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Logarithmic conformal field theory as a duality between Brauer-type algebras and quantum groups at roots of unity

Objective

This project concerns the little studied class of logarithmic two-dimensional conformal field theories (LCFTs) - with unusual properties such as non-unitarity and non-semisimplicity. LCFTs have applications in statistical mechanics (sand-piles, percolation, polymers, disordered electrons) and in the AdS/CFT conjecture. They have come under increased interest in the mathematical physics community . Yet, very few LCFTs have been understood at the writing of this proposal, and the subject is still in its infancy.

LCFTs involve difficult mathematical aspects, and bear a resemblance to non semi-simple Lie algebras. Their study involves such problems as the description of non-semisimple (modular) braided tensor categories of modules over vertex-operator algebras (VOAs), as well as representation theory of affine Hecke algebras and quantum groups at `roots of unity'.

We propose to bridge and further develop two recent promising approaches. One - put forward by the applicant - uses a new, mathematical, description of LCFTs based on VOAs, screening operators and ribbon Hopf algebras. The other, pioneered by the host, uses a more physical approach based on explicit lattice regularizations, and study of the corresponding lattice algebras. Both rely on similar mathematical structures, such as quantum groups, bimodules over mutual centralizers, non-semisimple tensor categories, and Morita equivalence. It is hoped that, by bridging the two approaches, several goals will be reached. On the mathematical side, these include better understanding and classification of indecomposable modules for chiral algebras in LCFTs, better understanding and definition of the scaling limit, and reconstruction of non chiral LCFTs from their chiral counterparts. On the physical side, these includes construction of the full consistent LCFT description of geometrical problems and of sigma models describing phase transitions in disordered electronic systems.

Field of science

  • /natural sciences/physical sciences/classical mechanics/statistical mechanics
  • /natural sciences/mathematics/pure mathematics/algebra
  • /natural sciences/mathematics/applied mathematics/mathematical physics

Call for proposal

FP7-PEOPLE-2010-IIF
See other projects for this call

Funding Scheme

MC-IIF - International Incoming Fellowships (IIF)

Coordinator

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Address
Rue Michel Ange 3
75794 Paris
France
Activity type
Research Organisations
EU contribution
€ 185 864
Administrative Contact
Michèle Saumon (Ms.)