In basic string theory Lagrangians (e.g. the Polyakov or the Nambu-Goto), the variables include a function $x:X\rightarrow M$ embedding a world-sheet $X$ into some target space $M$, which can be Minkowski space or some curved target space. Is it possible to reformulate these theories to eliminate the target space? Perhaps to replace it with sections of some bundle over $X$? Although this would seem to amount to just some kind of $(1+1)$-dimensional quantum field theory?

I realize this is a somewhat open-ended question; I'm just wondering if there has been any research done toward formulating string theories that might not involve any dependence on a target space.

  • $\begingroup$ Well, a function $X \to M$ is a section of the trivial bundle $X \times M$ over $X$... $\endgroup$ – Qiaochu Yuan Apr 6 '15 at 23:33
  • $\begingroup$ Sigma models where the fields are sections of a bundle $F$ on $X$ with fiber $M$ are usually studied as 'gauged sigma models' or 'twisted sigma models', depending on whether one plans to fix a connection on $F$. $\endgroup$ – user1504 Feb 12 '19 at 20:12

Sure you can. In fact, this usually goes by the name of strings without strings. The basic observation is that when you quantise the sigma model when $M$ is, say, Minkowski spacetime, what you end up with is a quantum conformal field theory with a given central charge ($c=26$ for the bosonic string, for instance) and you identify the physical states with the semi-infinite cohomology of (a suitable extension of) the Virasoro algebra relative to the centre with values in the underlying vector space of the conformal field theory.

Now it should be clear from the above that you could side-step the quantisation of the sigma model and take as your starting point a module of Virasoro (or the suitable extension) of the right central charge so that its semi-infinite cohomology relative to the centre is nontrivial.

Of course, it may happen (e.g., the so-called Gepner models) that quantum string theories constructed in this way, can then be interpreted as the quantisation of a sigma model with some target space. Having said that, I think it is safe to say that most such algebraically constructed quantum string theories do not have a sigma-model interpretation, though.

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  • $\begingroup$ Very interesting - do you have any specific sources you would recommend to learn more about this approach? $\endgroup$ – Blake Apr 8 '15 at 19:06

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