4.1 A first implementation

The first way to implement this is as follows: let us take (17) with the first term in the right hand side only, but let us also say that the variation with respect to $\Phi$ has to be performed taking into account that $\Phi$ is in some sense the analogous of a ``total derivative''. But then, what is a total derivative in the matrix formulation? One property of a total derivative is that its integral is is fixed from the boundaries, which are not varied. Let us say for simplicity that we take the integral to be zero ( fixing it to another constant will not change anything ). In this case we must proceed as follows:

1. Take the action as in (17), but only consider the first term in the right hand side.
2. Consider the variation of such action, but with the constraint that $\Phi$ is a total derivative, which means that $\mathrm{Tr}(\Phi) = 0$ .
3. To do this in practice we add to the action defined in (1) $c\, Tr(\Phi)$, where $c$ is an undetermined lagrange multiplier which implements condition $(2)$. The resulting theory will have then an arbitrary string tension if we continue from $(2)$. The constant $c$ , i.e. the undetermined lagrange multiplier playing the role of the constant of integration in (11).