Todorov, I. T.

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PLISKASTUDIA MATHEMATICA BULGARICAПЛИСКАБЪЛГАРСКИ МАТЕМАТИЧЕСКИ СТУДИИProvided for non-commercial research and educational use.Not for reproduction, distribution or commercial use.The attached copy is furnished for non-commercial research and education use only. Authors are permitted to post this version of the article to their personal websites or institutional repositories and to share with other researchers in the form of electronic reprints. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to third party websites are prohibited.For further information on Pliska Studia Mathematica Bulgarica visit the website of the journal http://www.math.bas.bg/-pliska/ or contact: Editorial Office Pliska Studia Mathematica Bulgarica Institute of Mathematics and Informatics Bulgarian Academy of Sciences Telephone: (+359-2)9792818, FAX:(+359-2)971-36-49 e-mail: pliska@math.bas.bgON  T H E  C LA S SIFIC A TIO N  O F  U NITARY L O W E S T  W E IG H T  R E P R E S E N T A T IO N S  O F  C O N F O R M A L  C U R R E N T  A LG EB R A SI. T. TODOROVTo the memory o f my teacher Yaroslav Tagamlitzki1. I n t r o d u c t i o n .  In searching for the extreme elements of a partially or­dered set К the good half of the work consists in devising a decomposition of a generic positive element of К  into a sum (or a convex linear combination) of positive elements [1]. The purpose of this note is to point out the use of such a decomposition in a context, in which К  is not a convex set.The problem is concerned with the classification of the unitary lowest weight representations of an infinite dimensional Lie algebra VG, which appears in conformal quantum field theory models on the circle. In the rest of this in­troduction I shall define the algebra and say some words about its physicalinterpretation (for more detail see, e. g. [2 ]).Let G be a simple compact Lie group and let dо be its Lie algebra. To  each element X$  do we make correspond a sequence of elements Xn, n$ Zof an infinite dimensional “ affine Kac-Moody Lie algebra” dg in such a way that the following commutation relations hold:[X „  Ym) =  [X, Y]n+m +  bn(X, Y)bnt_ mwhere (X, Y) is a suitably normalized invariant inner product (the positive “Killing form”) on do, and k is a “ central charge”^  \ky X n] ≈  0.Thus, do appears as a Z-graded infinite Lie algebra which contains do as asubalgebra (if we identify X  with X 0).The V i r a s o r o  a l g e b r a  Vir is a central extension of the algebra of first-order differential operators that generate the diffeomorphisms of the circle. It is another Z-graded algebra with generators Ln and с satisfying(1-3) л ’ [£*• Lm\ =  {n -m )L n+m +  ^  п{п*—(1.4) [c, Ln} ≈ 0.f ∙ i-.r. > ∕ ∙ ■' ∙ - ) .ViThe c o n f o r m a l  c u r r e n t  a l g e b r a  VG is defined as the semidirect sum of Vir with do, in which the commutation relations (1.1-1.4) are supplemented by(1.5) л. [X m, Ln] ≈ m X n+m.The physical context in which such an algebra appears is an 1-dimensional c o n f o r m a l  q u a n t u m  f i e l d  t h e o r y  on the circle with an internal sym­metry group G. The fields of such a theory are defined in general as ope- PLISKA Studia mathematica bulgarica, Vol. 11, 1991, p. 102-104.rator-valued distributions on a Hilbert space Ж  (satisfying certain requirements). In our case they include the s t r e s s  e n e r g y  t e n s o r(1.6) T(z) =  ∑nL nz- n- a, L ; ≈ L „ n(n(: Z) and the current(1.7) X(z) =  ∑nA uz - n~\ K  =( X  belongs to a dG dimensional space, where do-=dim G). The s t a t e s  o f  t h e  t h e o r y  include the v a c u u m  v e c t o r  \0)^Ж satisfying <0 1 0 > =  1 and(1.8) Z,„|0) =  0 for n ^  — 1, X m I 0) =  0 for m^O.We make further the following i r r e d u c i b i l i t y  a s s u m p t i o n :  if a boun­ded operator В in Ж  commutes with all the fields of the theory, then it is a multiple of the identity in Ж. In particular, the central charges с and k act as multiplication of real numbers.C o r o l l a r i e s .  1. The positivity of the inner product in Ж  and the her- miticity properties in (1.6) and (1.7) imply that с and k are positive(1.9) 0 <  II I O ≥  !t» =  <0 | i 2Z._210> =  - f -(1.10) 0 < II X_x I 0>|| 9 =  (01 A'1A’_ 1 10 )= k (X ,  Л ).2 . Eqs. (1.6-1.8) allow to find all correlation functions of the stress-energy tensor and the current:(1.11) <0|7'(z1)T(z2 )|0> =  -| -z -4, z12 =  z i—za.