Taking off the square root of Nambu-Goto action and obtaining Filippov-Lie algebra gauge theory action

We propose a novel prescription to take off the square root of NambuGoto action for a p-brane, which generalizes the Brink-Di VecchiaHowe-Tucker or also known as Polyakov method. With an arbitrary decomposition as d + n = p + 1, our resulting action is a modified d-dimensional Polyakov action which is gauged and possesses a Nambu n-bracket squared potential. We first spell out how the (p+1)dimensional diffeomorphism is realized in the lower dimensional action. Then we discuss a possible gauge fixing of it to a direct product of d-dimensional diffeomorphism and n-dimensional volume preserving diffeomorphism. We show that the latter naturally leads to a novel Filippov-Lie n-algebra based gauge theory action in d-dimensions.