Conformal Blocks, Generalized Theta Functions and the Verlinde Formula

In 1988, E. Verlinde gave a remarkable conjectural formula for the dimension of conformal blocks over a smooth curve in terms of representations of affine Lie algebras. Verlinde's formula arose from physical considerations, but it attracted further attention from mathematicians when it was realized that the space of conformal blocks admits an interpretation as the space of generalized theta functions. A proof followed through the work of many mathematicians in the 1990s. This book gives an authoritative treatment of all aspects of this theory. It presents a complete proof of the Verlinde formula and full details of the connection with generalized theta functions, including the construction of the relevant moduli spaces and stacks of G-bundles. Featuring numerous exercises of varying difficulty, guides to the wider literature and short appendices on essential concepts, it will be of interest to senior graduate students and researchers in geometry, representation theory and theoretical physics.

Exact Modular $S$ Matrix for ${\mathbb Z}_{k}$ Parafermion Quantum Hall Islands and Measurement of Non-Abelian Anyons

Using the decomposition of rational conformal filed theory characters for the $\Z_k$ parafermion quantum Hall droplets for general $k=2,3,\dots$, we derive analytically the full modular $S$ matrix for these states, including the $\uu$ parts corresponding to the charged sector of the full conformal field theory and the neutral parafermion contributions corresponding to the diagonal affine coset models. This precise neutral-part parafermion $S$ matrix is derived from the explicit relations between the coset matrix and those for the numerator and denominator of the coset and the latter is expressed in compact form due to the level-rank duality between the affine Lie algebras $\widehat{\frak{su}(k)_2}$ and $\widehat{\frak{su}(2)_k}$. The exact results obtained for the $S$ matrix elements are expected to play an important role for identifying interference patterns of fractional quantum Hall states in Fabry-P\'erot interferometers which can be used to distinguish between Abelian and non-Abelian statistics of quasiparticles localized in the bulk of fractional quantum Hall droplets as well as for nondestructive interference measurement of Fibonacci anyons which can be used for universal topological quantum computation