**Proceedings of Symposia in Pure Mathematics**

Volume: 78;
2008;
304 pp;
Hardcover

MSC: Primary 14; 53; 81;

Print ISBN: 978-0-8218-4430-4

Product Code: PSPUM/78

List Price: $81.00

Individual Member Price: $64.80

**Electronic ISBN: 978-0-8218-9385-2
Product Code: PSPUM/78.E**

List Price: $81.00

Individual Member Price: $64.80

# From Hodge Theory to Integrability and TQFT: tt*-geometry

Share this page *Edited by *
*Ron Y. Donagi; Katrin Wendland*

Ideas from quantum field theory and string theory have had an enormous impact on geometry over the last two decades. One extremely fruitful source of new mathematical ideas goes back to the works of Cecotti, Vafa, et al. around 1991 on the geometry of topological field theory. Their tt*-geometry (tt* stands for topological-antitopological) was motivated by physics, but it turned out to unify ideas from such separate branches of mathematics as singularity theory, Hodge theory, integrable systems, matrix models, and Hurwitz spaces. The interaction among these fields suggested by tt*-geometry has become a fast moving and exciting research area.

This book, loosely based on the 2007 Augsburg, Germany workshop "From tQFT to tt* and Integrability", is the perfect introduction to the range of mathematical topics relevant to tt*-geometry. It begins with several surveys of the main features of tt*-geometry, Frobenius manifolds, twistors, and related structures in algebraic and differential geometry, each starting from basic definitions and leading to current research. The volume moves on to explorations of current foundational issues in Hodge theory: higher weight phenomena in twistor theory and non-commutative Hodge structures and their relation to mirror symmetry. The book concludes with a series of applications to integrable systems and enumerative geometry, exploring further extensions and connections to physics.

With its progression through introductory, foundational, and exploratory material, this book is an indispensable companion for anyone working in the subject or wishing to enter it.

#### Table of Contents

# Table of Contents

## From Hodge Theory to Integrability and TQFT: tt*-geometry

- Contents iii4 free
- Introduction v6 free
- Universal unfoldings of Laurent polynomials and tt* structures 110 free
- From primitive forms to Frobenius manifolds 3140
- Twistor structures, tt*-geometry and singularity theory 4958
- Differential geometric aspects of the tt*-equations 7584
- Hodge theoretic aspects of mirror symmetry 8796
- A weight two phenomenon for the moduli of rank one local systems on open varieties 175184
- Associativity for the Neumann system 215224
- Two-dimensional Gauge Theories and Quantum Integrable Systems 239248
- Hurwitz numbers, matrix models and enumerative geometry 263272
- Background independence and the Open Topological String Wavefunction 285294

#### Readership

Graduate students and research mathematicians interested in mathematical physics.