[Paper Review] Reduction of the N=8 BLG and N=6 BL Theories to 2D Effective Field Theories

Paul Franche|ArXiv.org|Nov 10, 2008

Black Holes and Theoretical Physics15 references3 citations

TL;DR

This paper reduces the $σ$ =8 and $σ$ =6 Bagger-Lambert-Gustavsson (BLG) theories—3D superconformal field theories for M2-branes—to effective 2D field theories via compactification on a circle. Using dimensional reduction, the Chern-Simons term generates a $φ F$ coupling, and the resulting 2D actions retain full 3-algebra structure, with $φ$ inducing mass-like terms for scalars and fermions. The key result is a 2D effective field theory preserving 3-algebra structure, potentially describing F1 strings in type IIA string theory.

ABSTRACT

Starting from the three dimensional N=8 BLG and N=6 BL theories with CS level k=1, a dimensional reduction is performed on the M2-branes worldvolume. The IR limit of these 2D EFTs should represent the action of F1 strings in type IIA. The CS term reduces to a particular Phi*F term and the scalars and fermions get "mass" terms. The reduced actions still rely on three-algebra, which we did not require to define precisely.

Motivation & Objective

• To derive a 2D effective field theory (EFT) from the $σ$ =8 BLG theory via dimensional reduction on $S^1$.
• To extend the reduction to the $σ$ =6 BL theory, preserving its $σ$ =6 supersymmetry and SU(4) $×$ U(1) R-symmetry.
• To explore whether the reduced 2D actions can describe F1 strings in type IIA string theory, particularly in the infrared limit.
• To maintain the full 3-algebra structure in the 2D theory, avoiding reduction to a Lie algebra.
• To analyze the role of the scalar $φ$ in generating mass-like terms for scalars and fermions via the $φ F$ term.

Proposed method

• Perform dimensional reduction of the 3D $σ$ =8 BLG theory on $S^1$, compactifying along the M2-brane worldvolume.
• Use the fundamental identity of the 3-algebra to reduce the 3D Chern-Simons action to a $φ F$ term in 2D.
• Apply the same reduction procedure to the $σ$ =6 BL theory, using its complex 3-algebra structure at $k=1$.
• Retain the full 3-algebra indices in the 2D action, ensuring the structure is not broken to a Lie algebra.
• Use the 2D chiral decomposition of fermions via $̼̼̼̼𝕃̼𝕄̼𝕅̼𝕆̼𝕇$ matrices to express the reduced Lagrangian in terms of chiral components.
• Derive the 2D supersymmetry transformations and equations of motion from the reduced action, preserving the original SUSY algebra.

Experimental results

Research questions

• RQ1 Can the $σ$ =8 BLG theory be consistently dimensionally reduced to a 2D effective field theory while preserving its 3-algebra structure?
• RQ2 How does the Chern-Simons term in the 3D BLG theory reduce to a $φ F$ coupling in 2D, and what is the role of the scalar $φ$ in this reduction?
• RQ3 Does the reduced 2D theory for the $σ$ =6 BLG theory retain the full 3-algebra structure, or is it broken to a Lie algebra?
• RQ4 Can the resulting 2D EFTs describe F1 strings in type IIA string theory, particularly in the infrared limit?
• RQ5 How do the mass-like terms for scalars and fermions arise from the $φ$ field in the 2D action?

Key findings

• The dimensional reduction of the $σ$ =8 BLG theory on $S^1$ yields a 2D effective field theory with a $φ F$ term arising from the Chern-Simons action, where $φ$ is the compactified component of the gauge field.
• The reduced 2D action preserves the full 3-algebra structure, with both scalars and fermions acquiring mass-like terms through the $φ$ field.
• For the $σ$ =6 BL theory, the same reduction procedure leads to a 2D EFT with a $φ F$ coupling and mass terms, maintaining the complex 3-algebra structure.
• The 2D supersymmetry transformations are derived and expressed in terms of chiral fermion components, with $φ$ contributing to the fermion mass terms.
• The 2D actions are not equivalent to D2-brane actions but may be related via dualities, such as S-duality, mapping $1/R$ factors to $α'$ corrections.
• The results suggest that the 2D EFTs could describe F1 strings in type IIA string theory, particularly in the infrared limit, though the presence of ghosts remains a concern.