[Paper Review] On Realising N=1 Super Yang-Mills in M theory

B. S. Acharya|ArXiv.org|Nov 10, 2000

Black Holes and Theoretical Physics11 references126 citations

TL;DR

This paper proposes that N=1 super Yang-Mills theory in four dimensions arises as the low-energy limit of M-theory compactified on G₂-holonomy spaces with ADE singularities fibered over 3-cycles. It demonstrates that fractional M2-brane instantons generate the superpotential, identifies QCD strings and domain walls in the M-theory background, and proposes a gravity dual to the IIA theory on the resolved conifold with RR flux, linking M-theory on a Lens space to the strongly coupled limit of D6-brane configurations.

ABSTRACT

Pure N=1 super Yang-Mills theory can be realised as a certain low energy limit of M theory near certain singularities in $G_2$-holonomy spaces. For SU(n) and SO(2n) gauge groups these $M$ theory backgrounds can be regarded as strong coupling limits of wrapped D6-brane configurations in Type IIA theory on certain non-compact Calabi-Yau spaces such as the deformed conifold. Various aspects of such realisations are studied including the generation of the superpotential, domain walls, QCD strings and the relation to recent work of Vafa. In the spirit of this recent work we propose a `gravity dual' of M theory near these singularities.

Motivation & Objective

• To establish a strong coupling limit of Type IIA D6-brane configurations on non-compact Calabi-Yau spaces as an M-theory realization of N=1 super Yang-Mills.
• To provide a rigorous derivation of the superpotential via fractional M2-brane instantons in M-theory on singular G₂-manifolds.
• To identify and characterize domain walls and QCD strings in the M-theory background as dual to BPS states in the dual IIA theory.
• To propose a gravity dual of the IIA theory on the resolved conifold with RR 2-form flux, identifying it with M-theory on a Lens space with G₂-holonomy.
• To extend the correspondence to gauge groups SO(2n) and E_n by generalizing the ADE subgroup construction.

Proposed method

• Constructs M-theory backgrounds as singular G₂-holonomy 7-manifolds fibered over 3-cycles M, with fibers R⁴/Γ for ADE subgroups Γ ⊂ SU(2).
• Analyzes the low-energy effective theory on M × R^{3,1}, showing it reduces to N=1 super Yang-Mills with b₁(M) adjoint chiral multiplets.
• Computes the superpotential via fractional M2-brane instantons wrapping the 3-cycle M, matching field-theoretic results.
• Identifies QCD strings in M-theory as M2-branes wrapping non-trivial 1-cycles in the Lens space S³/ℤₙ, carrying ℤₙ charge.
• Proposes that M-theory on the spin bundle of the Lens space S³/ℤₙ (with G₂-holonomy) is the gravity dual of Type IIA on the resolved conifold with RR 2-form flux.
• Relates domain walls in the M-theory background to M5-branes wrapping the Lens space, mapping to D4-branes in the IIA limit.

Experimental results

Research questions

• RQ1 How can pure N=1 super Yang-Mills theory be realized as a low-energy limit of M-theory compactified on G₂-holonomy spaces with singularities?
• RQ2 What is the origin of the superpotential in such M-theory backgrounds, and how do fractional M2-brane instantons contribute to it?
• RQ3 How are QCD strings and domain walls in the N=1 theory realized in the proposed M-theory dual?
• RQ4 What is the gravity dual of Type IIA string theory on the resolved conifold with RR 2-form flux, and how does it relate to M-theory on a Lens space?
• RQ5 Why does the M-theory dual predict n domain walls while the N=1 SYM theory only has n−1 physical vacua, and how can this discrepancy be resolved?

Key findings

• The low-energy effective theory on M × R^{3,1} for M-theory on a G₂-holonomy space with singular fibers R⁴/Γ is N=1 super Yang-Mills with b₁(M) adjoint chiral multiplets.
• The superpotential in the theory is generated by fractional M2-brane instantons wrapping the 3-cycle M, and this matches the field-theoretic superpotential for pure N=1 SYM.
• QCD strings in the M-theory background are realized as M2-branes wrapping non-trivial 1-cycles in the Lens space S³/ℤₙ, carrying ℤₙ charge and matching the spectrum of super QCD strings.
• Domain walls in the M-theory background correspond to M5-branes wrapping the Lens space, with their worldvolume theory supporting flat B-fields, and map to D4-branes wrapping S² in the IIA limit.
• The proposed gravity dual of IIA theory on the resolved conifold with RR 2-form flux is M-theory on the G₂-holonomy space S(S³/ℤₙ), which shares the same symmetries as the original M-theory background on J.
• The discrepancy between n domain walls in the M-theory dual and n−1 physical vacua in N=1 SYM is attributed to a trivially charged domain wall that may separate regions of empty M-theory space, suggesting it does not correspond to a physical vacuum.