String/gauge duality and Penrose limit

Abstract

Berenstein, Maldacena, and Nastase have recently discovered a particular limit of AdS/CFT correspondence where string theory in a plane wave background is dual to a sector of $mathcal N =4$ SYM in a double scaling limit. It is based on the observation that a plane wave background can be obtained by taking Penrose limit of Anti de Sitter background. The corresponding gauge theory limit is identified via AdS/CFT dictionary. This proposal is especially exciting because string worldsheet theory in a plane wave background is exactly solvable, thereby opening a possibility that one can go beyond supergravity approximation. In the absence of string interactions, the duality made a remarkable prediction for anomalous dimension of gauge theory operators from exact free string spectrum, which was soon verified. In this thesis, we attempt to extend the duality to the interacting theory level. We propose that the correct holographic recipe is to identify the full string field theory Hamiltonian with the dilatation operator of gauge theory. In practice, we must find an identification map between string theory and gauge theory Hilbert spaces and evaluate matrix elements of the two operators accordingly. The requirement that the inner product should be preserved determines a unique identification map assuming that it is hermitian. We show that transition amplitudes of string field theory agree with matrix elements of dilatation operator under this preferred identification for states with two different impurities. We later extend it to states with arbitrary impurities. In doing so, we find a diagrammatic correspondence between string field theory and gauge theory Feynman diagrams thereby providing direct handles on the duality. Our proposal is $universal$ in the sense that it is applicable to any interaction type such as the open-closed interaction, and to all orders in $g_2$ and $l^prime$. Hopefully, this thesis will be a key step towards proving the novel duality and a beginning of an exciting journey to the stringy regime of string/gauge duality.