# Geometry and Representation Theory

# Working Seminar Winter 2015

The topic of the seminar is the WZW model in conformal field theory, focusing on geometrical aspects - in particular the construction of the (flat projective) WZW on the bundles over the moduli space of stable curves . One could describe this as “a gift of representation theory to geometry, prophesied by physics”. In the first half of the seminar we will focus on the (infinite-dimensional) representation theory that is needed, and then we will use this to tell the (finite-dimensional) geometric story. At the end we will discuss some of the physics behind this, and possibly give some recent applications in topology or geometry.

The seminar will meet weekly on Tuesdays from 5 till 6 pm in JCMB 5326, during S2 of 2014-15. In the first meeting we will give an introduction, and divide the work for the rest of the seminar.

## Seminar schedule:

Date | Topic | Speaker |

20 Jan | Introduction & organization of the seminar. Quick review of highest-weight representations of semi-simple Lie algebras. | Johan Martens |

27 Jan | Introduction to affine Kac-Moody algebras. Noah's notes. | Noah White |

3 Feb | Representation theory of affine Lie algebras. | David Jordan |

10 Feb | The Virasoro algebra and its representations. Segal-Sugawara construction. | Sue Sierra |

17 Feb | No seminar | |

24 Feb | Generalities on (projective) connections, Atiyah algebroids, twisted D-modules. | Chunyi Li |

3 Mar | Brief intro to stacks, M_g,n-bar and the Hodge bundle. | Sjoerd Beentjes |

10 Mar | Construction of WZW conformal blocks bundles. Factorization. | Salvatore Dolce |

17 Mar | Fusion Rules and the Verlinde Formula. | Salvatore Dolce |

24 Mar (4-6pm) | The WZW flat projective connection. | Johan Martens |

31 Mar | The WZW model in conformal field theory. | José Figueroa-O'Farrill |

7 Apr (4-5pm) | The WZW model in conformal field theory continued. | José Figueroa-O'Farrill |

7 Apr (5-6pm) | WZW model and modular functors. | Adrien Brochier |

## Some relevant literature & other resources:

- V.G. Kac, Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990. xxii+400 pp. Available online (from within University network).
- A. Tsuchiya, K. Ueno and Y, Yamada, Conformal field theory on universal family of stable curves with gauge symmetries. Integrable systems in quantum field theory and statistical mechanics, 459–566, Adv. Stud. Pure Math., 19, Academic Press, Boston, MA, 1989.
- E. Looijenga, From WZW models to modular functors. Handbook of moduli. Vol. II, 427–466, Adv. Lect. Math. (ALM), 25, Int. Press, Somerville, MA, 2013. Available online.
- Chapter 7 of B. Bakalov and A. Kirillov,Jr. Lectures on tensor categories and modular functors. University Lecture Series, 21. American Mathematical Society, Providence, RI, 2001. Available online.
- Y. Tsuchimoto, On the coordinate-free description of the conformal blocks. J. Math. Kyoto Univ. 33 (1993), no. 1, 29–49. Available online.
- A. Beauville, Conformal blocks, fusion rules and the Verlinde formula. Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), 75–96, Israel Math. Conf. Proc., 9, Bar-Ilan Univ., Ramat Gan, 1996. Available online.
- C. Sorger, La formule de Verlinde. Séminaire Bourbaki, Vol. 1994/95. Astérisque No. 237 (1996), Exp. No. 794, 3, 87–114. Available online.
- N. Fakhruddin, Chern classes of conformal blocks. Compact moduli spaces and vector bundles, 145–176, Contemp. Math., 564, Amer. Math. Soc., Providence, RI, 2012. Available online.
- A. Marian, D. Oprea, R. Pandharipande, A. Pixton and D. Zvonkine, The Chern character of the Verlinde bundle over the moduli space of stable curves, 2013, arXiv:1311.3028.
- E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Second Edition. Vol. 88 of Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 2004.
- A.A. Beĭlinson and V.V. Schechtman, Determinant bundles and Virasoro algebras. Comm. Math. Phys. 118 (1988), no. 4, 651–701. Available online.
- J.S. Nauta, Affine Lie Algebras and Affine Root Systems, MSc thesis, University of Amsterdam, 2012. (covers some of the chapters in Kac' book in somewhat greater detail). Available online.
- V.G. Kac, A. Raina and N. Rozhkovskaya, Bombay lectures on highest weight representations of infinite dimensional Lie algebras. Second edition. Advanced Series in Mathematical Physics, 29. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
- V. Chari, Infinite Dimensional Lie Algebras, video lectures, Available online.
- ...

The seminar is coordinated by Johan Martens. All are welcome to attend. If you want to be included in the mailing list please email johan.martens@ed.ac.uk