[Paper Review] Level Truncation Approach to Open String Field Theory

Matěj Kudrna|arXiv (Cornell University)|Jan 19, 2021

Black Holes and Theoretical Physics121 references4 citations

TL;DR

This paper applies the level truncation approach in bosonic open string field theory to study classical solutions on various D-brane backgrounds, including free bosons on circles and tori, and minimal models. It identifies both known D-brane solutions and novel exotic solutions, suggesting new boundary states beyond standard brane configurations, using numerical methods like Newton's method and homotopy continuation with consistency checks via Ellwood invariants and energy calculations.

ABSTRACT

Given a D-brane background in string theory (or equivalently boundary conditions in a two-dimensional conformal field theory), classical solutions of open string field theory equations of motion are conjectured to describe new D-brane backgrounds (boundary conditions). In this thesis, we study these solutions in bosonic open string field theory using the level truncation approach, which is a numerical approach where the string field is truncated to a finite number of degrees of freedom. We start with a review of the theoretical background and numerical methods which are needed in the level truncation approach and then we discuss solutions on several different backgrounds. First, we discuss universal solutions, which do not depend on the open string background, then we analyze solutions of the free boson theory compactified on a circle or on a torus, then marginal solutions in three different approaches and finally solutions in theories which include the A-series of Virasoro minimal models. In addition to known D-branes, we find so-called exotic solutions which potentially describe yet unknown boundary states. This paper is based on my doctoral thesis submitted to the Faculty of Mathematics and Physics at Charles University in Prague.

Motivation & Objective

• To investigate classical solutions of open string field theory on diverse D-brane backgrounds using numerical methods.
• To identify both known D-brane configurations and new exotic solutions that may correspond to unknown boundary states.
• To test the consistency of solutions using observables like energy, Ellwood invariants, and out-of-Siegel equations.
• To explore marginal deformations and tachyon condensation in free boson and minimal model backgrounds.
• To assess computational scalability and limitations of numerical algorithms like Newton's method and homotopy continuation in high-level truncations.

Proposed method

• Employs the level truncation approach, truncating the string field to finite degrees of freedom at increasing levels (up to L=26).
• Uses matrix representations of operators in the $KBc$ algebra and computes the kinetic term via Gram matrices.
• Evaluates the cubic vertex using recursive algorithms and SU(1,1) singlet projections, with parallelization for performance.
• Applies Newton's method and homotopy continuation to solve the equations of motion, with gauge fixing and Jacobian evaluation.
• Computes observables such as energy, Ellwood invariants, and out-of-Siegel equations to verify solution consistency.
• Uses Padé approximants and extrapolation techniques to estimate infinite-level behavior from finite-level results.

Experimental results

Research questions

• RQ1 Do classical solutions in open string field theory on free boson backgrounds reproduce known D-brane configurations like D0- and D1-branes?
• RQ2 Can the level truncation approach reveal new, exotic solutions beyond standard boundary states in free boson and minimal model theories?
• RQ3 How do marginal deformations manifest in the string field theory framework, and can they be consistently described using different approaches?
• RQ4 What is the role of twist symmetry and conservation laws in ensuring the consistency of numerical solutions?
• RQ5 How do computational constraints, especially memory and time, limit the achievable level in numerical solutions?

Key findings

• The study reproduces known D-brane solutions such as the tachyon vacuum and MSZ lump solution at $R = \sqrt{3}$, with consistent energy and Ellwood invariants.
• Exotic solutions are found in free boson on a circle and torus, particularly at $R=3$, $R=2$, and $R=\sqrt{3}$, suggesting new boundary states.
• Double lump solutions are observed at $R=3$ and $R=2$, with convergence of Ellwood invariants confirmed via Padé approximants.
• Wilson line solutions are found to be radius-independent, and marginal deformations are successfully computed at $R=\sqrt{3}$.
• In minimal models like Ising and Lee-Yang, both regular and exotic solutions are identified, including for the $\sigma \otimes \sigma$ brane in double Ising and Ising $\otimes$ tricritical Ising models.
• Computational limits are reached at level 26 due to memory and time constraints, especially for homotopy continuation, which scales as $2^{e^{\alpha\sqrt{L}}}$.