Topological field theory and singularity theory

The outset of this project is the program of an ERC Starting Independent Researcher Grant The Geometry of Topological Quantum Field Theories, which I held in 2009-2014 and which was devoted to a bottom-up approach to the geometry of topological quantum field theories (TQFTs).

TQFTs are useful for various reasons: firstly and originally, they can be viewed as toy models of string theory, dramatically simplifying the latter up to solvability. Secondly and even better, full-fledged superstring theories, which are usually built on a Calabi-Yau "target space" X, can be projected onto TQFTs, yielding a selection of original superstring data prone to simplified calculations. These data, in fact, are often not even accessible otherwise. Since they are closely related to the underlying target space X, such calculations regularly have an important impact on geometry.

The research project is based on fundamental concepts concerning the very geometry of moduli spaces of TQFTs, and it aims at a broad view of TQFTs, including D-branes and the role of generalized theta functions as well as BPS algebras and automorphic forms. Following a mathematical route we aim for a complete understanding of the geometric properties of moduli spaces of TQFTs. As a starting point, Hertling's "TERP structures" yield an abstract description of such moduli spaces, while TQFTs shall be viewed as arising from quantization of spaces with TERP structure. The approach combines the advantages of both a mathematician's and a physicist's viewpoint: it puts the proposed research on a solid mathematical foundation while, by exploiting their common roots in physics, it relates seemingly disjoint areas of mathematics which have evolved over a period of more than twenty years.

In preparation of the project I have organized two international conferences as well as a spring school on this topic,

From tQFT to tt* and integrability in Augsburg, Germany, May 25 - 29, 2007 (with Prof. Ron Donagi)


School and Workshop on The geometry and integrability of topological QFT and string theory ,
March 24 - April 5, 2008, at the University of Warwick (with Dr. Emanuel Scheidegger) under the TQFT subactivity (organised with Prof. Ron Donagi, Prof. Claus Hertling, Prof. Nigel Hitchin, Prof. Miles Reid, Dr. Emanuel Scheidegger) of the 2007-08 Warwick EPSRC Symposium on Algebraic Geometry.

For the Augsburg conference, Prof. Ron Donagi and I have edited a proceedings volume:

Ron Donagi, Katrin Wendland, eds.
From Hodge Theory to Integrability and TQFT: tt* geometry

A broad introduction to some of the ideas behind the project and to ongoing work can be found in

Katrin Wendland
On the geometry of singularities in quantum field theory
  • Proceedings of the International Congress of Mathematicians Hyderabad, August 19-27, 2010
  • Hindustan Book Agency (2010), 2144-2170

Among other topics, the article summarizes aspects of singularity theory with a view on its applications in quantum field theory. In particular, the ADE classification of simple singularities is revised, and the recovery of the ADE theme in the context of the classification of superconformal field theories is explained. Moreover, the full classification of N=(2,2) superconformal minimal models is addressed. An important step towards the latter has been achieved by my student Dr. Oliver Gray in his Ph.D thesis, where he shows that the partition function of each such model agrees with the one of an orbifold of an ADE type minimal model. We are currently working on the completion of the classification program along with its geometric interpretation, which is related to singularity theory, generalizing the result by Cecotti and Vafa.

As part of this project, and under my supervision jointly with PD Dr. Emanuel Scheidegger, my student Dr. Florian Beck completed his Ph.D. studying Hitchin and Calabi-Yau integrable systems via variations of Hodge structures. Here, he initiates a generalization of the results of the publication Intermediate Jacobians and ADE Hitchin Systems by Diaconescu, Donagi, Pantev, where the authors give a beautiful construction of families of Calabi-Yau threefolds arising from the semi-universal unfoldings of ADE type singularities. The Griffiths intermediate Jacobians for these families are the above-mentioned Calabi-Yau integrable systems. As algebraic integrable systems, they agree with the respective ADE Hitchin integrable systems. In his thesis, Beck uses the folding automorphisms of ADE type Lie groups to construct automorphisms on the Calabi-Yau threefolds obtained by Diaconescu, Donagi, Pantev. The resulting orbifold stacks yield intermediate Jacobian fibrations that, as Beck shows, are isomorphic to BCFG type Hitchin integrable systems.

In joint work with Dr. Florian Beck and Prof. Ron Donagi, in the preprint

Florian Beck, Ron Donagi, Katrin Wendland
Folding of Hitchin systems and crepant resolutions
we complete the picture: we implement folding for Hitchin integrable systems of ADE type. On the level of algebraic integrable systems and away from singular fibers, we show that the fixed point loci of the folding automorphisms are isomorphic to the Hitchin systems of the folded groups. For the quotient varieties arising from dividing the families obtained by Diaconescu, Donagi, Pantev by the induced folding automorphisms found in Beck's thesis, we construct simultaneous crepant resolutions. The resulting Calabi-Yau integrable systems do not agree with the folded Hitchin systems, a mismatch which I hope to interpret on the level of integrable systems in the future.