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Advances in Mathematical Physics

Volume 2010 (2010), Article ID 509538, 12 pages

http://dx.doi.org/10.1155/2010/509538

Review Article

## Gauge Symmetry and Howe Duality in 4D Conformal Field Theory Models

Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria

Received 17 September 2009; Accepted 28 October 2009

Academic Editor: Richard Kerner

Copyright © 2010 Ivan Todorov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- N. M. Nikolov and I. T. Todorov, “Rationality of conformally invariant local correlation functions on compactified Minkowski space,”
*Communications in Mathematical Physics*, vol. 218, no. 2, pp. 417–436, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. M. Nikolov, Ya. S. Stanev, and I. T. Todorov, “Four-dimensional conformal field theory models with rational correlation functions,”
*Journal of Physics A*, vol. 35, no. 12, pp. 2985–3007, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. M. Nikolov, Ya. S. Stanev, and I. T. Todorov, “Globally conformal invariant gauge field theory with rational correlation functions,”
*Nuclear Physics B*, vol. 670, no. 3, pp. 373–400, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. M. Nikolov, K.-H. Rehren, and I. T. Todorov, “Partial wave expansion and Wightman positivity in conformal field theory,”
*Nuclear Physics B*, vol. 722, no. 3, pp. 266–296, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Harmonic bilocal fields generated by globally conformal invariant scalar fields,”
*Communications in Mathematical Physics*, vol. 279, no. 1, pp. 225–250, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Unitary positive-energy representations of scalar bilocal quantum fields,”
*Communications in Mathematical Physics*, vol. 271, no. 1, pp. 223–246, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Infinite-dimensional Lie algebras in 4D conformal quantum field theory,”
*Journal of Physics A*, vol. 41, no. 19, Article ID 194002, 12 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. F. Streater and A. S. Wightman,
*PCT, Spin and Statistics, and All That*, Princeton Landmarks in Physics, Princeton University Press, Princeton, NJ, USA, 2000. View at MathSciNet - R. Howe, “Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons,” in
*Applications of Group Theory in Physics and Mathematical Physics*, M. Flato, P. Sally, and G. Zuckerman, Eds., vol. 21 of*Lectures in Applied Mathematics*, pp. 179–207, American Mathematical Society, Providence, RI, USA, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Howe, “Transcending classical invariant theory,”
*Journal of the American Mathematical Society*, vol. 2, no. 3, pp. 535–552, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Doplicher and J. E. Roberts, “Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics,”
*Communications in Mathematical Physics*, vol. 131, no. 1, pp. 51–107, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Haag,
*Local Quantum Physics: Fields, Particles, Algebras*, Texts and Monographs in Physics, Springer, Berlin, Germany, 1992. View at MathSciNet - R. Howe, “On the role of the Heisenberg group in harmonic analysis,”
*Bulletin of the American Mathematical Society*, vol. 3, no. 2, pp. 821–843, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Weil, “Sur certains groupes d'opérateurs unitaires,”
*Acta Mathematica*, vol. 111, no. 1, pp. 143–211, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Weil, “Sur la formule de Siegel dans la théorie des groupes classiques,”
*Acta Mathematica*, vol. 113, no. 1, pp. 1–87, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - G. Mack and I. T. Todorov, “Irreducibility of the ladder representations of $U(2,2)$ when restricted to the Poincaré subgroup,”
*Journal of Mathematical Physics*, vol. 10, pp. 2078–2085, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Joseph, “Minimal realizations and spectrum generating algebras,”
*Communications in Mathematical Physics*, vol. 36, pp. 325–338, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Joseph, “The minimal orbit in a simple Lie algebra and its associated maximal ideal,”
*Annales Scientifiques de l'École Normale Supérieure Série 4*, vol. 9, no. 1, pp. 1–29, 1976. View at Google Scholar - D. Kazhdan, B. Pioline, and A. Waldron, “Minimal representations, spherical vectors and exceptional theta series,”
*Communications in Mathematical Physics*, vol. 226, no. 1, pp. 1–40, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Günaydin and O. Pavlyk, “A unified approach to the minimal unitary realizations of noncompact groups and supergroups,”
*Journal of High Energy Physics*, vol. 2006, no. 9, article 050, 2006. View at Publisher · View at Google Scholar · View at Scopus - T. Kobayashi and G. Mano, “The Schrödinger model for the minimal representation of the indefinite orthogonal group $O(p,q)$,” to appear in
*Memoirs of the American Mathematical Society*. - G. C. Wick, A. S. Wightman, and E. P. Wigner, “The intrinsic parity of elementary particles,”
*Physical Review*, vol. 88, pp. 101–105, 1952. View at Google Scholar · View at MathSciNet - V. G. Kac,
*Infinite-Dimensional Lie Algebras*, Cambridge University Press, Cambridge, UK, 3rd edition, 1990. View at MathSciNet - V. G. Kac and A. K. Raina,
