Annulus Amplitudes and ZZ Branes in Minimal String Theory

TL;DR

This paper investigates the annulus amplitudes in (p,q) minimal string theory, revealing that ZZ branes are localized in strong coupling, and that ZZ-FZZT open strings behave as fermions, providing geometric insights into brane deformations.

Contribution

It offers a new geometric interpretation of annulus amplitudes in terms of Riemann surfaces and clarifies the fermionic nature of ZZ-FZZT open strings in a bosonic theory.

Findings

ZZ branes are localized in the strong-coupling region.

ZZ-FZZT open strings are fermions.

ZZ branes deform specific A-periods of the Riemann surface.

Abstract

We study the annulus amplitudes of (p,q) minimal string theory. Focusing on the ZZ-FZZT annulus amplitude as a target-space probe of the ZZ brane, we use it to confirm that the ZZ branes are localized in the strong-coupling region. Along the way we learn that the ZZ-FZZT open strings are fermions, even though our theory is bosonic! We also provide a geometrical interpretation of the annulus amplitudes in terms of the Riemann surface M_{p,q} that emerges from the FZZT branes. The ZZ-FZZT annulus amplitude measures the deformation of M_{p,q} due to the presence of background ZZ branes; each kind of ZZ-brane deforms only one A-period of the surface. Finally, we use the annulus amplitudes to argue that the ZZ branes can be regarded as "wrong-branch" tachyons which violate the bound \alpha<Q/2.