[Paper Review] Flux, Supersymmetry and M theory on 7-manifolds

B. S. Acharya, Bill Spence|ArXiv.org|Jul 26, 2000

Black Holes and Theoretical Physics11 references43 citations

TL;DR

This paper investigates M-theory compactifications on 7-manifolds, showing that unbroken supersymmetry in the low-energy limit requires the warp factor to be trivial, the four-form field strength to vanish, and the internal 7-manifold to possess G₂ holonomy. The result is derived both from direct supergravity analysis and via Gukov’s proposed superpotential, confirming consistency between the two approaches and highlighting the necessity of G₂ geometry for supersymmetric M-theory compactifications to four dimensions.

ABSTRACT

Various aspects of low energy M theory compactified to four dimensions are considered. If the supersymmetry parameter is parallel in the unwarped metric, then supersymmetry requires that the warp factor is trivial, the background four-form field strength is zero and that the internal 7-manifold has $G_2$ holonomy (we assume the absence of boundaries and other impurities). A proposal of Gukov - extended here to include M2-brane domain walls - for the superpotential of the compactified theory is shown to yield the same result. Finally, we make some speculative remarks concerning higher derivative corrections and supersymmetry breaking.

Motivation & Objective

• To determine the conditions under which M-theory compactified on a warped product of 4D Minkowski space and a 7-manifold preserves unbroken supersymmetry.
• To analyze the interplay between the four-form field strength, warp factor, and holonomy structure in eleven-dimensional supergravity backgrounds.
• To test the consistency of Gukov’s proposed superpotential for M-theory compactifications with the standard supergravity equations of motion.
• To explore the implications of higher derivative corrections for supersymmetry breaking in M-theory compactifications.

Proposed method

• The analysis uses a warped product metric ansatz: $ g_{11}(x,y) = riangle^{-1}(y)(g_4(x) + g_7(y)) $, with $ g_4 $ Lorentzian and $ g_7 $ Euclidean.
• Supersymmetry is imposed via the gravitino variation equation $ abla_M heta + Z_M heta = 0 $, where $ Z_M $ depends on the four-form field strength $ G_{PQRS} $.
• The four-form field strength is assumed to have non-zero components only in the 4D and 7D sectors: $ G_{etaetaetaeta} = 3m ilde{eta}etaetaeta $ and $ G_{mnpq} eq 0 $, with $ m $ constant.
• The spinor connection is decomposed using gamma-matrix structures $ ilde{ heta} = heta heta $, with $ heta $ a spinor on $ M^4 $ and $ M^7 $, and the 11D gamma matrices split as $ ilde{ heta}_\mu = \gamma_\mu \otimes \mathbb{1} $, $ ilde{ heta}_m = \gamma_5 \otimes \gamma_m $.
• The calculation uses identities involving $ G_n $, $ G $, and $ G_{pqrs} $, such as $ \gamma^{mn} G_m G_n = -7G_n^2 - G^2 $, to simplify the supersymmetry conditions.
• The consistency of Gukov’s superpotential proposal with the supergravity results is verified by showing that it leads to the same constraints: trivial warp factor, vanishing $ G $, and $ G_2 $ holonomy.

Experimental results

Research questions

• RQ1 Under what conditions does M-theory compactification on a warped $ M^4 \times M^7 $ background preserve unbroken supersymmetry?
• RQ2 Does Gukov’s proposed superpotential for $ M $-theory compactifications on $ G_2 $-holonomy 7-manifolds reproduce the standard supergravity conditions for supersymmetry?
• RQ3 Can non-trivial warp factors and non-zero four-form field strengths coexist in supersymmetric M-theory compactifications on 7-manifolds?
• RQ4 What role do higher derivative corrections play in potentially allowing non-trivial $ G $ and warp factors while preserving supersymmetry?
• RQ5 Can spacetime topology alone break supersymmetry in M-theory compactifications, even when parallel spinors exist?

Key findings

• Unbroken supersymmetry in M-theory compactified on a warped $ M^4 \times M^7 $ background requires the warp factor $ \triangle(y) $ to be trivial.
• The four-form field strength $ G_{PQRS} $ must vanish identically in such supersymmetric backgrounds.
• The internal 7-manifold $ M^7 $ must possess $ G_2 $ holonomy for supersymmetry to be preserved under the given assumptions.
• Gukov’s proposed superpotential for $ G_2 $-compactifications yields the same constraints as the supergravity analysis, confirming consistency.
• Higher derivative corrections may allow non-trivial $ G $ and warp factors while preserving supersymmetry, suggesting a possible mechanism for low-energy supersymmetry breaking.
• Spacetime topology, particularly non-trivial 8-cycles, may lead to supersymmetry breaking in the low-energy effective theory, even if the geometry admits parallel spinors.