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Institute for the Physics and Mathematics of the UniverseWPI

IPMU Komaba Lectures

Date: Starting October 14, 2008, every Tuesday, 10:30 - 12:00
Place: Room 128, Mathematical Sciences Building, Komaba Campus, The University of Tokyo.
Speaker: Akihiro Tsuchiya (IPMU)
Title: Conformal field theory and vertex operator algebras
Contents: In these series of lectures we develop conformal field theory (CFT) on Riemann surfaces using representation theory of vertex operator algebras (VOA). Among the most important notions in CFT are the locality of fields defined on Riemann surfaces, and their operator product expansion (OPE). In order to formulate them algebraically as a mathematical concept, vertex operator algebras have been introduced. Vertex operators provide a framework which allow for freely manipulating with OPE, without recourse to the language of the representation theory of Virasoro or affine Lie algebras which have been the main tools in the previous treatment. In these lectures, we begin as a warmup with explaining CFT based on free fields. After that, we develop CFT for the Virasoro minimal series, making use of free field representation and screening operators. Through these lectures the audience will get acquainted with OPE and representations of VOA.

In the latter half of the lectures, we explain the structure of VOA with C_2 finiteness condition, in particular the finiteness properties of the abelian categories of their representations. This abelian category is both Artinian and Noetherian, having a finite number of simple objects but itself cannot be semi-simple. For that reason the moding operator L_0 of the Virasoro algebra is non-diagonalizable, allowing for Jordan blocks in general. Within the framework of this VOA and its representation theory we will develop CFT on general Riemann surfaces. Since the category of representations is non-semisimple, problems arise when one tries to formulate the factorization property of conformal blocks. We show that this issue can be successfully resolved.

Finally we address the structure of the tensor category based on the notion of the fusion tensor product in CFT.

The class of CFT we are concerned with is known under the name of log CFT. It starts to draw attention in various fields. Our aim in these lectures is to introduce this subject to the audience.

  • (1) Introduction to conformal field theory (in Japanese), Yasuhiko Yamada, Baifuukan.
  • (2) Introduction to conformal field theory, Lectures given by Akihiro Tsuchiya, Notes taken by Toshiro Kuwabara, Mathematical Memoires vol 4, Japan Mathematical Society.
  • (3) E.Frenkel and D.Ben-Zvi, Vertex Algebras and Algebraic Curves, American Math.Soc.