Conceptual progress in fundamental theoretical physics is linked with the search
for suitable mathematical structures that model the physics in question. There
are a number indications that today we are in a period where the fundamental
mathematical nature of quantum ﬁeld theory (QFT) and of the worldvolume aspects
of string theory is being identiﬁed. It is not unlikely that future generations will
think of the turn of the millennium and the beginning of the 21st century as the
time when it was fully established that QFT in general and worldvolume theories
in particular are precisely the representations of higher categories of cobordisms
with structure or, dually, encoded by copresheaves of local algebras of observables,
vertex operator algebras, factorization algebras and their siblings.
While signiﬁcant insights on these matters have been gained in the last several
years, their full impact has possibly not yet received due attention, notably not
among most of the theoretical but pure physicists for whom it should be of utmost
relevance. At the same time, those who do appreciate the mathematical structures
involved may wonder how it all ﬁts into the big physical picture of quantum ﬁeld
and string theory.
This volume is aimed at trying to improve on this situation by collecting original
presentations as well as reviews and surveys of recent and substantial progress in
the unravelling of mathematical structures underlying the very nature of quantum
ﬁeld and worldvolume string theory. All contributions have been carefully refereed.
It is reassuring that some of the conferences on fundamental and mathematical
physics these days begin to witness a new, more substantial interaction between
theoretical physicists and mathematicians, where the latter no longer just extract
the isolated remarkable conjectures that the black box string theory has been pro-
ducing over the decades, but ﬁnally hold in their hands a workable axiom system
that allows one to genuinely consider core aspects of QFT in a formal manner. This
book has grown out of the experience of such meetings.
The editors express their thanks to the authors who kindly made their work
available for this volume. We also acknowledge the hard work of the referees. We
thank Sergei Gelfand, Christine Thivierge, and the dedicated staﬀ at the American
Mathematical Society for their eﬀort in publishing this volume. We also thank
Arthur Greenspoon for carefully proofreading the papers and for his input on the
volume as a whole.