20 research outputs found
Intersections of Loops and the Andersen-Mattes-Reshetikhin Algebra
Given two free homotopy classes of loops on an oriented
surface, it is natural to ask how to compute the minimum number of intersection
points of loops in these two classes.
We show that for the number of terms in the
Andersen-Mattes-Reshetikhin Poisson bracket of and is
equal to . Chas found examples showing that a similar
statement does not, in general, hold for the Goldman Lie bracket of
and .
The main result of this paper in the case where do not
contain different powers of the same loop first appeared in the unpublished
preprint of the second author. In order to prove the main result for all pairs
of we had to use the techniques developed by the first
author in her study of operations generalizing Turaev's cobracket of loops on a
surface.Comment: We added a Theorem on comuting the minimal number of self
intersection points using the Andersen-Mattes-Reshetikhin Poisson bracket. 20
pages, 5 figure
Topological Field Theory Interpretation of String Topology
The string bracket introduced by Chas and Sullivan [math.GT/9911159] is
reinterpreted from the point of view of topological field theories in the
Batalin-Vilkovisky or BRST formalisms. Namely, topological action functionals
for gauge fields (generalizing Chern-Simons and BF theories) are considered
together with generalized Wilson loops. The latter generate a (Poisson or
Gerstenhaber) algebra of functionals with values in the -equivariant
cohomology of the loop space of the manifold on which the theory is defined. It
is proved that, in the case of with standard representation, the
(Poisson or BV) bracket of two generalized Wilson loops applied to two cycles
is the same as the generalized Wilson loop applied to the string bracket of the
cycles. Generalizations to other groups are briefly described.Comment: 27 pages, 2 figure
Topological Field Theory Interpretation of String Topology
The string bracket introduced by Chas and Sullivan is reinterpreted from the point of view of topological field theories in the Batalin-Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern-Simons and BF theories) are considered together with generalized Wilson loops. The latter generate a (Poisson or Gerstenhaber) algebra of functionals with values in the S 1 -equivariant cohomology of the loop space of the manifold on which the theory is defined. It is proved that, in the case of GL(n,â) with standard representation, the (Poisson or BV) bracket of two generalized Wilson loops applied to two cycles is the same as the generalized Wilson loop applied to the string bracket of the cycles. Generalizations to other groups are briefly describe
A generalization of Turaev's virtual string cobracket and self-intersections of virtual strings
Previously we defined an operation ” that generalizes Turaevâs cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding question for virtual strings, and conjecture that ” gives a formula for the minimum number of self-intersection points of a virtual string in a given virtual homotopy class. To support the conjecture, we show that ” gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaevâs virtual string cobracket. We also use Turaevâs based matrices to describe a large set of strings α such that ” gives a formula for the minimal self-intersection number α. Finally, we construct an example that shows the bound on the minimal self-intersection number given by ” is always at least as good as, and sometimes stronger than, the bound Ï given by Turaevâs based matrix invariant