O Virasoro Constructions from Kac-Moody Algebras

This work attacks the problem of constructing energy-momentum tensors and hamiltonians from current algebras. The current algebras considered obey the commutation relations of a (twisted) Kac-Moody (KM) algebra, with the central term corresponding to the Schwinger anomaly. We generalise previous constructions by including the case of arbitrarily twisted KM algebras, using a new ordering function, and looking at some solutions of the Virasoro Master Equation, where one replaces the Cartan-Killing metric over which the KM fields are normally summed by an arbitrary symmetric matrix. The technique used throughout is that of calculating the commutation relations rather than the operator product expansions. We use mathematical results concerning the representation theory of Lie algebras (both finite and infinite dimensional) and their automorphisms to carry out our constructions. The main results are as follows. We exhibit a new ordering function--as distinct from the usual "normal ordering" in quantum field theory--which when used with arbitrarily twisted KM fields will always allow a Sommerfield -Sugawara construction which is bilinear in the fields. In some cases, the conventional normal ordering will still allow a bilinear construction using twisted fields, and we state a condition (involving the automorphism of the Lie algebra which defines the twist) for this to occur. For the construction from twisted free fields, both bosonic and fermionic, we derive the form of the linear part of the central term (which corresponds to a shift in the energy of the hamiltonian), and show that it is independent of conformal spin. We then compare the Virasoro constructions for arbitrarily and "canonically" twisted KM algebras, develop a condition for the centers to agree, and show some examples where this occurs. Finally we show that for the Virasoro Master Equation, a bilinear construction that sums over an arbitrary matrix rather than the Cartan-Killing form, the Sommerfield-Sugawara and Goddard-Kent-Olive solutions arise by restricting the form of the matrix in a special way. We also examine the effect of using the new ordering described above, and of the Weyl group symmetries of the KM algebra on this generalised construction.