Narrow Results

There are two physical actions that have a natural setting in terms of the coadjoint representation of the algebra of diffeomorphisms and of affine Lie algebras. One is the usual geometric action that comes from coadjoint orbits. The other action lives on the phase space that is transverse to the orbits and are called transverse actions, where Yang-Mills theory in two dimensions is an example. Here we show that the transverse action associated with the Virasoro algebra might contain clues for a theory for dark energy. These actions might also suggests a mechanism for symmetry changing.

We pursue our study of generalized Kac–Moody and Virasoro algebras defined on compact homogeneous manifolds. Extending the well-known vertex operator in the case of the two-torus or the two-sphere, we obtain explicit bosonic realizations of the semi-direct product of the extension of Kac–Moody and Virasoro algebras on ${𝕊}^1\times {𝕊}^1$ and ${𝕊}^2$, respectively. As for the fermionic realization previously constructed, in order to have well defined algebras, we introduce, beyond the usual normal ordering prescription, a regulator and regularize infinite sums by means of the Riemann $\zeta$-function.

By using the notion of extension of Kac–Moody algebras for higher-dimensional compact manifolds recently introduced, we show that for the two-torus ${𝕊}^1\times {𝕊}^1$ and the two-sphere ${𝕊}^2$, these extensions, as well as extensions of the Virasoro algebra can be obtained naturally from the usual Kac–Moody and Virasoro algebras. Explicit fermionic realizations are proposed. In order to have well-defined generators, beyond the usual normal ordering prescription, we introduce a regulator and regularize infinite sums by means of Riemann $\zeta$-function.

In this paper we introduce and study n-point Virasoro algebras, , which are natural generalizations of the classical Virasoro algebra and have as quotients multipoint genus zero Krichever–Novikov type algebras. We determine necessary and sufficient conditions for the latter two such Lie algebras to be isomorphic. Moreover we determine their automorphisms, their derivation algebras, their universal central extensions, and some other properties. The list of automorphism groups that occur is C_n, D_n, A₄, S₄ and A₅. We also construct a large class of modules which we call modules of densities, and determine necessary and sufficient conditions for them to be irreducible.

We introduce the extended Griess algebra for a vertex operator superalgebra and derive its Matsuo-Norton trace formula based on conformal design structure. This article is an exposition of the paper [Y12].