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Vertex algebras in higher dimensions and globally conformal invariant quantum field theory. (English) Zbl 1125.17010
The notion of vertex algebras [V. Kac, Vertex Algebras for Beginners, 2nd ed. University Lecture Series. 10. Providence, RI: American Mathematical Society (AMS) (1998; Zbl 0924.17023)] is closely related to the chiral two dimensional conformal field theory, where a vertex operator $Y\left(a,z\right)$ for a state $a$ is a formal power series in a formal or complex variable $z$ and its inverse ${z}^{-1}$. In the paper under review the author proposes an extension of the definition of vertex algebras in higher space-time dimensions. In this context, a vertex operator $Y\left(a,z\right)$ is a formal power series in $D$ variables $z=\left({z}^{1},\cdots ,{z}^{D}\right)$ including negative powers of ${z}^{2}={\left({z}^{1}\right)}^{2}+\cdots +{\left({z}^{D}\right)}^{2}$. The author begins with the axioms of vertex algebras in higher dimensions, which essentially consist of the locality of vertex operators together with the vacuum vector and translation endomorphisms ${T}_{1},...{T}_{D}$. The harmonic decomposition of homogeneous polynomials in $z$ is used for the description of vertex operators. Basic results, such as the existence theorem and the associativity of vertex operators are obtained. Moreover, the author discusses the conformal symmetry and a nondegenerate hermitian form and gives a one-to-one correspondence between the quantum field theory with globally conformal invariance and the vertex algebras in higher dimensions. Examples of free field vertex algebras based on Lie superalgebras of formal distributions are also presented.
##### MSC:
 17B69 Vertex operators; vertex operator algebras and related structures 81R10 Infinite-dimensional groups and algebras motivated by physics
##### Keywords:
vertex algebra; quantum field theory; conformal invariance
##### References:
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