[Paper Review] TASI Lectures on Solitons

David Tong|ArXiv.org|Sep 28, 2005

Black Holes and Theoretical Physics274 references136 citations

TL;DR

This paper presents a comprehensive review of solitons—instantons, monopoles, vortices, and domain walls—in the context of supersymmetric gauge theories and string theory. Using D-brane realizations and moduli space geometry, it establishes a web of connections between solitons of different types, revealing their roles in quantum dynamics, dualities, and holography, with key results including the D-brane description of domain walls and vortex strings, and the emergence of gauge theories on soliton worldvolumes.

ABSTRACT

These lectures cover aspects of solitons with focus on applications to the quantum dynamics of supersymmetric gauge theories and string theory. The lectures consist of four sections, each dealing with a different soliton. We start with instantons and work down in co-dimension to monopoles, vortices and, eventually, domain walls. Emphasis is placed on the moduli space of solitons and, in particular, on the web of connections that links solitons of different types. The D-brane realization of the ADHM and Nahm construction for instantons and monopoles is reviewed, together with related constructions for vortices and domain walls. Each lecture ends with a series of vignettes detailing the roles solitons play in the quantum dynamics of supersymmetric gauge theories in various dimensions. This includes applications to the AdS/CFT correspondence, little string theory, S-duality, cosmic strings, and the quantitative correspondence between 2d sigma models and 4d gauge theories.

Motivation & Objective

• To explore the interconnections between different types of solitons—instantons, monopoles, vortices, and domain walls—through their moduli spaces and D-brane realizations.
• To clarify the role of solitons in the quantum dynamics of supersymmetric gauge theories across various dimensions.
• To establish a unified framework linking solitons via geometric and duality structures, including ADHM and Nahm constructions.
• To demonstrate how solitons serve as dynamical probes in AdS/CFT, S-duality, and little string theory.
• To provide a pedagogical yet technically rigorous overview for advanced students and researchers in high-energy theory and mathematical physics.

Proposed method

• Use of D-brane techniques to realize and analyze the moduli spaces of instantons and monopoles via the ADHM and Nahm constructions.
• Application of moduli space formalism to describe multi-soliton configurations and their collective coordinates.
• Derivation of soliton equations in Euclidean and Minkowski spacetime, with focus on finite-action solutions and boundary conditions.
• Analysis of fermionic zero modes, dyonic excitations, and non-commutative solitons in supersymmetric settings.
• Use of BPS equations and central charge structures to identify D-brane configurations in field theory, including domain walls and vortex strings.
• Dualization of periodic scalars on domain walls to yield effective U(1) gauge theories on the worldvolume, linking to gauge field localization.

Experimental results

Research questions

• RQ1 How are the moduli spaces of instantons, monopoles, vortices, and domain walls interconnected through geometric and duality structures?
• RQ2 In what way do D-brane constructions realize the ADHM and Nahm formalisms for instantons and monopoles in field theory?
• RQ3 How do vortex strings end on domain walls, and what selection rules govern their attachment?
• RQ4 What is the role of the central charge in the BPS equations that describe domain walls and their D-brane nature?
• RQ5 How do quantum effects on domain wall worldvolumes, such as Chern-Simons terms, prevent wall penetration and stabilize soliton configurations?

Key findings

• The moduli space of a single domain wall is topologically R × S¹, supporting a scalar X and a periodic scalar θ, which can be dualized to a U(1) gauge field on the worldvolume.
• The low-energy effective action on the domain wall worldvolume is a free U(1) gauge theory coupled to a neutral scalar, with kinetic terms proportional to the domain wall tension and inverse gauge coupling.
• Vortex strings can end on domain walls, with the string associated to field q_i ending on the α_i wall from the left and q_{i+1} from the right, governed by selection rules.
• A negative binding energy—known as a boojum—arises when vortex strings attach to domain walls due to the monopole central charge in the BPS energy formula.
• The BPS equations for domain walls in multi-vacuum theories are solved analytically in the e² → ∞ limit, confirming the D-brane interpretation of these solitons.
• Chern-Simons interactions on the domain wall worldvolume dynamically prevent the passage of walls, stabilizing the system against collapse through quantum effects.