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.
Highlights
In [2], a rather different approach towards a quantum theory of membranes is pioneered, which is designed such that its ground-state wavefunction reproduces the partition function of bosonic string theory
This construction introduces a space and time anisotropy in the worldvolume, and the membranes are described by a three-dimensional nonlinear sigma model (NLSM) at a z = 2 Lifshitz point, where z denotes the critical dynamical exponent, implying that the worldvolume degrees of freedom enjoy a quadratic dispersion relation and are fundamentally nonrelativistic
We considered a novel type of two-dimensional NLSMs, defined on a nonrelativistic worldsheet that lacks any local (Lorentzian nor Galilean) boost symmetries
Summary
We construct a NLSM that maps a two-dimensional nonrelativistic worldsheet Σ to a d-dimensional spacetime manifold M equipped with two metric fields. We first define the desired nonrelativistic symmetries of the worldsheet, generically excluding any (Lorentzian nor Galilean) boost transformations. The dynamics of the worldsheet geometry is described by two-dimensional Hořava gravity at a z = 1 Lifshitz point, which is topological. We will couple scalar fields to this two-dimensional nonrelativistic gravity and build up a sigma model that describes classical strings moving in a bimetric geometry
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