Abstract
The elementary string interactions are the Reidemeister moves, birth/death, and fusion/fission. Such interactions have as their trace generically mapped surfaces, and these combine to form knotted surfaces in 4-space. The syzygies among these interactions are moves to such surfaces analogous to the Reidemeister moves for knots. ‘Movie’ parametrizations of these syzygies are given and interpreted in dimension 2+2. A Morse theoretic argument shows there are 15 such movie moves.
These moves, with appropriate choices of crossing information, are sufficient to construct any isotopy of an embedded surface on which a height function has been specified. The first seven of the movie moves are parametrized versions of those given by Roseman. The remaining eight are moves of Δ-type.
Amplitudes assigned to these interactions must satisfy relations that correspond to the movie moves. One such relation is a Zamolodchikov tetrahedral equation. We present some puzzles about the matrix formulations of these amplitudes.
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Supported by a grant from the University of South Alabama Research Committee.
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Carter, J.S., Saito, M. Syzygies among elementary string interactions in 2+1 dimensions. Lett Math Phys 23, 287–300 (1991). https://doi.org/10.1007/BF00398826
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DOI: https://doi.org/10.1007/BF00398826