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)

and

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.