Basics of M‐theory

Abstract

This is a review article of eleven dimensional supergravity in which we present all necessary calculations, namely the Noether procedure, the equations of motion (without neglecting the fermions), the Killing spinor equation, as well as some simple and less simple supersymmetric solutions to this theory. All calculations are printed in much detail and with explicit comments as to how they were done. Also contained is a simple approach to Clifford algebras to prepare the grounds for the harder calculations in spin space and Fierz identities.

Bilinear Forms and Fierz Identities for Real Spin Representations

“…A more specialized version of the above tensor is used in [3] as the starting point for their Fierz identities. Let ·, · = (·, ·) + or (·, ·) − and Γ = γ p or γ q accordingly.…”

“…Since T ± is a fruitful object (its complex version is used in [1] and a more specialized version is used for Fierz identities in [3]) this is seen as a hint that the bilinear forms induced by γ p and γ q are important structures.…”

“…Like those in [2], our Fierz identities are relations among spinors as opposed to spinor one-forms, which are the objects of interest in [3] and [6]. However, our identities are more general than those in [2]: we have four-spinor identities instead of just one-spinor identities and our constructions work in arbitrary dimension and signature.…”

“…In the next section we define two natural bilinear forms on the space of spinors and then examine how we can use these to relate spinors to elements of the exterior algebra. Finally, in the third section, we derive the various identities, considering the three cases where a Clifford algebra is isomorphic to a full matrix algebra over R, C, or H. We also consider two cases of special interest to physics: the Clifford algebras associated with signature (1,3) and (10, 1). We show that our identities reduce to some familiar Fierz identities upon specialization from four spinors to one spinor.…”

“…We have thus reduced showing that Ω is surjective (and therefore that M is an involutory matrix) for any corner algebra to checking it for signature (8,0) and the corner signatures in dimensions less than 8: (1,0),(2,1), (3,2), (4,3), and (0,7). The map Ω is indeed surjective in these cases, as we have verified computationally in Maple.…”