D ec 2 00 3 Supersymmetry breaking by Wilson lines in AdS 5

In the Randall-Sundrum compactification of AdS5 with detuned brane tensions, supersymmetry can be spontaneously broken by a non-trivial Wilson line for the graviphoton. The supersymmetry breaking vanishes in the tuned limit. This effect is equivalent to supersymmetry breaking by Scherk-Schwarz boundary conditions. email: bagger@jhu.edu email: redi@pha.jhu.edu In this note we show that in the supersymmetric Randall-Sundrum model [1] with detuned brane tensions [2], supersymmetry can be broken by the Hosotani (or Wilson line) mechanism [3]. The setup is based on five-dimensional anti de-Sitter (AdS5) supergravity, compactified on the orbifold S1/Z2, with branes located at the orbifold fixed points. The brane actions and supersymmetry transformations are adjusted so that the bulk-plus-brane theory is locally supersymmetric. This can always be done provided the magnitudes of the brane tensions do not exceed the tuned value [4]. In five-dimensional AdS supergravity, the dynamical fields are the vielbein eAM , the graviphoton BM , and a symplectic Majorana gravitino ΨMi. When the tensions are tuned, the groundstate metric on each brane is flat, and the distance between the branes is a modulus of the compactification. In the detuned case, the metric on each brane is AdS4, and the distance between the branes is fixed in terms of the brane tensions and the bulk cosmological constant. In each case, the vacuum expectation value (VEV) of B5 is not determined by the classical equations of motion. It is a modulus of the compactification, even in the detuned case. The graviphoton gauges a U(1) subgroup of the flat-space SU(2) automorphism group. Under this U(1), the gravitino is charged, with charge proportional to the AdS5 curvature. As in the ordinary Hosotani mechanism, the VEV of B5 induces a gravitino bilinear in the five-dimensional action. At first glance, one might think that supersymmetry is spontaneously broken. In fact, we will see that a VEV breaks supersymmetry when the branes are not tuned, but does not break supersymmetry when they are. This is related to the fact that the distance between the branes is a modulus in the tuned limit. In ref. [6], using the interval representation of the orbifold, boundary conditions consistent with local supersymmetry were found that spontaneously break global supersymmetry. The boundary conditions were proved to be equivalent to a Scherk-Schwarz twist on the orbifold covering space. In the final part of this paper, we will show that these results are equivalent to the breaking by a Wilson line. We start recalling the action for pureN = 2, D = 5 supergravity with cosmological constant, Sbulk = M 3 5 ∫ dx e5 [ − 1 2 R + 6k − 1 4 FFMN + i 2 Ψ̄MiΓ DNΨKi − 2 k Ψ̄MiΣ ~q · ~σijΨNj + √ 6 4 k BNΨ̄MiΓ ~q · ~σijΨKj − i √ 6 16 FMN ( 2Ψ̄Mi Ψ N i + Ψ̄PiΓ ΨQi ) − 1 6 √ 6 ǫFMNFPQBR ] , (1) Note that the VEV of B5 is gauge invariant for gauge transformations defined on the circle. A non-zero VEV of B5 gives rise to a non-trivial Wilson line for the graviphoton around the fifth dimension. This mechanism is reminiscent of one proposed in [5], where an auxiliary field was used to trigger supersymmetry breaking in flat five-dimensional supergravity.