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Here is what I know about Landau-Ginzburg models:

It is an important player in the story of mirror symmetry. It involves "potentials" which are functions of complex varibles, which have isolated singularities at the origin. There is the notion of universal unfolding(miniversal unfolding, or miniversal deformation) attached to each of these functions. One can attach the frobenius manifold structure to this unfolding. One version of mirror symmetry is the isomorphisms between frobenius manifolds.

My question is:

Is there a good book or paper where I can learn about LG models? Especially mysterious to me is about the potentials. What is the physical motivations behind these functions?

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Nice question. It would be great if some expert can briefly give a somewhat self-contained answer. – Alexander Chervov Jan 17 '12 at 5:07
If you know that in physics the energy is a sum of kinetic energy and potential energy, then the appearance of a potential in a physical theory should not be too surprising. The potential here is constructed from a holomorphic function, of which there is none for a compact manifold, but possibly many if the target space is noncompact. – Eric Zaslow Feb 18 '12 at 6:58
up vote 5 down vote accepted

I think the Clay Math books have nice descriptions about Landau-Ginzburg Models. The book called Mirror Symmetry and Dirichlet Branes and Mirror Symmetry both have nice physical and mathematical descriptions of these models. Mirror Symmetry is also available as a pdf. I would start with Chapter 13 of Mirror Symmetry.

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I would also suggest the article of Blok-Varchenko "Topological Conformal Field Theories and the Flat Coordinates".

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