Alday and Maldacena I

Luis Alday and Juan Maldacena opened the floodgates in 2007 with their work on scattering amplitudes at strong coupling. By finding the solution to the equation of motion for bosonic strings that move in $adS_{5}$, Alday and Maldacena were able to make contact with something that looked like the BDS ansatz for scattering amplitudes of gluons in $\mathcal{N} = 4$ super Yang-Mills. The connection between tree-level string theory in $adS_{5}$ and strongly-coupled super Yang-Mills is possible thanks to the anti de-Sitter / conformal field theory correspondence.

The work of Alday and Maldacena is important for many reasons. First, it makes contact with gauge theory scattering amplitudes at strong coupling, something that is naively inaccessible with perturbation theory. Second, the way the Alday-Maldacena amplitude was obtained uncovered a link between scattering amplitudes and expectation values of certain null Wilson loops. This in turn lead to uncovering the Yangian symmetry of the planar sector.

The bold claim is that the scattering amplitude at strong coupling in the gauge theory side of adS/CFT corresponds to the classical limit of a scattering amplitude in the string theory side.

String theory is usually first-quantized. This means that the action functional involves the geometric variables that describe (in some limit) the classical dynamics of a string and not some fields in spacetime. That is, \[S\left[ X \right] = \int d^{2}\sigma \, \mathcal{L}\left[X, \, \partial X \right] \] When a string moves through spacetime, it traces out a surface called the worldsheet. The interaction of strings can be described by considering a disk with a certain amount of punctures on the boundary of the disk. The number of punctures corresponds to the number of external states. It is at this punctures that vertex operators are inserted carrying the information of the external states. The take-away from this is that, classically, the boundary conditions of the string worldsheet contain the information about the external states of the process. The string action will lead to equations of motions. To solve this equations of motions one needs to specify boundary conditions. Alday and Maldacena solved the equations of motion for a string with boundary conditions such that the worldsheet described the scattering of four external bosons. Later, this was generalised to any number of external states.

But the strings that Alday and Maldacena studied moved in $adS_{5}$. The equations of motion are hard to solve given the boundary conditions. It proved useful to perform a coordinate transformation to simplify the boundary conditions. The transformation that AM performed had the same form as a T-duality transformation. The original problem had as string worldsheet a surface that was pinched at four points. Under the T-duality, the string worldsheet becomes a surface that ends on a four-sided polygon. The sides of the polygon are null since they are related to the momentum of the external states, which are massless. The problem of finding a surface with this boundary condition is apparently simpler and AM give a classical solution.

With the classical solution $X_{cl}(\sigma)$ at hand, one can obtain the value of the action functional at this configuration. Since the strings are relativistic, the action is proportional to the area of the worldsheet. The answer for AM was divergent. After appropriate regularization an answer for the classical action $S_{cl}$ was given. Since the amplitude is related to a path-integral of the form \[A_{4} = \int DX \left(V_{1} V_{2} V_{3} V_{4} e^{iS\left[X\right]}\right)\] the semiclassical approximation to this amplitude is of the form \[A_{4} \sim \exp{\left( i S_{cl} \right)}\] Alday and Maldacena found that the classical action $S_{cl}$ has the form \[S_{cl} = S_{div}(s) + S_{div}(t) + S_{finite}(s, \, t)\] with all of three terms in the right-hand side being functions of the `t Hooft coupling $\lambda$ too. Both the divergent and the finite part agree exactly with the BDS ansatz.