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Panorama Around the Virasoro Algebra

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Panorama Around the Virasoro Algebra PANORAMA AROUND THE VIRASORO ALGEBRA SEBASTIEN´ PALCOUX Our trip begins from the circle S1. Its space of vector fields Vect(S1) admits inθ d the base dn = ie dθ , with n 2 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. 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 algebras Lg: 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 algebra W is 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 2 S. PALCOUX O. Sheinman has a particular focus on the representation theory part [85]. This theory of Krichever-Novikov algebras is rooted to Kac-Moody algebras as Virasoro algebra is for loop algebras. The Virasoro algebra is linked to modern physical fields: statistical me- chanics (Ising model [67]), conformal field theory [19], and string theory, see Green-Schwarz-Witten [41]. It is also rooted to many many mathematical's fields: to probability theory, see Friedrich-Werner [29], [30]; to complex geom- etry via Teichm¨ullerand moduli spaces (of Riemann surfaces), see Kontsevich [62], Beilinson-Schechtman [5]; to algebraic geometry, see Frenkel-Ben-Zvi [23], Schechtman [83]; to arithmetic geometry via moduli space , see Manin [68], Vaintrob [93], and via geometric Langlands program, see Frenkel [24]; to rep- resentation theory of infinite dimensional group, see Goddman-Wallach [40], Toledano Laredo [88]. This contempory effervescence renewed is exposed in the book of Guieu-Roger [42]. Finally, our particular interest are the connections with operator algebra theory: K-theory, see Kitchloo [60] as the beginnings of a formulation of the Baum-Connes conjecture [94] for infinite dimensional groups; von Neumann algebras: III1 factors, Connes fusion, see Connes [17] (Chapter V, Appendix B), Wassermann [99], Jones [46]; subfactor theory, see Jones [45], Wassermann [98]; Galois correspondence, see Nakamura-Takeda [70], [71], Aubert [3], Oc- neanu [74]. The generalization to higher dimensional manifolds is close to the millenium prize problem called `quantum Yang-Mills theory' (see Jaffe-Witten [44]). It admits a serious cohomological obstruction (see Fuchs [31]): let M a compact k manifold over a field K then H (Vect(M); K) = 0 if k ≤ dimK(M), so that, the existence of a central extension imposes dimK(M) < 2, ie, curves over K. For K = R, there is only S1, and for K = C, the Riemann surfaces. In fact, to obtain higher dimensional manifolds by this way, it is necessary to take several extensions of the real field R via Cayley-Dickson construction: the complex C, the quaternions H, the octonions O, the sedenions S... (see Baez [4] ); but each extension lose an algebraic or topological structure: comparability, commutativity, associativity, to be normed..., so that, the higher generalization is not possible with current knowledge. Neverthelss, R. Borcherds introduces higher-dimensional analogues of vertex algebras [13]. See also Billing-Need [10], Gelfand-Fuchs [33]. For nonassociative original investigation see Jordan- von Neumann-Wigner [48], Albert [1], with a particular interest to their algebra 8 M3, the only \(irreducible) r-number algebra which does not arise from quasi- multiplication ". This dimensional obstruction admits another fruitful viewpoint via Segal's quantization criterion (see [101] p 34): Let G be a group acting unitarily on PANORAMA AROUND THE VIRASORO ALGEBRA 3 an Hilbert space H, then, it exists a canonical way to build projective rep- resentations, by the choice of a projection P on H, with the criterion that [G; P ] ⊂ `2(H) Hilbert-Schmidt operators. Now, let M be a compact smooth manifold. It is completly determined by a spectral triple (A; H; D) (see A. Connes[18]), with A = C1(M) represented on the Hilbert space H = L2(M) , and D a Dirac operator on H. Then M comes with a canonical projector P such that D = (2P − 1):jDj, and so G = Diff(M), admits a canonical pro- jective representation; but the criterion imposes on the spectral triple to have `dimension spectrum' strictely bounded by 2, so that we retrieve dim(M) < 2. But spectral triple framework permits to choose M with 1 < dim(M) < 2 (for example fractals or brownian curves), or to take several many noncom- mutative algebra A, respecting the Segal criterion. Note that, we can define the noncommutative equivalents of a fixed manifold M giving a completely operator algebraic definition of it, and classifying all the associate structure replacing commutative by noncommutative. All this would be the beginning of a completely transversal generalization of our trip. Finally, a more accessible way of generalization is supersymmetry (see [58]). Without going into details, if a manifold M of K-dimension n is defined by charts U ⊂ Kn, a supermanifold M (njm) of superdimension (njm) is defined by supercharts C1(U) ⊗ Λ(Km). The superextensions of a manifold M are obtained as the exterior algebra bundles associated to its finite dimensional vector bundles; so that, there are completely classified by the K-theory functor 0 0 1 1 K0(C(M)) = K (M). Now, K (S ) = Z, so that the superextensions of S (1jN) (1jN) are S and S+ , with N ≥ 1, whose underlying bundles are the trivial one EN and EN−1 ⊕ M, with M the M¨obiusbundle over S1. Now, S1 corresponds (1j1) (1j1) to N = 0; if we begin our trip from the supercircles S and S+ , then, we obtain the two so-called (N = 1)-superVirasoro algebras: the Neveu-Schwarz and the Ramond algebras, [73], [80]. On the litterature, we find partial proofs for the classification of the unitary discrete series and characters: [26], [53], [82], [35]; and a non-unitary proof [2], using Feigin-Fuchs resolutions [20]. References [1] A. A. Albert, A solution of the principal problem in the theory of Riemann matrices. Ann. of Math. (2) 35 (1934), no. 3, 500{515. [2] A. Astashkevich,On the structure of Verma modules over Virasoro and Neveu-Schwarz algebras. Comm. Math. Phys. 186 (1997), no. 3, 531{562. [3] P.L. Aubert, Th´eoriede Galois pour une W ?-alg`ebre. Comment. Math. Helv. 51 (1976), no. 3, 411{433. [4] J. C. Baez The octonions. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 145{205 [5] A. A. Beilinson, V. V. Schechtman, Determinant bundles and Virasoro algebras. Comm. Math. Phys. 118 (1988), no. 4, 651{701. 4 S. PALCOUX [6] L. A. Beklaryan, On analogues of the Tits alternative for groups of homeomorphisms of the circle and the line. (Russian) Mat. Zametki 71 (2002), no. 3, 334{347; translation in Math. Notes 71 (2002), no. 3-4, 305{315 [7] A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Infinite conformal symmetry of critical fluctuations in two dimensions. J. Statist. Phys. 34, 763{774 (1984) [8] A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B 241 (1984), no. 2, 333{380. [9] F. A. Berezin, The method of second quantization. Translated from the Russian by Nobumichi Mugibayashi and Alan Jeffrey. Pure and Applied Physics, Vol. 24 Academic Press, New York-London 1966 xii+228 pp. [10] Y. Billig, K.-H. Neeb, On the cohomology of vector fields on parallelizable manifolds. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 1937{1982. [11] R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068{3071. [12] R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109 (1992), no. 2, 405{444. [13] R. E. Borcherds, Vertex algebras. Topological field theory, primitive forms and related topics (Kyoto, 1996), 35{77, Progr.
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