[Paper Review] Aspects of Supersymmetry in Multiple Membrane Theories

Andrew M. Low|arXiv (Cornell University)|Dec 13, 2010

Black Holes and Theoretical Physics139 references3 citations

TL;DR

This thesis investigates the worldvolume supersymmetry algebras of multiple M2-brane theories, including Bagger-Lambert, ABJM, and Nambu-Poisson M5-brane models, and derives $l_p^3$ corrections to the Bagger-Lambert supersymmetry transformations using non-abelian duality in 2+1 dimensions. The key contribution is the explicit determination of $l_p^3$-corrected supersymmetry transformations for fermions and scalars in both abelian and non-abelian Bagger-Lambert theories via dualisation of $ar{ heta}ar{ heta}$ terms from the D2-brane effective action.

ABSTRACT

This thesis consists of two parts. In the first part we investigate the worldvolume supersymmetry algebra of multiple membrane theories. We begin with a description of M-theory branes and their intersections from the perspective of spacetime and worldvolume supersymmetry algebras. We then provide an overview of the recent work on multiple M2-branes focusing on the Bagger-Lambert theory and its relation to the Nambu-Poisson M5-brane and the ABJM theory. The worldvolume supersymmetry algebras of these theories are explicitly calculated and the charges interpreted in terms of spacetime intersections of M-branes. The second part of the thesis looks at $l_p^3$ corrections to the supersymmetry transformations of the Bagger-Lambert theory. We begin with a review of the dNS duality transformation which allows a gauge field to be dualised to a scalar field in 2+1 dimensions. Applying this duality to the $α'^2$ terms of the D2-brane supersymmetry transformations we are able to determine the $l_p^3$ corrections to the supersymmetry transformations of the scalar and fermion fields for the abelian theory. Generalising to the `non-abelian' Bagger-Lambert theory we are able to determine the $l_p^3$ correction to the supersymmetry transformation of the fermion field. Along the way make a number of observations relating to the implementation of the dNS duality transformation at the level of supersymmetry transformations.

Motivation & Objective

• To understand the worldvolume supersymmetry algebras of multiple M2-brane theories, including Bagger-Lambert, ABJM, and Nambu-Poisson M5-brane models.
• To interpret central charges in these algebras as arising from spacetime intersections of M-branes.
• To derive higher-order corrections to the Bagger-Lambert supersymmetry algebra, specifically $l_p^3$-corrections, using duality transformations.
• To generalize abelian duality results to the non-abelian case and determine the structure of $l_p^3$-corrected supersymmetry transformations.
• To analyze the consistency of duality transformations at the level of supersymmetry algebra and transformation rules.

Proposed method

• Applied the dNS duality transformation to $ar{ heta}ar{ heta}$ terms in the D2-brane effective action to map gauge fields to scalar fields in 2+1 dimensions.
• Used dimensional reduction and Hamiltonian analysis to relate the D2-brane theory to the Bagger-Lambert theory and extract higher-order corrections.
• Performed explicit calculations of supersymmetry transformations at $ar{ heta}ar{ heta}$ order, identifying terms involving fermions, scalars, and gauge fields.
• Used SO(8) invariance and tensorial structure to classify and simplify higher-order corrections in the abelian and non-abelian cases.
• Derived the $l_p^3$-corrected supersymmetry transformation for the fermion field in the non-abelian Bagger-Lambert theory via dualisation of the D2-brane supersymmetry algebra.
• Verified invariance of the higher-order Lagrangian under the derived supersymmetry transformations, ensuring consistency of the correction.

Experimental results

Research questions

• RQ1 How do the worldvolume supersymmetry algebras of multiple M2-brane theories relate to spacetime intersections of M-branes?
• RQ2 What is the structure of the $l_p^3$-corrected supersymmetry transformations in the abelian Bagger-Lambert theory?
• RQ3 How can the dNS duality transformation be consistently applied to higher-order corrections in the D2-brane action to derive corrections in the Bagger-Lambert theory?
• RQ4 What is the form of the $l_p^3$-corrected supersymmetry transformation for the fermion field in the non-abelian Bagger-Lambert theory?
• RQ5 How do the central charges in the Nambu-Poisson M5-brane superalgebra relate to the central charges in the M2-brane superalgebra?

Key findings

• The $l_p^3$-corrected supersymmetry transformation for the fermion field in the non-abelian Bagger-Lambert theory was uniquely determined via duality transformation of the D2-brane supersymmetry algebra.
• For the abelian Bagger-Lambert theory, the $l_p^3$ corrections to both the fermion and scalar field supersymmetry transformations were uniquely fixed by the duality procedure.
• The duality transformation successfully mapped $ar{ heta}ar{ heta}$ terms from the D2-brane action to the Bagger-Lambert theory, yielding consistent higher-order corrections.
• The central charges in the Nambu-Poisson M5-brane superalgebra were shown to be expressible in terms of M2-brane central charges, linking the two theories algebraically.
• The Hamiltonian analysis of the Bagger-Lambert theory confirmed the BPS equations associated with fuzzy funnel solutions, supporting the interpretation of the theory as describing multiple M2-branes.
• The $l_p^3$-corrections preserve the SO(8) R-symmetry of the original Bagger-Lambert theory, ensuring consistency with the underlying supersymmetry algebra.