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In string theory, we are told strings can split and merge if the string coupling is nonzero, even while the worldsheet action remains Nambu-Goto or Polyakov plus a topological term. However, a classical solution, in say the light cone gauge, shows that provided the worldsheet time increases with light cone cone time initially, adding topological terms will not change the solution, and the string will not split. So why and how do strings split?

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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.

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