Mathematical Physics

   

Could One Generalize Braid Invariant Defined by Vacuum Expecation of Wilson Loop to and Invariant of Braid Cobordisms and of 2-Knots?

Authors: Matti Pitkänen

Witten was awarded by Fields medal from a construction recipe of Jones polynomial based on topological QFT assigned with braids and based on Chern-Simons action. Recently Witten has been working with an attempt to understand in terms of quantum theory the so called Khovanov polynomial associated with a much more abstract link invariant whose interpretation and real understanding remains still open.

The attempts to understand Witten's thoughts lead to a series of questions unavoidably culminating to the frustrating "Why I do not have the brain of Witten making perhaps possible to answer these questions?". This one must just accept. In this article I summarize some thoughts inspired by the associations of the talk of Witten with quantum TGD and with the model of DNA as topological quantum computer. In my own childish manner I dare believe that these associations are interesting and dare also hope that some more brainy individual might take them seriously.

An idea inspired by TGD approach which also main streamer might find interesting is that the Jones invariant defined as vacuum expectation for a Wilson loop in 2+1-D space-time generalizes to a vacuum expectation for a collection of Wilson loops in 2+2-D space-time and could define an invariant for 2-D knots and for cobordisms of braids analogous to Jones polynomial. As a matter fact, it turns out that a generalization of gauge field known as gerbe is needed and that in TGD framework classical color gauge fields defined the gauge potentials of this field. Also topological string theory in 4-D space-time could define this kind of invariants. Of course, it might well be that this kind of ideas have been already discussed in literature.

Comments: 17 Pages.

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Submission history

[v1] 1 Nov 2011
[v2] 2012-01-30 08:34:19

Unique-IP document downloads: 162 times

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