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finally cross-linked Landau-Ginzburg model with TCFT and added a corresponding reference,
prompted by this Physics.SE question
removed the two references on elliptic genera and replaced them by !include
-ing the comprehensive list from elliptic genera as partition functions – references
It was definitely translated earlier, but I could not find a reference.
I am anyway quite confused with the entry, even after lightly reading great number of references. First of all, I do not understand the exact connection between the traditional physics and string and mathematical versions.
In condensed matter physics, people talk about a particular quantum mesoscopic equation of state governing the order parameter, the Landau-Ginzburg equation. As far as I understand the space of states is infinite-dimensional. In old times, physicists often used term Landau-Ginzburg theory, extremely rarely “model”. As far as I understand there is nothing supersymmetric nor topological (hence finite-dimensonal) in the theory.
Then we have appearance of certain sector in supersymmetric string theory, which leads to a couple of supersymmetric topological field theories, quite different and not generalizing the above, which are called topological Landau-Ginzburg models of types A and B. On the cohomological side there are corresponding infinity-categories of D-branes of types A and B, which present these. I can not see how is this at all related to theories generalizing Landau-Ginzburg equation in physically meaningful sense. (I would like to understand why it is named the same way, what is the connection, but even then I think then the supersymmetric topological “models” should be eventually separated from maybe a new page on Landau-Ginzburg equation talking the equation, order parameter, superconductivity etc.)
Furthermore, Hori and Vafa have this way of looking at mirror symmetry by a fibered picture. If we have a (super)potential then we unify all fibers into a fibered variety and then do mirror symmetry by doing some fiberwise operations plus some quantum corrections. So the mirorr symmetry is if I understood correctly for the total spaces of such fibrations rather than for the base modified with the potential. Kontsevich had that picture that potential twists the propetrty of being a differential for the categories of D-branes of type B, hence dg categories of coherent sheaves modify, this would however be point of view of the base only and potential is just a way to twist; the connection to matrix factorizations is then immediate. So which is the true dimension of the space viewed in the mirror symmetry, the base dimension or plus 1 because of the potential ? And is the fibered picture in Hori and Vafa any similar to SYZ viewpoint (which is however in quite different generality).
Can somebody clarify ? I am spending huge amount of time with little progress in understanding. Urs ?
Yeah, this is an odd terminological situation.
Of course, string theorists broadly use “LG-model” for (super-)sigma-models with a (super-)potential that has a degenerate critical point.
I am not aware that the original string theoretic references make a particularly strong (or any) case for the choice of reference to Landau-Ginzburg theories as understood in condensed matter theory.
The authors of
show some ambition to justify the terminology, but it remains rather vague, I find.
Our entry is currently no better. It should eventually say that the term “LG model” is de facto used in two different senses in solid state and in string theory, with only a rather broad-brush relation between them. Ideally we’d split up two separate entries for the two meanings.
Thank you Urs. It is a relief to get an insider point of view not to bump with a head into the wall too much.
One of the several reasons I look for the context now is that I put a big effort into trying to do a couple of good reviews of articles touching upon the subject, but of course I want to know more general.
The first review tangent to the subject is MR4456599 (zoranskoda). It is submitted already, but if somebody finds a useful correction I think I can still replace it.
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