*Bombay Lectures on Highest Weight Representations of Infinite-Dimensional Lie Algebras*, vol. 2 of*Advanced Series in Mathematical Physics*, World Scientific, Teaneck, NJ, USA, 1987. View at MathSciNet - D. W. Robinson, “On a soluble model of relativistic field theory,”
*Physics Letters B*, vol. 9, pp. 189–190, 1964. View at Publisher · View at Google Scholar · View at MathSciNet - J. H. Lowenstein, “The existence of scalar Lie fields,”
*Communications in Mathematical Physics*, vol. 6, pp. 49–60, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Baumann, “There are no scalar Lie fields in three or more dimensional space-time,”
*Communications in Mathematical Physics*, vol. 47, no. 1, pp. 69–74, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. B. Frenkel and V. G. Kac, “Basic representations of affine Lie algebras and dual resonance models,”
*Inventiones Mathematicae*, vol. 62, no. 1, pp. 23–66, 1980. View at Publisher · View at Google Scholar · View at MathSciNet - R. E. Borcherds, “Vertex algebras, Kac-Moody algebras, and the Monster,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 83, no. 10, pp. 3068–3071, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Kac,
*Vertex Algebras for Beginners*, vol. 10 of*University Lecture Series*, American Mathematical Society, Providence, RI, USA, 2nd edition, 1998. View at MathSciNet - E. Frenkel and D. Ben-Zvi,
*Vertex Algebras and Algebraic Curves*, vol. 88 of*Mathematical Surveys and Monographs*, American Mathematical Society, Providence, RI, USA, 2nd edition, 2004. - Y. Zhu, “Modular invariance of characters of vertex operator algebras,”
*Journal of the American Mathematical Society*, vol. 9, no. 1, pp. 237–302, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Uhlmann, “The closure of Minkowski space,”
*Acta Physica Polonica*, vol. 24, pp. 295–296, 1963. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. T. Todorov, “Infinite-dimensional Lie algebras in conformal QFT models,” in
*Conformal Groups and Related Symmetries: Physical Results and Mathematical Background*, A. O. Barut and H.-D. Doebner, Eds., vol. 261 of*Lecture Notes in Physics*, pp. 387–443, Springer, Berlin, Germany, 1986. View at Publisher · View at Google Scholar · View at MathSciNet - N. M. Nikolov, “Vertex algebras in higher dimensions and globally conformal invariant quantum field theory,”
*Communications in Mathematical Physics*, vol. 253, no. 2, pp. 283–322, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. M. Nikolov and I. T. Todorov, “Elliptic thermal correlation functions and modular forms in a globally conformal invariant QFT,”
*Reviews in Mathematical Physics*, vol. 17, no. 6, pp. 613–667, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. A. Dolan and H. Osborn, “Conformal four point functions and the operator product expansion,”
*Nuclear Physics B*, vol. 599, no. 1-2, pp. 459–496, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Bakalov and N. M. Nikolov, “Jacobi identity for vertex algebras in higher dimensions,”
*Journal of Mathematical Physics*, vol. 47, no. 5, Article ID 053505, 30 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. K. Dobrev, G. Mack, V. B. Petkova, S. G. Petrova, and I. T. Todorov,
*Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory*, vol. 63 of*Lecture Notes in Physics*, Springer, Berlin, Germany, 1977. - G. Mack, “All unitary ray representations of the conformal group $SU(2,2)$ with positive energy,”
*Communications in Mathematical Physics*, vol. 55, no. 1, pp. 1–28, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Lang,
*Algebra*, vol. 211 of*Graduate Texts in Mathematics*, Springer, New York, NY, USA, 3rd edition, 2002. View at MathSciNet - T. Enright, R. Howe, and N. Wallach, “A classification of unitary highest weight modules,” in
*Representation Theory of Reductive Groups*, vol. 40 of*Progress in Mathematics*, pp. 97–143, Birkhäuser, Boston, Mass, USA, 1983. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. U. Schmidt, “Lowest weight representations of some infinite-dimensional groups on Fock spaces,”
*Acta Applicandae Mathematicae*, vol. 18, no. 1, pp. 59–84, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Mack and M. de Riese, “Simple space-time symmetries: generalizing conformal field theory,”
*Journal of Mathematical Physics*, vol. 48, no. 5, Article ID 052304, 21 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Kac and A. Radul, “Representation theory of the vertex algebra ${W}_{1+\infty}$,”
*Transformation Groups*, vol. 1, no. 1-2, pp. 41–70, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Kashiwara and M. Vergne, “On the Segal-Shale-Weil representations and harmonic polynomials,”
*Inventiones Mathematicae*, vol. 44, no. 1, pp. 1–47, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. P. Jakobsen, “The last possible place of unitarity for certain highest weight modules,”
*Mathematische Annalen*, vol. 256, no. 4, pp. 439–447, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Günaydin and R. J. Scalise, “Unitary lowest weight representations of the noncompact supergroup $OSp(2{m}^{\ast}|2n)$,”
*Journal of Mathematical Physics*, vol. 32, no. 3, pp. 599–606, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet