This paper presents a planar diagram theory for strong interactions, focusing on the behavior of gauge theories with a colour gauge group $ U(N) $ in the limit $ N \to \infty $, with $ g^2N $ fixed. The theory shows that only planar diagrams dominate, and the perturbation series in $ 1/N $ has the same topological structure as dual models, with $ 1/N $ corresponding to the dual coupling constant. The paper suggests that for hadrons, $ N $ is likely equal to three. A mathematical framework is proposed to link these concepts with the functional integrals of Gervais, Sakita, and Mandelstam for the dual string. The paper discusses the infrared behavior of massless gauge theories, which makes perturbation expansions difficult. It introduces the Han-Nambu quark theory, which suggests that creating a physical state with non-zero "colour" quantum number requires high energy. The paper emphasizes a coincidence: when considering the parameter $ N $ of the colour gauge group $ SU(N) $, an expansion of the amplitudes at $ N \to \infty $ arranges the Feynman diagrams into sets with the topology of the quantized dual string with quarks at its ends. The analogy with the string is further pursued by writing the planar diagrams in the light cone reference frame. The paper formulates a gauge theory for strong interactions, with quarks forming three representations of the group $ U(N) $. The theory includes an anti-Hermitian gauge field and a Lagrangian with specific terms for the gauge and matter fields. The Feynman rules are described, and the paper discusses the propagators and vertices of the theory. In the $ N \to \infty $ limit, the paper shows that the number of planar diagrams is dominated by those with quarks at the edges. The topological structure of the perturbation series in $ 1/N $ is identical to that of dual models, with $ 1/N $ corresponding to the dual coupling constant. The paper also discusses the Hamiltonian formalism for planar diagrams, showing that the sum of all planar diagrams can be generated by a Hamiltonian that yields the correct propagators and vertices. The paper concludes that the topological structure of dual theories suggests a planar diagram field theory for bound quarks. If the eigenstates of a certain Hamiltonian crystallize into a discrete spectrum, the original particles will condensate into a string that keeps quarks together. For baryons, the situation is more complex, and the $ 1/N $ expansion is extremely delicate. The dual coupling constant is expected to be calculable and of order $ 1/3 $.This paper presents a planar diagram theory for strong interactions, focusing on the behavior of gauge theories with a colour gauge group $ U(N) $ in the limit $ N \to \infty $, with $ g^2N $ fixed. The theory shows that only planar diagrams dominate, and the perturbation series in $ 1/N $ has the same topological structure as dual models, with $ 1/N $ corresponding to the dual coupling constant. The paper suggests that for hadrons, $ N $ is likely equal to three. A mathematical framework is proposed to link these concepts with the functional integrals of Gervais, Sakita, and Mandelstam for the dual string. The paper discusses the infrared behavior of massless gauge theories, which makes perturbation expansions difficult. It introduces the Han-Nambu quark theory, which suggests that creating a physical state with non-zero "colour" quantum number requires high energy. The paper emphasizes a coincidence: when considering the parameter $ N $ of the colour gauge group $ SU(N) $, an expansion of the amplitudes at $ N \to \infty $ arranges the Feynman diagrams into sets with the topology of the quantized dual string with quarks at its ends. The analogy with the string is further pursued by writing the planar diagrams in the light cone reference frame. The paper formulates a gauge theory for strong interactions, with quarks forming three representations of the group $ U(N) $. The theory includes an anti-Hermitian gauge field and a Lagrangian with specific terms for the gauge and matter fields. The Feynman rules are described, and the paper discusses the propagators and vertices of the theory. In the $ N \to \infty $ limit, the paper shows that the number of planar diagrams is dominated by those with quarks at the edges. The topological structure of the perturbation series in $ 1/N $ is identical to that of dual models, with $ 1/N $ corresponding to the dual coupling constant. The paper also discusses the Hamiltonian formalism for planar diagrams, showing that the sum of all planar diagrams can be generated by a Hamiltonian that yields the correct propagators and vertices. The paper concludes that the topological structure of dual theories suggests a planar diagram field theory for bound quarks. If the eigenstates of a certain Hamiltonian crystallize into a discrete spectrum, the original particles will condensate into a string that keeps quarks together. For baryons, the situation is more complex, and the $ 1/N $ expansion is extremely delicate. The dual coupling constant is expected to be calculable and of order $ 1/3 $.