PANORAMA AROUND THE VIRASORO ALGEBRA

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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.

  • fuchs

  • vertex algebras

  • lie algebra

  • connes fusion

  • algebra vir

  • conformal invariance

  • dimensional quantum


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PANORAMA AROUND THE VIRASORO ALGEBRA
´ SEBASTIEN PALCOUX
1 1 Our trip begins from the circleS. Its space of vector fields Vect(S) admits inθ d the basedn=ie, withnZ, so that their commutator comes easily: [dm, dn] = (mn)dm+ncomplex Lie algebra they generate is called Witt. The algebraWwas first defined in 1909 by E. Cartan [15], (and admits. It 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 toWits unique central extension [32], called Virasoro algebraVir, after works of A. Virasoro [97]. Then,Virappears 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. The princi-pal mathematicians and physicists who participated in the demonstration are: Feigin-Fuchs [20], [21]; Friedan-Qiu-Shenker [25], [27], [28]; Fuchs-Gelfand [33]; Goddard-Kent-Olive [35]; Kac [49], [50]; Kac-Peterson [51]; Kent [59]; Lang-lands [64]. A key point were the knowledge of representations theory and characters (Weyl-Kac formula) of the loop algebrasLg: the first example of general Kac-Moody algebra [56]. All this effervescence was edited by Goddard-Olive in the book [36], and recasted by Kac-Raina in [54] and by Wassermann in [100]. The new concept emerging from these works is vertex operator alge-bra, discovered as current algebra for 2-dimensional quantum field theory, by Knizhnik-Zamolochikov [61]; and define in a rigorous mathematical context by Richards Borcherds [11], offering monstrous mathematical consequences [12]. See [37], [57] and [65] as three differents focus of vertex framework digestion. Now, the Witt algebraWis also the Lie algebra of meromorphic vector fields defined on the Riemann sphere, that are holomorphic except at two fixed poles; this other viewpoint has enabled Krichever-Novikov to begin the generalization of the Virasoro algebra, with general compact Riemann surfaces [63], and with more than two poles by M. Schlichenmaier [84]; its coautor 1