# Vacuum Structure of Charge k Two-Dimensional QED and Dynamics of an Anti D-String Near an O1- plane

## TL;DR

This paper explores the vacuum structure of two-dimensional QED with arbitrary charge, revealing spontaneous symmetry breaking and multiple vacua, and models the system using string theory with anti D-strings near an O1-plane, analyzing their interactions.

## Contribution

It provides an explicit construction of multiple vacua in charge k two-dimensional QED and connects this to a string theory realization involving anti D-strings and orientifold planes.

## Key findings

- Axial symmetry breaks from Z_(kN_f) to Z_(N_f) with k degenerate vacua.
- Potential calculations show repulsion between anti D-strings and O1-planes at various couplings.
- Non-perturbative potential for (Q,-1) strings near O1-plane is obtained.

## Abstract

We study the vacuum structure of N_f flavour two-dimensional QED with an arbitrary integer charge k. We find that the axial symmetry is spontaneously broken from Z_(kN_f) to Z_(N_f) due to the non-vanishing condensate of a flavour singlet operator, resulting in k degenerate vacua. An explicit construction of the k vacua is given by using a non-commutative algebra obtained as a central extension of the Z_(kN_f) discrete axial symmetry and Z_k 1-form (center) symmetry, which represents the mixed 't Hooft anomaly between them. We then give a string theory realization of such a system with k=2 and N_f=8 by putting an anti D-string in the vicinity of an orientifold O1- plane and study its dynamics using the two-dimensional gauge theory realized on it. We calculate the potential between the anti D-string and the O1- plane and find repulsion in both weak and strong coupling regimes of the two dimensional gauge theory, corresponding to long and short distances, respectively. We also calculate the potential for the (Q,-1) string (the bound state of an anti D-string and Q fundamental strings) located close to the O1- plane. The result is non-perturbative in the string coupling.