Vertex Algebras and Algebraic Curves (Mathematical Surveys and Monographs) (Mathematical Surveys & Monographs) 2nd Edition by Edward Frenkel (PDF)

Description

Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The notion of a vertex algebra is introduced in a coordinate-independent way, so that vertex operators become well defined on arbitrary smooth algebraic curves, possibly equipped with additional data, such as a vector bundle. Vertex algebras then appear as the algebraic objects encoding the geometric structure of various moduli spaces associated with algebraic curves. Therefore they may be used to give a geometric interpretation of various questions of representation theory. The book contains many original results, introduces important new concepts, and brings new insights into the theory of vertex algebras. The authors have made a great effort to make the book self-contained and accessible to readers of all backgrounds. Reviewers of the first edition anticipated that it would have a long-lasting influence on this exciting field of mathematics and would be very useful for graduate students and researchers interested in the subject. This second edition, substantially improved and expanded, includes several new topics, in particular an introduction to the Beilinson-Drinfeld theory of factorization algebras and the geometric Langlands correspondence. The book is suitable for graduate students and research mathematicians interested in representation theory, algebraic geometry, and mathematical physics.

User’s Reviews

Editorial Reviews: Review From a review of the First Edition: “The authors give a deep new insight into the theory of vertex algebras … many original results, important new concepts and very nice interpretations of structural results in the theory of vertex algebras … provides a natural link with earlier approaches to vertex algebras … The authors also present an excellent introduction to the theory of Wakimoto modules and $mathcal W$-algebras … contains many new concepts and results that are important for the modern theory of vertex algebras.” —- Mathematical Reviews, Featured Review

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Physicists view vertex algebras in many different ways, depending on their area of research. Some have used them without calling them as such, such as those dealing with the quantum operator equations of motion in the Heisenberg picture for potentials of higher order than quadratic. Researchers in string theory will view them in the context of two-dimensional conformal field theory, and in the use of the ubiquitous Virasoro algebra. Those in quantum field theory/elementary particle physics come across them in the context of the operator product expansion.Mathematicians view them in many different ways also. The exciting results in the construction of the Monster group by Richard Borcherds has an interpretation in the theory of vertex algebras. And, as the authors of this book point out, vertex algebras provide a rigorous formulation of two-dimensional conformal field theories. Most concepts in quantum field theory do not have such a rigorous formulation, and so any results that one can get in this regard are very useful to have.This book gives a highly detailed overview of the theory of vertex algebras, so much so that the book is highly dense and requires long blocks of time for readers (such as this reviewer) who are not experts. However, even though the authors are mathematicians, they motivate the subject very well, instead of presenting a “definition-theorem-proof” format. Indeed, at the beginning of the book the authors give a very good summary of the theory and at the beginning of every chapter the authors inform the reader just what they are going to do in that chapter.Loosely speaking, vertex operators can be viewed as generalizations of linear operators acting on a vector space, i.e. if V is a vector space, one can construct linear maps from V into the space of Laurent series with coefficients in V and from the space of bi-infinite Fourier series into the space of Laurent series with coefficients in V. One can also construct a bi-infinite formal series with coefficients in End(V) (the collection of endomorphisms of V), where these coefficients when acting on any element of V will vanish for large values of n. These different conceptions of a vertex operator are all equivalent, but the main issue at hand is the multiplication of vertex operators at the same point. The space End(V) is of course an associative algebra, and one can extend the products of elements of End(V) to the space of vertex operators via the “Wick product,” a construction that is very familiar to those readers working in quantum field theory. All of these considerations are made rigorous in the first chapter of the book, wherein also the axioms for vertex algebras are given explicitly. Noted in these axioms is the presence of a `vacuum vector’ in the vector space V, reflecting of course the connection of vertex operators with quantum field theory. The authors also point out that the axioms of a vertex algebra are natural generalizations of the axioms of an associative commutative algebra with a unit. When a vertex algebra happens to be commutative, it is essentially equivalent to a commutative algebra with a unit and a derivation.The physicist reader will probably appreciate this book more than the mathematician reader, since, again, so many of the constructions have their place (albeit with a rigorous foundation) in physics. Indeed, vertex algebras appear as the chiral symmetry algebras of two-dimensional conformal field theories, such as the Heisenberg vertex algebra, which appears as the free bosonic theory; the Kac-Moody algebra, which appears as Wess-Zumino model, and the Virasoro algebra, appearing as the Belavin-Polyakov-Zamolodchikov minimal model. These are all vertex algebras that are associated to (infinite-dimensional) Lie algebras, and which are discussed in great detail in chapter two of the book.The operator product expansion, again a familiar construction in quantum field theory, wherein the product of two fields at nearby points can be expanded in terms of other fields, is studied in chapter three. The most interesting fact that comes out of this chapter is the power of the locality axiom for vertex algebras, which allows the construction of a vertex operator from knowledge on how it acts on the vacuum vector. This is called the Goddard Uniqueness Theorem, and is proved in this chapter. The authors prove the associativity property of vertex algebras and give examples of the operator product expansion: the Heisenberg, Kac-Moody, and conformal vertex algebras.The mathematician reader, especially one that is working the field of algebraic geometry, will not be disappointed in this book, as the authors connect vertex algebras with vector bundles and algebraic curves. In particular the authors show how to give a geometric realization to a vertex operator and consequently give a global geometric meaning to vertex operators on arbitrary algebraic curves. Central to this discussion is the notion of a conformal and more generally a quasi-conformal vertex algebra, and the action of a group of transformations on the vertex algebra that change coordinates by `internal symmetries.’ This allows a coordinate-independent description of the vertex operation, and this is used to study spaces of coninvariants and conformal blocks associated with a quasi-conformal vertex algebra and a smooth projective curve. The conformal blocks form a vector space that give information on the algebraic curve, and the vector bundles on it. The authors spend a great deal of time discussing how the spaces of conformal blocks change as the complex structure on the curve changes. The authors do this by considering the space of conformal blocks as a sheaf on the moduli space of smooth pointed curves of some genus. This sheaf allows a kind of uniformization on the moduli space involving the action of the Lie algebra of derivations on this space. This is done both for Virasoro vertex algebras as well as affine Kac-Moody algebras. The book ends with a thorough discussion of chiral algebras, which give a coordinate-free approach to the operator product expansion on algebraic curves.

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