamakawa, 2015) one can consider their natural deformations, generalising both the mapping class group actions on the usual (tame) character varieties, and the G-braid groups already known to occur in the wild/irregular setting. Here we study these wild mapping class groups in the case of arbitrary formal structure in type A. As we will recall, this story is most naturally phrased in terms of admissible deformations of wild Riemann surfaces. The main results are: 1) the construction of configuration spaces containing all possible local deformations, 2) the definition of a combinatorial object, the ``fission forest'', of any wild Riemann surface and a proof that it gives a sharp parameterisation of all the admissible deformation classes. As an application of 1), by considering basic examples, we show that the braid groups of all the complex reflection groups known as the generalised symmetric groups appear as wild mapping class groups. As an application of 2), we compute the dimensions of all the (global) moduli spaces of type A wild Riemann surfaces (in fixed admissible deformation classes), a generalisation of the famous ``Riemann's count'' of the dimensions of the moduli spaces of compact Riemann surfaces.

math.AG↗

arXiv2022

Topology of irregular isomonodromy times on a fixed pointed curve

We will define and study (moduli) spaces of deformations of irregular classes on Riemann surfaces, which provide an intrinsic viewpoint on the `times' of irregular isomonodromy systems in general. Our aim is to study the deeper generalisation of the G-braid groups that occur as fundamental groups of such deformation spaces, with particular focus on the generalisation of the full G-braid groups.

math.AG↗

arXiv2022

Simply-laced quantum connections generalising KZ

We construct a new family of flat connections generalising the KZ connection, the Casimir connection and the dynamical connection. These new connections are attached to simply-laced graphs, and are obtained via quantisation of time-dependent Hamiltonian systems controlling the isomonodromic deformations of meromorphic connections on the sphere.

math.QA↗

arXiv2022

A colourful classification of (quasi) root systems and hyperplane arrangements

We introduce a class of graphs with coloured edges to encode subsystems of the classical root systems, which in particular classify them up to equivalence. We further use the graphs to describe root-kernel intersections, as well as restrictions of root (sub)systems on such intersections, generalising the regular part of a Cartan subalgebra. We also consider a slight variation to encode the hyperplane arrangements only, showing there is a unique noncrystallographic arrangement that arises. Finally, a variation of the main definition leads to elementary classifications of closed and Levi root subsystems.

math.RA↗

arXiv2022

Local wild mapping class groups and cabled braids

We will define and study some generalisations of pure $\mathfrak{g}$-braid groups that occur in the theory of connections on curves, for any complex reductive Lie algebra $\mathfrak{g}$. They make up local pieces of the wild mapping class groups, which are fundamental groups of (universal) deformations of wild Riemann surfaces, underlying the braiding of Stokes data and generalising the usual mapping class groups. We will establish a general product decomposition for the local wild mapping class groups, and in many cases define a fission tree controlling this decomposition. Further in type A we will show one obtains cabled versions of braid groups, related to braid operads.

math.GT↗

arXiv2022

Genus-one complex quantum Chern--Simons theory

We consider the geometric quantisation of Chern--Simons theory for closed genus-one surfaces and semisimple complex groups. First we introduce the natural complexified analogue of the Hitchin connection in Kähler quantisation, with polarisations coming from the nonabelian Hodge hyper-Kähler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, and we identify it with the complexified Hitchin connection using a version of the Bargmann transform on polarised sections over the moduli spaces.

math.QA↗

arXiv2021

$\operatorname{Sp}(1)$-symmetric hyperk\"ahler quantisation

We provide a new general scheme for the geometric quantisation of $\operatorname{Sp}(1)$-symmetric hyper-K\"ahler manifolds, considering Hilbert spaces of holomorphic sections with respect to the complex structures in the hyper-K\"ahler 2-sphere. Under properness of an associated moment map, or other finiteness assumptions, we construct unitary quantum (super) representations of central extensions of certain subgroups of Riemannian isometries preserving the 2-sphere, and we study their decomposition in irreducible components. We apply this quantisation scheme to hyper-K\"ahler vector spaces, the Taub--NUT metric on $\mathbb{R}^4$, moduli spaces of framed $\operatorname{SU}(r)$-instantons on $\mathbb{R}^4$, and partly to the Atiyah--Hitchin manifold of magnetic monopoles in $\mathbb{R}^3$

math.DG↗

arXiv2020

Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks

We give a mathematical definition of spaces of irregular vacua/covacua in genus zero, for any simple Lie algebra, working at generic noncritical level. This uses coinvariants of affine-Lie-algebra modules whose parameters match up with those of moduli spaces of irregular-singular meromorphic connections: the de Rham spaces. The Segal--Sugawara representation of the Virasoro algebra is used to show that the spaces of irregular conformal blocks assemble into a flat vector bundle over the space of somonodromy times \`a la Klar\`es, and we provide a universal version of the resulting flat connection generalising the irregular KZ connection of Reshetikhin and the dynamical KZ connection of Felder--Markov--Tarasov--Varchenko.

math.QA↗

arXiv2020

Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.

math.QA↗