Algebra di Virasoro

L'algebra di Virasoro è un'algebra di Lie complessa, data come estensione centrale del campo vettoriale dei polinomi complessi sulla circonferenza unitaria; questa algebra prende il nome dal fisico Miguel Angel Virasoro.

È molto usata in teoria delle stringhe e in teoria di campo conforme.

Definizione

L'algebra di Virasoro è una copertura lineare degli elementi:

per

con:

e c, che sono tutti elementi reali. Ogni c è un elemento centrale ovvero è una carica centrale.

L'algebra di Virasoro soddisfa alle seguenti due proprietà:

e

 ;

con:

1) il fattore di 1 / 12 è dovuto esclusivamente ad una questione di convenzione;

2) il simbolo se e se .

Si osservi che, la relazione:

 ;

può essere riscritta in termini del simbolo di Kronecker

.

Generalizzazioni

Ci sono due estensioni supersimmetriche (con N = 1) dell'algebra di Virasoro, chiamate algebra di Neveu-Schwarz e algebra di Ramond. Queste due teorie sono simili a quella dell'algebra Virasoro.

Bibliografia

  • A.A. Belavin, A.M. Polyakov e A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, in Nuclear Physics B, vol. 241, n. 2, --, pp. 333–380, DOI:10.1016/0550-3213(84)90052-X.
  • R.E. Block, On the Mills–Seligman axioms for Lie algebras of classical type Trans. Amer. Math. Soc. , 121 (1966) pp. 378–392
  • R. C. Brower, C. B. Thorn, Eliminating spurious states from the dual resonance model. Nucl. Phys. B31 163-182 (1971).
  • E. Cartan, Les groupes de transformations continus, infinis, simples. Ann. Sci. Ecole Norm. Sup. 26, 93-161 (1909).
  • B.L. Feigin, D.B. Fuks, Verma modules over the Virasoro algebra L.D. Faddeev (ed.) A.A. Mal'tsev (ed.) , Topology. Proc. Internat. Topol. Conf. Leningrad 1982 , Lect. notes in math. , 1060 , Springer (1984) pp. 230–245
  • Friedan, D., Qiu, Z. and Shenker, S.: https://prola.aps.org/abstract/PRL/v52/i18/p1575_1[collegamento interrotto], Phys. Rev. Lett. 52 (1984) 1575-1578.
  • I.M. Gel'fand, D.B. Fuks, The cohomology of the Lie algebra of vector fields in a circle Funct. Anal. Appl. , 2 (1968) pp. 342–343 Funkts. Anal. i Prilozh. , 2 : 4 (1968) pp. 92–93
  • P. Goddard, A. Kent and D. Olive Unitary representations of the Virasoro and super-Virasoro algebras Comm. Math. Phys. 103, no. 1 (1986), 105–119.
  • A. Kent, "Singular vectors of the Virasoro algebra", Physics Letters B, Volume 273, Issues 1-2, 12 December 1991, Pages 56-62.
  • (EN) Victor Kac, Virasoro algebra, in Encyclopaedia of Mathematics, Springer e European Mathematical Society, 2002.
  • V.G. Kac, Highest weight representations of infinite dimensional Lie algebras , Proc. Internat. Congress Mathematicians (Helsinki, 1978),
  • V.G. Kac, A.K. Raina, Bombay lectures on highest weight representations, World Sci. (1987) ISBN 9971503956.
  • V.K. Dobrev, Multiplet classification of the indecomposable highest weight modules over the Neveu-Schwarz and Ramond superalgebras, Lett. Math. Phys. {\bf 11} (1986) 225-234 & correction: ibid. {\bf 13} (1987) 260.
  • I.M. Krichever, S.P. Novikov, Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons, Funkts. Anal. Appl. , 21:2 (1987) p. 46–63.
  • V.K. Dobrev, Characters of the irreducible highest weight modules over the Virasoro and super-Virasoro algebras, Suppl. Rendiconti Circolo Matematici di Palermo, Serie II, Numero 14 (1987) 25-42.
  • 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
  • M. A. Virasoro, https://prola.aps.org/abstract/PRD/v1/i10/p2933_1[collegamento interrotto] Phys. Rev. , D1 (1970) pp. 2933–2936
  • A. J. Wassermann, Lecture notes on Kac-Moody and Virasoro algebras

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