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The geometry of topological quantum field theories

Objective

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.

Field of science

  • /natural sciences/mathematics/pure mathematics/algebra
  • /natural sciences/physical sciences/quantum physics/quantum field theory
  • /natural sciences/mathematics/pure mathematics/geometry
  • /natural sciences/mathematics

Call for proposal

ERC-2007-StG
See other projects for this call

Funding Scheme

ERC-SG - ERC Starting Grant

Host institution

ALBERT-LUDWIGS-UNIVERSITAET FREIBURG
Address
Fahnenbergplatz
79098 Freiburg
Germany
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 417 420
Principal investigator
Katrin Wendland (Prof.)
Administrative Contact
Katrin Wendland (Prof.)

Beneficiaries (2)

ALBERT-LUDWIGS-UNIVERSITAET FREIBURG
Germany
EU contribution
€ 417 420
Address
Fahnenbergplatz
79098 Freiburg
Activity type
Higher or Secondary Education Establishments
Principal investigator
Katrin Wendland (Prof.)
Administrative Contact
Katrin Wendland (Prof.)
UNIVERSITAET AUGSBURG
Germany
EU contribution
€ 332 580
Address
Universitaetsstrasse 2
86159 Augsburg
Activity type
Higher or Secondary Education Establishments
Administrative Contact
Alois Zimmermann (Mr.)