**Mathematical Surveys and Monographs**

Volume: 170;
2011;
251 pp;
Hardcover

MSC: Primary 81;

Print ISBN: 978-0-8218-5288-0

Product Code: SURV/170

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-1397-2
Product Code: SURV/170.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

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#### Supplemental Materials

# Renormalization and Effective Field Theory

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*Kevin Costello*

This book tells mathematicians about an amazing subject invented by physicists and it tells physicists how a master mathematician must proceed in order to understand it. Physicists who know quantum field theory can learn the powerful methodology of mathematical structure, while mathematicians can position themselves to use the magical ideas of quantum field theory in “mathematics” itself. The retelling of the tale mathematically by Kevin Costello is a beautiful tour de force.

—Dennis Sullivan

This book is quite a remarkable contribution. It should make perturbative quantum field theory accessible to mathematicians. There is a lot of insight in the way the author uses the renormalization group and effective field theory to analyze perturbative renormalization; this may serve as a springboard to a wider use of those topics, hopefully to an eventual nonperturbative understanding.

—Edward Witten

Quantum field theory has had a profound influence on mathematics,
and on geometry in particular. However, the notorious difficulties of
renormalization have made quantum field theory very inaccessible for
mathematicians. This book provides complete mathematical foundations
for the theory of perturbative quantum field theory, based on Wilson's
ideas of low-energy effective field theory and on the
Batalin–Vilkovisky formalism. As an example, a cohomological
proof of perturbative renormalizability of Yang–Mills theory is
presented.

An effort has been made to make the book accessible to mathematicians who
have had no prior exposure to quantum field theory. Graduate students who
have taken classes in basic functional analysis and homological algebra
should be able to read this book.

#### Readership

Graduate students and research mathematicians interested in quantum field theory and mathematical physics.

#### Table of Contents

# Table of Contents

## Renormalization and Effective Field Theory

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Introduction 110 free
- Theories, Lagrangians and counterterms 3140 free
- Field theories on ℝⁿ 91100
- Renormalizability 113122
- Gauge symmetry and the Batalin-Vilkovisky formalism 139148
- Renormalizability of Yang-Mills theory 207216
- Asymptotics of graph integrals 227236
- Nuclear spaces 243252
- Bibliography 249258
- Back Cover Back Cover1264