In string theory, a compactified manifold refers to a higher-dimensional space in which some of the dimensions are "curled up" or compactified. This process allows for a reduction in the number of observable dimensions, resulting in a lower-dimensional spacetime that we experience.
In the context of string theory, the theory requires extra dimensions beyond the usual four (three spatial dimensions and one time dimension) to be consistent mathematically. However, since we do not observe these extra dimensions directly in our everyday experience, they must be somehow hidden or compactified.
The idea of compactification involves assuming that the additional dimensions are not extended in size, but rather "wrapped" into tiny, compact shapes. These compactified dimensions are typically assumed to be small and curled up at scales much smaller than we can currently observe or measure. The size and shape of these compact dimensions are determined by various factors, including the specific formulation of string theory and the dynamics of the theory.
By compactifying the extra dimensions, string theory can reconcile the higher-dimensional framework with the observed four-dimensional spacetime of our universe. The properties and interactions of particles and forces in our four-dimensional world can be understood in terms of the vibrational modes and geometrical properties of the compactified higher-dimensional space.
It's important to note that the process of compactification in string theory is mathematically intricate and can result in a wide range of possible compactified manifolds with different topologies and geometries. The specific choice of compactification can have profound implications for the resulting physical phenomena and the predictions of the theory.