On Wed, 15 Sep 2004, Urs Schreiber wrote (in the "Background independence" thread):
But for something different: You mentioned recently the failure of SFT to capture certain non-perturbative degrees of freedom. Is it conceivable that one can somehow "augment" SFT in a nice way to include these?This is a very interesting question, I think. Let me say a couple of related opinions of mine plus facts.
SFT used to be a very natural candidate for a full formulation of string theory. It is the closest thing to "field theory" that one can have - a field theory with infinitely many fields, if you decompose the string field into component fields. From this point of view, it looks "background independent" to many critics - and it is "off-shell", which means that it sort of *has* local Green's functions, not just the S-matrix, which is something that has extensively been used in the study of tachyon condensation.
This ambitious program has only been partially successful. First of all, a non perturbatively complete theory should be defined with an exact form of the action etc., not just a perturbative approximation of it. However, closed string field theory requires to add correcting terms to the action at each order (as well as the BV machinery). The action is therefore an expansion itself, and it can have various non-perturbative completions. Not a good starting point for a non-perturbatively complete theory.
All these reasons led the people to focus on the open string field theory whose action can be well-defined - e.g. the cubic (polynomial) Witten's action; it is enough to get the full amplitudes and cover the full Riemann surface moduli spaces. Can one see all physics of string theory in it? Well, the first problem are the closed string states. They can be seen as poles in open string scattering, but as far as I know, no one has made a convincing construction of the closed strings as composites of the open string fields so far. The understanding of the closed strings would have to improve a lot so that one could also construct non-trivial geometric configurations including NS21-branes (or NS5-branes) etc. in open SFT.
Another question are the D-branes. Using the modern perspective, the open strings themselves describe dynamics of a spacetime filling D-brane. Sen's insights made it expected that one can construct the lower dimensional branes as classical solutions of open string field theory.
String field theory has nevertheless been made less natural by the results of the Duality revolution - its degrees of freedom are made of strings, but at a generic point in the moduli space, there should be brane democracy and strings are equally (non-)fundamental as other objects. If string field theory becomes a good starting point for a full formulation, one must ask several obvious questions.
Are the S-dualities and the strong coupling limit derivable from this generalized SFT? For example, can one derive that the strong coupling limit of a type IIA string theory is eleven-dimensional, and type IIB is S-self-dual?
The answers must be yes if the generalized SFT is gonna become non-perturbatively successful. Well, there are still two basic pictures how this could happen:
- One would still be using the same string fields, even at strong coupling, and there are non-trivial functions or transformations of these string fields that can be used to define the S-dual strings, or the 11-dimensional physics, and so on.
I think that this viewpoint has become a bit obsolete after the strong coupling revolutions of the 1990s. At strong coupling, the original degrees of freedom are strongly coupled, physics becomes obscure if we use them. They are not too useful, and moreover we have learned that there can be better degrees of freedom that become weakly coupled. They are typically infinitely heavy in the weakly coupled limit, and therefore they are absent.
In field theory it is legitimate to imagine - for example in QCD - that the fundamental UV fields are the gluons and quarks, and the IR physics is whatever is implied. The gluons are superuseful in the UV - because of the asymptotic freedom - but their physics can be extrapolated to low energies. But we know that there is an asymmetry - the IR can be derived from the UV, but not quite the other way around. Therefore the analogy with strings, that are good variables at the weak coupling, is not quite perfect - because the strong and weak coupling may be totally equivalent.
The lessons of the 1990s seem to indicate that we should not try to push the validity of some degrees of freedom to too strong couplings. - Of course, the second choice is that at generic coupling, there could be new generalized degrees of freedom, whose structure itself is determined, together with the action or whatever replaces it, by some self-consistent rules. These degrees of freedom, determined by the deeper rules, would have to reduce to the usual perturbative strings in each weakly-coupled stringy limit.
While this second option is highly unusual, I believe that it is plausible and attractive. It is unusual because we have not constructed a single theory whose degrees of freedom are themselves determined by deeper rules, dynamically. We always start with some well-defined degrees of freedom, with a well-defined action or something equivalent. Such theories can have many interesting regimes and behaviors, but they cannot be quite universal.
In the perturbative limit we kind of know what are the rules that determine the allowed degrees of freedom and the action: the rules are the usual axioms of conformal field theory. The conformal symmetry constrains both the worldsheet field content as well as the action. But is there a non-perturbative generalization of this nice structure?
What happens with the worldsheet as you increase the coupling? Well, it transmutes into a M-volume, which is the worldvolume of M, which is the non-perturbative generalization of a string. ;-) The worldsheet becomes a bit fuzzy, non-local, its dimension may effectively grow (strings become membranes, but don't imagine quite local membranes). I think that its internal dynamics is itself target space dynamics of some other string theory; I have the N=2 and N=(2,1) string in mind.
We know that this complicated structure of the worldsheet theory *does* occur in some context: the worldsheet of a D-string at weak coupling,in which the D-string is superheavy, is described by open string theory - all open strings attached to the D-string with the whole Hagedorn tower of excitations are relevant. Nevertheless this D-string can be continued to something we call the fundamental string.
There should be some more general description of the allowed worldvolume theories of objects, including non-geometrical ones - and the rules would non-perturbatively generalize conformal field theory.
I've spent some time with thinking about the form of such a possible generalization. Try to think about a more general theory that has a BRST operator and the state-operator correspondence, but you relax the assumption that it is a local two-dimensional theory. It can be a theory of any dimension, with fuzzy dynamics, matrices, whatever you want. Just try to require that something as strong as the requirement of conformal symmetry applies, and the conformal symmetry itself appears as a limit of this requirement for the special case of weakly coupled backgrounds...
... One more comment. There have been some Japanese papers that studied the behavior of the boundary states under the closed-string Kyoto-group-like SFT star product; the boundary states act as projectors, roughly speaking. This sort of thinking, even though it is formal, looks like an important step towards obtaining the non-perturbative generalization of CFT mentioned above. Today, our consistency requirements for closed strings and open strings follow similar logic, but technically they are different.
The allowed spectrum of D-branes must follow Cardy's rules, and so forth. What I would like to see is to derive Cardy's rules as something like the (generalized) closed string (M) equations of motion applied on the closed string field whose vev happens to be the (total) boundary state. Adding a D-brane is a deformation of the background, and it does correspond to a change of the two-dimensional CFT. Well, the change is that we allow some new boundaries. Formally, it is analogous to adding the vertex operator for the boundary state into the 2D action although I realize that there are technical difficulties in making this procedure well-defined (but definitely, this is how the D-brane is seen from far away, as a deformation of the closed string background; in this case, we can restrict the boundary state to its lowest components).
Now imagine that the coupling becomes larger. Adding a D1-brane in type IIB becomes equivalent to adding a light string if the coupling is really large - by S-duality. But adding a fundamental string is a local change of the 2D action. Recall that adding the D-brane was a non-local change: we allowed the worldsheets to have boundaries. The goal is to describe a structure that interpolates between this local modification of the 2D action (adding a fundamental string) and a non-local modification (allowing D-branes). The worldsheet itself should become fuzzy; the distinction between local and non-local must go away at the generic coupling. But what is exactly the theory at the generic point, and how do you constrain it?
This is a sort of bootstrap thinking, but maybe not so impossible - it may be just a generalization of CFTs.
