Bulk induced boundary perturbations

TL;DR

The work investigates how closed-string moduli perturbations affect D-brane moduli via boundary RG flows, showing that an exactly marginal bulk deformation can lose exact marginality when a boundary is present. It derives general bulk–boundary RG equations and demonstrates how a nonzero bulk–boundary coupling induces a boundary flow, whose end point is a conformal brane in the perturbed theory. The radius deformation of a free boson at the self-dual point serves as a central example, where the flow drives generic SU(2) branes to pure Dirichlet or Neumann configurations, with exact boundary-flow formulas available at self-duality. The results generalize to higher rank groups and current–current deformations, offering a geometric/dynamical picture of moduli stabilization across open and closed string sectors and suggesting time-dependent extensions and backreaction considerations.

Abstract

The influence of closed string moduli on the D-brane moduli space is studied from a worldsheet point of view. Whenever a D-brane cannot be adjusted to an infinitesimal change of the closed string background, the corresponding exactly marginal bulk operator ceases to be exactly marginal in the presence of the brane. The bulk perturbation then induces a renormalisation group flow on the boundary whose end-point describes a conformal D-brane of the perturbed theory. We derive the relevant RG equations in general and illustrate the phenomenon with a number of examples, in particular the radius deformation of a free boson on a circle. At the self-dual radius we can give closed formulae for the induced boundary flows which are exact in the boundary coupling constants.

Figures (1)

• Figure 1: The moduli space of D-branes on the self-dual circle, $SU(2)$, can be described as a product of two circles $S^{1}$ (given by the phases of $a$ and $b$ in \ref{['su2']}) fibred over an interval where $|a|$ runs between $0$ and $1$, and $|a|^{2}+|b|^{2}=1$. The ends of the interval where one of the circles shrinks to zero describe Dirichlet and Neumann branes, respectively. If we start with a generic boundary condition and increase (decrease) the radius, the boundary condition will flow to a Dirichlet (Neumann) boundary condition.