Exact lattice supersymmetry (0903.4881v2)

Abstract: We provide an introduction to recent lattice formulations of supersymmetric theories which are invariant under one or more real supersymmetries at nonzero lattice spacing. These include the especially interesting case of ${\cal N}=4$ SYM in four dimensions. We discuss approaches based both on twisted supersymmetry and orbifold-deconstruction techniques and show their equivalence in the case of gauge theories. The presence of an exact supersymmetry reduces and in some cases eliminates the need for fine tuning to achieve a continuum limit invariant under the full supersymmetry of the target theory. We discuss open problems.

• The paper demonstrates that using twisted and orbifold-deconstruction methods preserves an exact subset of supersymmetric charges, significantly mitigating fine-tuning requirements.
• It shows how scalar supercharges in twisted formulations enable robust lattice constructions that smoothly approach continuum limits while retaining critical symmetries.
• The research paves the way for effective numerical simulations in supersymmetric gauge theories, offering practical insights into strong-coupling dynamics and gauge/gravity duality.

Overview of "Exact lattice supersymmetry"

This paper presents an insightful exploration of lattice formulations for supersymmetric theories, particularly those maintaining invariance under one or more real supersymmetries at a finite lattice spacing. It encompasses recent advancements in this area, addressing formulations based on twisted supersymmetry and orbifold-deconstruction techniques, highlighting their equivalence for gauge theories. An exact supersymmetry on the lattice obviates, or significantly mitigates, the necessity of fine-tuning for achieving a continuum limit that upholds the full supersymmetry of the target theory.

Key Topics and Concepts

The paper underscores the theoretical significance of supersymmetric theories, alluding to their ability to illustrate mechanisms like confinement, chiral symmetry breaking, and links to string theory and supergravity. Historically, lattice formulations of most supersymmetric theories have been elusive due to discretization breaking the supersymmetry, preventing the target theory's continuum symmetry without substantial fine-tuning. However, by employing an exact supersymmetric subalgebra on the lattice action, these issues can be alleviated. Two major formulation approaches are:

1. Twisted Supersymmetry: This employs Dirac-Kähler fermions and twisting procedures, wherein Lorentz spinor supercharges decompose into integer spin tensors under a diagonal subgroup of Lorentz and R-symmetry groups. The twisted formulations, pioneered by Witten, utilize this decomposition to exhibit scalar supercharges on the lattice at finite spacings, making it possible to preserve certain supersymmetries.
2. Orbifold-Deconstruction Techniques: This method formulates a supersymmetric lattice theory by orbifolding a supersymmetric matrix model (the "mother" theory). The resulting "daughter" theory has a coherent lattice structure and some retained supersymmetries. The degenerate ground states form a moduli space, wherein the distance from the origin signifies the inverse lattice spacing, thereby defining a supersymmetric gauge theory in the continuum limit.

Significance and Implications

The paper holds particularly significant implications for numerical simulations, especially in theories like the four-dimensional $\mathcal{N}=4$ SYM, offering a promising point of departure. These formulations pave the way for computing aspects of strong-coupling dynamics in supersymmetric systems, potentially informing studies on gauge/gravity duality conjectures and probing the deep connections between gauge theories and string theory.

Future Directions and Challenges

While this research has laid crucial groundwork, there remain open questions surrounding:

• Full Supersymmetry Restoration: While these formulations show promise, determining the extent to which full supersymmetry can be restored as lattice spacing approaches zero is essential.
• Broader Theoretical Models: Extending these formulations to other interesting theoretical models (like $\mathcal{N}=2$ SYM or MSSM) remains a challenge, necessitating further exploration of appropriate lattice symmetries or extensions to these methods.
• Practical Computation: The development of computational strategies and algorithms for efficiently simulating these lattice theories remains crucial to their practical application and potential breakthroughs in understanding strongly coupled supersymmetric dynamics.

In conclusion, the paper by Catterall, Kaplan, and Ünsal provides a comprehensive account of the advancements in lattice formulations of supersymmetric theories, emphasizing that exact lattice supersymmetry can be pivotal in achieving robust lattice regularizations, which may revolutionize both theoretical explorations and practical computations in quantum field theories.