Gravity Dual to Pure N=1 SU(N) Gauge Theory

Abstract

A correspondence between type IIB string theory with N D7-branes on R^{1,3} x C^1/Z_2 x T^2/Z_2 x T^2/Z_2 and pure N=1 SU(N) gauge theory in four dimensions is proposed and argued. First the supergravity background of unwrapped and flat D7-branes with running axion and dilaton on R^{1,7} x C^1/Z_2 is studied together with the corresponding N=1 SU(N) gauge theory in eight dimensions. The D7-branes are then wrapped over a 4-cycle on T^2/Z_2 x T^2/Z_2 which turns on all F_1, F_3, H_3, and F_5 fluxes of type IIB theory and induces torsion. The supergravity solutions are explicitly constructed with exact analytic expressions for all components of the metric and the fluxes. The background geometry of the four-dimensional gauge theory is compact and conformally Calabi-Yau. The internal space normal to the wrapped D7-branes at the infrared boundary is S^1 whose radius is set by the nonperturbative scale of the gauge theory and spacetime is R^{1,3} at the ultraviolet boundary. The gauge coupling of the four-dimensional gauge theory is related to the gauge coupling of the eight-dimensional gauge theory and the volume of the 4-cycle. The gravity theory reproduces the renormalization group flow and the pattern of chiral symmetry breaking of the gauge theory and leads to confinement. The curvature is small and nearly constant and the supergravity flow is smooth in the infrared region where the gauge theory is strongly coupled and a dual gravity description is useful. String loop corrections are small for large N. The scale of string tension in four dimensions is of the same order as the scale of Kaluza-Klein masses.