Closed Strings as Imaginary D-branes

TL;DR

The paper studies imaginary-time D-brane configurations as a route to purely closed-string backgrounds, showing that disk amplitudes with imaginary-brane arrays are equivalent to sphere amplitudes with an extra closed-string insertion |W⟩. It develops a general prescription, linking disk amplitudes to sphere amplitudes via a discontinuity-based transform S(E) = F(E) Disc_E[ \widetilde{A}(i E) ] with F(E) = 1/[2 \sinh(a E/2)], and demonstrates this explicitly for two-point and higher-point disk amplitudes. The resulting closed-string state |W⟩ contains a massless dilaton wave and an infinite tower of massive modes, with intriguing behavior depending on the brane spacing a (notably a > 2π yields finite energy, while a = 2π leads to divergences). Open-string moduli reappear as deformations of the closed-string background, and the framework extends to superstrings and to general brane distributions; the authors also discuss potential links to tachyon condensation, tachyon matter, and gravitational phenomena like black hole formation and phase transitions, suggesting a deep open/closed duality in this non-supersymmetric setting.

Abstract

Sen has recently drawn attention to an exact time-dependent Boundary Conformal Field Theory with the space-time interpretation of brane creation and annihilation. An interesting limit of this BCFT is formally equivalent to an array of D-branes located in imaginary time. This raises the question: what is the meaning of D-branes in imaginary time? The answer we propose is that D-branes in imaginary time define purely closed string backgrounds. In particular we prove that the disk scattering amplitude of m closed strings off an arbitrary configuration of imaginary branes is equivalent to a sphere amplitude with m+1 closed string insertions. The extra puncture is a specific closed string state, generically normalizable, that depends on the details of the brane configuration. We study in some detail the special case of the array of imaginary D-branes related to Sen's BCFT and comment on its space-time interpretation. We point out that a certain limit of our set-up allows to study classical black hole creation and suggests a relation between Choptuik's critical behavior and a phase-transition a` la Gregory-Laflamme. We speculate that open string field theory on imaginary D-branes is dual to string theory on the corresponding closed string background.

Figures (4)

• Figure 1: A graph of $\widetilde{G}_{array}(X)$, which has the interpretation of the field produced by an infinite array of $\delta$-function sources ('D-branes') located at $X = a (n +\frac{1}{2})$ . The dashed line represents the analytic continuation to $|X| > \frac{a}{2}$ of the branch around the origin.
• Figure 2: Integration contours in the complex $P$ plane. The zeros of $\sin(a P/2)$ are denoted by the symbols '$x$' along the real $P$ axis. The black dots represent possible poles of $\widetilde{A}$ and the thick line represents a possible cut. We assume that the only singularities of $\widetilde{A}$ are on the imaginary $P$ axis.
• Figure 3: Before the double Wick rotation we have a standard disk amplitude. The disk can be viewed as the region ${\cal H}_\rho$, which is the complex plane with a hole of radius $\rho$. There are contributions to the scattering amplitude from all values of $\rho \leq \rho_0$, where $\rho_0$ is the distance of the closest puncture. After the double Wick rotation the only contribution is coming from $\rho=0$. The hole shrinks to a point leaving behind an extra puncture ${\cal W}$ inserted at the origin.
• Figure 4: The two possible motion modes of a pair of D-branes in the complex $X^0$ plane.