Surface Operators, holography and BPS equations [HBNI Th194]

Abstract

In this thesis, we do investigations related to surface operators in supersymmetric gauge theories in four dimensions. In the first part of the thesis, we study half-BPS surface operators in ${\mathcal N}=2$ SU(N) gauge theory following two different approaches. In the first approach we analyze the chiral ring equations for certain quiver theories in two dimensions coupled to four dimensional gauge theory. The chiral ring equations, which arise from extremizing a twisted chiral superpotential, are solved as power series in the infrared scales of the quiver theories. In the second approach we use equivariant localization and obtain the twisted chiral superpotential as a function of the Coulomb moduli of the four-dimensional SU(N) gauge theory, and find a perfect match with the results obtained from chiral ring equations. In the second-half of the thesis, we study singular time-dependent $\frac{1}{8}$-BPS configurations in the abelian sector of ${\mathcal N}=4$ Yang-Mills theory that represent BPS string-like defects in ${{\mathbb R}\times S^3}$ spacetime. Such BPS strings can be described as intersections of the zeros of holomorphic functions in two complex variables with a 3-sphere. We argue that these BPS strings map to $\frac{1}{8}$-BPS surface operators under state-operator correspondence of the conformal field theory.

We show that the string defects are holographically dual to non-compact probe D3 branes in global $AdS_5\times S^5$ that share supersymmetries with a class of dual-giant gravitons. For simple configurations, we demonstrate how to define a good variational problem for the associated action principle and propose a regularization scheme that leads to finite energy and global charges on both sides of the holographic correspondence.