This is a big topic I am now in love with. I'll try to expound on it here, over time.
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)
Charles Bloom / cb at my domain Send Me Email
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