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Topics in String Field Theory

L. Bonora, C. Maccaferri, D. Mamone, M. Salizzoni

TL;DR

This review analyzes bosonic open string field theory with a focus on Vacuum SFT and the ghost/matter factorization, deriving the three-string vertex and Neumann coefficients via conformal methods and solving ghost EOMs in an enlarged Fock space. It then connects matter projectors and lump solutions to D-branes, leveraging a constant background $B$ field to realize noncommutative solitons (GMS), and demonstrates that in the low-energy limit the VSFT star product factorizes into a Witten $*$-product and a Moyal $\star$-product, yielding a direct correspondence between VSFT projectors and noncommutative solitons. The paper constructs an infinite family of lumps $|\Lambda_n\rangle$, whose low-energy profiles reproduce GMS solitons as $|n\rangle\langle n|$, thereby providing a concrete bridge between D-branes in VSFT and solitons in noncommutative field theory. This framework clarifies how nonperturbative VSFT configurations encode D-brane physics and highlights the role of the $B$-field in smoothing singularities and facilitating a Moyal-Witten correspondence with tangible solitonic objects.

Abstract

This review of bosonic string field theory is concentrated on two main subjects. In the first part we revisit the construction of the three string vertex and rederive the relevant Neumann coefficients both for the matter and the ghost part following a conformal field theory approach. We use this formulation to solve the VSFT equation of motion for the ghost sector. This part of the paper is based on a new method which allows us to derive known results in a simpler way. In the second part we concentrate on the solution of the VSFT equation of motion for the matter part. We describe the construction of the three strings vertex in the presence of a background B field. We determine a large family of lump solutions, illustrate their interpretation as D-branes and study the low energy limit. We show that in this limit the lump solutions flow toward the so-called GMS solitons.

Topics in String Field Theory

TL;DR

This review analyzes bosonic open string field theory with a focus on Vacuum SFT and the ghost/matter factorization, deriving the three-string vertex and Neumann coefficients via conformal methods and solving ghost EOMs in an enlarged Fock space. It then connects matter projectors and lump solutions to D-branes, leveraging a constant background field to realize noncommutative solitons (GMS), and demonstrates that in the low-energy limit the VSFT star product factorizes into a Witten -product and a Moyal -product, yielding a direct correspondence between VSFT projectors and noncommutative solitons. The paper constructs an infinite family of lumps , whose low-energy profiles reproduce GMS solitons as , thereby providing a concrete bridge between D-branes in VSFT and solitons in noncommutative field theory. This framework clarifies how nonperturbative VSFT configurations encode D-brane physics and highlights the role of the -field in smoothing singularities and facilitating a Moyal-Witten correspondence with tangible solitonic objects.

Abstract

This review of bosonic string field theory is concentrated on two main subjects. In the first part we revisit the construction of the three string vertex and rederive the relevant Neumann coefficients both for the matter and the ghost part following a conformal field theory approach. We use this formulation to solve the VSFT equation of motion for the ghost sector. This part of the paper is based on a new method which allows us to derive known results in a simpler way. In the second part we concentrate on the solution of the VSFT equation of motion for the matter part. We describe the construction of the three strings vertex in the presence of a background B field. We determine a large family of lump solutions, illustrate their interpretation as D-branes and study the low energy limit. We show that in this limit the lump solutions flow toward the so-called GMS solitons.

Paper Structure

This paper contains 26 sections, 235 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A summary of String Field Theory
  3. Vacuum string field theory
  4. Organization of the paper. First part
  5. Three strings vertex and matter Neumann coefficients
  6. Non--zero modes
  7. Zero modes
  8. Ghost three strings vertex and $bc$ Neumann coefficients
  9. Neumann coefficients: definitions and properties
  10. Computation of the coefficients
  11. Two alternatives
  12. Matrix structure
  13. Enlarged Fock space
  14. Solving the ghost equation of motion in VSFT
  15. A comment on the eigenvalues of Neumann coefficients matrices
  16. ...and 11 more sections

Figures (1)

  • Figure 1: The conformal maps from the three unit semidisks to the three-wedges unit disk