[Paper Review] Introductory Lectures on Multiple Membranes

Neil B. Copland|arXiv (Cornell University)|Dec 2, 2010

Black Holes and Theoretical Physics46 references3 citations

TL;DR

This paper provides an introductory overview of multiple membrane theories, focusing on the BLG and ABJM models as candidates for describing multiple M2-branes in M-theory. It establishes their connection to 11-dimensional supergravity and type IIA string theory via dimensional reduction, showing how BLG uses a non-associative 3-algebra structure and ABJM generalizes this to a U(N)×U(N) Chern-Simons-matter theory with enhanced supersymmetry, leading to an AdS₄×S⁷/ℤₖ gravity dual with N³ᐟ² degrees of freedom.

ABSTRACT

These lecture notes introduce the multiple membrane theories known as BLG and ABJM. We assume the reader is familiar with string theory, but not with M-theory, 11-dimensional supergravity or membranes. We therefore start with a background on M-theory and its extended objects before discussing BLG and ABJM. The link to string theory via dimensional reduction will be maintained throughout.

Motivation & Objective

• To introduce the theoretical framework of multiple membranes in M-theory, focusing on BLG and ABJM as candidate quantum theories.
• To explain how these theories emerge from dimensional reduction of 11-dimensional supergravity and M-theory branes.
• To clarify the role of 3-algebras in BLG and their replacement by gauge-theoretic structures in ABJM.
• To establish the connection between the field theory on M2-branes and the gravitational AdS₄×S⁷/ℤₖ background via the AdS/CFT correspondence.
• To highlight unresolved issues, such as the N³ᐟ² scaling of degrees of freedom in the field theory, and the role of monopole operators in symmetry enhancement.

Proposed method

• Derive the BLG Lagrangian using a non-associative 3-bracket (the fundamental trilinear operation) and relate it to the Bagger-Lambert-Gustavsson formulation.
• Construct the ABJM theory as a U(N)×U(N) Chern-Simons-matter theory with bifundamental matter fields and a level-k Chern-Simons term.
• Use dimensional reduction from the M2-brane worldvolume action to connect to D2-branes and the type IIA string theory.
• Analyze the R-symmetry and global symmetries of ABJM, particularly the SU(4)R×U(1)b structure and the role of monopole operators in enhancing supersymmetry.
• Construct the gravity dual of ABJM using the near-horizon geometry of N M2-branes at a C⁴/ℤₖ orbifold singularity, yielding AdS₄×S⁷/ℤₖ.
• Match chiral operators and conserved currents between the ABJM CFT and the gravity side, including the necessity of monopole operators for full symmetry enhancement.

Experimental results

Research questions

• RQ1 How can multiple M2-branes be consistently described by a quantum field theory, given the lack of a known fundamental action for M-theory?
• RQ2 What is the role of the non-associative 3-algebra in the BLG theory, and why is it insufficient for generalizing beyond N=2?
• RQ3 How does the ABJM theory generalize the BLG model to arbitrary N and k, and what is the significance of the Chern-Simons level k?
• RQ4 What is the gravitational dual of the ABJM theory, and how does it reproduce the N³ᐟ² scaling of degrees of freedom?
• RQ5 How do monopole operators in ABJM contribute to the enhancement of the R-symmetry and completion of the current algebra?

Key findings

• The ABJM theory is a U(N)×U(N) Chern-Simons-matter theory with level k and bifundamental matter, providing a consistent description of N M2-branes at a C⁴/ℤₖ orbifold singularity.
• For k=1 and k=2, monopole operators combine with antisymmetric currents to enhance the global symmetry to 28 conserved currents, indicating supersymmetry enhancement.
• The near-horizon geometry of N M2-branes at the C⁴/ℤₖ singularity is AdS₄×S⁷/ℤₖ, with the radius R⁶ = 32π²N′lₚ⁶ and N′ = Nk.
• The field theory has an 't Hooft coupling λ = N/k, and is weakly coupled when k ≪ N, with the M-theory description valid only when k⁵ ≪ N.
• The gravity dual predicts N³ᐟ² degrees of freedom, which remains unexplained in the field theory side despite extensive study.
• Monopole operators are essential for matching the full current algebra and for realizing the full R-symmetry on the field theory side.