Strings in bimetric spacetimes

Abstract

We put forward a two-dimensional nonlinear sigma model that couples (bosonic) matter fields to topological Hořava gravity on a nonrelativistic worldsheet. In the target space, this sigma model describes classical strings propagating in a curved spacetime background, whose geometry is described by two distinct metric fields. We evaluate the renormalization group flows of this sigma model on a flat worldsheet and derive a set of beta-functionals for the bimetric fields. Imposing worldsheet Weyl invariance at the quantum level, we uncover a set of gravitational field equations that dictate the dynamics of the bimetric fields in the target space, where a unique massless spin-two excitation emerges. When the bimetric fields become identical, the sigma model gains an emergent Lorentz symmetry. In this single metric limit, the beta-functionals of the bimetric fields reduce to the Ricci flow equation that arises in bosonic string theory, and the bimetric gravitational field equations give rise to Einstein’s gravity.

Aspects of Nonrelativistic Strings

“…This limit gives rise to non-vibrating Galilean strings described by a NR worldsheet, propagating in a target-space geometry that is Newton-like, equipped with a codimension-one foliation structure. For another example, sigma models with various Lifshitz scalings have been applied to construct the worldvolume theories for membranes [123] and strings [124], in the absence of any worldvolume (Lorentzian nor Galilean) boost symmetries. Such endeavours beyond the relativistic framework may eventually lead to an alternative route towards the unification of different string theories, as well as a larger class of holographic dualities.…”

Renormalization of supersymmetric Lifshitz sigma models

“…Disregarding the mass term, which is relevant around the z = 2 Lifshitz point and becomes suppressed in the UV, the target space corresponding to the worldvolume action (3.4) develops an intriguing bimetric feature. This is reminiscent of the two-dimensional Lifshitz sigma model considered in [29], which is in contrast purely bosonic.…”

“…This brings interesting challenges to the evaluation of the beta-functions of Lifshitz sigma models. In section 3.2, we will generalize the covariant heat kernel method in [29] to incorporate supersymmetry, and then apply this method to compute the one-loop effective action for the Lifshitz sigma model defined by the action (3.13).…”

“…In connection to the studies of supermembranes, we are ultimately interested in supersymmetric sigma models, which bring significant simplifications in comparison to the pure bosonic case. The three-dimension N = 1 supersymmetric sigma model around a z = 2 Lifshitz point has been introduced in [27,28] (also see [29] for a review), which is power-counting renormalizable. The relevant target-space geometry is Lorentzian and has two independent metrics, but there is no extra higher-form tensorial structure as in the bosonic case.…”

“…In the deep IR, a relevant deformation controlled by the coupling constant associated with the worldvolume speed of light dominates, which drives the theory toward the relativistic fixed point satisfying a z = 1 Lifshitz scaling. We will generalize the heat kernel method in [29,30] to supersymmetric field theories on a foliated spacetime manifold, and use it to study the RG flows of the supersymmetric Lifshitz sigma model. This will form an essential step toward determining whether such a worldvolume theory is qualified…”