LPTHE-05-16 LPTENS-05-22 Aspects of Dirichlet S-branes

In this note, we discuss some features of the Dirichlet S-brane, defined as a Dirichlet boundary condition on a time-like embedding coordinate of open strings. We analyze the Euclidean theory on the S-brane world-volume, and trace its instability to the infinite fine-tuning of the initial conditions required to produce an infinitely extended space-like defect. Using their equivalence under T-duality with D-branes with supercritical electric field, we argue that under generic perturbation, S-branes turn into D-brane / anti-D-branes. We extract the imaginary part of the cylinder amplitude, and interpret its inverse as a “decay length”, beyond which a pair of S-branes annihilates. Finally, we reconsider the boundary state of the Dirichlet S-brane and find that it is either a solution of type II string theory with imaginary R-R fields, or a solution of type II∗ with real fields. This leaves the non-BPS S-branes as potentially physical solutions of type II string theory. S-branes are generically defined as defects localized on a space-like hypersurface. In string theory, this designation covers a variety of different constructions: the Dirichlet Sbrane, defined as a space-like locus where open strings can end, is the direct counterpart of the D-brane [1]. The “decaying D-brane” describes the rolling of the open string tachyon on an unstable D-brane to or from the closed string vacuum (see e.g. [2] for a review), and in certain cases may be viewed as an array of Dirichlet S-branes in imaginary time [3]. More generally, S-branes can be viewed as homogeneous time-dependent solutions in string theory or supergravity, primarily localized at a given time (see e.g. [4] for a review). Although they are expected to require very fine-tuned initial conditions, they may be useful tractable models of time-dependent backgrounds in string theory. Furthermore, S-branes may be important ingredients in constructing inflationary backgrounds in string theory [5], just as D-branes are useful sources of AdS spaces. Finally, they may be used as probes [6] or initial/final value surfaces [7] in more general time-dependent backgrounds. In this note, we study some aspects of Dirichlet S-branes, obtained by imposing a Dirichlet condition on the time coordinate (or a time-like combination) of open strings. We re-analyze the world-volume theory of a single S-brane, and relate the “instability” due to the “wrong” signature of the transverse fluctuations to the infinite fine-tuning which is necessary in order to maintain an infinitely extended, flat S-brane under radial evolution. We show that under T-duality with respect to a transverse spatial coordinate, S-branes are T-dual to D-branes carrying a supercritical electric field. By analogy with the discharge of the supercritical electric field due to the nucleation and stretching of open strings, this suggests that generic S-brane configurations may decay into a shower of D-brane antiD-brane pairs. We study the first quantization of the open strings stretched between two S-branes, and find that, in contrast to the D-brane case, there are only a finite number of physical states, which grows as the S-branes are further separated in time. We analyze the cylinder amplitude for two parallel S-branes, and interpret the infrared divergences due to these physical states as another signal of this instability. Although the flat S-brane configurations are infinitely fine-tuned, it is important to note that their mere existence may have a potentially disastrous effect: since S-branes can be viewed as the world-volume of a tachyonic particle, they may signal an instability of the theory at hand. Indeed, S-brane may sometimes be viewed as the thin-shell approximation of a tunneling event from a false to a true vacuum, although this requires a non-standard analytic continuation from the Coleman-de Luccia instanton [8]. It is thus important to ascertain whether the S-brane configurations mentioned above are indeed solutions of string theory, with the correct reality conditions on the fields. As we shall explain, the boundary state for the Dirichlet S-brane constructed in [1] is a solution of type II* [5] rather than type II string theory1. A S-brane type solution of type II string theory can be obtained by analytical continuation from the D-brane boundary state but it radiates Ramond-Ramond fields with the wrong reality property. As for the “decaying D-brane” mentioned above, although it is possible in the bosonic string case to arrange the deformation parameter so as to describe a Dirichlet S-brane at X0 = 0 (together with its translates in imaginary This issue was also raised by the authors of [9], in a different, plane wave background.