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Solving 't Hooft or Veneziano large-$N$ QCD by a string theory is a long-standing problem. The would-be canonical string solution for 't Hooft large-$N$ QCD is a theory of closed strings for the glueballs and of open strings for the mesons. Recently, we demonstrated that an inconsistency arises between the renormalization properties of the 't Hooft large-$N$ QCD S matrix, worked out in Phys. Rev. D 95, 054010, and the open/closed duality of the would-be string solution, which thus does not exist in the canonical string framework. Specifically, the glueball amplitudes in the would-be open-string one-loop effective action, which canonically only get contributions from surfaces with one hole, must be ultraviolet (UV) divergent by the aforementioned renormalization properties. Hence, naively, the inconsistency occurs because these amplitudes are dual to tree closed-string diagrams, which are universally believed to be both UV finite -- since they are closed-string tree diagrams -- and infrared finite because of the glueball mass gap. In fact, the inconsistency follows from a low-energy theorem of the Novikov-Shifman-Vainshtein-Zakharov type, derived in Phys. Rev. D 95, 054010, that controls the renormalization in QCD-like theories. Interestingly, the proof of the inconsistency, at least in its present form, does not extend to a would-be string solution for Veneziano large-$N$ QCD. In an a-posteriori-related development, we proposed a new class string theories on noncommutative twistor space, whose space-time one-loop effective action in the open-string sector gets contributions, contrary to the canonical framework, from string instantons wrapping geometrically planar surfaces with any number of holes, topologically as in the Veneziano-planar expansion, avoiding the inconsistency.

Open/closed string duality is incompatible with renormalization in ’t Hooft large- N QCD, but not in Veneziano large- N QCD