(1.12) <0| X (z1)V(z2)\0) =  A i ( X ,  Y)Z\2(1.13) (0 I Х (г г)У (г2)Т ( г3) I 0 )=  k{X, Y ) z ^ z £  etc.The operator L0 is interpreted physically as the e n e r g y  o p e r a t o r .  E n e r g y  p o s i t i v i t y  singles out l o w e s t  w e i g h t  (LW ) representations of VG. Any LW u n i t a r y  i r r e d u c i b l e  r e p r e s e n t a t i o n  (UlR) of VG is characterized by a LW  vector | *F) such that(1.14) Ln I 4/> =  0 =  Лгя j xF>for n ≈  l , 2 , . . . a 0 - ∆ ) | T > 0(which is usually also assumed to be a highest weight eigenvector of a Cartan subalgebra of do).The classification of LWUIRs of VG has only been completed recently [3], following developments in [4—7] among others. We shall review in Sec. 2 a central point of this derivation, related to a decomposition of T(z).2. LWUIRs of V G . P г о p о s i t i о n 1. Given any unitary LW  representation of dо, it can be extended to a LW  representation of VG by setting(2.1) Lj,* ≈  £ -|_C2  ^ Qn—mQn+m}> ^Si + l e  fc+C2 ^Lq ^ n~~m ’—>where Q —(Q i, . . . ,  Qt,G) is an orthonormal basis of do so that(2 .2 ) (Qa, Qh) =  4 -  №  QbJ  =  ‘fnbcQUm +  ~2"fin. -M'KbOn the classification of unitary lowest weight representations of conformal . . .  JQ3104 I. Todorovand Go is the eigenvalue of the second order Casimir operator for the adjoint representation of da(2.3) fstnfstb — C^ab∙The Virasoro central charge associated with L£ is(2.4) Ca( =  2[L°, L U  -  8 L°)  =  da .The proof (2.6) involves a straightforward (albeit somewhat lengthy) com­putation using (2.1-2.3) (One should first verify [Qam, L „ ] ≈ m  Q^+∏ anc* then(1.3), (2.4)).Given a LWUIR of do, the construction of Proposition 1 gives a m i n i m a lr e p r e s e n t a t i o n  of VG in the sense of the fotlowing result.P r o p o s i t i o n  2* Given an arbitrary LWUIR of VG the operators(2.5) . ln= Ln— L°give rise to a LWUIR of Vir with central charge(2 .6 ) ci =  c — ca≥O.P r o o f .  It follows from (1.5) and from Proposition 1 that(2-7) [Q*. g  =  0 =  [L£, Ц.The rest is straightforward. The inequality (2.6) is a consequence of the assumed unitarity of the representation under consideration.Propositions 1 and 2 reduce the study of the LWUIRs of VG to theknown classification of such representations for ∂ g and,Vir (2.4-2.7).R E F E R E N C E S, (1. Я. А. Т а г а м л и ц к и .  Изследваме на вектори, които са неразложими относно някои ко­нуси. Изв. Мат. Инст. БАН, 1 , кн. 1, 1953, 57— 6 8 .2. I. Т. T o d o r o v .  Infinite dimensional Lie algebras in 2-dirtiensiona! conformal field theory.—In : Differential Geometric Methods in Theoretical Physics, Ed. by H. D. Doebner, T. D. Palev, Singapore, 1986, 297-347.8 . V К a c. Highest weight representation of conformal current algebras. Proceeding o f  the Symposium on Topological and Geometrical Methods in Field Theory, Espoo, Singa­pore, 1986.4. V. К a c. Covariant form for infinite dimensional Lie algebras and superalgebras. LectureNotes in Physics, 94, 1979, 441-445 ; Infinite Dimensionai Lie Algebras. An Intro­duction (Birkhauser, London, 1983 ; 2nd ed. Cambridge UniV. Press, 1985) ane refe­rences therein.5. D. F r i e d a n ,  Z. Q i u, S. S h e n k e r. Conformal invariance, unitarity, and critical expo­nents in two dimensions. Phys. Rev. Lett., 52, 1984, 1575-1578, and in: VertexOperators in Mathematics and Physics. Ed. by J. Lepowski, S. Mandelstam,I. M. Singer, N. Y., 1985, 419-449; Details of the nonunitarity proof for highest weigt representations of the Virasoro algebra, Chicago preprint EFI, 86-92.6 . P. G o d d a r d ,  A. K e n t ,  D, O l i v e .  Virasoro algebra and coset space models. Phys. Lett.,152B, 1985, 88-93; University representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys., 103, 1986, 105-119,7. V. G. К a c, I. T. T o d o r o v .  Superconformal current algebras and their unitary represen­tations. Commun. Math. Phys., 102, 1985, 337-347; 104, 1986, 175; V. G. К a c, M. W a k i m o t o .  Unilarizable highest weight representations of the Virasoro, Neveu Schwarz and Ramond algebras. Proceedings o f the Symposium on Conformal Groups and Structures. Chausthal, 1985, Lecture Notes in Physics, 1986.Institute o f  Nuclear Research Received 18. 11. 198band Nuclear EnergyBulgarian Academy o f SciencesSofia 1184, Bulgaria

On the classifications of unitary lowest weight representations of conformal current algebras

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On the classifications of unitary lowest weight representations of conformal current algebras

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