Sen’s lump-solution conjecture for lower-dimensional D-branes

Construct and establish the existence of non-trivial, spatially dependent lump solutions of the classical open string field equations that interpolate between the tachyon vacuum and the perturbative vacuum and describe stable lower-dimensional D(p−n)-branes for n<p.

Background

The paper reviews Sen’s tachyon-condensation conjectures for Witten’s cubic open string field theory on unstable D-branes. The second conjecture concerns spatially localized solitonic configurations, or lump solutions, of the full classical string field equations. Such a solution should approach the stable tachyon minimum asymptotically, pass through the perturbative unstable vacuum at a localized codimension-n interface, and represent a stable lower-dimensional D-brane.

The paper states that Schnabl’s analytic solution proved Sen’s first and third conjectures, but it does not claim a corresponding proof of the lump-solution conjecture. Establishing these solutions and their precise D-brane interpretation therefore remains an unresolved problem within the discussion presented.

References

Conjecture 2: While the global minimum represents a completely homogeneous vacuum, the classical string field equations admit non-trivial, spatially dependent lump solutions (solitonic configurations). A lump solution \Psi_{\text{lump}(x) that transitions asymptotically along a spatial direction xi \to \infty to the stable minimum \phi_0, but passes through the unstable maximum \phi=0 at a localized codimensional interface, describes a stable lower-dimensional D(p-n)-brane, where n<p.

— Quantum Gravity in Asymptotically Flat Spacetimes from Unstable D-Branes  (2610.11493 - Pius, 8 Oct 2026) in Section 4.2, “Sen’s conjectures” (Conjecture 2)