##
**Vertex operator algebras and the monster.**
*(English)*
Zbl 0674.17001

A vertex operator algebra is a \(\mathbb{Z}\)-graded vector space \(V=\oplus_{n\in \mathbb{Z}}V_{(n)}\), such that the \(V_{(n)}\) are all finite-dimensional and are 0 for \(n\ll 0\), together with a linear map \(v\mapsto Y(v,z)\) from \(V\) to the space of formal Laurent series in an indeterminate \(z\) with coefficients in \(\operatorname{End}(V)\). The formal series \(Y(v,z)\) are called vertex operators. Vertex operator algebras are neither commutative nor associative in general; instead, the commutation relations are expressed in terms of a generalized “Jacobi identity”:
\[
z_0^{-1}\delta (\frac{z_1 - z_2}{z_0})Y(u,z_1)Y(v,z_2) - z_0^{- 1}\delta (\frac{z_2 - z_1}{-z_0})Y(v,z_2)Y(u,z_1)=
\]

\[ z_2^{-1}\delta (\frac{z_1-z_0}{z_ 2})Y(Y(u,z_0)v,z_2). \] Here, \(\delta\) (\(\frac{z_1-z_2}{z_0})=\sum_{n\in\mathbb{Z}}(\frac{z_1-z_2}{z_0})^n\), expanded in powers of \(z_2\), and similarly for the other \(\delta\)-functions. Further, there is an element \(w\in V\) such that the coefficients of \(Y(w,z)=\sum_{n\in {\mathbb{Z}}}Y_nz^{-n-2}\) are a representation of the Virasoro algebra \[ [Y_m,Y_n] = (m-n)Y_{m+n}+((m^3-m)/12)\delta_{m+n,0} c. \] The central charge \(c\) is called the rank of \(V\). The graded dimension of \(V\) is the formal series \(\sum_{n\in\mathbb{Z}}q^n \dim V_{(n)}.\)

One of the principal constructions described in the book is that of a vertex operator algebra \(V_L\) associated to any even integral unimodular lattice \(L\); one has \(\operatorname{rank} V_L = \operatorname{rank} L\). When \(L\) is the root lattice of a simple Lie algebra \(\mathfrak g\) of type A, D or E, the algebra \(V_L\) can be used to construct the basic representation of the affine Lie algebra \(\widehat{\mathfrak g}.\)

A “twisted” version of this construction gives a second vertex operator algebra \(V_L'\). When \(L\) is the Leech lattice, these constructions combine to give a vertex operator algebra \(V^{\#}=V^+_L\oplus V_L^{+'},\) where the superscripts \(+\) denote the subspaces fixed by certain involutions of \(V_L\) and \(V_L'\).

Theorem: One has (a) \(\operatorname{rank}V^{\#}=24\) and the graded dimension of \(V^{\#}\) is the elliptic modular function \[ J = q^{-1}+196884 q+21493760 q^2+...; \] (b) the group of grading-preserving automorphisms of the vertex operator algebra \(V^{\#}\) is the Monster simple group.

These results provide a realization of part of “monstrous moonshine” [J. H. Conway and S. P. Norton, Bull. Lond. Math. Soc. 11, 308–339 (1979; Zbl 0424.20010)]. We remark that the homogeneous component \(V^{\#}_{-1}\) of \(V^{\#}\) is the Griess algebra, which was used in the original construction of the Monster [R. L. Griess, Invent. Math. 69, 1–102 (1982; Zbl 0498.20013)].

This book provides the first detailed exposition of the above results on vertex operator algebras, which are mainly due to the authors of the book and independently to Borcherds [R. E. Borcherds, Proc. Natl. Acad. Sci. USA 84, 3068–3071 (1986; Zbl 0613.17012)]. Vertex operator algebras can be regarded as the algebraic foundation of conformal field theory. The authors discuss this connection, as well as its relationship to the complex analytic approach.

