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This book gives a modern presentation of modular operands and their
role in string field theory. The authors aim to outline the
arguments from the perspective of homotopy algebras and their
operadic origin. Part I reviews string field theory from the point
of view of homotopy algebras, including A-infinity algebras, loop
homotopy (quantum L-infinity) and IBL-infinity algebras governing
its structure. Within this framework, the covariant construction of
a string field theory naturally emerges as composition of two
morphisms of particular odd modular operads. This part is intended
primarily for researchers and graduate students who are interested
in applications of higher algebraic structures to strings and
quantum field theory. Part II contains a comprehensive treatment of
the mathematical background on operads and homotopy algebras in a
broader context, which should appeal also to mathematicians who are
not familiar with string theory.
This book brings together both the classical and current aspects of
deformation theory. The presentation is mostly self-contained,
assuming only basic knowledge of commutative algebra, homological
algebra and category theory. In the interest of readability, some
technically complicated proofs have been omitted when a suitable
reference was available. The relation between the uniform
continuity of algebraic maps and topologized tensor products is
explained in detail, however, as this subject does not seem to be
commonly known and the literature is scarce. The exposition begins
by recalling Gerstenhaber's classical theory for associative
algebras. The focus then shifts to a homotopy-invariant setup of
Maurer-Cartan moduli spaces. As an application, Kontsevich's
approach to deformation quantization of Poisson manifolds is
reviewed. Then, after a brief introduction to operads, a strongly
homotopy Lie algebra governing deformations of (diagrams of)
algebras of a given type is described, followed by examples and
generalizations.
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