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The study of W algebras began in 1985 in the context of
two-dimensional conf- mal field theories, the aim being to explore
higher-spin extensions of the Virasoro algebra. Given the
simultaneous growth in the understanding of two-dimensional metric
gravity inspired by analyses of string models, it was inevitable
that these algebras would be applied to give analogues of putative
higher-spin gravity t- ories. This book is an exposition of the
past few years of our work on such an application for the algebra:
in particular, the BRST quantization of the n- critical 4D string.
We calculate the physical spectrum as a problem in BRST cohomology.
The corresponding operator cohomology forms a BV algebra, for which
we provide a geometrical model. The algebra has one further
generator, of spin three, in addition to the (spin two)
energy-momentum tensor which generates the Virasoro algebra. C-
trary to the Virasoro algebra, it is an algebra defined by
nonlinear relations. In deriving our understanding of the resulting
gravity theories we have had to - velop a number of results on the
representation theory of W algebras, to replace the standard
techniques that were so successful in treating linear algebras.
In the last decade there has been an extraordinary confluence of
ideas in mathematics and theoretical physics brought about by
pioneering discoveries in geometry and analysis. The various
chapters in this volume, treating the interface of geometric
analysis and mathematical physics, represent current research
interests. No suitable succinct account of the material is
available elsewhere. Key topics include: * A self-contained
derivation of the partition function of Chern- Simons gauge theory
in the semiclassical approximation (D.H. Adams) * Algebraic and
geometric aspects of the Knizhnik-Zamolodchikov equations in
conformal field theory (P. Bouwknegt) * Application of the
representation theory of loop groups to simple models in quantum
field theory and to certain integrable systems (A.L. Carey and E.
Langmann) * A study of variational methods in Hermitian geometry
from the viewpoint of the critical points of action functionals
together with physical backgrounds (A. Harris) * A review of
monopoles in nonabelian gauge theories (M.K. Murray) * Exciting
developments in quantum cohomology (Y. Ruan) * The physics origin
of Seiberg-Witten equations in 4-manifold theory (S. Wu) Graduate
students, mathematicians and mathematical physicists in the
above-mentioned areas will benefit from the user-friendly
introductory style of each chapter as well as the comprehensive
bibliographies provided for each topic. Prerequisite knowledge is
minimal since sufficient background material motivates each
chapter.
In recent years, there has been tremendous progress on the
interface of geometry and mathematical physics. This book reflects
the expanded articles of several lectures in these areas delivered
at the University of Adelaide, with an audience of primarily
graduate students. The aim of this volume is to provide surveys of
recent progress without assuming too much prerequisite knowledge
and with a comprehensive bibliography, so that researchers and
graduate students in geometry and mathematical physics will
benefit. The contributors cover a number of areas in mathematical
physics. Chapter 1 offers a self-contained derivation of the
partition function of Chern-Simons gauge theory in the
semiclassical approximation. Chapter 2 considers the algebraic and
geometric aspects of the Knizhnik-Zamolodchikov equations in
conformal field theory, including their relation to the braid
group, quantum groups and infinite dimensional Lie algebras.
Chapter 3 surveys the application of the representation theory of
loop groups to simple models in quantum field theory and to certain
integrable systems. Chapter 4 examines the variational methods in
Hermitian geometry from the viewpoint of the critical points of
action functionals together with physical backgrounds. Chapter 5 is
a review of monopoles in non-Abelian gauge theories and the various
approaches to understanding them. Chapter 6 covers much of the
exciting recent developments in quantum cohomology, including
relative Gromov-Witten invariant, birational geometry, naturality
and mirror symmetry. Chapter 7 explains the physics origin of the
Seiberg-Witten equations in four-manifold theory and a number of
important concepts in quantum field theory, such asvacuum, mass
gap, (super)symmetry, anomalies and duality. Contributors: D.H.
Adam, P. Bouwknegt, A.L. Carey, A. Harris, E. Langmann, M.K.
Murray, Y. Ruan, S. Wu D. H. Adams: Semiclassical Approximation in
Chern-Simons Gauge Theory P. Bouwknegt: The Knizhnik-Zamolodchikov
Equations A. L. Carey and E. Langmann: Loop Groups and Quantum
Fields A. Harris: Some Applications of Variational Calculus in
Hermitian Geometry M. K. Murray: Monopoles Y. Ruan: On
Gromov-Witten Invariants and Quantum Cohomology S. Wu The Geometry
and Physics of the Seiberg-Witten Equations
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