Final Report Summary - TQFT (The geometry of topological quantum field theories)
Quantum field theory has had an enormous impact on geometry over the last decades. Examples are mirror symmetry, Gromov-Witten invariants, and the McKay correspondence, all related to topological quantum field theory (TQFT). Many exciting developments emerging from the original work of Cecotti, Vafa et al are by now almost disparate, while basic questions about the very geometry of TQFTs are still open. The broad aim of this project was to find a unified, conclusive picture of the geometry of TQFTs. Four main areas of mathematics where addressed: Symplectic geometry and integrable systems, singularity theory, category theory, and modular forms.
Basic aspects of the project have been the following: Relating topological quantum field theory, conformal field theory, singularity theory and integrable systems (Wendland), symplectic field theory, topological field theory and integrable systems (Fabert), the theory of matrix models and integrable systems (Alexandrov), category theory - specifically matrix factorizations - the geometry of topological field theory and singularity theory (Herbst, Shklyarov), and the application of modular forms in TQFT, specifically in the context of Gromov-Witten invariants (Scheidegger).
In more detail the team - with a number of international coauthors - has made the following achievements:
- development of algebraic foundations of symplectic field theory, including the definition of the full topological field theory version of local symplectic field theory (Fabert, Rossi), and confirmation of cohomology F-manifolds as the right framework to formulate classical mirror symmetry for open Calabi-Yau manifolds (Fabert)
- development of techniques in matrix model theory, in particular construction of integrable group operators clarifying the relation between matrix integrals and W-operators and an operator in the group of symmetries of the KP system which connects the Kontsevich matrix model with the generating function of simple Hurwitz numbers (Alexandrov, Mironov, Morozov, Natanzon)
- discovery of a hidden fermionic symmetry in the quantum Gaudin model (Alexandrov, Leurent, Tsuboi, Zaborodin)
- development of methods to investigate global aspects of the moduli space of topological field theories, for instance using certain monodromy actions to investigate and describe autoequivalences of the relevant categories (Herbst, Walcher)
- explicit proof of correspondences between structures known in singularity theory, on the one hand, and novel structures in category theory, on the other; in particular establishment of a Thom-Sebastiani type theorem for the Hodge-like structures on the periodic cyclic homology of differential graded categories, and construction of a canonical pairing on cyclic cohomology which corresponds to the higher residue pairing on the twisted de Rham cohomology of an isolated singularity (Shklyarov)
- proof of the Yau-Zaslov conjecture, which had been open since the 1990s, and which roughly states that rational curves in a K3 surface can be counted (in an appropriate sense) by calculating coefficients of certain modular forms (Klemm, Maulik, Pandharipande, Scheidegger)
- further development of the deformation theory of matrix factorizations (Knapp, Scheidegger)
- construction of a differential polynomial ring obtained from the special geometry of the moduli space for certain families of Calabi-Yau threefolds, which matches the ring of quasi-modular forms for a suitable duality group (Alim, Scheidegger, Yau, Zhou)
- development of methods towards a geometric interpretation for Mathieu Moonshine phenomena (Taormina, Wendland)
Basic aspects of the project have been the following: Relating topological quantum field theory, conformal field theory, singularity theory and integrable systems (Wendland), symplectic field theory, topological field theory and integrable systems (Fabert), the theory of matrix models and integrable systems (Alexandrov), category theory - specifically matrix factorizations - the geometry of topological field theory and singularity theory (Herbst, Shklyarov), and the application of modular forms in TQFT, specifically in the context of Gromov-Witten invariants (Scheidegger).
In more detail the team - with a number of international coauthors - has made the following achievements:
- development of algebraic foundations of symplectic field theory, including the definition of the full topological field theory version of local symplectic field theory (Fabert, Rossi), and confirmation of cohomology F-manifolds as the right framework to formulate classical mirror symmetry for open Calabi-Yau manifolds (Fabert)
- development of techniques in matrix model theory, in particular construction of integrable group operators clarifying the relation between matrix integrals and W-operators and an operator in the group of symmetries of the KP system which connects the Kontsevich matrix model with the generating function of simple Hurwitz numbers (Alexandrov, Mironov, Morozov, Natanzon)
- discovery of a hidden fermionic symmetry in the quantum Gaudin model (Alexandrov, Leurent, Tsuboi, Zaborodin)
- development of methods to investigate global aspects of the moduli space of topological field theories, for instance using certain monodromy actions to investigate and describe autoequivalences of the relevant categories (Herbst, Walcher)
- explicit proof of correspondences between structures known in singularity theory, on the one hand, and novel structures in category theory, on the other; in particular establishment of a Thom-Sebastiani type theorem for the Hodge-like structures on the periodic cyclic homology of differential graded categories, and construction of a canonical pairing on cyclic cohomology which corresponds to the higher residue pairing on the twisted de Rham cohomology of an isolated singularity (Shklyarov)
- proof of the Yau-Zaslov conjecture, which had been open since the 1990s, and which roughly states that rational curves in a K3 surface can be counted (in an appropriate sense) by calculating coefficients of certain modular forms (Klemm, Maulik, Pandharipande, Scheidegger)
- further development of the deformation theory of matrix factorizations (Knapp, Scheidegger)
- construction of a differential polynomial ring obtained from the special geometry of the moduli space for certain families of Calabi-Yau threefolds, which matches the ring of quasi-modular forms for a suitable duality group (Alim, Scheidegger, Yau, Zhou)
- development of methods towards a geometric interpretation for Mathieu Moonshine phenomena (Taormina, Wendland)