Witten’s vertex made simple

The infinite matrices in Witten's vertex are easy to diagonalize. It just requires some $SL(2,R)$ lore plus a Watson-Sommerfeld transformation. We calculate the eigenvalues of all Neumann matrices for all scale dimensions s, both for matter and ghosts, including fractional s which we use to regulate the difficult $s=0$ limit. We find that $s=1$ eigenfunctions just acquire a p term, and x gets replaced by the midpoint position.

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