Three-dimensional gauge theories and gravitational instantons from string theory

Various realizations of gauge theories in string theory allow an identification of their spaces of vacua with gravitational instantons. Also, they provide a correspondence of vacua of gauge theories with nonabelian monopole configurations and solutions of a system of integrable equations called Nahm equations. These identifications make it possible to apply powerful techniques of differential and algebraic geometry to solve the gauge theories in question. In other words, it becomes possible to find the exact metrics on their moduli spaces of vacua with all quantum corrections included. As another outcome we obtain for the first time the description of a series of all {dollar}Dsb{lcub}k{rcub}{dollar}-type gravitational instantons.