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

TL;DR

The paper analyzes vacuum structure in massless $N_f$-flavour 2D QED with charge $k$, showing spontaneous breaking of $Z_{kN_f}^{\rm axial}$ to $Z_{N_f}$ and a $k$-fold vacuum degeneracy tied to a mixed 't Hooft anomaly with the $Z_k^{\rm 1-form}$ symmetry. It uses bosonization ( Abelian and non-Abelian) to realize and match the anomaly, and studies mass deformations which introduce a chiral condensate and a theta-dependent vacuum energy, producing a theta-dependent string tension with nontrivial screening properties. The second part maps these field-theory results to a non-supersymmetric string theory setup—an $ar{D1}$-brane near an $O1^-$-plane—where the 2D QED sector governs brane dynamics, yielding repulsive O1$^-$–$\bar{D1}$ potentials at both short and long distances and revealing a non-perturbative $g_s^{1/9}$ contribution to the $(Q,-1)$-string tension. By connecting one-loop Coleman–Weinberg analysis and Möbius-strip amplitudes, the work provides a coherent field-theory/string-theory link and non-perturbative predictions for brane interactions with theta-angle and mass-deformation dependence.

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.

Figures (3)

• Figure 1: A plot of the function $f(\theta)$ for $k=5$ and $N_f=4$ in the range $-2\pi k\le\theta\le 2\pi k$. The dots in the left and right panels are the points at $\theta=2\pi n$ and $\theta=2\pi n+\pi$ ($n\in\hbox{Z}$), respectively. These dots are proportional to the expectation value of the energy density ${\cal E}(\theta+2\pi n)$ for the states $\left|\, \theta+2\pi n\,\right\rangle$ ($n\in\hbox{Z}$) with $\theta=0$ (left) and $\theta=\pi$ (right). The $Q$-string tension $\sigma(Q)$ (with $Q=n\in\hbox{Z}$) is proportional to the height of the dots measured from the lowest one.
• Figure 2: Reconnection of two F-string - $\overline{\rm F}$-string pairs.
• Figure 3: Potential for the $\hbox{O}1^-$-$\overline{\hbox{D ,}}\!\!\!1$ system: $V(Z)=\int_0^{\infty}\frac{dt}{2t^2}e^{-tZ^2}F_O(t)$