Invent. math. 133, 467±538 (1998) Operator algebras and conformal ®eld theory III. Fusion of positive energy representations of LSU(N) using bounded operators Antony Wassermann Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, UK Oblatum 1-III-1996 & 11-III-1997 Contents I. Positive energy representations of LSU(N) ................................................................. 477 II. Local loop groups and their von Neumann algebras ................................................. 491 III. The basic ordinary dierential equation ..................................................................... 505 IV. Vector and dual vector primary ®elds ........................................................................ 513 V. Connes fusion of positive energy representations ....................................................... 525 References ........................................................................................................................... 536 1. Introduction This is one of a series of papers devoted to the study of conformal ®eld theory from the point of view of operator algebras (see [41] and [42] for an overview of the whole series). In order to make the paper accessible and self-contained, we have not assumed a detailed knowledge of either operator algebras or conformal ®eld theory, including short-cuts and direct proofs wherever possible. This research programme was originally motivated by V. Jones' suggestion that there might be a deeper `operator algebraic' explanation of the coincidence between certain unitary representations of the in®nite braid group that had turned up independently in the theory of subfactors, exactly solvable models in statistical mechanics and conformal ®eld theory (CFT). To understand why there should be any link between these subjects, recall that, amongst other things, the classical `additive' theory of von Neumann algebras [26] was developed to provide a framework for studying unitary representations of Lie groups. In concrete examples, for example the Plancherel theorem for semisimple groups, this abstract framework had to be complemented by a considerably harder analysis of intertwining operators 468 A. Wassermann and associated dierential equations. The link between CFT and operator algebras comes from the recently developed `multiplicative' (quantum?) theory of von Neumann algebras. This theory has three basic sources: ®rstly the algebraic approach to quantum ®eld theory (QFT) of Doplicher, Haag and Roberts [10]; then in Connes' theory of bimodules and their tensor products of fusion [9]; and lastly in Jones' theory of subfactors [18]. Our work reconciles these ideas with the theory of primary ®elds, one of the funda- mental concepts in CFT. Our work has the following consequences, some of which will be taken up in subsequent papers: (1) Several new constructions of subfactors. (2) Non-trival algebraic QFT's in 1+1 dimensions with ®nitely many sectors and non-integer statistical (or quantum) dimension (``algebraic CFT''). (3) A de®nition of quantum invariant theory without using quantum groups at roots of unity. (4) A computable and manifestly unitary de®nition of fusion for positive energy representations (``Connes fusion'') making them into a tensor cate- gory. (5) Analytic properties of primary ®elds (``constructive CFT''). To our knowledge, no previous work has suceeded in integrating the theory of primary ®elds with the ideas of algebraic QFT nor in revealing the very simple analytic structure of primary ®elds. As we explain below, the main thrust of our work is the explicit computation of Connes fusion of positive energy representations. Finiteness of statistical dimension (or Jones index) is a natural consequence, not a technical mathematical inconvenience. It is perhaps worth emphasising that the theory of operator algebras only pro- vides a framework for studying CFT. As in the case of group representa- tions, it must be complemented by a detailed analysis of certain interwining operators, the primary ®elds, and their associated dierential equations. As we discuss later, however, the operator algebraic point of view can be used to reveal basic positivity and unitarity properties in CFT that have previ- ously seem to have been overlooked. Novel features of our treatment are the construction of representations and primary ®elds from fermions. This makes unitarity of the representa- tions and boundedness properties of smeared vector primary ®elds obvious. The only formal ``vertex algebra'' aspects of the theory of primary ®elds borrowed from [39] are the trivial proof of uniqueness and the statement of the Knizhnik-Zamolodchikov equation; our short derivation of the KZ equation circumvents the well-known contour integral proof implicit but not given in [39]. The proof that the axioms of algebraic QFT are satis®ed in the non-vacuum sectors is new and relies heavily on our fermionic