Exploring the existence of extra dimension(s)

Abstract

Whether there exists extra dimensions is an interesting and fundamental question. In this article, certain aspects of this have been explored theoretically. For this purpose, we first discuss the role of an extra dimension in the early unification of the classical Einstein gravity and electromagnetic interaction, namely the Kaluza-Klein (KK-) theory, and the issues we encounter in carrying this out. We then move to discuss the relationship between supersymmetry and extra dimension(s) and point out that there exist well-defined quantum theories without gravity in six dimensions such as (2, 0) superconformal theory and little string theories, the former is a local field theory but has no Lagrangian description while the latter are non-local non-gravitational string theories. We discuss also the similar issue for supergravities in diverse dimensions and point out that the largest number of supersymmetries with propagating gravity degrees of freedom is $N$=8 in four dimensions and the largest number of spacetime dimensions for supergravity is eleven. We would like to stress that extra dimensions in classical supersymmetric theories such as supergravities in diverse dimensions appear only as an option. When gravity is included, the four-dimensional $N$=8 supergravity appears to be the only one so far which has the potential to be a good quantum gravity theory in the usual field theory sense. All the rest are sure to be UV incomplete ones whose UV completions require the existence of an unified theory which is most likely to be the string/M-theory. In this sense, a quantum consistent theory containing gravity needs extra dimensions in general. Good examples include both bosonic strings and superstrings. For the former, we need the spacetime dimensions to be twenty six while for the latter we need spacetime dimensions to be ten such that the corresponding theories are quantum consistent at least perturbatively. So for these quantum consistent theories containing gravity, extra dimensions are no longer optional but must be there. In this paper, we give also an elementary introduction to both bosonic string and superstring theories. We mention that in addition to the usual one dimensional strings, there exist also non-perturbative $p$-branes with $p~=~0,~1,~\cdots~9$ in non-perturbative superstring theories. These $p$-branes are the so-called 1/2 BPS extended objects, preserving one half of the spacetime supersymmetries. They play the key role in establishing various dualities such as S-duality and U-duality and help to explain how the existence of various superstring theories leads to that of an even bigger unified theory called M-theory, which unifies 5 superstrings in ten dimensions and the 11-dimensional supergravity, with its largest spacetime dimension to be eleven. To bring the string/M-theory to our four-dimensional real world, we usually need to compactify six or seven extra dimensions, which should be small enough to avoid experimental constraints. Phenomenologically, people also explore the existence of large extra dimensions, which can be used to solve gauge hierarchy problem or gravity localization, etc. We give some discussions on these topics as well.