Is Minimal CFT Essential for Understanding String Theory?

[crackjack]

How useful is the framework of minimal Conformal Field Theory (ie. CFTs with finite primary fields) in String Theory?
From what I have come across, I have only seen its usefulness in studying toy models of minimal string theory.

 

[suprised]

How useful is the framework of minimal Conformal Field Theory (ie. CFTs with finite primary fields) in String Theory?
From what I have come across, I have only seen its usefulness in studying toy models of minimal string theory.

Right.. non-critical toy models in less than one dimension. Tensor products of (supersymmetric) minimal models were used to decribe exactly solvable models on Calabi-Yau manifolds, but these tensor products aren't minimal any more. Minimal models per se are not too interesting or important in string theory.

 

[Thomas Larsson]

The real value of minimal models is in 2D statphys, where they have been realized experimentally e.g. in monolayers of noble gases on a graphite substrate.

 

[xepma]

Just as a sidenote, the class of CFT's with a finite number of primary fields are called Rational Conformal Field Theories (RCFT). The word 'rational' comes from the fact that all fields carry a conformal dimension equal to some fractional number (the same goes for the central charge).

The minimal models are a special subset within these CFT's, namely the central charge falls in the regime 0 < c < 1.

But I don't know anything about it's application to string theory (apart from the fact that CFT's arise as theories which describe the worldsheet dynamics).

 

[crackjack]

Oh ok.
Is this (not-of-much-use-in-strings) the case even for extended minimal (or RCFT) models? - like those with finite conformal blocks (rather than finite individual primary fields)?

 

[suprised]

Oh ok.
Is this (not-of-much-use-in-strings) the case even for extended minimal (or RCFT) models? - like those with finite conformal blocks (rather than finite individual primary fields)?

Sure, just the same. Each RCFT is minimal with respect to its maximal chiral algebra. So the question essentially is whether RCFTs play an important role. Probably the answer depends whom you ask. I would say, RCFTs are non-generic and emphasize an algebraic structure (namely the one of the extended chiral algebra) that goes away the moment you deform the theory, even slighly. That goes against the spirit of studying continuous parameter families of string vacua.

So there are two schools of thought/taste: the RCFT people study isolated points in the full parameter space (typically with extra symmetries), the benefit being an exact solvability of the CFT and thus in principle, of all correlation functions at a given point. Opposite to this spirit is topological string theory, where one solves only a subsector of the theory (roughly speaking the massless one, which is the relevant one), but as continuous deformation family over the moduli (vacuum parameters) of the theory. Most people find the latter more interesting and important, as for example questions about dualities can be addressed there, while algebraic considerations (RCFT, that is) tend not to be useful. One might loosely say that geometry beats algebra, but that's surely also a matter of taste.

 

[crackjack]

Ok. I think I get you.
When you say one solves the massless case (as against RCFT for a particular parameter), I assume you meant solving for the correlation functions using CFT. So, how does masslessness make CFT solvable? (maybe a sketch or a reference to some book)
From an earlier post above I gather that this masslessness cannot result in finite primary fields.

PS: I recently started on string theory.

 

[suprised]

So, how does masslessness make CFT solvable?

It is supersymmetry what is used, CFT only indirectly. More precisely, there is a special subsector in the theory which has special properties, which allows to solve the theory for this subsector (ie, compute the correlation functions, depending on continuous moduli). This "massless" subsector may be called topological, or BPS, or chiral primary, or holomorphic, etc. There is some special geometric structure which makes it "integrable", ie, one can determine the correlators by geometry, which boils down to solve certain differential equations, and/or perfom integrals. As said, this philosophy is in a sense opposite to the one of RCFT, whose structure discontinuously jumps under the smallest perturbations.

 

[crackjack]

Ah ok. Thanks.