I started this series of posts about the simple pendulum from humble beginnings, talking about the exact solutions of a bob oscillating from a light string. In the second part, the “field theory” of the exact simple pendulum led to solitons and such interesting stuff. But of course, by that point, the simple in the title started becoming obscured. Unfortunately, that trend will continue on this post.
The motivation for this post is twofold – firstly, I started reading up on some Seiberg-Witten theory and found it enchanting and so beautiful, which took me through duality and other related esoterica; secondly, I fell ill to blogging-sickness, I had to post again!!
It has always been a source of fascination for me to hear about the ever increasing complexity of the mathematical machinery underlying much of today’s theoretical physics (at least, high-energy theory, which has slowly percolated to condensed matter theory as well). But then there are flashes like Seiberg-Witten, that provides a continuous and expansive canvas which interpolates between various disparate vertices of physics. And this unity, when it does become apparent, makes all the mathematical toil worthwhile in the end.
Steering back to the actual point of the post, the sine-Gordon Lagrangian, with a slight re-parametrization,
has the equation of motion,
The equation has soliton, anti-soliton and multi-soliton solutions. In fact, the general (1+1)-d N-soliton solution is given by,
![\cos \beta \theta (x,t) = 1+2 \partial_\mu \partial^\mu \ln \det [M_{ij}]](http://l.wordpress.com/latex.php?latex=%5Ccos+%5Cbeta+%5Ctheta+%28x%2Ct%29+%3D+1%2B2+%5Cpartial_%5Cmu+%5Cpartial%5E%5Cmu+%5Cln+%5Cdet+%5BM_%7Bij%7D%5D+&bg=ffffff&fg=000000&s=0 "\cos \beta \theta (x,t) = 1+2 \partial_\mu \partial^\mu \ln \det [M_{ij}]")
where the N-dimensional matrix 
One of the fundamental characteristics of the sine-Gordon equation (and of very many other PDE’s) is that its small excitations (in the 
But it so happens that at times, the strong coupled regime of one theory is equivalent to the weak coupling regime of another theory. Perhaps the earliest reported use of this duality principle was in the determination of the critical temperature of the 2D Ising model via the so-called Kramers-Wannier duality transform [cf. P. Ruelle, Phys. Rev. Lett. 95, 225701 (2005)]. This strong-weak duality is termed S-duality and is a central theme to the development of the Seiberg-Witten theory.
Now, it so happens that the sG theory, which is basically a theory of bosons, is S-dual to the Thirring model, which is a theory of fermions! The Thirring model is an extension of the Dirac theory of the massive fermion to include self-interactions; the specific Lagrangian being,
where 
The proof of the duality starts from considering the bosonisation ansatz,
Now, the bosonic sG theory has the (equal time) bosonic canonical commutation relations,
![[\theta(x),\theta(y)]=0=[\dot{\theta}(x),\dot{\theta}(y)]](http://l.wordpress.com/latex.php?latex=%5B%5Ctheta%28x%29%2C%5Ctheta%28y%29%5D%3D0%3D%5B%5Cdot%7B%5Ctheta%7D%28x%29%2C%5Cdot%7B%5Ctheta%7D%28y%29%5D+&bg=ffffff&fg=000000&s=0 "[\theta(x),\theta(y)]=0=[\dot{\theta}(x),\dot{\theta}(y)]")
![[\theta (x), \dot{\theta}(y)]=i\delta(x-y)](http://l.wordpress.com/latex.php?latex=%5B%5Ctheta+%28x%29%2C+%5Cdot%7B%5Ctheta%7D%28y%29%5D%3Di%5Cdelta%28x-y%29&bg=ffffff&fg=000000&s=0 "[\theta (x), \dot{\theta}(y)]=i\delta(x-y)")
Then, a few steps of algebra later, employing the bosonisation ansatz, one can calculate the products of the spinor components. This gives the familiar fermionic anti-commutators,
These results prove the essential duality of the theories via bosonisation. It also shows the intimate symmetry between certain bosonic and fermionic theories. Of course, these days, this symmetry is better understood in the form of super-symmetry.
(To be contd.)
