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Symplectic Four‐Manifolds and Conformal Blocks
Author(s) -
Smith Ivan
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610705006307
Subject(s) - symplectic geometry , conformal map , mathematics , pure mathematics , conformal field theory , noether's theorem , vector bundle , field (mathematics) , functor , algebra over a field , mathematical analysis , lagrangian
Ideas from conformal field theory are applied to symplectic four‐manifolds through the use of modular functors to ‘linearise’ Lefschetz fibrations. In Chern–Simons theory, this leads to the study of parabolic vector bundles of conformal blocks. Motivated by the Hard Lefschetz theorem, the author shows that the bundles of SU(2) conformal blocks associated to Kähler surfaces are Brill–Noether special, although the associated flat connexions may be irreducible if the surface is simply connected and not spin.