[Paper Review] New superspaces/algebras for superparticles/strings

Warren Siegel|arXiv (Cornell University)|Jun 8, 2011

Black Holes and Theoretical Physics22 references11 citations

TL;DR

This paper introduces extended superspaces and algebras for superparticles and superstrings by covariantly extending gauge derivatives to include dual coordinates for spin and supersymmetry, enabling prepotentials to appear as undifferentiated potentials. The framework unifies field strengths and prepotentials via dimensional reduction and lightcone analysis, generalizing gauge couplings and extending the superstring's affine Lie algebra with new dual symmetries and compensating gauge structures.

ABSTRACT

We describe covariant derivatives with respect to the coordinates of the full superPoincaré group and dual coordinates, for Yang-Mills and supergravity. The derivatives have engineering dimension running from 0 to 2. Prepotentials appear as potentials for Lorentz derivatives (spin). Their role is clarified in a lightcone analysis, where they also act as compensating gauge parameters and Hertz potentials. Field strengths appear as potentials for dual-coordinate derivatives, until dimensional reduction. These generalizations extend the superstring's affine Lie algebra, and generalize gauge couplings for the superparticle.

Motivation & Objective

• To generalize covariant derivatives in super Yang-Mills and supergravity to include dual coordinates for spin and supersymmetry.
• To clarify the role of prepotentials as undifferentiated potentials in covariant derivatives by introducing extra coordinates.
• To unify field strengths and prepotentials through dimensional reduction and dual derivative structures.
• To extend the superstring's affine Lie algebra by incorporating dual Lorentz and R-symmetry coordinates.
• To provide a covariant first-quantization framework that naturally generates currents and preserves gauge invariance via compensating parameters.

Proposed method

• Introduce dual coordinates for Lorentz and supersymmetry generators via coset construction of the super-Poincaré group.
• Gauge-covariantize derivatives with respect to full super-Poincaré group, including dual derivatives for spin and supersymmetry.
• Use isotropy (coset) constraints to eliminate extra coordinates while preserving gauge invariance and defining spin as a differential operator.
• Apply dimensional reduction to reinterpret field strengths as gauge fields in extra dimensions, establishing duality between prepotentials and field strengths.
• Employ lightcone analysis to derive standard formulations and show how prepotentials emerge naturally in covariant derivatives without solving differential constraints.
• Treat R-symmetry on equal footing with spin via dual coordinates, with R-symmetry arising from dimensional reduction of spin generators.

Experimental results

Research questions

• RQ1 How can prepotentials be consistently introduced as undifferentiated potentials in covariant derivatives for super Yang-Mills and supergravity?
• RQ2 What is the physical role of dual coordinates associated with spin and supersymmetry generators in gauge theories?
• RQ3 How does dimensional reduction relate field strengths to prepotentials and enable a duality between them?
• RQ4 In what way do dual Lorentz and R-symmetry coordinates extend the affine Lie algebra of the superstring?
• RQ5 Can the lightcone formulation be derived covariantly from the extended superspace framework, and does it clarify the structure of self-dual theories?

Key findings

• Prepotentials appear as undifferentiated potentials in covariant derivatives due to the inclusion of dual coordinates, resolving dimensionality issues in their appearance.
• In self-dual 4D Yang-Mills, the usual prepotential emerges automatically in the covariant derivative without solving differential constraints.
• In self-dual (gauged) supergravity, the prepotential with negative engineering dimension acts as a connection for the dual spin derivative.
• Field strengths in the original theory become gauge fields in the extended theory after dimensional reduction, establishing a duality between prepotentials and field strengths.
• The lightcone formulation of 10D super Yang-Mills is derived directly from the covariant framework, suggesting a covariant approach to 4D N=4 super Yang-Mills using complex null directions in R-space.
• The new dual coordinates allow the vierbein and Lorentz connection to be treated on equal footing in gravity, with spin defined as a differential operator via coset constraints.