Though string theory implies extra dimensions, that doesn’t mean that the extra dimensions need to exist as dimensions of space. Some work has been done to formulate a 4-dimensional string theory where the extra degrees of freedom aren’t physical space dimensions; but the results are incredibly complex, and it doesn’t seem to have caught on.
Several groups have performed this sort of work, because some physicists are uncomfortable with the extra space dimensions that seem to be required by string theory. In the late 1980s, a group worked on an approach called free fermions. Other approaches that avoid introducing additional dimensions include the covariant lattice technique, asymmetric orbifolds, the 4-D N=2 string (what’s in a name?), and non-geometric compactifications.
These are technically complex formulations of string theory (aren’t they all?) that seem to be ignored by virtually all popular books on the subject, which focus on the idea of extra dimensions to the exclusion of these alternative approaches. Even among string theorists, the geometric approach of compactifying extra dimensions is the dominant approach.
One early, technically complex (and largely ignored) approach to 4-dimensional string theory is work performed by S. James Gates Jr., of the University of Maryland at College Park (along with assistance from Warren Siegel of Stony Brook University’s C. N. Yang Institute for Theoretical Physics).
This work is by no means the dominant approach to 4-dimensional string theory, but it’s benefit is that it can be explained and understood (in highly simplified terms) without a doctorate in theoretical physics.