[Paper Review] TASI Lectures on Solitons Instantons, Monopoles, Vortices and Kinks

David Tong|arXiv (Cornell University)|Jun 1, 2005

Black Holes and Theoretical PhysicsPhysics and Astronomy257 references95 citations

TL;DR

This paper provides a comprehensive overview of solitons—instantons, monopoles, vortices, and kinks—in the context of supersymmetric gauge theories and string theory, emphasizing their moduli spaces and interconnections. It reviews D-brane realizations of the ADHM and Nahm constructions and explores their roles in quantum dynamics, including applications to AdS/CFT, S-duality, and dimensional dualities between 2d sigma models and 4d gauge theories.

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 role of solitons in the quantum dynamics of supersymmetric gauge theories across different dimensions.
• To clarify the web of connections linking solitons of varying co-dimensions, from instantons to domain walls.
• To review the D-brane realization of the ADHM and Nahm constructions for instantons and monopoles.
• To extend these constructions to vortices and domain walls, highlighting their geometric and dynamical properties.
• To connect soliton physics to broader theoretical frameworks such as AdS/CFT, little string theory, and S-duality.

Proposed method

• Systematic analysis of solitons ordered by co-dimension, starting from instantons and progressing to domain walls.
• Use of moduli space techniques to describe collective coordinates and low-energy dynamics of solitons.
• Application of D-brane engineering to realize the ADHM and Nahm constructions for instantons and monopoles.
• Extension of these constructions to vortices and domain walls via brane configurations and dualities.
• Incorporation of vignettes at the end of each lecture to link soliton dynamics to quantum phenomena in 4d and 2d theories.
• Use of dimensional reduction and dualities to connect 2d sigma models to 4d gauge theories.

Experimental results

Research questions

• RQ1 How do the moduli spaces of different solitons—instantons, monopoles, vortices, and kinks—relate to one another in supersymmetric theories?
• RQ2 What is the role of D-brane configurations in realizing the ADHM and Nahm constructions for instantons and monopoles?
• RQ3 How do solitons contribute to the quantum dynamics of supersymmetric gauge theories in various dimensions?
• RQ4 In what ways do solitons provide insights into the AdS/CFT correspondence and S-duality?
• RQ5 What is the quantitative correspondence between 2d sigma models and 4d gauge theories via soliton solutions?

Key findings

• The moduli space of solitons forms a web of interconnections, revealing deep geometric and dynamical relationships across different soliton types.
• D-brane realizations provide a physical and geometric interpretation of the ADHM and Nahm constructions for instantons and monopoles.
• Vortices and domain walls are successfully described through analogous brane-based constructions, extending the web of dualities.
• Solitons play a central role in the quantum dynamics of supersymmetric gauge theories, particularly in the emergence of dualities and non-perturbative effects.
• The correspondence between 2d sigma models and 4d gauge theories is quantitatively supported by soliton solutions and their moduli spaces.
• Applications to cosmic strings, little string theory, and S-duality are illuminated through the soliton framework, revealing new insights into non-perturbative dynamics.