Yang-Mills algebra

Michel Dubois-Violette (1) , Alain Connes (2, 3, 4)

1 LPT - Laboratoire de Physique Théorique d'Orsay [Orsay]
2 IHES - Institut des Hautes Études Scientifiques
3 Vanderbilt University [Nashville]
4 Collège de France - Chaire Analyse et géométrie

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang-Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Gorenstein and Koszul of global dimension 3 but except for s=1 (i.e. in the 2-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin-Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincare series of this algebra A and for the dimension in degree n of the graded Lie algebra of which A is the universal enveloping algebra. In the 4-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie-algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.

Domaines: Algèbres quantiques [math.QA]; K-théorie et homologie [math.KT]; Physique mathématique [math-ph]; Physique mathématique [math-ph]; Physique des Hautes Energies - Théorie [hep-th]

Michel Dubois-Violette, Alain Connes. Yang-Mills algebra. Letters in Mathematical Physics, 2002, 61, pp.149-158. ⟨hal-00013415⟩