Description
We use String Field Theory (SFT) techniques to investigate bulk RG flows in two-dimensional (B)CFTs triggered by slightly relevant operators of weight $(1-y,1-y)$, with $y$ small and positive. Specifically, we compute the change in the $g$-function up to the second order in $y$. In Conformal Perturbation Theory, this involves a notoriously difficult two-loop computation. On the other hand, in SFT, we accomplish this by identifying the solution to the classical open-closed string field equations of motion corresponding to the background associated with the IR fixed point. On this solution, we compute the disk action, a gauge-invariant quantity that gives the disk partition function
of the new open-closed background, which can be expressed in terms of the $g$-function and the string coupling constant $g_s$.
When we apply this procedure to well-known RG flows triggered by an exactly marginal operator or a slightly relevant operator, we make a surprising discovery. We find that the desired
$g$-function can only be obtained by assuming a change in $g_s$. Therefore, from the SFT point of view, a bulk deformation generally induces both a change in the (B)CFT background and a variation in the string coupling constant. In particular, we demonstrate that this change in
$g_s$ is universally proportional to the sphere two-point function of the perturbing bulk operator and is independent of the boundary conditions.
