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   field theories, and Witten's explanation of the new knot polynomials in   
   terms of topological quantum field theories.   
      
   Let's say we have two-dimensional magnet.  (Ferro- or antiferro- doesn't   
   really matter at the level of vagueness I'll be working at; the high-Tc   
   superconductors are layered crystals that are antiferromagnets in each   
   layer.)  We'll just naively assume each atom has a spin which is a unit   
   vector, i.e. a point on S^2.  And we'll just model the state of the   
   magnet as a spin field, that is, a map from space, R^2, to S^2.  (I.e.   
   we're doing a continuum limit: for antiferromagnets we flip over the   
   spin of every other atom (in our model) to get a nice continuous map   
   from R^2 to S^2.)  Let's assume that all the spins are lined up at   
   spatial infinity.  Thus we can add a point at infinity to R^2 (getting a   
   sphere S^2) and describe the state of our magnet as a map from S^2 to   
   S^2.  Physicists like to use the delightfully uninformative term   
   "nonlinear sigma model" to describe a field theory in which the field is   
   a map from one manifold (e.g. S^2) to another (e.g. S^2) - a   
   generalization of the usual vector or tensor field.  So we've got   
   ourselves a simple nonlinear sigma model to describe the 2d magnet.  I   
   should tell you the Lagrangian but I'm carefully avoiding any   
   equations, so I'll just say (for those in the know) that it's the one   
   that gives harmonic maps.   
      
   Now, maps from S^2 to S^2 come in various homotopy classes,   
   that is, different maps from S^2 to S^2 may not be able to be   
   continuously deformed into each other.  It is a little hard for me to   
   draw these things on this crummy text-based news system, but they really   
   aren't hard to visualize with some work.  Just as the homotopy classes   
   of maps from S^1 to itself are indexed by an integer, the winding   
   number, so are the maps from S^2 to itself: there's a kind of "winding   
   number" that counts (with sign) how many times you've wrapped the   
   sphere over itself.   These twists in the field act sort of like   
   localized particles (for a lower-dimensional analogy imagine them as   
   twists in a ribbon) and are called topological solitons.  For   
   physicists, the "winding number" I mentioned above is called the   
   soliton number.  It acts like a conserved charge.  One can start with a   
   field configuration with zero soliton number -- all spins lined straight   
   up -- and then have a soliton-antisoliton pair form, move around, and   
   then annihilate, for example.  Note that if we track the birth and death   
   of soliton-antisoliton pairs over time by drawing their worldlines, we   
   get a link!  This is where knots and braids sneak into the picture:   
      
      
               /\   
              /  \ /\   
             /    \  \   
            /    / \  \   
            \    \ /  /   
             \    \  /   
              \  / \/   
               \/   
      
   In this picture time goes up the page, and we see first one pair formed,   
   then another, then they move around each other and then they annihilate.   
   We have a *link* with linking number 1 (let's say - the sign actually   
   depends on a right-hand rule, but since I'm left-handed I object to   
   using the usual right-hand rule).   
      
   Polyakov's trick was to add a term to the Lagrangian which equals a   
   constant theta times the linking number.  It's a bit more technical so before   
   describing how he gets a local expression for this term I'll just say   
   what it's effect is on the physics.  Classically, it has no effect   
   whatsoever!  Since a small variation in the field configuation doesn't   
   change the linking number (which after all is a topological invariant),   
   the Euler-Lagrange equations (which come from differentiating the   
   Lagrangian) don't notice this term at all.  Quantum mechanically,   
   however, one doesn't just look for an extremum of the action.  Instead   
   one forms a path integral a la Feynman, integrating exp(i Action) over   
   all histories.   So if two histories have different linking numbers,   
   their contribution to the integral will differ by a phase.  For example,   
   the configuration above has the exact same action as this one:   
      
      
               /\   
              /  \ /\   
             /    /  \   
            /    / \  \   
            \    \ /  /   
             \    /  /   
              \  / \/   
               \/   
      
   by symmetry, except that the first, "right-handed", history has linking   
   number 1, while the second, "left-handed" one has linking number -1.   
   Thus the first will appear in the path integral with a factor of exp(i   
   theta), while the second will appear with a factor of exp(-i theta).   
   Thus if one soliton goes around another we get a phase factor, so --   
   here the reader needs a bit of faith -- they act like anyons.   
      
   Now let me describe Polyakov's term in the Lagrangian in a bit more   
   detail.  Here I'll allow myself to be a tad more technical.   Let us   
   assume that (as in the pictures above) we are considering histories   
   which begin and end with all spins lined up.  Thus our map from   
   spacetime (2d space, 1d time) to S^2 may be regarded as a map from S^3   
   to S^2, using the old "point at infinity" trick again.  The homotopy   
   classes of maps from S^3 to S^2 are also indexed by an integer, this   
   being called the Hopf invariant.  The first way to calculate the Hopf   
   invariant shows why Polyakov uses it as an extra term in the Lagrangian.   
   Take a map from S^3 to S^2.  By Sard's theorem almost every point in S^2   
   will have as its inverse image in S^3 a collection of nonintersecting   
   closed curves (i.e., a link).  The Hopf invariant may be calculated as   
   follows: take two such points in S^2 and call their inverse images in   
   S^3 L and L'.  The Hopf invariant is the linking number link(L,L')   
   (which doesn't depend on which points you picked).  To see how   
   this relates to the story above, take two nearby points in S^2 and draw   
   an arc between them.  The inverse image of this arc in S^3 is a "ribbon"   
   or "framed link," and the Hopf invariant, link(L,L'), is   
   also called the "self-linking number" of the framed link, since it includes   
   information about how the ribbon twists, as well as how it links itself   
   when it has more than one connected component.  Physically, the contribution   
   to the Hopf invariant due to ribbon twisting is interpreted as due to   
   the rotation of individual anyons.  Since spin as well as   
   statistics contributes to the phase exp(i Action), to be precise one   
   must model the anyon trajectories not by a link in spacetime, but by a   
   framed link, which keeps track of how they rotate.   
      
   The second way to calculate the Hopf   
   invariant shows how to write it down as an integral over S^3 of a local   
   expression (Lagrangian density).  Take the volume form on S^2 and pull   
   it back to S^3 by our map.  We now have a closed 2-form on S^3 so we can   
   write it as dA for some 1-form.  Now integrate A^dA over S^2 and divide   
   by something like 4 pi.  This is the Hopf invariant!  I leave it as an   
   easy exercise to show that it didn't depend on our choice of A, and as a   
   slightly harder exercise to show that it really is a diffeomorphism   
   invariant, and as a harder exercise to show that this definition of the   
   Hopf invariant agrees with the linking number one.   (For more info read   
   Bott and Tu's "Differential Forms and Algebraic Topology".)   
      
   Note that the freedom of choice of A here is none other than what   
   physicists call "gauge freedom."  What we have here, in other words, is   
   a gauge theory with a diffeomorphism invariant Lagrangian.  (That is, if   
   we keep the Hopf term and drop the harmonic action.)  Such theories give   
      
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