[Paper Review] Erice lectures on "The status of local supersymmetry"

M. J. Duff|ArXiv.org|Mar 15, 2004

Black Holes and Theoretical Physics80 references6 citations

TL;DR

This paper reviews the status of local supersymmetry in supergravity, string theory, and M-theory, focusing on D=11 supergravity as the low-energy limit of M-theory. It investigates how many supersymmetries M-theory vacua can preserve using generalized holonomy and G-structures, showing that the number of unbroken supersymmetries corresponds to the number of singlets in spinor decompositions under holonomy groups, with key examples like M2 and M5 branes yielding eight preserved supersymmetries.

ABSTRACT

In the first lecture we review the current status of local supersymmetry. In the second lecture we focus on D=11 supergravity as the low-energy limit of M-theory and pose the questions: (1) What are the D=11 symmetries? (2) How many supersymmetries can M-theory vacua preserve?

Motivation & Objective

• To assess the current status of local supersymmetry in supergravity, superstrings, and M-theory.
• To determine the number of supersymmetries preserved by M-theory vacua using geometric and group-theoretic methods.
• To explore the role of generalized holonomy and G-structures in classifying supersymmetric solutions in D=11 supergravity.
• To address the challenge of supersymmetry without supersymmetry, where supergravity approximations may miss full M-theory supersymmetry.
• To propose a framework for extending supergravity-based classifications to the full M-theory, including higher-order corrections and non-perturbative effects.

Proposed method

• Analyzes D=11 supergravity as the low-energy limit of M-theory, using the 32-component gravitino spinor representation of SL(32,R).
• Applies generalized holonomy to classify supersymmetric vacua by decomposing the 32-spinor under the structure group H ⊂ SL(32,R).
• Uses transverse holonomy (e.g., SO(5) for M5, SO(8) for M2) to compute the number of unbroken supersymmetries via singlet counting.
• Considers specific brane solutions (M2, M5, M-wave) and their intersections, computing decompositions under subgroups like SO(16)×SO(16) and USp(8).
• Compares generalized holonomy with G-structures, suggesting the latter may be better suited for finding solutions.
• Considers signatures (9,2) and (6,5) for M’ and M* theories, noting that group structures differ but counting principles may persist.

Experimental results

Research questions

• RQ1 What are the spacetime symmetries of D=11 supergravity and how do they constrain M-theory vacua?
• RQ2 How many supersymmetries can M-theory vacua preserve, and how can this be computed from geometric data?
• RQ3 Can generalized holonomy fully classify supersymmetric solutions in D=11 supergravity, especially for plane waves and non-compact spaces?
• RQ4 How do T-duality and S-duality relate to supersymmetry counting, and in what cases might supergravity miss the true number of unbroken supersymmetries?
• RQ5 To what extent can the counting of unbroken supersymmetries via spinor decompositions under H ⊂ SL(32,R) be extended to the full M-theory, including non-perturbative and higher-order corrections?

Key findings

• The M2-brane solution preserves eight supersymmetries, arising from the decomposition of the 32-spinor under SO(16), yielding eight singlets.
• The M5-brane solution also preserves eight supersymmetries, with the 32-spinor decomposing as 2(8,1,1)+2(1,2,2)+8(1,1,1) under SO(8)×SU(2)×SU(2).
• For the M5/MW intersection, the decomposition under SO(5)×USp(8) yields eight unbroken supersymmetries, consistent with a 2/9 split.
• The plane wave solution fails the generalized holonomy analysis due to the absence of compact subgroups in R^9, indicating limitations of the method.
• The maximal compact subgroup of the generalized holonomy group H is often sufficient to determine the number of preserved supersymmetries, as seen in the M2/MK/MK solution.
• Supersymmetry without supersymmetry can occur in supergravity approximations, where non-perturbative states like solitons or winding modes may restore full supersymmetry not visible in the effective theory.