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Philip Boalch

Publications and source records attributed to Philip Boalch.

At least 19 recordsLinked to original sources
arXiv2024

Counting the fission trees and nonabelian Hodge graphs (untwisted case)

Any algebraic connection on a vector bundle on a smooth complex algebraic curve determines an irregular class and in turn a fission tree at each puncture. The fission trees are the discrete data classifying the admissible deformation classes. Here we explain how to count the fission trees with given slope and number of leaves, in the untwisted case. This also leads to a clearer picture of the ``periodic table'' of the atoms that play the role of building blocks in 2d gauge theory.

math.AG↗
arXiv2023

Polystability of Stokes representations and differential Galois groups

Polystability of (twisted) Stokes representations (i.e. wild monodromy representations) will be characterised, in terms of the corresponding differential Galois group (generalising the Zariski closure of the monodromy group in the tame case). This extends some results of Richardson. Further, the intrinsic approach to such results will be established, in terms of reductions of Stokes local systems.

math.AG↗
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↗
arXiv2020

Symplectic Manifolds and Isomonodromic Deformations

We study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint (generalising the Atiyah-Bott approach). This enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of Jimbo, Miwa and Ueno, thereby putting the existing results for the six Painleve equations and Schlesinger's equations into a uniform framework.

math.DG↗
arXiv2019

Topology of the Stokes phenomenon

Several intrinsic topological ways to encode connections on vector bundles on smooth complex algebraic curves will be described. In particular the notion of {\em Stokes decompositions} will be formalised, as a convenient intermediate category between the Stokes filtrations and the Stokes local systems/wild monodromy representations. The main result establishes a new simple characterisation of the Stokes decompositions.

math.AG↗
arXiv2017

Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams

The theory of Hitchin systems is something like a "global theory of Lie groups", where one works over a Riemann surface rather than just at a point. We'll describe how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the non-compact case), and discuss their classification, outlining a theory of "Dynkin diagrams" as a step towards classifying some examples of such objects.

math.AG↗
arXiv2013

Poisson varieties from Riemann surfaces

Short survey based on talk at the Poisson 2012 conference. The main aim is to describe and give some examples of wild character varieties (naturally generalising the character varieties of Riemann surfaces by allowing more complicated behaviour at the boundary), their Poisson/symplectic structures (generalising both the Atiyah-Bott approach and the quasi-Hamiltonian approach), and the wild mapping class groups.

math.AG↗
arXiv2013

Global Weyl groups and a new theory of multiplicative quiver varieties

In previous work a relation between a large class of Kac-Moody algebras and meromorphic connections on global curves was established---notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many Nakajima quiver varieties, as moduli spaces of connections, whenever the underlying graph was a complete k-partite graph (or more generally a supernova graph). However in the isomonodromy story, or wild nonabelian Hodge theory, slightly larger moduli spaces of connections are considered. This raises the question of whether the full moduli spaces admit Weyl group isomorphisms, rather than just the open parts isomorphic to quiver varieties. This question will be solved here, by developing a "multiplicative version" of the previous approach. This amounts to constructing many algebraic symplectic isomorphisms between wild character varieties. This approach also enables us to state a conjecture for certain irregular Deligne--Simpson problems and introduce some noncommutative algebras, generalising the "generalised double affine Hecke algebras".

math.AG↗
arXiv2013

Painleve Equations and Complex Reflections

We will explain how some new algebraic solutions of the sixth Painleve equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-Mazzocco for real reflection groups. The problem of finding explicit formulae for these solutions will be addressed elsewhere.

math.CA↗
arXiv2013

Through the analytic halo: Fission via irregular singularities

This article is concerned with moduli spaces of connections on bundles on Riemann surfaces, where the structure group of the bundle may vary in different regions of the surface. Here we will describe such moduli spaces as complex symplectic manifolds, generalising the complex character varieties of Riemann surfaces.

math.AG↗
arXiv2013

Symmetric cubic surfaces and G_2 character varieties

We will consider a two dimensional "symmetric" subfamily of the four dimensional family of Fricke cubic surfaces. The main result is that such symmetric cubic surfaces arise as character varieties for the exceptional group of type G_2. Further, this symmetric family will be related to the fixed points of the triality automorphism of Spin(8), and an example involving the finite simple group of order 6048 inside G_2 will be considered.

math.AG↗
arXiv2012

Hyperkahler manifolds and nonabelian Hodge theory of (irregular) curves

Short survey based on talk given at the Institut Henri Poincare January 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail (in subsequent talks) the results of [Boa11c] related to wild character varieties and irregular mapping class groups.

math.AG↗
arXiv2011

Simply-laced isomonodromy systems

A new class of isomonodromy equations will be introduced and shown to admit Kac-Moody Weyl group symmetries. This puts into a general context some results of Okamoto on the 4th, 5th and 6th Painleve equations, and shows where such Kac-Moody root systems occur "in nature". A key point is that one may go beyond the class of affine Kac-Moody root systems. As examples, by considering certain hyperbolic Kac-Moody Dynkin diagrams, we find there is a sequence of higher order Painleve systems lying over each of the classical Painleve equations. This leads to a conjecture about the Hilbert scheme of points on some Hitchin systems.

math.CA↗
arXiv2010

Riemann-Hilbert for tame complex parahoric connections

A local Riemann-Hilbert correspondence for tame meromorphic connections on a curve compatible with a parahoric level structure will be established. Special cases include logarithmic connections on G-bundles and on parabolic G-bundles, where G is a complex reductive group. The corresponding Betti data involves pairs (M,P) consisting of the local monodromy M in G and a (weighted) parabolic subgroup P of G such that M is in P, as in the multiplicative Brieskorn-Grothendieck-Springer resolution (extended to the parabolic case). The natural quasi-Hamiltonian structures that arise on such spaces of enriched monodromy data will also be constructed.

math.DG↗