Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory (0804.2902v2)

Abstract: We study boundary conditions in N=4 super Yang-Mills theory that preserve one-half the supersymmetry. The obvious Dirichlet boundary conditions can be modified to allow some of the scalar fields to have a ``pole'' at the boundary. The obvious Neumann boundary conditions can be modified by coupling to additional fields supported at the boundary. The obvious boundary conditions associated with orientifolds can also be generalized. In preparation for a separate study of how electric-magnetic duality acts on these boundary conditions, we explore moduli spaces of solutions of Nahm's equations that appear in the presence of a boundary. Though our main interest is in boundary conditions that are Lorentz-invariant (to the extent possible in the presence of a boundary), we also explore non-Lorentz-invariant but half-BPS deformations of Neumann boundary conditions. We make preliminary comments on the action of electric-magnetic duality, deferring a more serious study to a later paper.

• The paper presents a systematic construction of half-BPS boundary conditions in N=4 SYM, modifying standard Dirichlet and Neumann conditions with scalar field poles.
• It employs Nahm’s equations to analyze moduli spaces and couples bulk vector multiplets with boundary hypermultiplets to bridge bulk-boundary dynamics.
• The study explores electric-magnetic duality and domain wall configurations, laying the groundwork for further research in gauge theories and string theory.

Supersymmetric Boundary Conditions in $\mathcal{N}=4$ Super Yang-Mills Theory

This paper by Davide Gaiotto and Edward Witten is a comprehensive paper of supersymmetric boundary conditions in $\mathcal{N}=4$ Super Yang-Mills (SYM) theory. The focus is on boundary conditions that preserve one-half of the supersymmetry, their modifications, and the implications for electric-magnetic duality. The authors explore the various boundary conditions via a systematic approach involving scalar fields and gauge theory symmetries, providing a rigorous mathematical and theoretical framework for understanding these supersymmetric configurations.

Summary of Key Contributions

1. Supersymmetric Boundary Condition Construction

The authors consider boundary conditions that are compatible with half of the supersymmetry in $\mathcal{N}=4$ SYM theory. These conditions include standard Dirichlet and Neumann boundary conditions which are modified by allowing scalar fields to exhibit poles at the boundary. They also generalize Neumann conditions with additional boundary fields and explore non-Lorentz-invariant but half-BPS deformations.

Key constructs in this analysis include:

• Dirichlet and Neumann Boundary Conditions: Fundamental approaches with modifications allowing scalar poles and boundary couplings, thus generalizing $\mathcal{N}=4$ supersymmetric gauge theory constructs.
• Nahm’s Equations: Utilized to analyze moduli spaces of vacua. These equations are crucial for boundary conditions where scalar fields exhibit a pole at the boundary, indicating an inherent coupling with boundary degrees of freedom.

2. Electric-Magnetic Duality

The paper makes preliminary comments on the action of electric-magnetic duality on these boundary conditions, setting the foundation for future investigations. It suggests the existence of dual versions of boundary conditions, which speak to the core of S-duality in field theories and are essential for understanding strong-weak coupling dynamics.

3. Coupling to Boundary Degrees of Freedom

The work extends to studying the coupling of vector multiplets to hypermultiplets at the boundary, forming a critical bridge between bulk dynamics and boundary phenomena. This leads to:

• Introduction of Boundary Degrees of Freedom: This allows for richer vacuums when bulk vector multiplets are under Neumann conditions.
• Boundary Hyper-Kähler Moment Map: Developed to describe boundary interactions and their role in preserving the supersymmetry.

4. Domain Walls and Defects

Gaiotto and Witten examine domain walls and defects within the framework of $\mathcal{N}=4$ SYM theory, using the constructs developed for boundary conditions. The paper of half-BPS domain walls via the folding trick provides an elegant method to analyze transitions between different gauge groups.

Implications and Future Directions

The implications of this work are significant for both theoretical and practical aspects of supersymmetry and gauge theories. The boundary condition modifications introduce new theoretical models that can be used to probe uncharted territories in high-energy physics, offering insights into non-perturbative effects and duality symmetries.

• Theoretical Impact: The approach provides a robust mathematical structure, enhancing our understanding of boundary phenomena in superconformal field theories and their associated dualities.
• Gauge Theory Applications: Practical applications could be seen in topological quantum field theories and string theory, where understanding boundary dynamics is crucial.

The exploration of boundary conditions expands potential research in string theory, condensed matter systems, and readily provides a platform for computational approaches to gauge theories on domains with boundaries. Future research may focus on extensive numerical simulations and further analytic methods, linking the constructed boundary conditions with experimental realities where applicable. The paper sets the stage for deeper explorations into boundary phenomena and dualities in extended supersymmetric models.