Making Poisson Sigma Again

par Rafal Suszek

vendredi 5 juin 2026, 14:00 → 15:00 Europe/Paris

Fokko du Cloux (Bat. Braconnier)

Groupoids G=>M have long been known to furnish models of symmetry in differential
geometry---and so also in geometric field theory---which are substantially more robust than groups.
And yet the formulation of the corresponding gauge principle---understood, in the spirit of Cartan
and Borel, as a procedure of effective descent of dynamics to configuration bundles with typical
fibres given by orbispaces M//G---has employed rather unnatural structures over the spacetime of
the field theory, known as principal groupoid bundles---generically devoid of a typical fibre (i.e., of
Ehresmann's space of local observers, or gauges), and with cumbersome notions of connection and
covariant differentiation. A geometric construction that circumnavigates the shortcomings of the old
structure was proposed by Strobl and the speaker in arXiv:2503.09886 [math.DG], and given the
name Principaloid G-Bundle at (secular) baptism. In the first part of the talk, the main elements of
the construction shall be reviewed, including the emergence of Lie-groupoidal gauge
transformations and that of a Lie algebroid-valued gauge field.
An ultimate test of the validity of the proposal is its application in the construction of a field theory
with a Lie-groupoidal symmetry gauged. In the second half of the talk, the recent attainment of this
goal, reported by the speaker in arXiv:2603.20914 [hep-th], shall be discusses in the setting of the
Polyakov-Alvarez-Gawędzki (PAG) σ-model of charged-loop dynamics, with a nontensorial
coupling in the Dirac amplitude functional given by a Cheeger-Simons differential character of
degree 3, aka gerbe holonomy. In the demonstration---inspired by the previous work on the gauging
of group-modelled symmetry by Gawędzki, Waldorf and the speaker---the key rôle of
multiplicatively twisted G-equivariant extensions of gerbes in the 2-stacky descent of the dynamics
to Godement quotients of principaloid G-bundles shall be indicated, and Crainic's description of the
underlying multiplicative Bott-Shulman-Stasheff extensions of gerbes' curvatures in terms of
Spencer data shall be used amply---leading to a natural mechanism of structure-group reduction for
principaloid G-bundles in the presence of higher-geometric objects on their typical fibre G.
As an unexpected spinoff of the latter bicategorial construction, internalised in the symplectic
category, we stumble upon a novel conceptualisation of the good old Poisson σ-model, and its
gerbish extension, which shall be shown to afford an unobstructed integration of its infinitesimal
Lie-algebroidal gauge symmetries to the groupoidal (lagrangean) level, and to lead to a far-reaching
extension of its definition to Godement quotients of (symplectic) principaloid G-bundles of
arbitrary topology as configuration bundles.
Time permitting, we shall also give a complete account of gauge anomalies and classify
inequivalent gaugings in both: the dynamical PAG σ-model and the topological Poisson σ-model, in
the framework of a Dupont extension of the standard Beilinson-Deligne hypercohomology over the
simplicial nerve of the symmetry model G=>M, and/or discuss a reinterpretation of the G-
equivariant structure on the σ-model gerbe as data of a topological gauge-symmetry defect in the σ-
model's worldsheet, in the approach to gauging through bicategorially decorated defect networks
developed by Runkel and the speaker.