Originally, Landau-Ginzburg models were used to describe phase transitions such as spontaneous magnetization. A field, F, is introduced together with a temperature-dependent potential which then controls the expectation value of the field, <F>. Above a critical temperature <F> takes one (usually zero) value, while below the critical temperature <F> becomes non-zero.
A (2,2)-supersymmetric generalization of such models was studied in the context of superstring compactification. The expectation value of an n-tuple of fields is controlled by a potential which, in supersymmetric models, is the square of the magnitude of the gradient of a holomorphic function. If this function is chosen quasi-homogeneous, the resulting theory is invariant with respect to a discrete symmetry, the target space a quotient (orbifold) with respect to this discrete group, and the Hilbert space has a component for every element of that discrete group.
Somewhat surprisingly, for many Landau-Ginzburg orbifold models, there turned out to exist a corresponding Calabi-Yau variety , such that the superstring theory compactified on one and the other turns out remarkably similar. In particular, the collection of supersymmetric vacua of the Landau-Ginzburg orbifold, which represents the light matter particles of the effective spacetime theory (quarks, leptons, etc.), turn out to agree very well with the light matter spectrum in the corresponding Calabi-Yau compactification. Moreover, the couplings and correlation functions in these respective models turn out to agree very well too.
Finally, it was the invention of gauged linear sigma models that explains this correspondence. The locus of minimal energy configurations in these models typically has several ‘branches’, one of which is in fact a Landau-Ginzburg orbifold (or its generalization), another of which a Calabi-Yau variety. Moreover, such models come with ‘radial’ parameters which can interpolate between the various branches. In turn, the space of these ‘radial’ parameters is then a moduli space where the Landau-Ginzburg and Calabi-Yau models appear in certain limiting ‘corners’ of their respective open regions - the various ‘phases’, akin to water and vapor.
© Tristan Hübsch, 2008