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 ∈ 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 models, 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 theory, with Belavin-Polyakov-Zamolodchikov’s seminal papers as starting point [8], [7], where the discrete series classification is first conjectured. The principal 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 algebra, 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 mechanics (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 geometry via Teichmüllerand 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 representation 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 representations, 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 = C∞(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).|D|, and so G = Diff(M), admits a canonical projective 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 noncommutative 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 (n|m) of superdimension (n|m) is defined by supercharts C∞(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 (1|N) (1|N) are S and S+ , with N ≥ 1, whose underlying bundles are the trivial one EN and EN−1 ⊕ M, with M the Möbiusbundle over S1. Now, S1 corresponds (1|1) (1|1) 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. Math., 160, BirkhäuserBoston, Boston, MA, 1998. [14] R. E. Borcherds, Quantum vertex algebras. Taniguchi Conference on Mathematics Nara ’98, 51–74, Adv. Stud. Pure Math., 31, Math. Soc. Japan, Tokyo, 2001. [15] E. Cartan, Les groupes de transformations continus, infinis, simples. (French) Ann. Sci. cole Norm. Sup. (3) 26 (1909), 93–161. [16] A. Connes, J. Feldman, B. Weiss, An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynamical Systems 1 (1981), no. 4, 431–450 (1982). [17] A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. [18] A. Connes, On the spectral characterization of manifolds , http://www.alainconnes.org/docs/reconstructionshort.pdf (2008) [19] P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal field theory. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997. xxii+890 pp. [20] B. L. Feigin, D. B. Fuchs, Verma modules over the Virasoro algebra. Topology (Leningrad, 1982), 230–245, Lecture Notes in Math., 1060, Springer, Berlin, 1984. [21] B. L. Feigin, D. B. Fuchs, Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody Lie algebras. J. Geom. Phys. 5 (1988), no. 2, 209–235. [22] I. B. Frenkel, Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory. J. Funct. Anal. 44 (1981), no. 3, 259–327 [23] E. Frenkel, D. Ben-Zvi, Vertex algebras and algebraic curves. Second edition. Mathe- matical Surveys and Monographs, 88. American Mathematical Society, Providence, RI, 2004. xiv+400 pp. [24] E. Frenkel, Lectures on the Langlands program and conformal field theory. Frontiers in number theory, physics, and geometry. II, 387–533, Springer, Berlin, 2007. [25] D. Friedan, Z. Qiu, S. Shenker, Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett. 52 (1984), no. 18, 1575–1578. [26] D. Friedan, Z. Qiu, S. Shenker, Superconformal invariance in two dimensions and the tricritical Ising model. Phys. Lett. B 151 (1985), no. 1, 37–43. PANORAMA AROUND THE VIRASORO ALGEBRA 5

[27] D. Friedan, Z. Qiu, S. Shenker, Conformal invariance, unitarity and two-dimensional critical exponents. Vertex operators in mathematics and physics (Berkeley, Calif., 1983), 419–449, Math. Sci. Res. Inst. Publ., 3, Springer, New York, 1985. [28] D. Friedan, Z. Qiu, S. Shenker, Details of the nonunitarity proof for highest weight representations of the Virasoro algebra. Comm. Math. Phys. 107 (1986), no. 4, 535– 542. [29] R. Friedrich, W. Werner, Conformal fields, restriction properties, degenerate representations and SLE. C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, 947–952. [30] R. Friedrich, W. Werner, Conformal restriction, highest-weight representations and SLE. Comm. Math. Phys. 243 (2003), no. 1, 105–122. [31] D. B. Fuchs, Cohomology of infinite-dimensional Lie algebras. Translated from the Russian by A. B. Sosinski?. Contemporary Soviet Mathematics. Consultants Bureau, New York, 1986. [32] D. B. Fuchs, I. M. Gelfand, Cohomologies of the Lie algebra of vector fields on the circle. (Russian) Funkcional. Anal. i Prilozhen. 