Abstract
Nonrelativistic string theory is a self-contained corner of string theory, with its string spectrum enjoying a Galilean-invariant dispersion relation. This theory is unitary and ultraviolet complete, and can be studied from first principles. In these notes, we focus on the bosonic closed string sector. In curved spacetime, nonrelativistic string theory is defined by a renormalizable quantum nonlinear sigma model in background fields, following certain symmetry principles that disallow any deformation towards relativistic string theory. We review previous proposals of such symmetry principles and propose a modified version that might be useful for supersymmetrizations. The appropriate target-space geometry determined by these local spacetime symmetries is string Newton-Cartan geometry. This geometry is equipped with a two-dimensional foliation structure that is restricted by torsional constraints. Breaking the symmetries that give rise to such torsional constraints in the target space will in general generate quantum corrections to a marginal deformation in the worldsheet quantum field theory. Such a deformation induces a renormalization group flow towards sigma models that describe relativistic strings.
Highlights
That are conjugate to string windings, and they are responsible for the consistency and salient features in nonrelativistic string theory
Nonrelativistic string theory is defined by a renormalizable quantum nonlinear sigma model in background fields, following certain symmetry principles that disallow any deformation towards relativistic string theory
This geometry is equipped with a two-dimensional foliation structure that is restricted by torsional constraints
Summary
We start with collecting ingredients in nonrelativistic string theory that will be essential in later discussions. Nonrelativistic string theory is defined on a two-dimensional Riemann surface Σ that acts as the worldsheet, parametrized by σα = (τ, σ) and equipped with a worldsheet metric hαβ. Which are solved by X = X(τ + iσ) and X = X(τ − iσ) These constraints are responsible for salient features of nonrelativistic string theory, including a string spectrum that enjoys a Galilean-invariant dispersion relation and intriguing localization theorems in the moduli space [2]. Another direct consequence of these one-form fields is that the free theory (2.2) is invariant under an infinite number of spacetime isometries [22]. These transformations form the extended Galilean symmetry algebra that contains two copies of the Witt algebra [22]
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