# Topological string theory - how useful is it?

1. Oct 5, 2009

### crackjack

Topological string theory is a description devoid of metric and hence is background independent and everything emerges from pure topological considerations. This should put it at the front of all other candidate string theories, but that is not the case (it is certainly considered important, but not the most important).
Why is that so?

Is one reason that of a general feeling that there is only so much that can emerge from pure topology and so its good for studying just toy models?

2. Oct 5, 2009

### crackjack

@ https://www.physicsforums.com/member.php?u=166976"

:tongue:

I am asking for opinion from those who are well versed with the field and not from some...

I think you should concentrate more on getting new, verifiable prediction from your theory before starting to bash everything else (that too without any knowledge of the theory that you are bashing). I don't know if this is the situation that Woit and Smolin wanted to bring in by writing such books.

edit: why did you delete your post Bob??

Last edited by a moderator: Apr 24, 2017
3. Oct 5, 2009

### atyy

Well, I am of course not an expert, but anyway - why should we need background independence? AdS/CFT has a fixed CFT background, but the bulk is almost background independent - where "background" means different things in each case. Even LQG may turn out to be background dependent if it ends up being formulated using group field theory, which has a fixed metric on the group manifold. (Incidentally, one of the times LQG inspired stuff got Einstein's equations out was in LQC, by putting in a fixed background of spherical symmetry.)

I don't know about reality, but I find the crystal melting in topological string theory really cute

4. Oct 6, 2009

### suprised

Well I can answer that. In fact there are various notions of topological strings and background independence. Mostly one means with this a certain modification, or "twist", of the ordinary superstring that projects the string down to its "topological" subsector.

This subsector has only a finite number of states, essentially given by zero modes of the fields, and the correlation functions are "topological" in the sense that they are given by mathematically exact expressions. They depend on parameters (essentially) only holomorphically and are largely protected from perturbative quantum corrections. This allows to expose non-perturbative corrections like instanton corrections and compute them exactly.

Because of this solvabilty and tractability, topological strings are an ideal "theoretical laboratory" for testing ideas, develop computational methods and make explicit contact to nice clean mathematics (eg due to mirror symmetry). So they are mainly a toy model for full-fledged strings, where many properties can be studied in a simpler and often exact way. For example, one can get a handle on "space-time foam" near the Planck scale and see very explicitly the kind of generalized geometries that contribute to the path integral. Or one can determine explicitly the microscopic states of quantum black holes. All this is very concrete and mathematically well-defined, so one can go much further as with ordinary methods.

But there is a price to pay: as said, topological strings are tied to a "simple" subsector of the full-fledged superstring, and they can't say much about quantities which are not in this subsector. So they provide only a partial solution, but the hope is that one can get a lot of relevant conceptional insights by studying just this subsector. Fortunately it is also the physically most interesting subsector, dealing with the low-energy or massless states; the correlation functions one can exactly compute are, for example, the Yukawa couplings or gauge coupling threshold corrections, which are of direct phenomenologial interest. Also the solution N=2 gauge theories of Seiberg and Witten can be direcly related to topological strings, and a lot of insights in the non-perturbative sector of general supersymmetric gauge theories had been gained from this perspective (this is different from AdS/CFT). On the other hand, not surprisingly, topological strings do not say much about the gravitational sector.

One kind of conceptual insights, for example, concerns background independence. This has a specific meaning in this context, and this should not be confused with other notions of background independence. As said, topological strings deal only with a sbsector of the theory, and this means that correlation functions depend only on a subset of all parameters (=vacuum expectation values that determine the compactification geometry). Thus they are independent of the "other" parameters in the theory (including anti-holomorphic parameters), and since these are also part of the background geometry, the theory is thus "partially" background independent. Actually the story is much more complicated in that there IS in fact a dependence on the "wrong" kind of parameters, due to anomalies, but it is a mild one that is very well definded and governed by certain differential equations. It is these equations what is interesting in this context. See http://arxiv.org/abs/hep-th/9306122 for a concrete discussion.

So far I was mainly discussing the space-time aspects of topological strings, but there are also wold-sheet aspects, and indeed the correlation functions I was talking before are metric-independent also from the world-sheet, 2d point of view. From this perspective there are many relations to integrable systems, matrix models etc.

All-in-all, to repeat: the main message is that the (usually considered) topologial string is a simplified toy model and testing ground for full-fledged strings, for which one can go pretty far, both conceptionally and computationally. As for "new, [experimentally] verifiable predictions", this is not the right thing to ask for in this context.

5. Oct 6, 2009

### tom.stoer

Regarding background independence: as we know from ART that the metric (or connection + vierbein in first order formalism) is fully dynamical we expect that any reasonable theory for QG incorporates this background independence. It must e.g. be possible to describe the dynamical evolution of the "entity which QG will call spacetime" to study the formation of a black hole (or whatever this black hole will be in the final theory).

I think that in principle it's "only" a technical issue to recover fully background independence by starting with a background. If somebody has an idea how this can work then everything is fine.

I don't think that we must not insist on background independence for other structures than spacetime. If e.g. AdS/CFT ist the correct theory for QG then it's fine to work with a background dependent CFT as long the emerging spacetime behaves physically correct. If you can't observe or measure the background in CFT (because you are not able to see this CFT at all :-) then nothing is lost.

6. Oct 19, 2009

### crackjack

Thanks everyone!

@surprised
Apart from gravity, what are the other 'desired' states that get projected out when we do these topological twists?
Also, what do you refer to when you say holomorphic (or anti-holomorphic) parameters?

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