Why is string theory in 10 or 26 dimension not divergent? Due to the high number of spacetime dimension (10 or 26) it should have a lot of UV divergencies of the form $ \int k^{n}dk $ and gravity within the approach of the string theory should be non-renormalizable too, or shouldn't it?
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asked Feb 15, 2012 at 20:46
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4$\begingroup$ Lubos Motl has a blog post about the finiteness of pertubative superstring amplitudes here that includes several references to the subject. $\endgroup$– Qmechanic ♦Feb 15, 2012 at 21:08
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1$\begingroup$ ... I assume You are honestly interested in what You are asking such that my +1 was not a premature slip of my mouse ;-) $\endgroup$– DilatonFeb 15, 2012 at 22:17
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1$\begingroup$ You must remember that string theory trades in an infinite tower of particles for a worldsheet, and the final world-sheet sum is much milder and more nonlocal than any of the particle sums that go into it. The momentum integration is not unbounded, because high k fluctuations become worldsheet fluctuations, and at high energy they are infrared big. $\endgroup$– Ron MaimonApr 8, 2012 at 6:31
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Bosonic closed oriented string theory is divergent in flat space time, see for example Lecture 3 of D'Hoker in "Quantum Fields and Strings" Volume 2. The reason is the presence of the tachyon. To my knowledge for the NS-NS string finiteness is only known up to one loop and already difficult.
Edit: Apparently a little bit more is known for the superstring, see Lectures on Two-Loop Superstrings by D'Hoker and Phong. This seems to be the most recent result.
