Seminar : Affine Lie Algebras

This is the page for the seminar on QFT at the University of Queensland. We focus on Affine Lie Algebras for Semester 1, 2015. The main goal is to understand various aspects of the theory of affine Lie algebras in mathematics and physics.

when : Thursdays 3-4:30 pm
where : Priestly Building Seminar Room 67-442

topics

Some references are given for each topic. But we won't necessarily follow them strictly or cover them all.

introduction (12/3/2015, Chul-hee Lee)

overview
distribution of the remaining talks

affine Lie algebras as central extensions of loop algebras

chapter 7 of [Kac 1994]
chapter 18 of [Carter 2005]

Kac-Moody algebras

chapter 1&4 of [Kac 1994]
chapter 14&15 of [Carter 2005]

Sugawara construction of Virasoro algebra (Chris Raymond)

chapter 12 of [Kac 1994]
chapter 15 of [DFMS 1997]
Bardakci, K., and M. B. Halpern. 2009. ‘The Dual Quark Models’. arXiv:0907.2705 [hep-Th, Physics:math-Ph], July. http://arxiv.org/abs/0907.2705.
Virasoro, M. A. “The Little Story of an Algebra.” In String Theory and Fundamental Interactions, edited by Maurizio Gasperini and Jnan Maharana, 137–44. Lecture Notes in Physics 737. Springer Berlin Heidelberg, 2008. http://link.springer.com/chapter/10.1007/978-3-540-74233-3_6.

integrable highest weight representations of affine Lie algebras

chapter 12 of [Kac 1994]
chapter 3 of [Wakimoto 2001]
chapter 20 of [Carter 2005]

Wess-Zumino-Witten model

chapter 15 of [DFMS 1997]
Walton, Mark. ‘Affine Kac-Moody Algebras and the Wess-Zumino-Witten Model’. arXiv:hep-th/9911187, 23 November 1999. http://arxiv.org/abs/hep-th/9911187.
Goddard, Peter, and David Olive. “Kac-Moody and Virasoro Algebras in Relation to Quantum Physics.” International Journal of Modern Physics A 01, no. 02 (July 1, 1986): 303–414. doi:10.1142/S0217751X86000149.

modular transformations of characters of affine Lie algebras

chapter 4 of [Wakimoto 2001]
chapter 13 of [Kac 1994]
Macdonald, I. G. 1981. “Affine Lie Algebras and Modular Forms.” In Séminaire Bourbaki Vol. 1980/81 Exposés 561–578, 258–276. Lecture Notes in Mathematics 901. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0097202.

fusion rules and Verlinde formula

chapter 5 of [Wakimoto 2001]
chapter 16 of [DFMS 1997]
Gannon, Terry. 2005. “Modular Data: The Algebraic Combinatorics of Conformal Field Theory.” Journal of Algebraic Combinatorics. An International Journal 22 (2): 211–250. doi:10.1007/s10801-005-2514-2.
Feingold, Alex J. 2004. ‘Fusion Rules for Affine Kac-Moody Algebras’. In Kac-Moody Lie Algebras and Related Topics, 343:53–96. Contemp. Math. Amer. Math. Soc., Providence, RI. http://arxiv.org/abs/math/0212387
Fuchs, J. 1994. ‘Fusion Rules in Conformal Field Theory’. Fortschritte Der Physik/Progress of Physics 42 (1): 1–48. doi:10.1002/prop.2190420102. http://arxiv.org/abs/hep-th/9306162
Verlinde, Erik. 1988. “Fusion Rules and Modular Transformations in 2D Conformal Field Theory.” Nuclear Physics B 300: 360–376. doi:10.1016/0550-3213(88)90603-7.

vertex operator constructions of basic representations

chapter 14 of [Kac 1994]
chapter 20 of [Carter 2005]
Frenkel, I. B., and V. G. Kac. “Basic Representations of Affine Lie Algebras and Dual Resonance Models.” Inventiones Mathematicae 62, no. 1 (February 1, 1980): 23–66. doi:10.1007/BF01391662.
Lepowsky, James, and Robert Lee Wilson. “Construction of the Affine Lie algebra $A_1^{(1)}$.” Communications in Mathematical Physics 62, no. 1 (1 August 1978): 43–53. doi:10.1007/BF01940329.

links

UQ Quantum Field Theory Seminar 2013-2014

references

[*Kac 1994] Kac, Victor G. Infinite-Dimensional Lie Algebras. Third. Cambridge University Press, Cambridge, 1990. http://www.ams.org/mathscinet-getitem?mr=1104219.
[*DFMS 1997] Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997. http://www.ams.org/mathscinet-getitem?mr=1424041.
[*Wakimoto 2001] Wakimoto, Minoru. Infinite-Dimensional Lie Algebras. Vol. 195. Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 2001. http://www.ams.org/mathscinet-getitem?mr=1793723.
[*Carter 2005] Carter, R. W. 2005. Lie Algebras of Finite and Affine Type. Vol. 96. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge. http://www.ams.org/mathscinet-getitem?mr=2188930.

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reading for fun

Helgason, Sigurdur. ‘A Centennial: Wilhelm Killing and the Exceptional Groups’. The Mathematical Intelligencer 12 (3): 54–57. doi:10.1007/BF03024019.
Coleman, A. J. ‘The Greatest Mathematical Paper of All Time’. The Mathematical Intelligencer 11 (3): 29–38. doi:10.1007/BF03025189.
Berman, Stephen, and Karen Hunger Parshall. ‘Victor Kac and Robert Moody: Their Paths to Kac-Moody Lie Algebras’. The Mathematical Intelligencer 24, no. 1 (13 January 2009): 50–60. doi:10.1007/BF03025312.
Dolan, Louise. ‘The Beacon of Kac-Moody Symmetry for Physics’. Notices of the American Mathematical Society 42, no. 12 (1995): 1489–95. http://www.ams.org/notices/199512/dolan.pdf
O’Raifeartaigh, L. ‘The Intertwining of Affine Kac–moody and Current Algebras’. International Journal of Modern Physics B 13, no. 24n25 (10 October 1999): 3009–20. doi:10.1142/S0217979299002824.

advanced reading

Goddard, Peter, and David Olive, eds. Kac-Moody and Virasoro Algebras. Vol. 3. Advanced Series in Mathematical Physics. World Scientific Publishing Co., Singapore, 1988. http://www.ams.org/mathscinet-getitem?mr=966668.
Frenkel, Igor, James Lepowsky, and Arne Meurman. Vertex Operator Algebras and the Monster. Vol. 134. Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1988. http://www.ams.org/mathscinet-getitem?mr=996026.