[Paper Review] Strings, boundary fermions and coincident D-branes

Linus Wulff|ArXiv.org|Jan 13, 2007

Black Holes and Theoretical Physics76 references3 citations

TL;DR

This thesis proposes a novel formulation of the dynamics of coincident D-branes using boundary fermions instead of matrices, demonstrating that the resulting action reproduces the Myers action and possesses kappa-symmetry—offering a more fundamental, supersymmetric description of non-abelian D-brane dynamics. The approach reveals a non-commutative geometric structure emerging from Lorentz transformations in transverse directions, suggesting deeper connections to non-commutative geometry.

ABSTRACT

This thesis describes an attempt to write down covariant actions for coincident D-branes using so-called boundary fermions instead of matrices to describe the non-abelian fields. These fermions can be thought of as Chan-Paton degrees of freedom for the open string. It is shown that by gauge-fixing and by suitably quantizing these boundary fermions the non-abelian action that is known, the Myers action, can be reproduced. Furthermore it is shown that under natural assumptions, unlike the Myers action, the action formulated using boundary fermions also posseses kappa-symmetry when formulated on superspace. Another aspect of string theory discussed in this thesis is that of tensionless strings. These are of great interest for example because of their possible relation to higher spin gauge theories via the AdS/CFT-correspondence. The tensionless superstring in a plane wave background, a Penrose limit of the near-horizon geometry of a stack of D3-branes, is considered and compared to the tensile case.

Motivation & Objective

• To develop a more fundamental, supersymmetric description of coincident D-branes beyond the matrix-based Myers action.
• To investigate whether boundary fermions can yield a kappa-symmetric action, a necessary condition for a correct supersymmetric theory of D-branes.
• To explore the emergence of non-commutative geometry from the dynamics of boundary fermions in the context of D-brane stacks.
• To clarify the geometric and algebraic structure underlying the non-abelian gauge theory on coincident D-branes.

Proposed method

• Formulate the D-brane action using boundary fermions derived from open string theory, treating them as fundamental degrees of freedom.
• Apply gauge-fixing and canonical quantization to the boundary fermions to derive the effective non-abelian action.
• Show that the resulting action matches the known Myers action in the classical limit.
• Demonstrate that the fermionic action preserves kappa-symmetry under natural assumptions, unlike the Myers action.
• Identify the Lorentz transformation structure in transverse directions as a non-abelian phenomenon, with matrix-valued Lorentz transformations arising from commutators of transverse coordinates.
• Use symmetrized products in the matrix representation to maintain Lorentz invariance, highlighting the role of non-commutativity.

Experimental results

Research questions

• RQ1 Can a boundary fermion formulation reproduce the Myers action for coincident D-branes?
• RQ2 Does the boundary fermion action possess kappa-symmetry, a hallmark of a correct supersymmetric D-brane action?
• RQ3 How does non-commutative geometry emerge from the dynamics of boundary fermions in the transverse directions?
• RQ4 What is the geometric interpretation of the matrix-valued Lorentz transformations in the transverse space?
• RQ5 How does the symmetrized product in the matrix representation preserve Lorentz invariance in the non-abelian setting?

Key findings

• The boundary fermion formulation successfully reproduces the Myers action upon gauge-fixing and quantization, confirming consistency with known results.
• The derived action explicitly possesses kappa-symmetry, a crucial advantage over the original Myers action which lacks this property.
• The transverse Lorentz transformation matrix $\Lambda_{a'}^{\phantom{a'}b'}$ becomes matrix-valued in the non-abelian case, with $M^{a'b'} \sim -i[z^{a'}, z^{b'}]$ representing the Poisson bracket in the matrix limit.
• The matrix Lorentz transformation preserves the Minkowski metric only when using a symmetrized product, indicating a non-commutative geometric structure.
• The non-abelian nature of the transverse Lorentz transformation arises from the presence of multiple D-branes, with the commutator $[z^{a'}, z^{b'}]$ encoding the non-commutativity.
• The framework suggests a natural emergence of non-commutative geometry from the underlying D-brane dynamics, with potential for deeper geometric interpretation.