In string theory, compactification refers to the process of reducing the dimensions of spacetime from the full ten or eleven dimensions, as suggested by the theory, to a lower-dimensional manifold. This is done by assuming that the extra dimensions are "compactified" or curled up into very small, compact shapes, often referred to as a compactified manifold.
A manifold is a mathematical concept that describes a space that locally looks like Euclidean space. It can have various shapes and topologies. In the context of string theory, the compactified manifold represents the geometry of the extra dimensions that are not directly observable at large scales.
The choice of the specific compactified manifold in string theory is important because it determines the properties of the resulting low-energy physics in our observable four-dimensional spacetime. Different compactifications can lead to different particle spectra, symmetries, and interactions in the resulting four-dimensional effective theory.
The mathematics involved in compactifying the extra dimensions can be highly complex and often requires advanced techniques from differential geometry and algebraic geometry. The study of compactified manifolds in string theory is a crucial area of research, as it provides insights into the possible connections between string theory and observed phenomena in our four-dimensional world.