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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(a,z) 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(a,z) is a formal power series in D variables z=(z 1 ,,z D ) including negative powers of z 2 =(z 1 ) 2 ++(z D ) 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:
17B69Vertex operators; vertex operator algebras and related structures
81R10Infinite-dimensional groups and algebras motivated by physics
References:
[1]Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Reading, MA: Addison?Wesley, 1969
[2]Borcherds, R.E.: Vertex Algebras. In: Topological field theory, primitive forms and related topics, (Kyoto, 1996), Progr. Math., 160, Boston, MA: Birkhäuser Boston, 1998, pp. 35-77
[3]Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Modern Geometry: Methods and Applications. New York: Springer Verlag, 1992
[4]Haag, R.: Local Quantum Physics: Fields, Particles, Algebras, 2nd revised edition. Berlin: Springer?Verlag, 1996
[5]Jost, R.: The General Theory of Quantized Fields. Providence, R.I.: Amer. Math. Soc., 1965
[6]Kac, V.: Vertex Algebras for Beginners. Providence, R.I.: AMS, 1996
[7]Nikolov, N.M., Stanev, Ya.S., Todorov, I.T.: Globally conformal invariant gauge field theory with rational correlation functions. Nucl. Phys. B 670 [FS], 373-400 (2003)
[8]Nikolov, N.M., Todorov, I.T.: Rationality of Conformally Invariant Local Correlation Functions on Compactified Minkowski Space. Commun. Math. Phys. 218, 417-436 (2001) · doi:10.1007/s002200100414
[9]Nikolov, N.M., Todorov, I.T.: Conformal Quantum Field Theory in Two and Four Dimensions. In: Dragovich, B., Sazdovi?, B. (eds.), Proceedings of the Summer School in Modern Mathematical Physics. Belgrade: Belgrade Institute of Physics, 2002, pp. 1-49
[10]Nikolov, N. M., Todorov, I. T.: Finite Temperature Correlation Functions and Modular Forms in a Globally Conformal Invariant QFT, hep-th/0403191.
[11]Robinson, D.W.: On a soluble model of relativistic field theory. Phys. Lett. 9, 189-191 (1964) · doi:10.1016/0031-9163(64)90139-8
[12]Todorov, I.T.: Infinite dimensional Lie algebras in conformal QFT models. In: Barut, A.O., Doebner, H.-D. (eds.), Conformal Groups and Related Symmetries, Lecture Notes in Physics 261, Berlin-Heidelberg-New York: Springer, 1986, pp. 387-443
[13]Uhlmann, A.: Remarks on the future tube. Acta Phys. Pol. 24, 293 (1963); The closure of Minkowski space, ibid. pp. 295-296; Some properties of the future tube, preprint KMU-HEP 7209 (Leipzig, 1972)