Niveau: Supérieur, Doctorat, Bac+8
PANORAMA AROUND THE VIRASORO ALGEBRA SEBASTIEN PALCOUX Our trip begins from the circle S1. Its space of vector fields Vect(S1) admits the base dn = iein? dd? , with n ? Z, so that their commutator comes easily: [dm, dn] = (m?n)dm+n. The complex Lie algebra they generate is called Witt algebra W. It was first defined in 1909 by E. Cartan [15], (and admits p-adic analogues after Witt's works [104]). Then in 1966 , this object won the interest of physics [9], but it appears with a little anomaly, for the needs of ‘second quantization'. This anomaly admits the concrete interpretation to be mathe- matically responsible of the energy of the vacuum (see [42] p 764). Next, in 1968, it appears in mathematics as a 2-cocycle, giving to W its unique central extension [32], called Virasoro algebra Vir, after works of A. Virasoro [97]. Then, Vir appears in many statistical mechanics contexts (Potts, Ising mod- els, see [67]), in fact related to differents representations of a particular kind: unitary and highest weight. And so these representations enjoyed to be study for themselves: it's the birth of the mathematical physics conformal field the- ory, with Belavin-Polyakov-Zamolodchikov's seminal papers as starting point [8], [7], where the discrete series classification is first conjectured.
- fuchs
- vertex algebras
- lie algebra
- connes fusion
- algebra vir
- conformal invariance
- dimensional quantum