Strong and weak coupling

    What is a coupling constant? This is some number that tells us how strong an interaction is. Newton's constant G_N, which appears in both Newton's law of gravity and the Einstein equation, is the coupling constant for gravitational interactions. For electromagnetism, the coupling constant is related to the electric charge through the fine structure constant a

    In both particle physics and string theory, usually the scattering amplitudes and other quantities have to be computed as an expansion in powers of the coupling constant or loop expansion parameter, which we've called g² below:

At low energies in electromagnetism, the dimensionless coupling constant a is very small compared to unity, and the higher powers in a become too small to matter. The first few terms in the series make a good approximation to the real answer, which often can't be calculated at all because the mathematical technology doesn't exist to solve the whole theory at once.
    If the coupling constant gets very large compared to unity, perturbation theory becomes useless, because higher powers of the expansion parameter are bigger, not smaller, than lower powers. This is called a strongly coupled theory. Coupling constants in quantum field theory end up depending on energy because of quantum vacuum effects. A quantum field theory can be weakly coupled at low energies and strongly coupled at high energies, as is true with the fine structure constant a in QED, or strongly coupled at low energies and weakly coupled at high energies, as is true with the coupling constant for quark and gluon interactions in QCD.
    Some quantities in a theory cannot be caluclated at all using perturbation theory, espcially not for weak coupling. For example, the amplitude below cannot be expanded around the value g²=0

because the amplitude is singular there. This is typical of a tunneling transition, which is forbidden by energy conservation in classical physics and hence has no expansion around a classical limit.
   String theories feature two kinds of perturbative expansions: an expansion in powers of the string parameter a' in the conformal field theory on the two-dimensional string worldsheet, and a quantum loop expansion for string scattering amplitudes in d-dimensional spacetime. But unlike in particle theories, the string quantum loop expansion parameter is not just a number, but depends on one of the dynamic modes of the string, called the dilaton field f(x)

   This relationship between the dilaton and the string loop expansion parameter is important in understanding the duality relation known as S-duality. S-duality can be examined most easily in type IIB string theory, because this theory happens to be S-dual to itself. The low energy limit of type IIB theory (meaning the lowest nontrivial order in the string parameter a') is a type IIB supergavity field theory, which features a complex scalar field r(x) whose real part is the axion field c(x) and whose imaginary part is the exponential of the dilaton field f(x):

This field theory is invariant under a global transformation by the group SL(2,R) (broken by quantum effects down to SL(2,Z)), with the field r(x) transforming as

    If there is no contribution from the axion field, then the expectation value of the field r(x) is given by the dilaton alone. Because the dilaton is identified with g_st, the SL(2,Z) transformation with b=-1,c=1

tells us that the theory at coupling g_st is the same as the theory at coupling 1/g_st!
    This transformation is called S-duality. If two string theories are related by S-duality, then one theory with a strong coupling constant is the same as the other theory with weak coupling constant. Type IIB superstring theory is S-dual to itself, so the strong and weak coupling limits are the same. This duality allows an understanding of the strong coupling limit of the theory that would not be possible by any other means.
    Something more surprising is that type I superstring theory is S-dual to heterotic SO(32) superstring theory. This is surprising because type I theories contain open and closed strings, where as heterotic theories contain only open strings. What's the explanation? At very strong coupling, heterotic SO(32) string theory has excitations that are open strings, but these open strings are highly unstable in the weakly coupled limit of the theory, which is the limit in which heterotic string theory is commonly understood.