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December 2009 The $so(d + 2, 2)$ minimal representation and ambient tractors: the conformal geometry of momentum space
Ashwin Rod Gover, Andrew Waldron
Adv. Theor. Math. Phys. 13(6): 1875-1894 (December 2009).


Tractor calculus is a powerful tool for analyzing Weyl invariance; although fundamentally linked to the Cartan connection, it may also be arrived at geometrically by viewing a conformal manifold as the space of null rays in a Lorentzian ambient space. For dimension d conformally flat manifolds we show that the $(d + 2)$-dimensional Fefferman–Graham ambient space corresponds to the momentum space of a massless scalar field. Hence on the one hand the null cone parameterizes physical momentum excitations, while on the other hand, null rays correspond to points in the underlying conformal manifold. This allows us to identify a fundamental set of tractor operators with the generators of conformal symmetries of a scalar field theory in a momentum representation. Moreover, these constitute the minimal representation of the non-compact conformal Lie symmetry algebra of the scalar field with Kostant–Kirillov dimension $d + 1$. Relaxing the conformally flat requirement, we find that while the conformal Lie algebra of tractor operators is deformed by curvature corrections, higher relations in the enveloping algebra corresponding to the minimal representation persist. We also discuss potential applications of these results to physics and conformal geometry.


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Ashwin Rod Gover. Andrew Waldron. "The $so(d + 2, 2)$ minimal representation and ambient tractors: the conformal geometry of momentum space." Adv. Theor. Math. Phys. 13 (6) 1875 - 1894, December 2009.


Published: December 2009
First available in Project Euclid: 17 August 2010

zbMATH: 1208.81098
MathSciNet: MR2678999

Rights: Copyright © 2009 International Press of Boston

Vol.13 • No. 6 • December 2009
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