Thermodynamic bubble ansatz

For computations of scattering amplitudes of gluons at strong coupling, one considers minimal area surfaces in AdS_5 which end on a null polygonal contour at the boundary. A new method was developed by Alday, Gaiotto and Maldacena in which one doesn't need to solve for exact embedding of worldsheet but instead one maps the problem to a Hitchin system. One then writes an integral equation which determines the area as a function of the shape of the polygon. The equations are identical to those of the Thermodynamic Bethe ansatz. The area is given by the free energy of this TBA system. The working of this method will be shown for the case of hexagon closely following the work of above authors in hep-th/0911.4708.