(in Polish) Duality, descent and defects: The higher geometry and category theory of charged dynamics 1100-DDD

The goal of this two-semester lecture course is an in-depth introduction into the conceptual framework and methodology of higher geometry and algebra employed in the modern study of phenomena involving dynamical (extended) distributions of (topological) charge -— from the definition of Dirac-Feynman amplitudes for charged dynamics in nontrivial spacetime and configurational topologies in terms of differential characters and a hierarchical geometrisation of the integral cohomology classes of the corresponding gauge fields (in the form of n-gerbes), through categorification of quantum-mechanically consistent symmetries and the universal (higher) gauge principle for symmetries modelled on group and groupoid actions and on more general correspondences (with a homological description of anomalies and classification of inequivalent reductions), all the way up to a realisation of dualities by topological defects, construction of simplicial field theories over defect-stratified spacetimes and the corresponding higher categorial structures on the configuration bundle.

Below, we give a (more) detailed outline of the course:

• A lightning introduction to/review of
  
  
  
  • the theory of Lie groupoids and algebroids (axiomatics, examples, the group of bisections and its actions), the Cartan-type calculus, groupoid modules and orbispaces;
  
  
  • the theory of fibre bundles with connective structure: vector bundles with Koszul connections, principal bundles with principal connection 1-forms and associated bundles with Crittenden connections, reduction and prolongation (generalised Stieffel-Whitney classes);
  
  
  • rudiments of category theory (universality, representability and internalisation) and homological algebra (Čech and de Rham cohomology, sheaf cohomology, Dupont's simplicial cohomology, hypercohomology, Lie-algebra and group cohomology).
  
  


• Murray's gerbes — from the lifting gerbe and the Beilinson-Deligne hypercohomology to (weak) higher categories, via (simplicial) higher geometry. The (higher) Aharonov-Bohm effect and beyond.


• Classical field theory with tensorial resp. simplicial couplings (using the Beilinson-Deligne hypercohomology and its Cheeger-Simons model). Dirac-Feynman amplitudes, the Tulczyjew-Gawędzki-Kijowski-Szczyrba(s) first-order formalism and prequantisation through cohomological transgression.


• Symmetry analysis: configurational and (semi-)gauge symmetries, Noether-Poisson realisations and (classical) central extensions, comomenta and rigid-symmetry algebroids, categorification of symmetries.


• Orbital field dynamics and non-linear realisations of internal and spacetime symmetries. The Ivanov-Ogievietskii (aka inverse Higgs) mechanism. Constructions in the invariant de Rham cohomology, gerbe objects in the category of groups, Nieuwenhuizen's FDA techniques and the rise of Stasheff's L_∞-structures via the Baez-Crans correspondence).


• The universal gauge principle and gauge anomalies for symmetries modelled on group actions:
  
  
  
  • the standard formulation (association) and the underlying universal mixing construction of Cartan and Borel;
  
  
  • the Lie groupoid behind the standard formulation and its Higgs module — a MacKenzie-Moerdijk-Mrčun principal groupoid bundle of the Higgs model;
  
  
  • one Lie groupoid to rule them all — the Atiyah gauge groupoid and the associated short exact sequence of Lie groupoids;
  
  
  • the Kobayashi-Nomizu induction scheme for associated connections and covariant derivatives, the minimal-coupling recipe for tensorial couplings, descent to configurational orbispaces;
  
  
  • equivariant-cohomology models and their geometrisations for simplicial couplings, Dirac anomalies in Courant algebroids for symmetries under gauging;
  
  
  • the gauge defect, twisted sectors and the ensuing simplicial field theory with defects.
  
  
  
  


• Groupoidal symmetries and their gauging (a report from the frontline):
  
  
  
  • the special rôle of bisections and their relation to the tangent Lie algebroid;
  
  
  • circumnavigating Fréchet: principaloid bundles and their foliated connections;
  
  
  • the inextricable entwinement of groupoidal matter and radiation in the augmented Atiyah short exact sequence for the groupoidal gauge symmetry;
  
  
  • the universal gauge principle for tensorial and simplicial field theories;
  
  
  • the curved Yang-Mills-Higgs theory;
  
  
  • Q-bundles and the Poisson-σ-model*.
  
  
  
  


• Some advanced gauge field theory:
  
  
  
  • the Anderson-Brout-Englert-Higgs-Guralnik-Hagen-Kibble-... mechanism;
  
  
  • orbital field dynamics and non-linear realisations of internal and spacetime symmetries, the Ivanov-Ogievietskii (aka inverse Higgs) mechanism, constructions in the invariant de Rham cohomology, gerbe objects in the category of groups, Nieuwenhuizen's FDA techniques and the rise of Stasheff's L_∞-structures via the Baez-Crans correspondence;
  
  
  • standard gauge fields vs gerbe-module connections;
  
  
  • bi-chiral Kač-Moody symmetries in the Wess-Zumino-Novikov-Witten-Gawędzki model of the two-dimensional Rational Conformal Field Theory*;
  
  
  • topological gauge field theory — a case study of the Chern-Simons theory in three dimensions in the presence of Wilson lines, of its reduction á la Alekseev and Malkin and of... its intricate relation to the WZNWG model (a hands-on example of `holography').*
  
  
  
  


• Field theories with the statistical gradation:
  
  
  
  • rudiments of supergeometry with supersymmetry (Z/2Z-graded manifolds, super-Harish-Chandra pairs, super-Cartan calculus and Lie-supergroup actions);
  
  
  • a recapitulation of the theory of spinor bundles with spin connections;
  
  
  • Freed's inner-Hom superfield theories and their (quasi-)supersymmetry;
  
  
  • super-σ-models and geometrisations in the Cartan-Eilenberg cohomology of Lie supergroups — supergerbes.*
  
  
  
  


• Dualities:
  
  
  
  • the (pre)symplectic description;
  
  
  • the defect-duality correspondence;
  
  
  • useful instantiations: T-duality via gauging of toroidal actions and the ensuing field-bundle topology change, the Hughes-Polchinski duality, S-duality in a weakly abelian gauge field theory*, a sector-restricted duality between the 2d WZNWG RCFT and the 3d CS TGFT — a first step towards functorial quantisation*, and more*...