# Nonperturbative Methods for Quantum Field Theory: Holographic Wilson Loops and S-matrix Bootstrap

#### Abstract

In this thesis, we explore nonperturbative methods of quantum field theory through two topics: holographic Wilson loops and S-matrix bootstrap. In the first part, we study the holographic calculation of Wilson loops using AdS/CFT correspondence, which is a duality between a string theory in AdS space and a gauge theory living on the conformal boundary of the AdS space. Under this duality, the expectation value of the Wilson loop operator in the gauge theory at strong t'Hooft coupling limit is given by the area of a string worldsheet ending on a boundary curve defined by the shape of the Wilson loop. We exploit the conformal symmetry and integrability properties of this problem and study the equivalent problem of finding the conformal reparametrization of a boundary curve. In (Euclidean) AdS3, we find analytic solutions in terms of Mathieu functions and implement numerical procedures for finding the conformal reparametrization in general. We also generalize the formalism to higher dimensional case and identify conformal invariants of the boundary curve which provide boundary conditions for the Pohlmeyer reduction of the string sigma model. In the second part, we study the S-matrix bootstrap program. In particular, we consider generic S-matrices in 2d relativistic quantum field theory with O(N) global symmetry and under crossing, real analyticity and unitarity constraints. We search for a maximization problem with these constraints that defines the 2d O(N) nonlinear sigma model which is an integrable theory. We find that the defining feature of this theory is that it resides at a vertex of the convex space defined by the constraints. Our numerical results reproduce the exact analytic S-matrix without assuming integrability.

#### Degree

Ph.D.

#### Advisors

Kruczenski, Purdue University.

#### Subject Area

Physics

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