Evolution of Fixed-End Strings and the Off-Shell Disk Amplitude

TL;DR

This paper derives an exact integral expression for the off-shell amplitude of a fixed-end Bosonic string, revealing issues with covariant subtraction and implications for string theory models beyond one dimension.

Contribution

It provides a new integral formulation for off-shell string amplitudes with fixed ends, highlighting the non-decoupling of the Liouville field and challenges in regularized models.

Findings

Exact integral expression for off-shell string amplitude.

Identification of non-zero amplitude issues due to covariant subtraction.

Recovery of static potential for R > R_c and divergence for R < R_c.

Abstract

An exact integral expression is found for the amplitude of a Bosonic string with ends separated by a fixed distance $R$ evolving over a time $T$ between arbitrary initial and final configurations. It is impossible to make a covariant subtraction of a covariant quantity which would render the amplitude non-zero. It is suggested that this fact (and not the tachyon) is responsible for the lack of a continuum limit of regularized random-surface models with target-space dimension greater than one. It appears consistent, however, to remove this quantity by hand. The static potential of Alvarez and Arvis $V(R)$ is recovered from the resulting finite amplitude for $R>R_{c}$. For $R<R_{c}$, we find $V(R)=-\infty$, instead of the usual tachyonic result. A rotation-invariant expression is proposed for special cases of the off-shell disk amplitude. {\it None} of the finite amplitudes discussed are Nambu or Polyakov functional integrals, except through an unphysical analytic continuation. We argue that the Liouville field does not decouple in off-shell amplitudes, even when the space-time dimension is twenty-six.