[Paper Review] On the Embedding of Space-Time Symmetries into Simple Superalgebras

Riccardo D’Auria, S. Ferrara+1 more|ArXiv.org|Feb 12, 2001

Black Holes and Theoretical Physics27 references11 citations

TL;DR

This paper investigates the embedding of space-time Lorentz and R-symmetry groups into simple orthosymplectic superalgebras for extended supersymmetry in arbitrary dimensions and signatures. It establishes that all physical conformal algebras so(D−2,2) mod 8 are consistently embedded into osp(1|64,R), with a universal so(1,1) grading, and shows that compact R-symmetry is possible only for signatures congruent to D−2 mod 8, unifying M-theory and related phases under a single superalgebra framework.

ABSTRACT

We explore the embedding of Spin groups of arbitrary dimension and signature into simple superalgebras in the case of extended supersymmetry. The R-symmetry, which generically is not compact, can be chosen compact for all the cases that are congruent mod 8 to the physical conformal algebra so($D-2$,2), $D\geq 3$. An $ m{so}(1,1)$ grading of the superalgebra is found in all cases. Central extensions of super translation algebras are studied in this framework.

Motivation & Objective

• To generalize the classification of space-time superalgebras to extended supersymmetry with R-symmetry in arbitrary dimensions and signatures.
• To determine under which conditions the R-symmetry group can be compact, particularly for physical conformal algebras so(D−2,2).
• To show that all extended superalgebras in D≤11, including M, M*, and M′ theories, arise as contractions or subalgebras of osp(1|64,R).
• To establish a universal so(1,1) grading in the superconformal algebra framework across all dimensions and signatures.
• To clarify the role of conjugations and pseudoconjugations in embedding real forms of classical Lie algebras into simple superalgebras.

Proposed method

• Uses conjugation and pseudoconjugation structures on complex spinor spaces to derive real forms of classical Lie algebras, including Spin(s,t) and R-symmetry groups.
• Applies the classification of real spinor representations based on dimension D mod 8 to determine reality types (R, C, H) and real dimensions.
• Constructs graded embeddings of orthosymplectic superalgebras osp(1|2m,R) into sl(2m,R) with an so(1,1) grading, decomposing the algebra into graded components.
• Analyzes the embedding of the superconformal algebra into osp(1|64,R) via contraction and subalgebra relations, particularly for D=10 and D=11.
• Derives the structure of maximal central extensions of supertranslation algebras as the positive-grade part of the superalgebra under the so(1,1) grading.
• Uses table-based classification of real forms of the bosonic part of the superalgebra, distinguishing between compact and non-compact R-symmetry cases.

Experimental results

Research questions

• RQ1 Can the R-symmetry group be compact for all space-time signatures in extended superconformal algebras?
• RQ2 How are the space-time Lorentz groups Spin(s,t) embedded into simple superalgebras for arbitrary D and signature?
• RQ3 What is the role of the so(1,1) grading in organizing the structure of superconformal algebras across dimensions?
• RQ4 To what extent can different M-theory-related superalgebras (M, M*, M′) be unified under a single superalgebra framework?
• RQ5 Under which conditions does the superconformal algebra osp(1|64,R) contain the super Poincaré algebra with maximal central extension?

Key findings

• All physical conformal algebras so(D−2,2) with D≥3 can be embedded into the simple superalgebra osp(1|64,R), which unifies M-theory and related phases.
• A universal so(1,1) grading is found in all cases, decomposing the superalgebra into graded components that include the superconformal and supertranslation algebras.
• Compact R-symmetry is possible only when the space-time signature satisfies (s,t) ≡ (D−2,2) mod 8, which includes the physical D=10 and D=11 cases.
• The super Poincaré algebra with maximal central extension (two- and five-brane charges) arises as a non-semisimple subalgebra of the superconformal algebra.
• For Euclidean signatures (s,t)=(D−1,1), only non-compact R-symmetry groups are allowed, consistent with earlier observations by Zumino and others.
• The real dimension of the spinor representation depends only on D mod 8, with values ranging from 2^{(D−1)/2} to 2^{D/2} depending on the reality type (R, C, H).