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SUMMARY:Making Poisson Sigma Again
DTSTART:20260605T120000Z
DTEND:20260605T130000Z
DTSTAMP:20260614T090200Z
UID:indico-event-16353@indico.math.cnrs.fr
CONTACT:kellendonk@math.univ-lyon1.fr\;athomas@math.univ-lyon1.fr
DESCRIPTION:Speakers: Rafal Suszek\n\nGroupoids G=>M have long been known 
 to furnish models of symmetry in differentialgeometry---and so also in geo
 metric field theory---which are substantially more robust than groups.And 
 yet the formulation of the corresponding gauge principle---understood\, in
  the spirit of Cartanand Borel\, as a procedure of effective descent of dy
 namics to configuration bundles with typicalfibres given by orbispaces M//
 G---has employed rather unnatural structures over the spacetime ofthe fiel
 d theory\, known as principal groupoid bundles---generically devoid of a t
 ypical fibre (i.e.\, ofEhresmann's space of local observers\, or gauges)\,
  and with cumbersome notions of connection andcovariant differentiation. A
  geometric construction that circumnavigates the shortcomings of the oldst
 ructure was proposed by Strobl and the speaker in arXiv:2503.09886 [math.D
 G]\, and given thename Principaloid G-Bundle at (secular) baptism. In the 
 first part of the talk\, the main elements ofthe construction shall be rev
 iewed\, including the emergence of Lie-groupoidal gaugetransformations and
  that of a Lie algebroid-valued gauge field.An ultimate test of the validi
 ty of the proposal is its application in the construction of a field theor
 ywith a Lie-groupoidal symmetry gauged. In the second half of the talk\, t
 he recent attainment of thisgoal\, reported by the speaker in arXiv:2603.2
 0914 [hep-th]\, shall be discusses in the setting of thePolyakov-Alvarez-G
 awędzki (PAG) σ-model of charged-loop dynamics\, with a nontensorialcoup
 ling in the Dirac amplitude functional given by a Cheeger-Simons different
 ial character ofdegree 3\, aka gerbe holonomy. In the demonstration---insp
 ired by the previous work on the gaugingof group-modelled symmetry by Gaw
 ędzki\, Waldorf and the speaker---the key rôle ofmultiplicatively twiste
 d G-equivariant extensions of gerbes in the 2-stacky descent of the dynami
 csto Godement quotients of principaloid G-bundles shall be indicated\, and
  Crainic's description of theunderlying multiplicative Bott-Shulman-Stashe
 ff extensions of gerbes' curvatures in terms ofSpencer data shall be used 
 amply---leading to a natural mechanism of structure-group reduction forpri
 ncipaloid G-bundles in the presence of higher-geometric objects on their t
 ypical fibre G.As an unexpected spinoff of the latter bicategorial constru
 ction\, internalised in the symplecticcategory\, we stumble upon a novel c
 onceptualisation of the good old Poisson σ-model\, and itsgerbish extensi
 on\, which shall be shown to afford an unobstructed integration of its inf
 initesimalLie-algebroidal gauge symmetries to the groupoidal (lagrangean) 
 level\, and to lead to a far-reachingextension of its definition to Godeme
 nt quotients of (symplectic) principaloid G-bundles ofarbitrary topology a
 s configuration bundles.Time permitting\, we shall also give a complete ac
 count of gauge anomalies and classifyinequivalent gaugings in both: the dy
 namical PAG σ-model and the topological Poisson σ-model\, inthe framewor
 k of a Dupont extension of the standard Beilinson-Deligne hypercohomology 
 over thesimplicial nerve of the symmetry model G=>M\, and/or discuss a rei
 nterpretation 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 networksde
 veloped by Runkel and the speaker.\n\nhttps://indico.math.cnrs.fr/event/16
 353/
LOCATION:Fokko du Cloux (Bat. Braconnier)
URL:https://indico.math.cnrs.fr/event/16353/
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