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The Shape of Branes Pulled by Strings

Akikazu Hashimoto

TL;DR

This work investigates how a string attached to a D-brane deforms the brane, producing a spike whose shape and energy reflect the string tension. It develops two complementary frameworks: (i) a M-theory lift for D2-branes where the brane geometry is encoded by holomorphic membrane embeddings, and (ii) a Yang–Mills/Higgs approach for D3-branes in type IIB, using the Prasad–Sommerfield monopole to read off the brane embedding. In both settings, the computed energies agree with the expected string and D-string tensions, and nonperturbative effects reveal richer, coupling-dependent brane geometries. The results connect abelian DBI solutions to non-abelian and holographic descriptions, and suggest avenues for generalizations to dyons, multi-monopoles, and larger gauge groups, while highlighting the need for a complete non-abelian Born–Infeld action.

Abstract

We examine the system where a string stretches between pair of D-branes, and study the bending of the D-brane caused by the tension of the string. If the distance between the pair of D-branes is sent to infinity, the tension of the string stretching between them is strong enough to pull the spike all the way to infinity. We study the shape of these spikes when the branes are finite distance apart using two different methods. First, we consider a string stretched between a pair of D2-branes in type IIA theory by going to the M-theory limit in which all of these branes are M-theory 2-branes embedded along a holomorphic curve. Second, we consider a D-string stretched between a pair of D3-branes in type IIB theory and infer the geometry of the D3-brane embeddings from the configuration of the adjoint scalar field in the magnetic monopole solution of Prasad and Sommerfield. The case of fundamental string stretching between a pair of D3-branes follows from S-duality. The energy of these configurations matches the expected value based on fundamental string and D-string tensions.

The Shape of Branes Pulled by Strings

TL;DR

This work investigates how a string attached to a D-brane deforms the brane, producing a spike whose shape and energy reflect the string tension. It develops two complementary frameworks: (i) a M-theory lift for D2-branes where the brane geometry is encoded by holomorphic membrane embeddings, and (ii) a Yang–Mills/Higgs approach for D3-branes in type IIB, using the Prasad–Sommerfield monopole to read off the brane embedding. In both settings, the computed energies agree with the expected string and D-string tensions, and nonperturbative effects reveal richer, coupling-dependent brane geometries. The results connect abelian DBI solutions to non-abelian and holographic descriptions, and suggest avenues for generalizations to dyons, multi-monopoles, and larger gauge groups, while highlighting the need for a complete non-abelian Born–Infeld action.

Abstract

We examine the system where a string stretches between pair of D-branes, and study the bending of the D-brane caused by the tension of the string. If the distance between the pair of D-branes is sent to infinity, the tension of the string stretching between them is strong enough to pull the spike all the way to infinity. We study the shape of these spikes when the branes are finite distance apart using two different methods. First, we consider a string stretched between a pair of D2-branes in type IIA theory by going to the M-theory limit in which all of these branes are M-theory 2-branes embedded along a holomorphic curve. Second, we consider a D-string stretched between a pair of D3-branes in type IIB theory and infer the geometry of the D3-brane embeddings from the configuration of the adjoint scalar field in the magnetic monopole solution of Prasad and Sommerfield. The case of fundamental string stretching between a pair of D3-branes follows from S-duality. The energy of these configurations matches the expected value based on fundamental string and D-string tensions.

Paper Structure

This paper contains 6 sections, 59 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Static solutions of Dirac-Born-Infeld action
  3. A view from M-theory
  4. Brane configurations in type IIB theory
  5. Discussion
  6. Monopole solutions of non-abelian Born-Infeld action

Figures (5)

  • Figure 1: A string stretches between pair of parallel D3-branes, causing the D3-branes to bend.
  • Figure 2: Electrostatic potential (in units of $[(p-2)c_p]^{1 \over p-1}$) for a BIon solution in 3+1 dimensional Born-Infeld theory.
  • Figure 3: Combined energy in units of $T \sqrt{c_3}$ of bent D3-brane and stretched strings for one parameter family of configuration (\ref{['family']}) as a function of the parameter $a$.
  • Figure 4: Projection to $(X_1,X_2,X_3)$ space for the holomorphic curve for the M-theory description of a string stretching between a pair of D2-branes
  • Figure 5: Deformation of D3-brane by the tension of a string stretching between them. The solid line is the shape inferred from the field configuration of the Higgs scalar in the Prasad Sommerfeld solution. The dotted line is the BPS configuration $X = c_p/r^{p-2}$ for the abelian Born-Infeld theory. The scale factor $r_0$ is $g/Tb$ for the fundamental string and $1/Tb$ for the D-string.