Pure spinor formalism and gauge/string duality

A worldsheet interpretation of AdS/CFT duality from the point of view of a topological open/closed string duality using the power of pure spinor formalism will be studied. We will show that the pure spinor superstring on some maximally supersymmetric backgrounds which admit a particular Z4 automorphism can be recasted as a topological A-model action on a fermionic coset plus a BRST exact term. This topological model will be interpreted as the superstring theory at zero radius. Using this decomposition we will prove the exactness of the ¾-model on these backgrounds. We then show that corresponding to this topological model, there exist a gauged linear sigma model which makes it possible to sketch the superstring theory in the small radius limit as the dual limit of the perturbative gauge theory. Studying the branch geometry of this gauged sigma model in di®erent phases gives information about how the gauge/string duality is realized at small radius from a similar point of view of the topological open/closed conifold duality studied by Gopakumar, Ooguri and Vafa. Moreover, we will discuss possible D-brane boundary conditions in this model. Using this D-branes, we will make an exact check in the N = 4 SYM/AdS5 £ S5 duality. We will show that the exact computation of the expectation value of the circular Wilson loops in the gauge theory side can be obtained from the amplitudes of some particular D-branes as the dual of the Wilson loops in the superstring side. The next step will be to construct a BV action for G=G principal chiral model with G 2 PSU(2; 2j4), we will show that after applying different gauge ¯xings of the model, we will get either a topological A-model theory or a topological thery whose supersymmetric charge is equal to the pure spinor BRST charge. Using this model one can explore the cohomology of the pure spinor action from the topological BV model. Then we show that there exist a particular consistent deformation of the G=G action equal to the pure spinor superstring action. In this way we generate the superstring action on a non-zero radius AdS background as a perturbation around a topological model corresponding to zero radius limit of the superstring theory. Using this picture we will give an argument based on the worldsheet interpretation of open/closed duality to give a worldsheet explanation for AdS/CFT duality. A better understanding of this picture might give a prove of Maldacena's conjecture. At the end we will show that using the topological A-model, one can also give a prescription for computing multiloop amplitudes in the superstring theory on AdS5 £ S5 background.