To indicate the organization of the book, we conclude with a list of chapter headings: (1) Lie algebras. (2) Formal calculus. (3) Realization of \({\mathfrak sl}{\hat{\;}}(2)\) by twisted vertex operators. (4) Realization of \(\mathfrak{sl}{\hat{\;}}(2)\) by untwisted vertex operators. (5) Central extensions. (6) The simple Lie algebras \(A_n\), \(D_n\), \(E_n\). (7) Vertex operator realizations of \(\hat A_n\), \(\hat D_n\), \(\hat E_n\). (8) General theory of untwisted vertex operators. (9) General theory of twisted vertex operators. (10) The Moonshine module. (11) Triality. (12) The main theorem. (13) Completion of the proof. Appendix: complex realization of vertex operator algebras.

\[ z_2^{-1}\delta (\frac{z_1-z_0}{z_ 2})Y(Y(u,z_0)v,z_2). \] Here, \(\delta\) (\(\frac{z_1-z_2}{z_0})=\sum_{n\in\mathbb{Z}}(\frac{z_1-z_2}{z_0})^n\), expanded in powers of \(z_2\), and similarly for the other \(\delta\)-functions. Further, there is an element \(w\in V\) such that the coefficients of \(Y(w,z)=\sum_{n\in {\mathbb{Z}}}Y_nz^{-n-2}\) are a representation of the Virasoro algebra \[ [Y_m,Y_n] = (m-n)Y_{m+n}+((m^3-m)/12)\delta_{m+n,0} c. \] The central charge \(c\) is called the rank of \(V\). The graded dimension of \(V\) is the formal series \(\sum_{n\in\mathbb{Z}}q^n \dim V_{(n)}.\)

One of the principal constructions described in the book is that of a vertex operator algebra \(V_L\) associated to any even integral unimodular lattice \(L\); one has \(\operatorname{rank} V_L = \operatorname{rank} L\). When \(L\) is the root lattice of a simple Lie algebra \(\mathfrak g\) of type A, D or E, the algebra \(V_L\) can be used to construct the basic representation of the affine Lie algebra \(\widehat{\mathfrak g}.\)

A “twisted” version of this construction gives a second vertex operator algebra \(V_L'\). When \(L\) is the Leech lattice, these constructions combine to give a vertex operator algebra \(V^{\#}=V^+_L\oplus V_L^{+'},\) where the superscripts \(+\) denote the subspaces fixed by certain involutions of \(V_L\) and \(V_L'\).

Theorem: One has (a) \(\operatorname{rank}V^{\#}=24\) and the graded dimension of \(V^{\#}\) is the elliptic modular function \[ J = q^{-1}+196884 q+21493760 q^2+...; \] (b) the group of grading-preserving automorphisms of the vertex operator algebra \(V^{\#}\) is the Monster simple group.

These results provide a realization of part of “monstrous moonshine” [J. H. Conway and S. P. Norton, Bull. Lond. Math. Soc. 11, 308–339 (1979; Zbl 0424.20010)]. We remark that the homogeneous component \(V^{\#}_{-1}\) of \(V^{\#}\) is the Griess algebra, which was used in the original construction of the Monster [R. L. Griess, Invent. Math. 69, 1–102 (1982; Zbl 0498.20013)].

This book provides the first detailed exposition of the above results on vertex operator algebras, which are mainly due to the authors of the book and independently to Borcherds [R. E. Borcherds, Proc. Natl. Acad. Sci. USA 84, 3068–3071 (1986; Zbl 0613.17012)]. Vertex operator algebras can be regarded as the algebraic foundation of conformal field theory. The authors discuss this connection, as well as its relationship to the complex analytic approach.

To indicate the organization of the book, we conclude with a list of chapter headings: (1) Lie algebras. (2) Formal calculus. (3) Realization of \({\mathfrak sl}{\hat{\;}}(2)\) by twisted vertex operators. (4) Realization of \(\mathfrak{sl}{\hat{\;}}(2)\) by untwisted vertex operators. (5) Central extensions. (6) The simple Lie algebras \(A_n\), \(D_n\), \(E_n\). (7) Vertex operator realizations of \(\hat A_n\), \(\hat D_n\), \(\hat E_n\). (8) General theory of untwisted vertex operators. (9) General theory of twisted vertex operators. (10) The Moonshine module. (11) Triality. (12) The main theorem. (13) Completion of the proof. Appendix: complex realization of vertex operator algebras.

Reviewer: Andrew N. Pressley (London)

### MSC:

17B69 | Vertex operators; vertex operator algebras and related structures |

17B68 | Virasoro and related algebras |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

20D08 | Simple groups: sporadic groups |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |