Preface
The subject of these lecture notes is modular operads and their rôle in string field
theory. We believe that the approach to string field theory based on homotopy
algebras and their operadic origin can be of interest to both theoretical physicists and
mathematicians. A mathematician can perhaps, starting from operads and homotopy
algebras related to them, find some conceptual explanation for the algebraic
structure of string field theory. A mathematically oriented physicist, starting from
string field theory, may find useful and appealing mathematical language and tools
of the theory of operads. We hope that this text will provide some inspiration to both
of them and thus enhance mutual interaction and progress in both areas.
Our presentation of string field theory is neither a textbook nor it provides a
proper introduction to the subject. This was not our aim. As already mentioned
above, we tried to present it, without going to details, from the perspective of
homotopy algebras and their operadic origin. Concerning operads, the presentation
can be used as an introduction to modular operads, their Feynman transform and
corresponding homotopy algebras.
In Part I, we start with a description of string field theory that should be familiar
to physicists and develop it in a way which makes the appearance of homotopy
algebras transparent. In this approach, Zwiebach’s construction of a string field
theory naturally emerges as a composition of two morphisms of particular odd
modular operads. In more traditional terms, these morphisms correspond to a
decomposition of a moduli space of Riemann surfaces and to conformal field theory,
respectively. A mathematically rigorous description of operads is presented in detail
in Part II. Whenever possible we comment on the connection between the two parts
by at the end of the respective sections.
A more detailed description of the structure of the book and the content of the
individual chapters is given in the introduction. In Part I, written by B. Jurˇ co and
I. Sachs, we comment at the end of each chapter on literature, which either we
followed in our exposition or where an interested reader can find further details. In
Part II, written by M. Doubek and M. Markl, the references are listed at the end of
each chapter.
I.S. and B.J. would like to thank Maxim Grigoriev for helpful comments
on Part I of the book and Kai Cieliebak, Ted Erler, Korbinian Muenster, and
Sebastian Konopka for valuable discussions that helped to develop the ideas that
are presented in this book. I.S. would like to acknowledge the hospitality of Clare
vii
The subject of these lecture notes is modular operads and their rôle in string field
theory. We believe that the approach to string field theory based on homotopy
algebras and their operadic origin can be of interest to both theoretical physicists and
mathematicians. A mathematician can perhaps, starting from operads and homotopy
algebras related to them, find some conceptual explanation for the algebraic
structure of string field theory. A mathematically oriented physicist, starting from
string field theory, may find useful and appealing mathematical language and tools
of the theory of operads. We hope that this text will provide some inspiration to both
of them and thus enhance mutual interaction and progress in both areas.
Our presentation of string field theory is neither a textbook nor it provides a
proper introduction to the subject. This was not our aim. As already mentioned
above, we tried to present it, without going to details, from the perspective of
homotopy algebras and their operadic origin. Concerning operads, the presentation
can be used as an introduction to modular operads, their Feynman transform and
corresponding homotopy algebras.
In Part I, we start with a description of string field theory that should be familiar
to physicists and develop it in a way which makes the appearance of homotopy
algebras transparent. In this approach, Zwiebach’s construction of a string field
theory naturally emerges as a composition of two morphisms of particular odd
modular operads. In more traditional terms, these morphisms correspond to a
decomposition of a moduli space of Riemann surfaces and to conformal field theory,
respectively. A mathematically rigorous description of operads is presented in detail
in Part II. Whenever possible we comment on the connection between the two parts
by at the end of the respective sections.
A more detailed description of the structure of the book and the content of the
individual chapters is given in the introduction. In Part I, written by B. Jurˇ co and
I. Sachs, we comment at the end of each chapter on literature, which either we
followed in our exposition or where an interested reader can find further details. In
Part II, written by M. Doubek and M. Markl, the references are listed at the end of
each chapter.
I.S. and B.J. would like to thank Maxim Grigoriev for helpful comments
on Part I of the book and Kai Cieliebak, Ted Erler, Korbinian Muenster, and
Sebastian Konopka for valuable discussions that helped to develop the ideas that
are presented in this book. I.S. would like to acknowledge the hospitality of Clare
vii