Abstract
Inspired by the recent advances in multiple M2-brane theory, we consider the generalizations of Nahm equations for arbitrary p-algebras. We construct the topological p-algebra quantum mechanics associated to them and we show that this can be obtained as a truncation of the topological p-brane theory previously studied by the authors. The resulting topological p-algebra quantum mechanics is discussed in detail and the relation with the M2–M5 system is pointed out in the p=3 case, providing a geometrical argument for the emergence of the 3-algebra structure in the Bagger–Lambert–Gustavsson theory.
Highlights
The appearance of new classes of gauge theories stimulated a growing interest in p-algebras as possible generalizations of the standard Lie algebras in the description of gauge interactions
A crucial input for this construction came from the study of the M2-M5 system in the Basu-Harvey’s work [3] where an equation describing the BPS bound state of multiple M2-branes ending on an M5 was formulated
The Basu-Harvey equation is a generalization of the Nahm equation [27] involving an algebraic structure modeled on the Nambu bracket [28]. p-algebras can be obtained as linearizations of Nambu algebras [32]
Summary
The appearance of new classes of gauge theories stimulated a growing interest in p-algebras as possible generalizations of the standard Lie algebras in the description of gauge interactions. In a series of papers [11, 12, 13], the authors formulated a class of topological theories for p-branes These are based on the realization of p-brane instantons wrapping calibrated cycles as solution of BPS bounds for the Nambu-Goto action. For p = 3 we will show that the discretization of the 3-brane instanton coincides with the generalized Basu-Harvey equation as studied in [8, 24] This corresponds to M2-M5 system compactified on a Spin(7) holonomy manifold, with the M5 wrapping a calibrated Cayley four-cycle. From the M2-brane viewpoint, this can be recovered as the large N limit of the generalized Basu-Harvey equation This provides a geometrical argument for the emergence of the 3-algebra structure in the Bagger-Lambert theory. In order to make this paper more readable we collect in Appendix A some facts about the cross vector products and in Appendix B we review the Nambu brackets and (Nambu-Lie) p-algebras
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