Surface Defects from Fractional Branes [HBNI Th203]

Abstract

Surface defects in gauge theories are non-local operators supported on co-dimension two sub manifold. Defects in quantum field theory can provide us with important non- perturbative information eg. di↵erent phase structures present in the system. They were first studied by Gukov-Witten in the context of N=4 super Yang-Mills theories. The Gukov-Witten surface defects are characterized by sets of discrete and continuous param- eters. In this thesis, we geometrically engineer Gukov-Witten surface defects in maxi- mally supersymmetric N = 4 Yang-Mills theory with gauge group U(N) within the setup of perturbative Type IIB string theory. In particular, we refine the proposal of Kanno and Tachikawa and realize the defect by a configuration of fractional D3 branes on an orbifold background that preserves two dimensional Poincaré invariance. On this particular orbifold target space in which the D3 world volume is extended partially along the orbifold, we consider closed string fields that act as a background. Moreover, the relevant closed string states are the twisted sector ones that are special to the orbifold space. Due to the presence of the fractional D3 branes which introduce a boundary on the worldsheet, the left and right moving sectors of the closed string fields are identified under some reflection rules. In addition, we consider the open string vertex operators that are invariant under the action of the orbifold group. After providing the necessary details about twisted closed string sectors, Reflection rules, and open string spectra, we calculate open/closed disk correlation functions on the worldsheet involving one massless closed string field from a twisted sector and one massless open string field. By giving a constant background vacuum expectation value to the twisted field, we interpret non- vanishing correlators as sources for the open string field. By Fourier transform to position space, we obtain a space-time profile for the open string field that matches exactly with the expected singular profiles of the four dimensional fields of the gauge theory in the presence of the Gukov-Witten defect. The background values of the twisted closed string fields are identified with the continuous parameters that define the defect in the gauge theory description. We provide an important check of our proposal by verifying that this identification is consistent with the expected S-duality properties of these continuous parameters. In the first part, we consider the simplest possible surface defect with two discrete pa- rameters realized via fractional D3 branes on a background with Z 2 orbifolding. In the later part, we generalize the above construction to the most generic type of Gukov-Witten defects with M discrete parameters realized via fractional D3 branes on a Z M orbifold.