Conformal Field Theory and moduli of G-bundles

This is the web-page of a graduate course that I tought at the University "Roma Tre" during the accademic year 2014/15. This is the official web-page.

Plan:

I will explain in details the Wess-Zumino-Witten model. This is one of the main example of Conformal Field Theory.  From a mathematical point of view, it gives a deep and fruitful link between Lie theory and moduli of principal G-bundle over Riemann surfaces.

Pre-requisites
Finite dimensional Representation Theory and basic Algebraic Geometry. I will discuss some background material.

Lectures summary:

10 - Feb - 2015: Overview. Background material on algebraic groups and flag varieties.

12- Feb  : Background on finite dimensional Lie algebras.

17 - Feb : Construction of Virasoro and affine untwisted Kac-Moody algebras. First examples of representations.

19- Feb: Integrable representation of a Kac-Moody algebra. Seagel-Sugawara form.

24- Feb: Examples of Ind-varieties. Kac-Moody groups and first link with vector bundles

26- Feb: More about Kac-Moody groups and vector bundles. First glimpse of conformal blocks.

3 - March: Construction of the coherent sheaf of conformal blocks. Local freeness for smooth curves.

5 March: Projective connection on the bundle of conformal blocks. Atiyah algebras and computation of the slope.

10 March: Parabolic bundles, generalized conformal blocks and propagation of vacua.

12 March:  Description of the conformal blocks in the genus zero case. Statement of the Verlinde Formula and idea of the proof.

References:

• Lie algebras of finite and affine type - Roger Carter;
• Loop Groups - Andrew Pressley and Graeme Segal;
• Formule de caracteres pour les algebres de Kac-Moody generales - O. Mathieu;
• Moduli Spaces of Curves and Representation Theory - E. Arbarello, C. De Concini, V.G. Kac,e C. Procesi
• La formule de Verlinde, Seminaire Bourbaki 794 - C. Sorger;
• Vector bundle on Riemann surfaces and Conformal Field Theory (1996) - A. Beauville
• Conformal blocks, Fusion rings and Verlinde formula,  Proc. of the Hirzebruch 65 Conf. on Algebraic Geometry, (1996) - A. Beauville
• Borel-Weil-Bott Theory on the moduli of G-bundles over a curve - Invent. Math. 134 (1998) - C. Teleman;
• Conformal Field Theory with Gauge Symmetry - K. Ueno
• Notes on 2D Conformal Field Theory and String Theory - Dennis Gaitsgory;
• Vertex Algebras and Algebraic Curves - Edward Frenkel and David Ben-Zvi;
• Topology, Geometry and Quantum Field Theory. Proc., 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal.