Topology of Irregular Isomonodromy Times on a Fixed Pointed Curve

Transformation Groups https://doi.org/10.1007/s00031-023-09800-9 Transformation Groups Topology of Irregular Isomonodromy Times on a Fixed Pointed Curve Jean Douçot1,2 · Gabriele Rembado3 Received: 4 August 2022 / Accepted: 7 March 2023 © The Author(s) 2023 Abstract 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. Keywords Generalised braid groups · Action operads · Isomonodromic deformations · Weyl groups Mathematics Subject Classification (2010) 20F55 · 20F36 · 55R10 1 Introduction Classically, the theory of isomonodromy constitutes a collection of nonlinear integrable differential equations, whose unknown is a (linear) meromorphic connection J. D. is funded by the NCCR SwissMAP: the Mathematics of Physics (SNF project 141869); G. R. is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - GZ 2047/1, Projekt-ID 390685813. B Gabriele Rembado Jean Douçot 1 Section de mathématiques, Université de Genève, Rue du Conseil-Général 7-9, 1205 Genève, Switzerland 2 Current address: Department of Mathematics, University of Lisbon, Edifício C6, PT-1749-016 Campo Grande, Lisboa, Portugal 3 Hausdorff Centre for Mathematics, University of Bonn, 60 Endenicher Allee, D-53115 Bonn, Germany J. Douçot, G. Rembado on a vector bundle over the Riemann sphere. Geometrically, these are flat Ehresmann connections on a bundle whose fibres are moduli spaces of such meromorphic connections. The underlying deformation parameters, the ‘times’, have recently been given an intrinsic formulation, leading to a generalisation of the moduli of pointed curves (in any genus). This framework is especially useful when considering the generalised deformations, beyond the generic case: recall [42] set up a theory of ‘generic’ isomonodromic deformations of meromorphic connections on vector bundles over a Riemann surface , where the leading coefficient at each pole has distinct eigenvalues (building on [9]; cf. [6, 44]). This has been extended in two directions: i) replacing vector bundles by principal G-bundles, leading in particular to the appearance of G-braid groups for complex reductive groups G [12], and ii) considering nongeneric admissible deformations, e.g. [15, 16], where the (untwisted/unramified) irregular type of the connection is arbitrary, leading to cabled braid groups [30]. In particular the data, the wild character varieties (a.k.a. wild Betti spaces), have been proved to form a local system of Poisson varieties [16]1 MB −→ B , (1) over any space B of admissible deformations. These give a purely topological description of the nonlinear isomonodromy differential equations, via the Riemann– Hilbert–Birkhoff correspondence. Our purpose in this paper is to study the fundamental groups of the base spaces B of such admissible deformations, the groups that will act by algebraic Poisson automorphisms on the wild character varieties (the fibres of (1)) from the parallel transport of the isomonodromy connection—i.e. the monodromy of the nonlinear differential equations. This builds on our previous paper [30], which used a fixed marking: here we will quotient by the Weyl group action and get to the full version of ‘wild’ mapping class groups, in analogy to forgetting the ordering of marked points on the underlying pointed curve. This encompasses the much-studied case of regular singular connections, involving the complex character varieties, which is the entry point for the standard mapping-class- and braid-group-actions in classical/quantum 2d gauge theories—via deformations of pointed curves, e.g. [4, 31, 43, 45] in the quantum case. The case of poles of order 2, however, has been extensively studied by various authors: there are relations to quantum groups [12, 66, 67], and already there the boundary of the space of times has a rich structure (corresponding to the ‘coalescence’ of irregular times, cf. [27, 64, 65]). In particular the simplest irregular singular case has been understood in rigorous analytic way, while this paper focuses on the algebro-geometric aspects of the general nongeneric case. In this series of ‘local’ papers we fix the underlying pointed curve, and vary the rest of the wild Riemann surface structure [16], i.e. the irregular types/classes, controlling principal parts of irregular singular connections beyond their (formal) residues. 1 Basically speaking, a bundle of Poisson manifolds equipped with a complete flat connection: the (Betti) isomonodromy connection, a.k.a. the wild nonabelian Gauβ–Manin connection. Topology of Irregular Isomonodromy Times... More precisely [30] constructs a fine moduli space of untwisted/unramified irregular types for the split Lie algebra (g, t):= (Lie(G), Lie(T)), where T ⊆ G is a maximal (algebraic) torus, while here we consider irregular classes. Recall in brief an untwisted irregular type Q at a point a ∈  is the germ of a t-valued meromorphic function based there, defined up to holomorphic terms: Q= p  Aj z −j ∈ t((z))/t[[z]] , Aj ∈ t , (2) j=1 in a local coordinate vanishing at the marked point. Then the Weyl group Wg = N(T)/ T acts on the left tensor factor of t ⊗C (C((z))/C[[z]])  t((z))/t[[z]] , and the irregular class underlying (2) is its projection Q in the quotient, i.e. the Weylorbit through Q [16, Rk. 10.6].2 The important fact is the fibres of (1) only depend on the collection of irregular classes underlying the irregular types at each marked point, and thus any (admissible) space of irregular classes provides an intrinsic topological description of the corresponding isomonodromy times. In the generic case, where the leading coefficient of (2) is out of all root hyperplanes, the homotopy type of the deformation space brings about the G-braid group: in this paper we shall encounter a generalisation in the nongeneric case, which we relate to braid cabling in type A. 1.1 Layout of the paper and main results In Section 2 we give the main definition: to a one-pointed (bare) wild Riemann surface  = (, a, Q) we associate a full/nonpure local ‘wild’ mapping class group Q (WMCG), viz. the fundamental group of a space BQ of admissible deformations of the irregular class Q (cf. Definition 2.1). The latter is a topological quotient of the (universal) admissible deformation space BQ of Q, where Q is any irregular type lifting Q. In Section 3 we describe the subgroup of Wg preserving BQ ⊆ tp , and further the quotient thereof that acts freely; the resulting subquotient is denoted Wg,h . The relevant statements are proven inductively along the sequence of fission/Levi (root) subsystems of g associated with (...truncated)