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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, \begin{equation} 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} \end{equation} 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 $\phi$, \begin{equation} \delta\phi \,\partial_\sigma\phi = \partial_\sigma\delta\phi.\tag{4} \end{equation} 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?

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