Braid theory

Abstract: A theory of braids leading to a classification was given in my paper "Theorie der Zopfe" in vol. 4 of the Hamburger Abhandlungen (quoted as Z). Most of the proofs are entirely intuitive. That of the main theorem in ?7 is not even convincing. It is possible to correct the proofs. The difficulties that one encounters if one tries to do so come from the fact that projection of the braid, which is an excellent tool for intuitive investigations, is a very clumsy one for rigorous proofs. This has lead me to abandon projections altogether. We shall use the more powerful tool of braid coordinates and obtain thereby farther reaching results of greater generality. A few words about the initial definitions. The fact that we assume of a braid string that it ends in a straight line is of course unimportant. It could be replaced by limit assumptions or introduction of infinite points. The present definition was selected because it makes some of the discussions easier and may be replaced any time by another one. I also wish to stress the fact that the definition of s-isotopy is of a provisional character only and is replaced later (Definition 3) by a general notion of isotopy. More than half of the paper is of a geometric nature. In this part we develop some results that may escape an intuitive investigation (Theorem 7 to 10). We do not prove (as has been done in Z) that the relations (18) (19) are defining relations for the braid group. We refer the reader to a paper by F. Bohnenblust1 where a proof of this fact and of many of our results is given by purely group theoretical methods. Later the proofs become more algebraic. With the developed tools we are able to give a unique normal form for every braid2 (Theorem 17, fig. 4 and remark following Theorem 18). In Theorem 19 we determine the center of the braid group and finally we give a characterisation of braids of braids. I would like to mention in this introduction a few of the more important of the unsolved problems: 1) Assume that two braids can be deformed into each other by a deformation of the most general nature including self intersection of each string but avoiding intersection of two different strings. Are they isotopic? One would be inclined to doubt it. Theorem 8 solves, however, a special case of this problem. 2) In Definition 3, we introduce a notion of isotopy that is already very general. What conditions must be put on a many to many mapping so that the result of Theorem 9 still holds?

Abstract: This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is M. Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim $X\geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

Abstract: The theory of superselection sectors is generalized to situations in which normal statistics has to be replaced by braid group statistics. The essential role of the positive Markov trace of algebraic quantum field theory for this analysis is explained, and the relation to exchange algebras is established.