2 1968 no. 4, 92–93. (English translation in Functional Anal. Appl. 2 (1968), no. 4, 342–343). [33] D. B. Fuchs, I. M. Gelfand, Cohomologies of the Lie algebra of vector fields on a manifold. (Russian) Funkcional. Anal. i Prilozhen. 3 1969 no. 2, 87. [34] E. Ghys, Groups acting on the circle. Enseign. Math. (2) 47 (2001), no. 3-4, 329–407. [35] P. Goddard, A. Kent, D. Olive, Unitary representations of the Virasoro and super- Virasoro algebras. Comm. Math. Phys. 103 (1986), no. 1, 105–119. [36] P. Goddard, D. Olive, eds, Kac-Moody and Virasoro algebras. Advanced Series in Math- ematical Physics, 3. World Scientific Publishing Co., Singapore, 1988. [37] P. Goddard, Meromorphic conformal field theory. Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), 556–587, Adv. Ser. Math. Phys., 7, World Sci. Publ., Teaneck, NJ, 1989. [38] V. J. Golodec, Conditional expectations and modular automorphisms of von Neumann algebras. (Russian) Funkcional. Anal. i Prilozhen. 6 (1972), no. 3, 68–69. (English translation: Functional Anal. Appl. 6 (1972), no. 3, 231–232). [39] R. Goodman, N. R. Wallach, Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle. J. Reine Angew. Math. 347 (1984), 69–133. [40] R. Goodman, N. R. Wallach, Projective unitary positive-energy representations of Diff(S1). J. Funct. Anal. 63 (1985), no. 3, 299–321. [41] M. B. Green, J. H. Schwarz, E. Witten, Superstring theory. Vol.1 and Vol.2; Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1987. [42] L. Guieu, C. Roger, L’algèbre et le groupe de Virasoro. Aspects géométriques et algébriques,généralisations. Les Publications CRM, Montreal, QC, 2007. [43] K. Iohara, Y. Koga, Fusion algebras for N = 1 superconformal field theories through coinvariants. II. N = 1 super-Virasoro-symmetry. J. Lie Theory 11 (2001), no. 2, 305– 337 [44] A. Jaffe, E. Witten, Quantum Yang-Mills theory. The millennium prize problems, 129– 152, Clay Math. Inst., Cambridge, MA, 2006. [45] V.F.R. Jones, Index for subfactors. Invent. Math. 72 (1983), no. 1, 1–25. [46] V.F.R. Jones, Fusion en algèbres de von Neumann et groupes de lacets (d’aprèsA. Wassermann)., Sminaire Bourbaki, Vol. 1994/95. AstérisqueNo. 237 (1996), Exp. No. 800, 5, 251–273. 6 S. PALCOUX

[47] V.F.R. Jones, V. S. Sunder, Introduction to subfactors. London Mathematical Society Lecture Note Series, 234. Cambridge University Press, 1997. [48] P. Jordan, J. von Neumann, E. Wigner, On an algebraic generalization of the quantum mechanical formalism. Ann. of Math. (2) 35 (1934), no. 1, 29–64. [49] V. G. Kac, Infinite-dimensional Lie algebras, and the Dedekind η-function. (Russian) Funkcional. Anal. i Prilozen. 8 (1974), no. 1, 77–78 (English translation: Functional Anal. Appl. 8 (1974), 68–70). [50] V. G. Kac, Contravariant form for infinite-dimensional Lie algebras and superalgebras Lecture Notes in Physics, vol. 94, 1979, p.441-445. [51] V. G. Kac, D. H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms. Adv. in Math. 53 (1984), no. 2, 125–264. [52] V. G. Kac, I. T. Todorov, Superconformal current algebras and their unitary representations. Comm. Math. Phys. 102 (1985), no. 2, 337–347. [53] V. G. Kac, M. Wakimoto, Unitarizable highest weight representations of the Virasoro, Neveu-Schwarz and Ramond algebras. Conformal groups and related symmetries: physical results and mathematical