Analytic solution for tachyon condensation in open string field theory

TL;DR

This work constructs an exact analytic tachyon-vacuum solution in Witten's cubic open string field theory by introducing a new basis and gauge, enabling a tractable, largely analytic treatment of tachyon condensation. The solution is expressed in Bernoulli-number coefficients within the $\mathcal{L}_0$-based framework and is shown to be the Euler–Maclaurin asymptotic of a sum over wedge states with insertions, yielding a fully regular level-truncation behavior. A rigorous analytic proof of Sen's first conjecture is provided by computing the energy density and demonstrating it matches the D-brane tension. By translating the solution to the Virasoro basis and applying Padé/Borel methods, the paper reinforces the robustness of the result and offers pathways toward off-shell computations and extensions to supersymmetric and closed string contexts.

Abstract

We propose a new basis in Witten's open string field theory, in which the star product simplifies considerably. For a convenient choice of gauge the classical string field equation of motion yields straightforwardly an exact analytic solution that represents the nonperturbative tachyon vacuum. The solution is given in terms of Bernoulli numbers and the equation of motion can be viewed as novel Euler--Ramanujan-type identity. It turns out that the solution is the Euler--Maclaurin asymptotic expansion of a sum over wedge states with certain insertions. This new form is fully regular from the point of view of level truncation. By computing the energy difference between the perturbative and nonperturbative vacua, we prove analytically Sen's first conjecture.

Figures (4)

• Figure 1: String worldsheet in three different coordinate systems related by $z = -e^{-i w}$ and $\tilde{z} = \arctan z$. In the $\tilde{z}$ coordinate the lines marked with an arrow are identified, so that the worldsheet forms semi-infinite cylinder $C_\pi$. Fock states are given by the insertion of local operators at the puncture $P$. Inserting operators also at $\tau = +\infty$, i.e. $z=\infty$ or $\tilde{z} = -\pi/2 = \pi/2 \mod \pi$ would correspond to taking the BPZ inner product. We have also marked the left and right (looking backwards in time) parts of the string at $\tau =0$ separated by the midpoint $M$.
• Figure 2: Worldsheet of disk topology (after adding the midpoint $M$ at infinity) glued together out of three semi-infinite strips. The lines marked with an arrow are identified. The three vertex $\langle\, \phi_1, \phi_2, \phi_3 \,\rangle$ is defined as a correlator of three local operators $\phi_i$ inserted in the punctures $P_i$.
• Figure 3: Star product of two states $|\tilde{\phi}_1 \rangle * |\tilde{\phi}_2 \rangle$ represented by local operator insertions at punctures $P_1$ and $P_2$. A local operator $\chi$ corresponding to the 'test state' $|\tilde{\chi} \rangle$ can be inserted at the puncture $P_3$. The correlator is evaluated on a semi-infinite cylinder of circumference $3\pi/2$.
• Figure 4: Multiple star product $|\tilde{\phi}_1 \rangle * |\tilde{\phi}_2 \rangle* \cdots * |\tilde{\phi}_n \rangle$, the so called wedge state with insertions. Without insertions it would be denoted as $|n+1 \rangle$. The correlator is evaluated on a semi-infinite cylinder of circumference $(n+1)\pi/2$.