Abstract
We study the vacuum structure of Nf flavour two-dimensional QED with an arbitrary integer charge k. We find that the axial symmetry is spontaneously broken from {mathbb{Z}}_{k{N}_f} to {mathbb{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 anon-commutative algebra obtained as a central extension of the {mathbb{Z}}_{k{N}_f} discrete axial symmetry and ℤk 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 Nf = 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.
Highlights
Multi flavour Schwinger model.1 the k dependence in the action of 2 dim QED can be eliminated by rescaling the gauge field and the gauge coupling, it enters in the flux quantization condition and the charge k is physically relevant
We argue that the ZkNf discrete axial symmetry is spontaneously broken to ZakxNiafl . (ZNf) and, as a result, there are k distinct vacua, generalizing the result for Nf = 1 given in [1]
We check that all the global symmetry as well as the mixed ’t Hooft anomaly are realized in the bosonized description and the results for Nf = 1 given in [1] are reproduced in a simplified way
Summary
The partition function is invariant only when α π l kNf with l = 1, 2, · · · 2kNf. U(1)A is broken explicitly to (Z2kNf )A by anomaly and the global symmetry G is given by replacing U(1)A/Z2 in (2.4) with (Z2kNf )A/Z2 ≡ ZakxNiafl:. When we choose ξ = lx1/R, it gives a constant shift of A1 as l A1 → A1 + kR Note that this transformation should not be considered as a part of the gauge transformation (2.2) with λ = ξ/k, unless l ∈ kZ, because (2.11) is not compatible with the 2π periodicity of λ in (2.2). The anomaly free part of (Z2kNf )A in such backgrounds is a Z2Nf subgroup, whose elements are given by eiα ∈ U(1)A with α π l Nf with l = 1, 2, · · · , 2Nf. ZakxNiafl(= (Z2kNf )A/Z2) is broken to ZNf by the mixed anomaly. Note that the unbroken subgroup ZNf of ZakxNiafl is equivalent, under the identification (2.7), to the center ZNf of SU(Nf )L (or SU(Nf )R)
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