background (Clausthal-Zellerfeld, 1985), 345–371, Lecture Notes in Phys., 261, Springer, Berlin, 1986. [54] V. G. Kac, A. K. Raina, Bombay lectures on highest weight representations of infinite- dimensional Lie algebras. Advanced Series in Mathematical Physics, 2. World Scientific Publishing Co., Inc., Teaneck, NJ, 1987. [55] V. G. Kac, J. W. van de Leur, On classification of superconformal algebras. Strings ’88 (College Park, MD, 1988), 77–106, World Sci. Publ., Teaneck, NJ, 1989. [56] V. G. Kac, Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990. [57] V. G. Kac, Vertex algebras for beginners. University Lecture Series, 10. American Math- ematical Society, Providence, RI, 1997. [58] V. G. Kac, Classification of supersymmetries. Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 319–344, Higher Ed. Press, Beijing, 2002. [59] A. Kent, Singular vectors of the Virasoro algebra, Physics Letters B, Volume 273, Issues 1-2, 1991, p 56-62. [60] N. Kitchloo, Dominant K-theory and integrable highest weight represenatations of Kac- Moody groups . provisionally accepted for publication in Advances in Math., 2009. [61] V.G. Knizhnik, A.B. Zamolodchikov, Current Algebra and Wess-Zumino Model in Two- Dimensions Nucl. Phys. (1984) B 247: 83103 [62] M. L. Kontsevich, The Virasoro algebra and Teichmüller spaces. (Russian) Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 78–79 (English translation: Functional Anal. Appl. 21 (1987), no. 2, 156–157). [63] I.M. Krichever, S.P. Novikov, Algebras of Virasoro type, Riemann surfaces and struc- tures of the theory of solitons, Funkts. Anal. Appl. , 21:2 (1987) p. 4663. [64] R. P. Langlands, On unitary representations of the Virasoro algebra. Infinite- dimensional Lie algebras and their applications (Montreal, PQ, 1986), 141–159, World Sci. Publ., Teaneck, NJ, 1988. [65] J. Lepowsky, S. Mandelstam, I. M. Singer, eds, Vertex operators in mathematics and physics. Proceedings of a conference held at the Mathematical Sciences Research Insti- tute, Berkeley, Calif., November 10–17, 1983. [66] T. Loke, Operator algebras and conformal eld theory for the discrete series representations of Diff(S1), thesis, Cambridge 1994. PANORAMA AROUND THE VIRASORO ALGEBRA 7

[67] B. M. McCoy, T. T. Wu, The Two-Dimensional Ising Model. Harvard University Press (1973). [68] Yu. I. Manin, Critical dimensions of string theories and a dualizing sheaf on the moduli space of (super) curves. (Russian) Funktsional. Anal. i Prilozhen. 20 (1986), no. 3, 88–89 (English translation: Functional Anal. Appl. 20 (1986), no. 3, 244–246). [69] J. Mickelsson, Current algebras and groups. Plenum Monographs in Nonlinear Physics. Plenum Press, New York, 1989. xviii+313 pp. [70] M. Nakamura, Z. Takeda, A Galois theory for finite factors. Proc. Japan Acad. 36 1960 258–260. [71] M. Nakamura, Z. Takeda, On the fundamental theorem of the Galois theory for finite factors. Proc. Japan Acad. 36 1960 313–318. [72] E. Nelson, Analytic vectors. Ann. of Math. (2) 70 1959 572–615. [73] A. Neveu, J. H. Schwarz, Factorizable dual model of pions. Nucl. Phys. B31, 86–112 (1971) [74] A. Ocneanu, Quantized groups, string algebras and Galois theory for algebras. Operator algebras and applications, Vol. 2, 119–172, London Math. Soc. Lecture Note Ser., 136, Cambridge Univ. Press, Cambridge, 1988. [75] S. Palcoux, Unitary series and characters for the Neveu-Schwarz algebra, to appear. [76] S. Palcoux, Connes fusion and subfators for the Neveu-Schwarz algebra, in preparation. [77] S. Popa, Classification of subfactors and their endomorphisms. CBMS Regional Con- ference Series in Mathematics, 86 , 1995. [78] S. Popa, S. Vaes On the fundamental group of II1 factors and equivalence relations arising from group actions, to appear. [79] L. Pukanszky, The Plancherel formula for the universal covering group of SL(R, 2). Math. Ann. 156 1964 96–143. [80] P. Ramond, Dual theory for free fermions. Phys. Rev. D (3) 3 (1971), 2415–2418. [81] M. Reed, B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich, Publishers], New York- London, 1975. xv+361 pp [82] F. Sauvageot, Représentationsunitaires des super-algèbres de Ramond et de Neveu- Schwarz. Comm. Math. Phys. 121 (1989), no. 4, 639–657. [83] V. Schechtman, Sur les algèbres vertex attachéesaux variétésalgébriques. