# How to deal with crossing duality and modular invariance in string field theory?

Some cases to ponder over.

A closed string splits into two closed strings, which then merge again into a single closed string. The overall string worldsheet has the topology of a torus. There is an SL(2,Z) group of large diffeomorphisms acting upon this worldsheet. The contribution to the partition function comes from summing up over all contributions with this topology. Suppose you insist upon a canonical description of this process. In the loop part in the middle, there is a different slicing associated with each of the SL(2,Z) elements. We have to sum up over all such contributions. Each such choice gives the same contribution, but there is no canonical choice of which slicing. If you consider summing up over all possible slicings, there is the possibility of "interference" between different choices of slicings because the configurations you get from a different slicing might be continuously deformable into that of a different slicing. So, you can't just insist that we sum up over one possible slicing only. However, if you sum up over all possible intermediate slicings, you pick up an infinite multiplicative factor compared to the no string interaction case.

The other case is a "tree level" worldsheet with two incoming closed strings, and two outgoing ones. There are the s-, t- and u-channels. They correspond to different possible slicings. Each by itself gives the same contribution. Each tells a different splitting story. You don't sum up over all channels. The problem comes when the two outgoing strings are identical. Then, you can't even distinguish between the s- and t-channels.

The moral of the story is, there is no canonical description of string interactions.

How to deal with crossing duality and modular invariance in string field theory?

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String field theory can't deal with the modular invariance. Indeed, that's a reason that there is no known consistent string field theory which contains and describes external physical closed string states.

This is a paradoxical proposition from me because the first string theory paper in my life that I studied in detail, one about the Kyoto group string field theory, claimed to have achieved exactly what I say is impossible. However, when done properly, all such attempts end up with an infinite multiple counting of the fundamental domain or something worse.

Because a description of processes with external closed string states can't be done properly in string field theory, the question about the crossing duality is meaningless for closed string external states. Those processes can't be calculated directly in string field theory.

(Let me mention, however, that all closed string amplitudes may actually be extracted indirectly from open string field theory, namely from poles of purely open-string scattering whose momenta are chosen so that there are closed string resonances.)

However, there's an equally interesting crossing duality for external open string states – it is the original $s$-$t$ duality known from the Veneziano amplitude. This and its multi-particle and multi-loop extensions are actually manifested in string field theory in an interesting way. String field theory doesn't say that the $s$-channel and $t$-channel diagrams are "completely the same" so that they would only be counted once. Instead, the moduli space of the marked disk is divided to a region whose contribution is calculated from the $s$-channel, and a region that is contributed by the $t$-channel in string field theory. So string field theory "explains" the origin of this duality just like any other quantum field theory, by an interchange of two channels (diagrams).

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Are you claiming string field theory for closed strings is fraudulent? – Richard Antonio Oct 15 '12 at 15:09
Is string theory even a field theory??? – Richard Antonio Oct 15 '12 at 15:16
Dear Richard, concerning the first question, the answer is essentially Yes, but a better adjective is probably "inconsistent" or "wrong", not "fraudulent". Every expert knows that - and this also answers the second question - that string theory may only be rewritten as a field theory with infinitely many fields, string field theory, in the case of open strings where it produces the S-matrix to all orders in the string theory expansion. – Luboš Motl Oct 15 '12 at 16:47
So in this sense - requiring a spacetime Lagrangian (in the same spacetime where strings live) that follows the usual rules of QFT - string theory (the whole thing) is not a field theory; it is a nontrivial generalization of it. That's not too surprising a statement - the closed string sector contains quantum gravity which is holographic (and slightly nonlocal: needs to preserve the information in the Hawking radiation) so it should better be impossible to exactly rewrite it as a "local QFT" in the bulk, and indeed, it's impossible. – Luboš Motl Oct 15 '12 at 16:49
Open strings may be rewritten in this SFT way because they're perturbations of D-branes and D-branes are lighter and at weak coupling, they don't create too curved black-hole-like gravitational field. So in this sense, D-branes and all the open string matter is just like a "non-gravitational matter", and may be described much like normal matter QFTs as a theory living on a background spacetime whose dynamics itself is more complicated than a local or similar field theory - it's quantum gravity i.e. the whole string theory. Some vacua of ST are equivalent to some QFTs, however, e.g. AdS/CFT. – Luboš Motl Oct 15 '12 at 16:50