Abstract
We study holographically Lifshitz-scaling theories with broken symmetries. In order to do this, we set up a bulk action with a complex scalar and a massless vector on a background which consists in a Lifshitz metric and a massive vector. We first study separately the complex scalar and the massless vector, finding a similar pattern in the twopoint functions that we can compute analytically. By coupling the probe complex scalar to the background massive vector we can construct probe actions that are more general than the usual Klein-Gordon action. Some of these actions have Galilean boost symmetry. Finally, in the presence of a symmetry breaking scalar profile in the bulk, we reproduce the expected Ward identities of a Lifshitz-scaling theory with a broken global continuous symmetry. In the spontaneous case, the latter imply the presence of a gapless mode, the Goldstone boson, which will have dispersion relations dictated by the Lifshitz scaling.
Highlights
We will consider the dynamics of a massless vector and a charged scalar in the bulk
In order to do this, we set up a bulk action with a complex scalar and a massless vector on a background which consists in a Lifshitz metric and a massive vector
In the presence of a symmetry breaking scalar profile in the bulk, we reproduce the expected Ward identities of a Lifshitz-scaling theory with a broken global continuous symmetry. The latter imply the presence of a gapless mode, the Goldstone boson, which will have dispersion relations dictated by the Lifshitz scaling
Summary
The basic setup of a free scalar field in Lifshitz spacetime has been considered previously in the literature [5,6,7] This simple case already displays some peculiar features with respect to the AdS version, namely the appearance of imaginary poles in the two-point function. The locations of the poles accumulate as one approaches the origin along the negative imaginary axis One may expect this behavior to be a consequence of the singularity at r = ∞ where for z > 1 Lifshitz spacetimes display diverging tidal forces [5]. For other considerations on the appearance of imaginary poles in Lifshitz holography, together with a field theoretic perspective on the issue, see [12, 14, 19]
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