Status  Confirmed 
Seminar Series  MATHIHES 
Subjects  math 
Date  Monday 2 October 2023 
Time  10:30 
Institute  IHES 
Seminar Room  Amphithéâtre Léon Motchane 
Speaker's Last Name  Kumar 
Speaker's First Name  Shrawan 
Speaker's Email Address  
Speaker's Institution  University of North Carolina, Chapel Hill & IHES 
Title  Verlinde Dimension Formula for the Space of Conformal Blocks and the moduli of Gbundles 
Abstract  Let G be a simplyconnected complex simple algebraic group and let C be a smooth projective curve of any genus. Then, the moduli space of semistable Gbundles on C admits so called determinant line bundles. E. Verlinde conjectured a remarkable formula to calculate the dimension of the space of generalized theta functions, which is by definition the space of global sections of a determinant line bundle. This space is also identified with the space of conformal blocks arising in Conformal Field Theory, which is by definition the space of coinvariants in integrable highest weight modules of affine KacMoody Lie algebras. Various works notably by TsuchiyaUenoYamada, KumarNarasimhanRamanathan, Faltings, BeauvilleLaszlo, Sorger and Teleman culminated into a proof of the Verlinde formula. The main aim of this talk is to give a basic outline of the proof of this formula derived from the Propogation of Vacua and the Factorization Theorem among others. The proof requires techniques from algebraic geometry, geometric invariant theory, representation theory of affine KacMoody Lie algebras, topology, and Lie algebra cohomology. Some basic knowledge of algebraic geometry and representation theory of semisimple Lie algebras will be helpful; but not required. This lecture should be suitable for any one interested in interaction between algebraic geometry, representation theory, topology and mathematical physics. 
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