Physics – Mathematical Physics
Scientific paper
2005-09-30
G\'{e}om\'{e}trie Complexe II Aspects contemporains dans les math\'{e}matiques et la physique (2004) ??
Physics
Mathematical Physics
Actes du Colloque "Complex Geometry '98"
Scientific paper
This talk gives a review on how complex geometry and a Lagrangian formulation of 2-d conformal field theory are deeply related. In particular, how the use of the Beltrami parametrization of complex structures on a compact Riemann surface fits perfectly with the celebrated locality principle of field theory, the latter requiring the use infinite dimensional spaces. It also allows a direct application of the local index theorem for families of elliptic operators due to J.-M. Bismut, H. Gillet and C. Soul\'{e}. The link between determinant line bundles equipped with the Quillen\'s metric and the so-called holomorphic factorization property will be addressed in the case of free spin $j$ b-c systems or more generally of free fields with values sections of a holomorphic vector bundles over a compact Riemann surface.
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