[Paper Review] Dimensional Enhancement via Supersymmetry

Michael Faux, Kevin Iga+1 more|ArXiv.org|Jul 21, 2009

Radio Astronomy Observations and Technology1 references5 citations

TL;DR

This paper introduces a formalism to determine which one-dimensional supersymmetric quantum mechanics models can be 'enhanced' to higher-dimensional field theories by encoding higher-dimensional spin and Lorentz structure within one-dimensional supersymmetry algebras. Using algebraic criteria derived from $rak{spin}(1,D-1)$ invariance and extended supercharge rules, the authors identify 'shadow' multiplets and introduce 'phantom' fields to realize higher-dimensional gauge invariance, successfully reproducing known results via one-dimensional reasoning alone.

ABSTRACT

We explain how the representation theory associated with supersymmetry in diverse dimensions is encoded within the representation theory of supersymmetry in one time-like dimension. This is enabled by algebraic criteria, derived, exhibited, and utilized in this paper, which indicate which subset of one-dimensional supersymmetric models describe "shadows" of higher-dimensional models. This formalism delineates that minority of one-dimensional supersymmetric models which can "enhance" to accommodate extra dimensions. As a consistency test, we use our formalism to reproduce well-known conclusions about supersymmetric field theories using one-dimensional reasoning exclusively. And we introduce the notion of "phantoms" which usefully accommodate higher-dimensional gauge invariance in the context of shadow multiplets in supersymmetric quantum mechanics.

Motivation & Objective

• To determine which one-dimensional supersymmetric quantum mechanics models can be enhanced into higher-dimensional supersymmetric field theories.
• To develop algebraic criteria that identify when a 1D model is a 'shadow' of a higher-dimensional supersymmetric theory.
• To reconstruct higher-dimensional spin and Lorentz representations from one-dimensional supersymmetry algebras.
• To introduce 'phantom' fields that realize higher-dimensional gauge invariance within 1D shadow multiplets.
• To demonstrate that known results in higher-dimensional supersymmetry can be derived using only one-dimensional reasoning and representation theory.

Proposed method

• Derive algebraic conditions under which a one-dimensional supersymmetric model can be enhanced to a higher-dimensional theory by requiring compatibility with $rak{spin}(1,D-1)$ invariance.
• Use extended one-dimensional supercharge transformation rules to encode the structure of higher-dimensional spinor representations.
• Apply the Majorana basis for $ m{Cl}(1,D-1)$ Clifford algebras to express gamma matrices, charge conjugation matrices, and $G$-matrices explicitly.
• Introduce 'phantom' fields as auxiliary degrees of freedom that restore higher-dimensional gauge invariance in 1D shadow multiplets.
• Construct boost and rotation operators (${ m f B}^a$, ${ m f R}_a$) from gamma matrices to realize the Lorentz algebra in 1D.
• Use the Lorentz algebra $[M_{ ho au}, M^{ ueta}] = ext{...}$ to ensure consistency of the enhanced higher-dimensional structure.

Experimental results

Research questions

• RQ1 Which one-dimensional supersymmetric quantum mechanics models can be enhanced to describe consistent higher-dimensional field theories?
• RQ2 How can the spinor representation content of a higher-dimensional supersymmetric theory be reconstructed from its one-dimensional shadow?
• RQ3 What algebraic constraints must a 1D supersymmetric model satisfy to be a shadow of a higher-dimensional supersymmetric theory?
• RQ4 How can higher-dimensional gauge invariance be realized in a 1D framework where such invariance is not manifest?
• RQ5 Can known results in higher-dimensional supersymmetry (e.g., on $N=4$ SYM or 10D supergravity) be derived using only one-dimensional reasoning?

Key findings

• A subset of one-dimensional supersymmetric models—those satisfying specific algebraic criteria—can be enhanced to higher-dimensional theories by restoring $rak{spin}(1,D-1)$ invariance in the supercharge structure.
• The formalism successfully reproduces well-known results in higher-dimensional supersymmetry using only one-dimensional supermultiplets and their transformation rules.
• The introduction of 'phantom' fields allows for the realization of higher-dimensional gauge invariance within 1D shadow multiplets, even though such invariance is not manifest in the 1D theory.
• The $G$-matrices $G^a = 2{ m f B}^a$ are symmetric, real, and traceless, and satisfy the Lorentz algebra when combined with boost and rotation operators.
• The boost and rotation operators ${ m f B}^a$ and ${ m f R}_a$ satisfy the Lorentz algebra $[ m{f B}^a, m{f B}^b] = ilde{ ho}^{abc} m{f R}_c$, etc., ensuring consistency of the enhanced higher-dimensional structure.
• The representation theory of 1D supersymmetry, though simpler, encodes the full spin content of higher-dimensional theories via the structure of Adinkra graphs and their associated algebraic constraints.