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From the Representation Theory of Vertex Operator Algebras to Modular Tensor Categories in Conformal Field Theory

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From the representation theory of vertex operator algebras to modular tensor categories in conformal field theory James Lepowsky* Department of Mathematics, Rutgers, The State University of New Jersey, Piscataway, NJ 08854 wo-dimensional conformal constructed tensor category. This work, of these intertwining operators are the quantum field theory (CFT) along with results used in this work, in- fusion rules referred to above. Then has inspired an immense cludes, in particular, the (mathematical) one has to construct a tensor product amount of mathematics and construction of a significant portion of theory that ‘‘implements’’ these inter- Thas interacted with mathematics in very CFT—structures that actually do satisfy twining operators. After work of rich ways, in great part through the the axioms—using the representation Kazhdan–Lusztig (11) for certain struc- mathematically dynamic world of string theory of vertex operator algebras. tures based on affine Lie algebras, a theory. One notable example of this in- Huang (6) invokes a great deal of ear- tensor product theory for modules for teraction is provided by Verlinde’s con- lier work in vertex (operator) algebra a suitable, general vertex operator al- jecture: E. Verlinde (1) conjectured that theory. The mathematical foundation of gebra was constructed in a series of certain matrices formed by numbers CFT may be viewed as resting on the papers summarized in ref. 12. Elabo- called the ‘‘fusion rules’’ in a ‘‘rational’’ theory of vertex operator algebras (ref. rate use of ‘‘formal calculus’’ (dis- CFT are diagonalized by the matrix 7; see also ref. 8), which reflect the cussed in ref. 8) was required in this given by a certain natural action of a physical features codified by Belavin work. The main paper in this series fundamental modular transformation et al. (9). Mathematically, vertex opera- (13) establishes Huang’s associativity (essentially, a certain distinguished ele- tor algebra theory is extremely rich. For theorem, which leads quickly to the ment of the group of two-by-two matri- (categorical) coherence of the resulting ces of determinant one with integer braided tensor category. In CFT termi- entries). His conjecture led him to the To construct a nology, this associativity theorem as- ‘‘Verlinde formula’’ for the fusion rules serts the existence and associativity of and, more generally, for the dimensions tensor product theory the operator product expansion for in- of spaces of ‘‘conformal blocks’’ on Ri- tertwining operators, an assertion that emann surfaces of arbitrary genera. A of modules for a vertex was a key assumption (not theorem) in great deal of progress has been achieved ref. 3. In addition to braided tensor in interpreting and proving Verlinde’s operator algebra, one category structure, this series of papers (physical) conjecture and the Verlinde constructs the much richer ‘‘vertex ten- formula in mathematical settings, in is forced to proceed sor category’’ structure, which involves the case of the Wess–Zumino–Novikov- the conformal-geometric structure es- Witten models in CFT, which are based ‘‘backwards.’’ tablished in ref. 14, on the module cat- on affine Lie algebras. On the other egory of a suitable vertex operator hand, Moore and Seiberg (2, 3) showed, algebra. on a physical level of rigor, that the gen- the work discussed here, one needs the A fundamental theorem establishing eral form of the Verlinde conjecture is a representation theory of vertex operator natural modular transformation prop- consequence of the axioms for rational algebras, especially a tensor product the- erties of ‘‘characters’’ of modules for a CFTs, thereby providing a conceptual ory for modules for a suitable vertex suitable vertex operator algebra was understanding of the conjecture. In the operator algebra. In classical tensor proved by Zhu (15). Requiring all of process, they formulated a CFT ana- product theories for modules for a the theory mentioned here, as well as logue, later termed ‘‘modular tensor cat- group or for a suitable algebra such as a results in refs. 16–18, Huang formu- egory’’ (discussed in refs. 4 and 5) by I. Lie algebra, one automatically has the lates a general, mathematically precise, Frenkel, of the classical notion of tensor tensor product vector space available, statement of the Verlinde conjecture category for representations of (modules and one endows it with tensor product in the framework of the theory of ver- for) a group or a Lie algebra. It re- module structure by means of a natural tex operator algebras. Assuming only mained a very deep problem to con- coproduct operation. A module map such purely algebraic, natural hypothe- struct, in a mathematical as opposed to from the tensor product of two modules ses as simplicity of the vertex operator physical sense, structures (‘‘theories’’) to a third module then amounts to an algebra, complete reducibility of suit- satisfying these axioms for rational CFT. ‘‘intertwining operator’’ satisfying a nat- able modules, natural grading restric- These axioms are, in fact, much stronger ural condition coming from the group tions, and cofiniteness, hypotheses that than the Verlinde conjecture and modu- or algebra actions on the three modules. are relatively easily checked and have lar tensor category structure, and, in- However, vertex operator algebra indeed been previously verified in a deed, the mathematical construction of theory is imbued with considerable wide range of important families of CFTs (as opposed to the physical as- ‘‘nonclassical’’ subtleties, intimately examples, Huang sketches his proof sumption that they should exist) is a related to the nonclassical nature of (see ref. 6). The proof is heavily based very rich field of study to which many string theory in physics, and to con- on the results of his recent papers (19, mathematicians have contributed. In this struct a tensor product theory of mod- 20), in which natural duality and mod- issue of PNAS, Huang (6) announces a ules for a vertex operator algebra, one (mathematical) proof of the Verlinde is forced to proceed ‘‘backwards’’: conjecture in a very general form, along First, one defines suitable ‘‘intertwin- See companion article on page 5352. with two notable consequences: the ri- ing operators’’ (3, 10) among triples of *E-mail: [email protected]. gidity and modularity of a previously modules. The dimensions of the spaces © 2005 by The National Academy of Sciences of the USA 5304–5305 ͉ PNAS ͉ April 12, 2005 ͉ vol. 102 ͉ no. 15 www.pnas.org͞cgi͞doi͞10.1073͞pnas.0501135102 Downloaded by guest on September 25, 2021 COMMENTARY ular invariance properties for genus-zero fact, the main work is to establish two used by Huang (6) will have further and genus-one multipoint correlation formulas of Moore and Seiberg that consequences. In fact, this tensor cate- functions constructed from intertwining they had derived from strong assump- gory theory has already been applied operators for a vertex operator algebra tions: the axioms for rational CFT. The to a variety of fields in mathematics satisfying the general hypotheses are difficulties lie in the sequence of math- and physics, including string theory or established; the multiple-valuedness of ematical developments briefly men- M-theory, in particular, D-branes. The the multipoint correlation functions tioned here. insight that continues to flow from the leads to considerable subtleties that As has been the case with many combined and respective efforts of had to be handled analytically and other major developments in the math- many physicists and mathematicians in geometrically, rather than just algebra- ematical study of string theory and this remarkable age of string theory ically. The strategy of the proof re- conformal field theory over the years, and its mathematical counterparts will flects the pattern of refs. 2 and 3; in it is to be expected that the methods surely produce new surprises. 1. Verlinde, E. (1988) Nucl. Phys. B 300, 360–376. 8. Frenkel, I. B., Lepowsky, J. & Meurman, A. (1988) 13. Huang, Y.-Z. (1995) J. Pure Appl. Alg. 100, 173– 2. Moore, G. & Seiberg, N. (1988) Phys. Lett. B 212, Vertex Operator Algebras and the Monster (Aca- 216. 451–460. demic, New York). 14. Huang, Y.-Z. (1998) Two-Dimensional Conformal 3. Moore, G. & Seiberg, N. (1988) Comm. Math. 9. Belavin, A. A., Polyakov, A. M. & Zamolod- Geometry and Vertex Operator Algebras Phys. 123, 177–254. chikov, A. B. (1984) Nucl. Phys. B 241, 333–380. (Birkhauser, Boston). 4. Bakalov, B. & Kirillov, A., Jr. (2001) Lectures on Tensor 10. Frenkel, I. B., Huang, Y.-Z. & Lepowsky, J. (1993) On 15. Zhu, Y. (1996) J. Am. Math. Soc. 9, 237–307. Categories and Modular Functors, University Lecture Axiomatic Approaches to Vertex Operator Algebras and 16. Huang, Y.-Z. (1996) J. Alg. 182, 201–234. Series (Am. Math. Soc., Providence, RI), Vol. 21. Modules, Memoirs of the American Mathematical So- 17. Huang, Y.-Z. (2000) Selecta Math. 6, 225– 5. Turaev, V. G. (1994) Quantum Invariants of Knots ciety (Am. Math. Soc., Providence, RI), Vol. 104. 267. and 3-Manifolds, de Gruyter Studies in Mathemat- 11. Kazhdan, D. & Lusztig, G. (1991) Int. Math. Res. 18. Dong, C., Li, H. & Mason, G. (2000) Comm. Math. ics (de Gruyter, Berlin), Vol. 18. Notices 2, 21–29. Phys. 214, 1–56. 6. Huang, Y.-Z. (2005) Proc. Natl. Acad. Sci. USA 12. Huang, Y.-Z. & Lepowsky, J. (1994) in Lie Theory 19. Huang, Y.-Z. (2005) Comm. Contemp. Math.,in 102, 5352–5356. and Geometry, in Honor of Bertram Kostant, eds. press. 7. Borcherds, R. E. (1986) Proc. Natl. Acad. Sci. USA Brylinski, J.-L., Brylinski, R., Guillemin, V. & 20. Huang, Y.-Z. (2005) Comm. Contemp. Math.,in 83, 3068–3071. Kac, V. (Birkhauser, Boston), pp. 349–383. press. Lepowsky PNAS ͉ April 12, 2005 ͉ vol. 102 ͉ no. 15 ͉ 5305 Downloaded by guest on September 25, 2021.
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