Edward Witten explores the solutions of four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories by formulating them as quantum field theories derived from configurations of fourbranes, fivebranes, and sixbranes in Type IIA superstrings, and then reinterpreting these configurations in $M$ theory. This approach leads to explicit solutions for the Coulomb branch of a large family of four-dimensional $\mathcal{N}=2$ field theories with zero or negative beta function. The paper begins by discussing the construction of models using Type IIA fourbranes and fivebranes on $\mathbf{R}^{10}$. The analysis is extended to include sixbranes, and then to models obtained by considering Type IIA fourbranes and fivebranes on $\mathbf{R}^{9} \times \mathbf{S}^{1}$. The key techniques involve using $SL(2,\mathbf{Z})$ duality of Type IIB superstrings to predict mirror symmetry and the strong coupling limit of Type IIA superstrings in ten dimensions, which is determined by an equivalence to eleven-dimensional $M$ theory. The main results are derived by interpreting the brane configurations in $M$ theory. For Type IIA fourbranes and fivebranes on $\mathbf{R}^{10}$, the gauge group is identified as $\prod_{\alpha=1}^n SU(k_{\alpha})$, with the beta function coefficients determined by the number of fourbranes between each pair of fivebranes. The low-energy effective action for the vector fields is determined by the Jacobian of the compactification of the Riemann surface $\Sigma$. For models with zero or negative beta function, the effective action is described by an integrable Hamiltonian system, and the Coulomb branch is identified with a complex torus. For models with positive beta function, the fivebrane configuration still describes something, but it is interpreted in terms of a different ultraviolet fixed point. The paper discusses two cases: when $N_f \geq 2k + 2$, the theory is conformally invariant at short distances and flows to the $SU(k)$ theory with $N_f$ flavors in the infrared; when $N_f = 2k' + 1$, the theory has no obvious Lagrangian description and flows to a strongly coupled fixed point. The paper concludes with a generalization to a chain of $n+1$ fivebranes, where the gauge group is $\prod_{\alpha=1}^n SU(k_{\alpha})$ and the beta function coefficients are determined by the number of fourbranes between each pair of fivebranes. The solution is expressed as a polynomial $F(t, v) = 0$, where the coefficients of $t^n$ and $t^0$ are non-zero constants to ensure the absence of semi-inEdward Witten explores the solutions of four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories by formulating them as quantum field theories derived from configurations of fourbranes, fivebranes, and sixbranes in Type IIA superstrings, and then reinterpreting these configurations in $M$ theory. This approach leads to explicit solutions for the Coulomb branch of a large family of four-dimensional $\mathcal{N}=2$ field theories with zero or negative beta function. The paper begins by discussing the construction of models using Type IIA fourbranes and fivebranes on $\mathbf{R}^{10}$. The analysis is extended to include sixbranes, and then to models obtained by considering Type IIA fourbranes and fivebranes on $\mathbf{R}^{9} \times \mathbf{S}^{1}$. The key techniques involve using $SL(2,\mathbf{Z})$ duality of Type IIB superstrings to predict mirror symmetry and the strong coupling limit of Type IIA superstrings in ten dimensions, which is determined by an equivalence to eleven-dimensional $M$ theory. The main results are derived by interpreting the brane configurations in $M$ theory. For Type IIA fourbranes and fivebranes on $\mathbf{R}^{10}$, the gauge group is identified as $\prod_{\alpha=1}^n SU(k_{\alpha})$, with the beta function coefficients determined by the number of fourbranes between each pair of fivebranes. The low-energy effective action for the vector fields is determined by the Jacobian of the compactification of the Riemann surface $\Sigma$. For models with zero or negative beta function, the effective action is described by an integrable Hamiltonian system, and the Coulomb branch is identified with a complex torus. For models with positive beta function, the fivebrane configuration still describes something, but it is interpreted in terms of a different ultraviolet fixed point. The paper discusses two cases: when $N_f \geq 2k + 2$, the theory is conformally invariant at short distances and flows to the $SU(k)$ theory with $N_f$ flavors in the infrared; when $N_f = 2k' + 1$, the theory has no obvious Lagrangian description and flows to a strongly coupled fixed point. The paper concludes with a generalization to a chain of $n+1$ fivebranes, where the gauge group is $\prod_{\alpha=1}^n SU(k_{\alpha})$ and the beta function coefficients are determined by the number of fourbranes between each pair of fivebranes. The solution is expressed as a polynomial $F(t, v) = 0$, where the coefficients of $t^n$ and $t^0$ are non-zero constants to ensure the absence of semi-in