Approaches to Quantum Gravity

String field theory 223similar complications of gauge choice, field redefinitions, and quantum definitionwill need to be resolved to make progress in this direction.Because of the closed string bulk tachyon in the bosonic theory, which is notyet known to condense in any natural way, the bosonic theory may not be welldefinedquantum mechanically. Again, we must turn to the supersymmetric theory.Until recently, there was no complete description of even a classical supersymmetricclosed string field theory. The recent work of Okawa and Zwiebach [25] andof Berkovits, Okawa and Zwiebach [4], however, has led to an apparently completeformulation of a classical string field theory for the heterotic string. Thisformulation combines the principles underlying the construction by Berkovits ofthe open superstring field theory with the moduli space decomposition developedby Zwiebach for the bosonic closed string field theory. Interestingly, for apparentlysomewhat technical reasons, the approach used in constructing this theorydoes not work in any natural way in the simpler type II theory. The action of theheterotic superstring field theory has a Wess–Zumino–Witten form, and containsan infinite number of interactions at arbitrarily high orders. The development of aSUSY CSFT makes it plausible for the first time that we could use a backgroundindependentclosed string field theory to address questions of string backgroundsand cosmology. Like the open bosonic theory discussed in the previous section, thisclosed string field theory can be defined in level truncation to give a well-definedset of interaction terms for a finite number of fields, but it is not known in any preciseway what the allowed space of fields should be or how to quantize the theory.These are important problems for future work in this area.12.4 OutlookWe have reviewed here the current state of understanding of string field theoryand some recent developments in this area. String field theory is currently the onlytruly background-independent approach to string theory. We have reviewed somerecent successes of this approach, in which it was explicitly shown that distinctvacua of open string field theory, corresponding to dramatically different stringbackgrounds, appear as solutions of a single theory in terms of a single set ofdegrees of freedom. While much of the work concretely confirming this picturein string field theory was numerical, it seems likely that further work in the nearfuture will provide a better analytic framework for analyzing these vacua, and forunderstanding how open string field theory can be more precisely defined, at leastat the classical level.We described open string field theory in some detail, and briefly reviewed thesituation for closed string field theory. While gravity certainly requires closedstrings, it is not yet clear whether we are better off attempting to directly construct