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 525–532 [84] M. Schlichenmaier, Differential operator algebras on compact Riemann surfaces H.-D. Doebner (ed.) V.K. Dobrev (ed.) A.G Ushveridze (ed.) , Generalized Symmetries in Physics, Clausthal 1993 , World Sci. (1994) p. 425–435 [85] O. K. Sheinman, Krichever-Novikov algebras and their representations. Noncommuta- tive geometry and representation theory in mathematical physics, 313–321, Contemp. Math., 391, Amer. Math. Soc., Providence, RI, 2005. [86] R. J. Szabo, Superconnections, anomalies and non-BPS brane charges. J. Geom. Phys. 43 (2002), no. 2-3, 241–292. [87] V. Toledano Laredo, Fusion of Positive Energy Representations of LSpin(2n) , thesis, Cambridge 1997, (on the arxiv). [88] V. Toledano Laredo, Integrating unitary representations of infinite-dimensional Lie groups. J. Funct. Anal. 161 (1999), no. 2, 478–508. [89] M. Takesaki, Conditional expectations in von Neumann algebras. J. Functional Analysis 9 (1972), 306–321. 8 S. PALCOUX

[90] M. Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III. Acta Math. 131 (1973), 249–310. [91] A. Tsuchiya, Y. Kanie, Vertex operators in conformal field theory on P 1 and monodromy representations of braid group. Conformal field theory and solvable lattice models (Ky- oto, 1986), 297–372, Adv. Stud. Pure Math., 16, Academic Press, Boston, MA, 1988. [92] A. Tsuchiya, T. Nakanishi, Level-rank duality of WZW models in conformal field theory. Comm. Math. Phys. 144 (1992), no. 2, 351–372. [93] A. Vaintrob, Conformal Lie superalgebras and moduli spaces. J. Geom. Phys. 15 (1995), no. 2, 109–122. [94] A. Valette, Introduction to the Baum-Connes conjecture. Lectures in Mathematics ETH Zrich. Birkhuser Verlag, Basel, 2002. x+104 pp. [95] E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory. Nuclear Phys. B 300 (1988), no. 3, 360–376. [96] R. W. Verrill, Positive energy representations of LσSU(2r) and orbifold fusion. thesis, Cambridge 2001. [97] M. A. Virasoro, Subsidiary conditions and ghosts in dual-resonance models, Phys. Rev. , D1 (1970) pp. 29332936 [98] A. J. Wassermann, Operator algebras and conformal field theory. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zrich, 1994), 966–979, Birkhuser, Basel, 1995. [99] A. J. Wassermann, Operator algebras and conformal field theory. III. Fusion of positive energy representations of LSU(N) using bounded operators. Invent. Math. 133 (1998), no. 3, 467–538. [100] A. J. Wassermann, Lecture Notes on the Kac-Moody and Virasoro algebras, 1998, http://iml.univ-mrs.fr/∼wasserm/kmv.ps [101] A. J. Wassermann, Analysis of operators, 2006, http://iml.univ-mrs.fr/∼wasserm/AO.ps [102] A. J. Wassermann, Subfactors and Connes fusion for twisted loop groups, to appear. [103] H. Wenzl, Hecke algebras of type An and subfactors. Invent. Math. 92 (1988), no. 2, 349–383. [104] E. Witt, Collected papers. Gesammelte Abhandlungen. (German) With an essay by Günter Harder on Witt vectors. Edited and with a preface in English and German by Ina Kersten. Springer-Verlag, Berlin, 1998. xvi+420 pp. [105] R. J. Zimmer, Ergodic theory and semisimple groups. Monographs in Mathematics, 81. Birkhuser Verlag, Basel, 1984.

Institut de Mathematiques´ de Luminy, Marseille, France. E-mail address: [email protected], http://iml.univ-mrs.fr/∼palcoux