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Ghost Structure and Closed Strings in Vacuum String Field Theory

Davide Gaiotto, Leonardo Rastelli, Ashoke Sen, Barton Zwiebach

TL;DR

$We propose a canonical purely ghost kinetic term for Vacuum String Field Theory (VSFT) given by a midpoint insertion ${\cal Q} = (c(i)-c(-i))/(2i)$, leading to an infinite normalization that requires careful regularization. Using Siegel-gauge level expansion and BCFT techniques, the ghost sector reduces to projector equations, yielding the twisted sliver and a new class of butterfly surface states as dominant solutions; gauge-invariant open-closed operators $O_V(\Psi)$ provide a route to pure closed-string amplitudes within VSFT, and a regulated action with a finite energy brane solution is constructed via ${\cal Q}$ replaced by $c_0(1+a^{-1}L_0)$. The analysis shows that closed-string moduli emerge from open-string moduli through a minimal-area construction, and that the regulated, level-truncated VSFT solutions converge to rank-one projectors, with the butterfly state emerging as a simple, well-defined projector. Together, these results connect OSFT tachyon condensation to a consistent VSFT framework with a clear ghost sector and a mechanism for accessing closed-string physics from open-string data.$

Abstract

We complete the construction of vacuum string field theory by proposing a canonical choice of ghost kinetic term -- a local insertion of the ghost field at the string midpoint with an infinite normalization. This choice, supported by level expansion studies in the Siegel gauge, allows a simple analytic treatment of the ghost sector of the string field equations. As a result, solutions are just projectors, such as the sliver, of an auxiliary CFT built by combining the matter part with a twisted version of the ghost conformal theory. Level expansion experiments lead to surprising new projectors -- butterfly surface states, whose analytical expressions are obtained. With the help of a suitable open-closed string vertex we define open-string gauge invariant operators parametrized by on-shell closed string states. We use regulated vacuum string field theory to sketch how pure closed string amplitudes on surfaces without boundaries arise as correlators of such gauge invariant operators.

Ghost Structure and Closed Strings in Vacuum String Field Theory

TL;DR

{\cal Q} = (c(i)-c(-i))/(2i)O_V(\Psi){\cal Q}c_0(1+a^{-1}L_0)

Abstract

We complete the construction of vacuum string field theory by proposing a canonical choice of ghost kinetic term -- a local insertion of the ghost field at the string midpoint with an infinite normalization. This choice, supported by level expansion studies in the Siegel gauge, allows a simple analytic treatment of the ghost sector of the string field equations. As a result, solutions are just projectors, such as the sliver, of an auxiliary CFT built by combining the matter part with a twisted version of the ghost conformal theory. Level expansion experiments lead to surprising new projectors -- butterfly surface states, whose analytical expressions are obtained. With the help of a suitable open-closed string vertex we define open-string gauge invariant operators parametrized by on-shell closed string states. We use regulated vacuum string field theory to sketch how pure closed string amplitudes on surfaces without boundaries arise as correlators of such gauge invariant operators.

Paper Structure

This paper contains 13 sections, 83 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction and Summary
  2. The Proposal for a Ghost Kinetic Operator
  3. Ghost kinetic operator and gauge structure
  4. Possible origin of a singular ${\cal Q}$
  5. Action of ${\cal Q}$ on the identity state
  6. Algebraic Analysis of the Classical Equations
  7. BCFT Analysis of Classical Equations of Motion
  8. Twisted Ghost Conformal Theory
  9. Relating Star Products and the analytic solution
  10. The twisted sliver state from CFT and a comparison
  11. Regularizing the VSFT action
  12. The proposal for regulated gauge fixed VSFT
  13. Level truncation analysis

Figures (1)

  • Figure 1: This figure shows the plot of the function $a^3 f(a,L)$, computed at level $(L,2L)$ approximation, as a function of $a$. Starting from the topmost graph, the six continuous curves correspond to $L=2$, 4, 6, 8, 10 and 12 respectively. The lowermost dotted curve is an $L =\infty$ extrapolation of the data obtained with a fit of the form $a_0 + a_1/L + a_2/L^2 + a_3/L^3$.