Gauss law constraint in A-theory branes

{\cal A} -theory realizes U-duality symmetry by extending the string worldsheet to a higher dimensional brane worldvolume, in which the worldvolume and the spacetime belong to different representations of the exceptional group. The closure of the brane Virasoro algebra requires the Gauss law constraint. The Gauss law constraint promotes spacetime coordinates to gauge fields and extends the string worldsheet into the brane worldvolume. While the Virasoro constraint is used to reduce the spacetime coordinate, the Gauss law constraint is used to reduce both the worldvolume and the spacetime coordinates. As in conventional gauge theories, the treatment of the Gauss law constraint is a technically important aspect of the quantization of {\cal A} -theory. We show that the string solution is the only consistent solution of the Gauss law dimensional reduction condition for D=3 and 4 cases as a perturbative solution, while leaving open the possibility of higher dimensional nonperturbative branes described by nonlinear actions. This result implies that the physical symmetry of the theory is two-dimensional conformal symmetry, suggesting that the theory admits a string-like quantization. We further construct a string solution that is covariant under the exceptional group symmetry. The relation between this solution and the constant charge parameter appearing in the exceptional \sigma σ -model is also discussed.