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This book collects various perspectives, contributed by both
mathematicians and physicists, on the B-model and its role in
mirror symmetry. Mirror symmetry is an active topic of research in
both the mathematics and physics communities, but among
mathematicians, the "A-model" half of the story remains much
better-understood than the B-model. This book aims to address that
imbalance. It begins with an overview of several methods by which
mirrors have been constructed, and from there, gives a thorough
account of the "BCOV" B-model theory from a physical perspective;
this includes the appearance of such phenomena as the holomorphic
anomaly equation and connections to number theory via modularity.
Following a mathematical exposition of the subject of quantization,
the remainder of the book is devoted to the B-model from a
mathematician's point-of-view, including such topics as polyvector
fields and primitive forms, Givental's ancestor potential, and
integrable systems.
An introduction to the theory of orbifolds from a modern
perspective, combining techniques from geometry, algebraic topology
and algebraic geometry. One of the main motivations, and a major
source of examples, is string theory, where orbifolds play an
important role. The subject is first developed following the
classical description analogous to manifold theory, after which the
book branches out to include the useful description of orbifolds
provided by groupoids, as well as many examples in the context of
algebraic geometry. Classical invariants such as de Rham cohomology
and bundle theory are developed, a careful study of orbifold
morphisms is provided, and the topic of orbifold K-theory is
covered. The heart of this book, however, is a detailed description
of the Chen-Ruan cohomology, which introduces a new product for
orbifolds and has had significant impact in recent years. The final
chapter includes explicit computations for a number of interesting
examples.
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