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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.

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

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Published by | mijec |

Reads | 32 |

Language | English |

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