Vacuum structure of charge k two-dimensional QED and dynamics of an anti D-string near an O1 − -plane

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.