construction; the easier properties in the vacuum sector have been known for some time [7, 15]. The treatment of braiding relations for smeared primary ®elds is new but inspired by the Bargmann-Hall-Wightman theorem [20, 36]. To our knowledge, the application of Connes fusion to a non-trivial model in QFT Operator algebras and conformal ®eld theory 469 is quite new. Our de®nition is a slightly simpli®ed version of Connes' original de®nition, tailor-made for CFT because of the ``four-point function formula''; no general theory is required. The ®nite-dimensional irreducible unitary representations of SU(N)and their tensor product rules are well known to mathematicians and physicists. The representations Vf are classi®ed by signatures or Young diagrams k N f1 f2 fN and, if V k k C , we have the tensor product rule ÁÁÁ Vf V Vg, where g ranges over all diagrams that can be obtained k ag>k f by adding k boxes to f with no two in the same row. For the in®nite- 1 dimensional loop group LSU N C1 S ; SU N , the appropriate unitary representations to consider in place of ®nite-dimensional representations are the projective unitary representations of positive energy. Positive energy representations form one of the most important foundation stones of con- formal ®eld theory [5, 12, 23]. The classi®cation of positive energy repre- sentations is straightforward and has been known for some time now. A positive energy representation Hf is classi®ed by its level `, a positive integer, and its signature f , which must satisfy the permissibility condition f1 fN `. Extending the tensor product rules to representations of a ®xed level, however, presents a problem. It is already extremely dicult just giving a coherent de®nition of the tensor product, since the naive one fails hopelessly because it does not preserve the level. On the other hand physi- cists have known for years how to `fuse' representations in terms of short range expansions of products of associated quantum ®elds (primary ®elds). We provide one solution to this `problem of fusion' in conformal ®eld theory by giving a mathematically sound de®nition of the tensor product that ties up with the intuitive picture of physicists. Our solution relates positive energy representations of loop groups to bimodules over von Neumann algebras. Connes de®ned a tensor product operation on such bimodules ± ``Connes fusion'' ± which translates directly into a de®nition of fusion for positive energy representations. The general fusion rules follow from the particular rules Hf 2 H Hg, where g must now also be k ag>k f permissible. In this way the level ` representations of LSU N exhibit a structure similar to that of the irreducible representations of a ®nite group. There are several other approaches to fusion of positive energy represen- tations, notably those of Segal [35] and Kazhdan & Lusztig [22]. Our picture seems to be a unitary boundary value of Segal's holomorphic proposal for fusion, based on a disc with two smaller discs removed. When the discs shrink to points on the Riemann sphere, Segal's de®nition should degen- erate to the algebraic geometric fusion of Kazhdan & Lusztig. We now give an informal summary of the paper. Fermions. Let Cli(H) be the Cliord algebra of a complex Hilbert space H, generated by a linear map f a f (f H) satisfying a f a g 7! 2 a g a f 0 and a f a g à a g Ãa f f ; g . It acts irreducibly on Fock space KH via a f x f x. Other representations of Cliff H arise ^ by considering the real linear map c f a f a f à which satis®es 470 A. Wassermann c f c g c g c f 2Re f ; g ; note that a f 1 c f ic if . Since c 2 relies only on the underlying real Hilbert space HR, complex structures on HR commuting with i give new irreducible representations of Cliff H . The structures correspond to projections P with multiplication by i given by i on PH and i on PH ?. The corresponding representation pP is given by 1 pP a f 2 c f ic i 2P I f . Using ideas that go back to Dirac and von Neumann, we give our own short proof of I. Segal's equivalence cri- terion: if P Q is a Hilbert-Schmidt operator, then pP and pQ are unitarily equivalent. On the other hand if u U H , then a uf and a ug also satisfy the complex Cliord algebra relations.2 Thus a f a uf gives an auto- morphism of Cliff H . We say that this ``Bogoliubov'' ! automorphism is implemented in pP i pP a uf UpP a f U à for some unitary U. This gives a projective representation of the subgroup of implementable unitaries UP H . Segal's equivalence criterion leads immediately to a quantisation criterion: if u; P is a Hilbert-Schmidt operator, then u U H . 2 P 1 Positive energy representations. Let G SU N and let LG C1 S ; G be the loop group, with the rotation group Rot S1acting as automorphisms. If H L2 S1; CN and P is the projection onto Hardy space H 2 S1; CN , 1 LSU N oRot S UP H so we get a projective representation ` 1 ` pP : LU N oRot S PU F where F denotes Fock space KHP .