Geometry and Duality

This is a big topic I am now in love with. I'll try to expound on it here, over time.

• Fractals, Stokes, and 3-d duality

  in 3d we can have a special sort of "duality",
  
          A = *dA
  
  (sometimes called a topological chern-simons
  mass (in units where the mass is 1) since it
  comes from the lagrangian
  
          L = dA/\*dA + A/\dA
  
  Now, I've been trying to interpret this using
  some geometrical pictures, and they seem to
  imply very strange things.  Here we go:
  
  let's consider a "Wilson probe" , the integral
  of A along some curve C :
  
          W = I[C] { A }
  
  to establish notation, I use stokes, with C = b S
  ( 'b' means bdry of ), so 
  
          W = I[S] { dA }
  
  Instead, I could have used duality :
  
          W = I[C] { *dA }
  
  Now, I use the fact that "*" in 3-d turns a vector into
  the plane perpindicular to it, and vice-a-versa (to what
  extent this is more than intuitive, I cannot say), so
  let
  
          F = family of surfaces, each satisfying *F = C
                  (a foliation of perpendicular surfaces
                  along the path C)
  
  then:
  
          W = I[F] { dA }
  
  Now we can use Stokes : bF = C' :
  
          W = I[C'] { A }
  
  Compare to where we started :
  
          W = I[C] { A }
  
  This is a purely topological equivalence between the curves
  C and C' - which are, however, non-trivially different.
  C' is the boundary of the family of surfaces perp. to C ;
  by drawing a little picture it is easy to see that C' is a
  helix which spirals around C (very tight, and infinitesimally
  close).
  
  So, it seems the "duality" condition means that any Wilson
  probe is equivalent to a thickened or framed Wilson probe.
  
  Of course, we can apply this "duality thickening" recursively,
  
  C' -> C'' -> C''' -> C''''
  
  Each time, the curve is replaced by the helix which wraps around
  that curve. (this is reminiscent of DNA hyper-coiling)  So,
  apparently C'''''''(infinity) = Ci is become fractally-thick !
  that is, the space becomes so full of coilings, that Ci becomes
  a solid object!
  
  if C is ~ S1 (reads homotopic)
  
  then Ci is S1 x S1 = T2 !
  
  duality has replaces the circle by the torus !
  (more precisly, if S1(r) is a circle of radius roughly r, then
  
  Ci = S1(r) x S1(epsilon) = T2(r,eps)
  
  is a "barely" thickened torus covering C.
  
  Have I done this right?  Can it be made more precise?
  (in particular, the step of replacing C by F was not rigorous)
  

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