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
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. …
منابع مشابه
ar X iv : m at h / 02 01 31 2 v 1 [ m at h . G R ] 3 1 Ja n 20 02 GROUP SPLITTINGS AND ASYMPTOTIC TOPOLOGY
It is a consequence of the theorem of Stallings on groups with many ends that splittings over finite groups are preserved by quasi-isometries. In this paper we use asymptotic topology to show that group splittings are preserved by quasi-isometries in many cases. Roughly speaking we show that splittings are preserved under quasi-isometries when the vertex groups are fundamental groups of aspheri...
متن کاملar X iv : m at h / 02 01 15 9 v 1 [ m at h . D G ] 1 7 Ja n 20 02 Reduction of HKT - Structures
KT-geometry is the geometry of a Hermitian connection whose torsion is a 3-form. HKT-geometry is the geometry of a hyper-Hermitian connection whose torsion is a 3-form. We identify non-trivial conditions for a reduction theory for these types of geometry.
متن کاملar X iv : h ep - t h / 02 01 09 7 v 1 1 5 Ja n 20 02 LORENTZ VIOLATION AT ONE LOOP
The proof of one-loop renormalizability of the general Lorentz-and CPT-violating extension of quantum electrodynamics is described. Application of the renormalization-group method is discussed and implications for theory and experiment are considered.
متن کاملar X iv : h ep - t h / 02 01 18 6 v 1 2 4 Ja n 20 02 Classical Solutions of SU ( 3 ) Pure Yang - Mills Theory
Regular classical solutions of pure SU(3) gauge theories, in Minkowsky spacetime, are computed in the Landau gauge. The classical fields have an intrinsic energy scale and produce quark confinement if interpreted in the sense of a nonrelativistic potential. Moreover, the quark propagator in the background of these fields vanishes at large positive and negative time and space.
متن کاملar X iv : m at h / 02 01 11 6 v 1 [ m at h . A T ] 1 4 Ja n 20 02 GERBES AND HOMOTOPY QUANTUM FIELD THEORIES
For manifolds with freely generated first homology, we characterize gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev's 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes on such spaces as above are the same as a specific class of rank one homotopy quantum field theories.
متن کامل