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This book is written for both physicists and mathematicians. It contains a detailed investigation of nonlinear extensions of two-dimensional conformal symmetries, i.e. the Virasoro symmetries. These extended conformal symmetries, generically called symmetries, generate so-called algebras which are defined by nonlinear relations. The algebras have attracted the attention of both physicists and mathematicians. From the mathematician's point of view they form interesting extensions of the usual linear algebras. From the physicist's point of view they could be used for the construction of new string theories in the same way that the Virasoro symmetries underlie the usual string theories. The book contains an overview of the author's work on the simplest example of a algebra: the algebra. For this special case many results are derived thereby clarifying the general case. Many explicit expressions are given, often via tables in the different appendices, that have sometimes been obtained by using a computer. In the introduction the authors first discuss the case of the linear Virasoro algebra to clarify and emphasize the common features of and the differences with the case of the nonlinear algebra. They discuss, in the first part of the book, the structure of the modules and use these results to compute (partly via conjectures) the semi-infinite cohomology of the algebra with values in the tensor product of c = 2 and c = 98 Fock modules. This case is relevant for the study of a noncritical string propagating in four dimensions. The second part of the book focuses on the geometrical structure underlying the operator cohomology. It turns out that this structure is described by Batalin - Vilkovisky (BV) algebras which have been introduced independently by mathematicians and physicists at around the same time. The authors first derive some general results on BV algebras. This part of the book can be read independently of the rest. They next discuss the BV algebra of the string that occurred in the first part of the book. The precise connection between the cohomology and the BV algebras is made via a claim which the authors themselves consider as perhaps the most important result in the book. In short, the authors are well able to present their results in a clear and precise way. I highly recommend this book both to the physicist who is interested in the precise mathematical properties of symmetries and to the mathematician who wants to learn more about the physical applications of algebras.

The book starts with a nice historical introduction to the subject of zeta functions, zeta function regularization and its applications in physics. The author emphasizes some of the advantages of the zeta function regularization method. In particular, the zeta function regularization can conveniently be applied in a curved spacetime. Anybody who wants to get a quick overview of the state of the art in the zeta function regularization method is recommended to read this introduction. Apart from the introduction the main virtue of this book is to provide help to any physics student and/or researcher who encounters zeta functions in his/her work. For this purpose the book provides the reader first of all with many technical details about zeta functions and some of their most important properties. Crucial amongst these properties is the existence of the so-called zeta function regularization theorem. This theorem is discussed and explained in the first part of the book. At a more advanced level, the author emphasizes the importance of a non-trivial extension of the celebrated Chowla--Selberg formula. This extension was invented by the author and plays an important role in some of the physics applications discussed later in the book. The second and main part of the book is devoted to illustrate the zeta function regularization method by giving ten physical applications. These applications cover such a wide range of topics as the Casimir effect, Kaluza--Klein compactification, two- and three-dimensional quantum gravity, extended objects like strings and membranes, critical behaviour of a field theory at non-zero temperature and topological mass generation. It is needless to say that all these applications give an impressive overview of the universality and power of the zeta function regularization method. I like in particular the discussion of the Casimir effect which is done in quite some detail. Apart from the technical discussion the author also explains why the Casimir effect only received so much attention many years after its discovery. Finally, I should note that it is interesting that the book discusses an application to the theory of membranes and, more generally p-branes, which has received such renewed interest in the past year. In short, I can recommend this book to anybody who encounters zeta functions in research and/or education. Undoubtedly, this book will be of great help in taking away some of the confusion on this topic.