DDF operators are transverse physical string oscillators constructed as contour integrals of weight-one string vertex operators. Choose a null reference momentum and transverse polarizations orthogonal to it and to a reference on-shell momentum. The integrand is proportional to ; its normalization and are fixed so the operators obey the oscillator algebra in the title and commute with the positive-mode Virasoro constraints. For an open-string tachyon seed, one can use . Acting with negative modes shifts momentum and raises oscillator level while preserving the physical mass-shell condition. Their transverse Fock space has positive norm and makes the physical content of the no-ghost theorem for the critical bosonic string explicit.
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