Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Compactification in the Lightlike Limit

Simeon Hellerman, Joseph Polchinski

TL;DR

The paper investigates field theories under compactification to a lightlike circle, focusing on whether perturbative amplitudes remain finite as the lightlike limit is approached. It finds that zero modes drive divergences in perturbation theory by coupling to fixed nonzero longitudinal momentum sectors, and that the limit is generally non-smooth unless the theory has special infrared structure; nonperturbative analysis suggests possible finite limits in certain UV-complete theories with infrared fixed points. The authors construct a supersymmetric toy model where the lightlike limit is finite at all orders due to cancellations between bosonic and fermionic zero-mode contributions, illustrating a concrete positive case. They discuss the implications for discrete light-cone quantization and the matrix theory proposal, arguing that reconciling DLCQ with eleven-dimensional physics requires additional ingredients, such as large-N dynamics, to define a consistent continuum limit.

Abstract

We study field theories in the limit that a compactified dimension becomes lightlike. In almost all cases the amplitudes at each order of perturbation theory diverge in the limit, due to strong interactions among the longitudinal zero modes. The lightlike limit generally exists nonperturbatively, but is more complicated than might have been assumed. Some implications for the matrix theory conjecture are discussed.

Compactification in the Lightlike Limit

TL;DR

The paper investigates field theories under compactification to a lightlike circle, focusing on whether perturbative amplitudes remain finite as the lightlike limit is approached. It finds that zero modes drive divergences in perturbation theory by coupling to fixed nonzero longitudinal momentum sectors, and that the limit is generally non-smooth unless the theory has special infrared structure; nonperturbative analysis suggests possible finite limits in certain UV-complete theories with infrared fixed points. The authors construct a supersymmetric toy model where the lightlike limit is finite at all orders due to cancellations between bosonic and fermionic zero-mode contributions, illustrating a concrete positive case. They discuss the implications for discrete light-cone quantization and the matrix theory proposal, arguing that reconciling DLCQ with eleven-dimensional physics requires additional ingredients, such as large-N dynamics, to define a consistent continuum limit.

Abstract

We study field theories in the limit that a compactified dimension becomes lightlike. In almost all cases the amplitudes at each order of perturbation theory diverge in the limit, due to strong interactions among the longitudinal zero modes. The lightlike limit generally exists nonperturbatively, but is more complicated than might have been assumed. Some implications for the matrix theory conjecture are discussed.

Paper Structure

This paper contains 4 sections, 22 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scalar Field Theory
  3. The $+-0$ Model
  4. Discussion

Figures (2)

  • Figure 1: One loop scalar graph. Lines are labeled by $p_-R$, flowing in the direction of the arrow. The dangerous term is $n=0$.
  • Figure 2: a) A divergent scalar graph. Lines are labeled by $p_-R$ and by the superfield. Other divergent graphs are obtained by replacing a pair of $+$ superfields with $-$. b) Canceling fermion loop.