We present the ghost-free infinite-derivative extensions of the
Spherically-Reduced Gravity (SRG) and Callan-Giddings-Harvey-Strominger (CGHS)
theories in two space-time dimensions. For the case of SRG, we specify the
Schwarzschild-type gauge and diagonalise the quadratic action for field
perturbations after taking the background fields to be those of the flat-space
solution with a linear dilaton. Using the obtained diagonalisation, we
construct ghost-free infinite-derivative modifications of the SRG theory. In
the context of this modified SRG theory we derive a non-local modification of
the linearised spherically-reduced Schwarzschild solution. For the case of CGHS
gravity, we work in the conformal gauge and diagonalise the quadratic action
associated with this theory for a general background solution. Using these
results, we construct the ghost-free infinite-derivative modifications of the
CGHS theory and examine non-local modifications to the linearised CGHS
black-hole solution.
