Anisotropic Compactification of Nonrelativistic M-Theory

TL;DR

This paper explores a nonrelativistic limit of M-theory with anisotropic compactification, revealing new dualities, geometric structures, and a nonrelativistic brane action that makes SL(2,Z) symmetry explicit.

Contribution

It introduces a frame covariant formalism for anisotropic toroidal compactification of nonrelativistic M-theory, connecting it to nonrelativistic type IIB string theory and manifesting SL(2,Z) symmetry.

Findings

Developed a geometric interpretation of SL(2,Z) duality in nonrelativistic string theory.

Constructed an action for a nonrelativistic M5-brane and its compactification to a D3-brane.

Revealed polynomial transformation properties of background fields under dualities.

Abstract

We study a decoupling limit of M-theory where the three-form gauge potential becomes critical. This limit leads to nonrelativistic M-theory coupled to a non-Lorentzian spacetime geometry. Nonrelativistic M-theory is U-dual to M-theory in the discrete light cone quantization, a non-perturbative approach related to the Matrix theory description of M-theory. We focus on the compactification of nonrelativistic M-theory over a two-torus that exhibits anisotropic behaviors due to the foliation structure of the spacetime geometry. We develop a frame covariant formalism of the toroidal geometry, which provides a geometrical interpretation of the recently discovered polynomial realization of SL(2,Z) duality in nonrelativistic type IIB superstring theory. We will show that the nonrelativistic IIB string background fields transform as polynomials of an effective Galilean "boost velocity" on the two-torus. As an application, we construct an action principle describing a single M5-brane in nonrelativistic M-theory and study its compactification over the anisotropic two-torus. This procedure leads to a D3-brane action in nonrelativistic IIB string theory that makes the SL(2,Z) invariance manifest in the polynomial realization.