ar X iv : m at h / 02 01 31 3 v 1 [ m at h . R T ] 3 1 Ja n 20 02 To Robert Moody Energy - momentum tensor for the toroidal

نویسنده

  • Yuly Billig
چکیده

Energy-momentum tensor for the toroidal Lie algebras. Abstract. We construct vertex operator representations for the full (N + 1)-toroidal Lie algebra g. We associate with g a toroidal vertex operator algebra, which is a tensor product of an affine VOA, a sub-VOA of a hyperbolic lattice VOA, affine sl N VOA and a twisted Heisenberg-Virasoro VOA. The modules for the toroidal VOA are also modules for the toroidal Lie algebra g. We also construct irreducible modules for an important subalgebra g div of the toroidal Lie algebra that corresponds to the divergence free vector fields. This subalgebra carries a non-degenerate invariant bilinear form. The VOA that controls the representation theory of g div is a tensor product of an affine VOA V ˙ g (c) at level c, a sub-VOA of a hyperbolic lattice VOA, affine sl N VOA and a Virasoro VOA at level c L with the following condition on the central charges: 2(N + 1) + rank V ˙ g (c) + c L = 26. Toroidal Lie algebras are very natural multi-variable generalizations of affine Kac-Moody algebras. The theory of affine Lie algebras is rich and beautiful, and has many important applications in physics. By large, applications of toroidal Lie algebras in physics are still to be discovered. We should mention however the papers [IKUX], [IKU], where the toroidal symmetry is discussed in the context of a 4-dimensional conformal field theory. We hope that the development of the representation theory of toroidal Lie algebras will help to find the proper place for these algebras in physical theories. The construction of a toroidal Lie algebra is totally parallel to the well-known construction of an (untwisted) affine Kac-Moody algebra [K1]. One starts with a finite-dimensional simple Lie algebra ˙ g and considers maps from an N + 1-dimensional torus into ˙ g. We may identify the algebra of functions on a torus with the Laurent polynomial algebra R = C[t ± 0 , t ± 1 ,. .. , t ± N ], by taking the Fourier basis, setting t k = e ix k. The Lie algebra of the ˙ g-valued maps from a torus will then become C[t ± 0 , t ± 1 Just as in affine case, one builds the universal central extension (R ⊗ ˙ g) ⊕ K of R ⊗ ˙ g. However when N ≥ 1, the center K is infinite-dimensional. …

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تاریخ انتشار 2002