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Nahm's equations, $N=1^*$ domain walls, and D-strings in $AdS_5 x S_5$

C. Bachas, J. Hoppe, B. Pioline

TL;DR

This work shows that BPS domain walls in the ${ m N}=1^*$ deformation and D-strings in $AdS_5 imes S^5$ are governed by Nahm equations with Kronheimer boundary conditions, linking field-theoretic vacua to brane constructions. The moduli spaces of walls are described as intersections ${ m N}( ho_-)\cap{ m S}( ho_+)$ of nilpotent orbits with shifted centralizers, yielding an additivity rule for wall dimensions and a clear notion of elementary walls. The radial D-strings furnish a holographic realization where fuzzy-sphere configurations correspond to higher-representation Wilson--'t Hooft lines, and kink transitions along the D-string worldsheet act as braiding operators in the dual CFT. The results provide a robust, index-theoretic framework for counting moduli and shed light on the nonperturbative vacuum structure, with potential implications for holography and matrix-model dynamics.

Abstract

We consider the following two problems: classical domain walls in the $N=1^*$ mass deformation of the maximally supersymmetric Yang Mills theory, and D-strings as external magnetic sources in the context of the AdS/CFT correspondence. We show that they are both described by Nahm's equations with unconventional boundary conditions, and analyze the relevant moduli space of solutions. We argue that general `fuzzy sphere' configurations of D-strings in AdS$_5$ correspond to Wilson-'t Hooft lines in higher representations of the dual SU(n) gauge theory.

Nahm's equations, $N=1^*$ domain walls, and D-strings in $AdS_5 x S_5$

TL;DR

This work shows that BPS domain walls in the deformation and D-strings in are governed by Nahm equations with Kronheimer boundary conditions, linking field-theoretic vacua to brane constructions. The moduli spaces of walls are described as intersections of nilpotent orbits with shifted centralizers, yielding an additivity rule for wall dimensions and a clear notion of elementary walls. The radial D-strings furnish a holographic realization where fuzzy-sphere configurations correspond to higher-representation Wilson--'t Hooft lines, and kink transitions along the D-string worldsheet act as braiding operators in the dual CFT. The results provide a robust, index-theoretic framework for counting moduli and shed light on the nonperturbative vacuum structure, with potential implications for holography and matrix-model dynamics.

Abstract

We consider the following two problems: classical domain walls in the mass deformation of the maximally supersymmetric Yang Mills theory, and D-strings as external magnetic sources in the context of the AdS/CFT correspondence. We show that they are both described by Nahm's equations with unconventional boundary conditions, and analyze the relevant moduli space of solutions. We argue that general `fuzzy sphere' configurations of D-strings in AdS correspond to Wilson-'t Hooft lines in higher representations of the dual SU(n) gauge theory.

Paper Structure

This paper contains 12 sections, 65 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction, summary and directory
  2. Supersymmetric domain walls in ${\cal N}=1^*$ SYM
  3. Vacua of mass deformed ${\cal N}=4$ Yang-Mills
  4. Supersymmetric domain walls
  5. Some explicit solutions
  6. Nahm's equations and non-abelian monopoles
  7. The moduli space of domain walls
  8. Lax pair, Moment map and Hyperkähler structure
  9. Nilpotent orbits and the moduli space of domain walls
  10. Additivity rule and elementary domain walls
  11. Morse theory and dimension of moduli space
  12. D strings in $AdS_5\times S^5$ and holography

Figures (1)

  • Figure 1: Partial order between $SU(2)$ representation. The number of squares contained in the first $p$ columns of the Young tableau has to increase for all $p$. The two representations on the right cannot be related by a BPS nor an anti-BPS domain wall.