Experimental String Field Theory

TL;DR

This paper demonstrates that level-truncation in open bosonic OSFT is a convergent, predictive approximation by computing the tachyon condensate within the universal subspace up to level $(18,54)$. The authors develop efficient algorithms using conservation laws to determine the universal action and perform direct $L$-level solutions ($L\le 18$) while also applying extrapolation in $1/L$ via the tachyon effective action $V_L(T)$ to forecast higher-level behavior. They show the stable vacuum energy converges toward the expected value $-1$ (minimizing near $L\sim 28$ with $E_{min}=-1.00063$) and that the full gauge-invariant equations of motion are satisfied in the infinite-level limit, lending strong support to level truncation as a robust approximation method. The work also uncovers analytic patterns in the tachyon string field and reveals a remarkable, though not exact, universality in certain ghost coefficients, offering both numerical validation and potential analytic avenues for OSFT solutions. Overall, the results reinforce the viability of level-truncation for exploring nonperturbative tachyon dynamics and may inform future analytic progress and connections to closed-string physics.

Abstract

We develop efficient algorithms for level-truncation computations in open bosonic string field theory. We determine the classical action in the universal subspace to level (18,54) and apply this knowledge to numerical evaluations of the tachyon condensate string field. We obtain two main sets of results. First, we directly compute the solutions up to level L=18 by extremizing the level-truncated action. Second, we obtain predictions for the solutions for L > 18 from an extrapolation to higher levels of the functional form of the tachyon effective action. We find that the energy of the stable vacuum overshoots -1 (in units of the brane tension) at L=14, reaches a minimum E_min = -1.00063 at L ~ 28 and approaches with spectacular accuracy the predicted answer of -1 as L -> infinity. Our data are entirely consistent with the recent perturbative analysis of Taylor and strongly support the idea that level-truncation is a convergent approximation scheme. We also check systematically that our numerical solution, which obeys the Siegel gauge condition, actually satisfies the full gauge-invariant equations of motion. Finally we investigate the presence of analytic patterns in the coefficients of the tachyon string field, which we are able to reliably estimate in the L -> infinity limit.

Figures (6)

• Figure 1: Curves of the vacuum energy as a function of level, as predicted by our extrapolation scheme for various values of $M$ (maximum level of the data used in the extrapolation). The figure shows the curves $E^{(M)}(L)$ on a logarithmic plot, for $M$ between 8 (lowermost curve) and 16 (uppermost curve). Data in the $(L,3L)$ scheme.
• Figure 2: Plots of the tachyon effective potential $V_L(T)$ at level $L$, for $L$ between zero (uppermost curve) and 16 (lowermost curve). The curves for $L=6,10,12,14,16$ appear superimposed in the figure.
• Figure 3: Plots of the order 16 estimates $V_L^{(16)}(T)$ for the effective tachyon potential, for some sample values of $L \geq 10$. The minimum of each curve is indicated by a black dot, which by definition has coordinates $(T^{(16)}_L, E_L^{(16)})$. The isolated uppermost plot corresponds to $L=10$. To follow the curves from $L=10$ to $L=\infty$, focus on the position of the minima: as $L$ increases, the dot moves from right to left ( i.e., the tachyon vev decreases). As $L \to \infty$, the curves crowd towards an asymptotic function with minimum at $(T_\infty^{(16)}, E_{\infty}^{(16)}) =(0.5405, -1.00003)$.
• Figure 4: Plot of $E_L^{(16)}$ as a function of $1/L$. The black dots represent the exact values up to $L=18$ computed by direct level-truncation (Table 1, $(L,3L)$ scheme). To first approximation, the curve in the figure is roughly a parabola: since the energy overshoots -1 at $1/L =1/14 \simeq 0.07$, we have a visual understanding of the position of the minimum of the energy around $1/L =(1/14)/2 = 1/28 \simeq 0.036$.
• Figure 5: Plot of the results in Table \ref{['L2Lnaive']}. The continuous line represents the $(L,3L)$ results, while the dashed line represents the $(L,2L)$ results.