Hamilton-Jacobi formulation for topological gauge theories

We develop a Hamilton–Jacobi formulation for topological gauge theories, focusing on the Chern–Simons and Maxwell–Chern–Simons models. A complete constraint analysis is performed, allowing the identification of involutive and non-involutive Hamiltonians, the construction of the associated characteristic equations, and the determination of the generators of gauge transformations. Particular emphasis is placed on the incorporation of gauge-fixing conditions into the Hamilton-Jacobi framework and on their role in the integrability structure of the theory. We show that this implementation preserves the integrability of the system and leads to a consistent generalized bracket structure, in full agreement with the results obtained through the Dirac formalism. These results provide a unified and self-consistent description of constrained gauge systems within the Hamilton–Jacobi approach.