# Importance of an extra total derivative term in Liouville theory

In this paper on boundary Liouville theory, the authors have introduced an extra term, $$-\partial_{\sigma}^2\phi$$, (the last term in the equation below) in defining the stress tensor of the Liouville theory as, \begin{align} \mathcal E = T+\bar T = \frac12\left(\partial_\tau\phi\right)^2+\frac12\left(\partial_\sigma\phi\right)^2+2m^2e^{2\phi}-\partial_{\sigma}^2\phi. \tag{1} \end{align} Moreoever, in writing the action for the theory this means that this extra term also appears in the action, $$$$S=\int\limits_M\!d^2x \left[ \frac12\left(\partial_\tau\phi\right)^2-\frac12\left(\partial_\sigma\phi\right)^2-2m^2e^{2\phi}+\partial_\sigma^2\phi\right].\tag{2}$$$$ Here it is just a total derivative boundary term. In particular, if the boundary is at $$\sigma=0,\pi$$ (as is the case in the paper), then the boundary term is $$-\int \!d\tau\left(\frac12\phi\partial_\sigma\phi-\partial_\sigma\phi\right).\tag{3}$$ Under the variation of the field $$\phi$$, $$$$\delta\phi \,\partial_\sigma\phi = \partial_\sigma\delta\phi.\tag{4}$$$$ on the boundary, requiring one to impose both Dirichlet, $$\delta\phi = 0$$ as well as Neumann boundary condition, $$\partial_\sigma\delta\phi=0$$ simultaneously. In the paper, the authors are studying Dirichlet problem. This looks weird and unfamiliar to me. In the standard scalar field theory where you don't have any term like $$\partial_\sigma^2\phi$$ the boundary variation is simply, $$-\delta\phi \,\partial_\sigma\phi =0$$, which is zero under Dirichlet boundary conditions.

So why is this extra term required in the Liouville theory?