Xu, Xiaoping Introduction to vertex operator superalgebras and their modules. (English) Zbl 0929.17030 Mathematics and its Applications (Dordrecht). 456. Dordrecht: Kluwer. xvi, 356 p. (1998). Ever since the first book on vertex operator algebras by I. B. Frenkel, J. Lepowsky and A. Meurman was published in 1988 [Academic Press, Zbl 0674.17001 ], and a year later (as a preprint) a book by I. B. Frenkel, Y.-Z. Huang and J. Lepowsky [On axiomatic approaches to vertex operator algebras and modules, Mem. Am. Math. Soc. 494 (1993; Zbl 0789.17022)], there has been a rapid development of the theory with numerous papers and several books. The objective of the book under review is to give a systematic update approach to the basic concepts, techniques and examples in vertex operator superalgebras and their twisted modules.The book is divided into two parts. The first part is about self-dual codes (Chapter 1) and self-dual lattices (Chapter 2). The second part is about vertex operator superalgebras and their modules, and contains five chapters: 3. Definitions and general properties, 4. Conformal superalgebras, affine Kac-Moody algebras and KZ equations, 5. Analogue of the highest weight theory, 6. Lattice vertex operator superalgebras, 7. VOSAs generated by their subspaces of small weights.Although the selection and the exposition of the material reflects the author’s own work and research interests, a number of basic results of the theory are presented, generally stated and proved for twisted modules. So Chapter 3 is, in a way, a twisted version of the above mentioned axiomatic approach of Frenkel, Huang and Lepowsky, including the definition and properties of intertwining operators among twisted modules. This chapter also includes H. Li’s theorem on the change of twisted vertex operators by the vertex operator of certain “weight-1” element, the author’s generator theorem, H. Li’s work on invariant bilinear forms on twisted modules, and the results of C. Dong, H. Li and G. Mason on the subalgebra of a vertex operator superalgebra consisting of the invariants under a finite-dimensional compact automorphism subgroup.The main examples of vertex operator algebras are VOAs associated to affine Kac-Moody algebras (Chapter 4), Virasoro vertex operator algebra (Chapter 5) and lattice vertex operator superalgebras (Chapter 6). In each case the irreducible modules are described and the intertwining operators are determined: in the case of affine algebras the theorem of I. B. Frenkel and Y. Zhu on fusion rules is proved by an explicit approach, which may be viewed as elicitation of the general theory of Frenkel and Zhu.In Chapter 5 the author generalizes Zhu’s theory of the associative algebras to twisted modules of a vertex operator superalgebra, and briefly discusses the results of W. Wang on irreducible representations and fusion rules for Virasoro algebra. Finally, in the case of lattice vertex operator algebras, the author classifies all irreducible twisted modules and, by generalizing the results of C. Dong and J. Lepowsky, determines the fusion rules.In Chapter 4 the author also presents the connection of conformal superalgebras, introduced and studied in [V. G. Kac, Vertex algebras for beginners, University Lecture Series, Vol. 10, AMS, Providence (1996; Zbl 0861.17017), 2nd ed. (1998; Zbl 0924.17023)], with vertex operator superalgebras. These results are used in the construction of VOAs associated to affine and Virasoro algebras, in the discussion on Zhu’s algebra in Chapter 5, and finally in Chapter 7, which contains a classification of simple vertex operator superalgebras generated by their subspaces of weight 1, due to B. Lian, and weight 1/2, due to the author, V. G. Kac’s description of the \(N=2\), \(3\) and \(4\) superconformal algebras, and the author’s results on a new family of infinite-dimensional Lie superalgebras.In order to give the reader a better picture of lattice vertex operator superalgebras, the author included his work on self-dual codes and lattices [see X. Xu, Commun. Algebra 22, 1281-1303 (1994; Zbl 0820.17028)] as the first part of this book. The introduction gives a good overview of the material covered and the bibliography contains almost two hundred items referred to in the text. The book is well written, with many improvements of known results and existing proofs. The researchers and the graduate students will use this book both as a graduate textbook and as a useful reference on twisted modules of vertex operator superalgebras. Reviewer: Mirko Primc (Zagreb) Cited in 4 ReviewsCited in 56 Documents MSC: 17B69 Vertex operators; vertex operator algebras and related structures 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras 94B05 Linear codes (general theory) 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations Keywords:vertex operator superalgebras; self-dual codes; self-dual lattices; conformal superalgebras; affine Kac-Moody algebras; Knizhnik-Zamolodchikov equations; highest weight analogues; lattice vertex operator superalgebras; twisted modules; intertwining operators; Virasoro vertex operator algebra; fusion rules Citations:Zbl 0789.17022; Zbl 0674.17001; Zbl 0861.17017; Zbl 0924.17023; Zbl 0820.17028 PDFBibTeX XMLCite \textit{X. Xu}, Introduction to vertex operator superalgebras and their modules. Dordrecht: Kluwer (1998; Zbl 0929.17030)