Showing papers by "Jürg Fröhlich published in 1990"

Braid statistics in local quantum theory

Abstract: We present details of a mathematical theory of superselection sectors and their statistics in local quantum theory over (two- and) three-dimensional space-time. The framework for our analysis is algebraic quantum field theory. Statistics of superselection sectors in three-dimensional local quantum theory with charges not localizable in bounded space-time regions and in two-dimensional chiral theories is described in terms of unitary representations of the braid groups generated by certain Yang-Baxter matrices. We describe the beginnings of a systematic classification of those representations. Our analysis makes contact with the classification theory of subfactors initiated by Jones. We prove a general theorem on the connection between spin and statistics in theories with braid statistics. We also show that every theory with braid statistics gives rise to a “Verlinde algebra”. It determines a projective representation of SL(2, ℤ) and, presumably, of the mapping class group of any Riemann surface, even if t...

On the structure of unitary conformal field theory II: Representation theoretic approach

Abstract: Two-dimensional, unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras. Chiral algebras are equipped with a family of co-multiplications which serve to define tensor product representations. Chiral vertices arise as Clebsch-Gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra. The algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by Yang-Baxter (braid) matrices. Chiral fusion is analyzed. It is shown that the braid- and fusion matrices determine invariants of knots and links. Connections between the representation theories of chiral algebras and of quantum groups are sketched. Finally, it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras.

Braid Statistics in Three-Dimensional Local Quantum Theory

Abstract: The general theory of superselection sectors and their statistics in three-dimensional local quantum theory is outlined. It is shown that abelian and nonabelian braid statistics can arise, provided that reflections at lines in two-dimensional space (parity) are not symmetries of the theory. Braid statistics is completely described by a family of braid- and fusion matrices satisfying polynomial equations. These braid- and fusion matrices have properties very similar to those of the corresponding matrices in two-dimensional conformal field theory and determine invariants for coloured links in S 3 . The role of quantum group theory in three-dimensional local quantum theory is elucidated. Excitations with braid statistics are believed to play an important role in fractional quantum Hall systems and two-dimensional high-Tc superconductors. Three-dimensional gauge theories with Chern-Simons term exhibiting such excitations are briefly described.

Variational problems on vector bundles

Abstract: A variety of problems in quantum physics and classical statistical mechanics, in particular the quantization of topological solitons and the statistical mechanics of defects in ordered media, are described. These problems can be studied within a semi-classical approximation, or with the help of low-temperature expansions, respectively. The calculation of the leading term in such expansions gives rise to variational problems for sections of vector bundles characterized by certain topological constraints. Examples of such problems are the quantization of kinks in the two-dimensional λϕ4-theory and the analysis of Bloch walls in a Landau-Ginzburg model of a threedimensional anisotropic ferromagnet. We state a general existence result for variational problems of this kind and develop regularity and decay estimates for solutions of the Landau-Ginzburg model describing Bloch walls with prescribed boundaries. For certain boundary configurations stability results are established. The relation between the minimizers of the Landau-Ginzburg model in a certain strong-coupling limit and minimal surfaces is pursued in some detail. An open question is whether, asymptotically, the stability of the limit (minimal) surface will imply the stability of the minimizers of the Landau-Ginzburg model.