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Contenuto archiviato il 2024-05-30

The geometry of topological quantum field theories

Obiettivo

The predictive power of quantum field theory (QFT) is a perpetual driving force in geometry. Examples include the invention of Frobenius manifolds, mixed twistor structures, primitive forms, and harmonic bundles, up to the discovery of the McKay correspondence, mirror symmetry, and Gromov-Witten invariants. Still seemingly disparate, in fact these all are related to topological (T) QFT and thereby to the work by Cecotti, Vafa et al of more than 20 years ago. The broad aim of the proposed research is to pull the strands together which have evolved from TQFT, by implementing insights from mathematics and physics. The goal is a unified, conclusive picture of the geometry of TQFTs. Solving the fundamental questions on the underlying common structure will open new horizons for all disciplines built on TQFT. Hertling’s “TERP” structures, formally unifying the geometric ingredients, will be key. The work plan is textured into four independent strands which gain full power from their intricate interrelations. (1) To implement TQFT, a construction by Hitchin will be generalised to perform geometric quantisation for spaces with TERP structure. Quasi-classical limits and conformal blocks will be studied as well as TERP structures in the Barannikov-Kontsevich construction of Frobenius manifolds. (2) Relating to singularity theory, a complete picture is aspired, including matrix factorisation and allowing singularities of functions on complete intersections. A main new ingredient are QFT results by Martinec and Moore. (3) Incorporating D-branes, spaces of stability conditions in triangulated categories will be equipped with TERP structures. To use geometric quantisation is a novel approach which should solve the expected convergence issues. (4) For Borcherds automorphic forms and GKM algebras their as yet cryptic relation to “generalised indices” shall be demystified: In a geometric quantisation of TERP structures, generalised theta functions should appear naturally.

Invito a presentare proposte

ERC-2007-StG
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Meccanismo di finanziamento

ERC-SG - ERC Starting Grant

Istituzione ospitante

ALBERT-LUDWIGS-UNIVERSITAET FREIBURG
Contributo UE
€ 417 420,00
Indirizzo
FAHNENBERGPLATZ
79098 Freiburg
Germania

Mostra sulla mappa

Regione
Baden-Württemberg Freiburg Freiburg im Breisgau, Stadtkreis
Tipo di attività
Higher or Secondary Education Establishments
Contatto amministrativo
Katrin Wendland (Prof.)
Ricercatore principale
Katrin Wendland (Prof.)
Collegamenti
Costo totale
Nessun dato

Beneficiari (2)