[Paper Review] Twisted lattice supersymmetry and applications to AdS/CFT

Simon Catterall|arXiv (Cornell University)|Oct 29, 2010

Black Holes and Theoretical Physics34 references3 citations

TL;DR

This paper presents a lattice formulation of ${\cal N}=4$ super Yang-Mills theory using topological twisting to preserve one exact supersymmetry, enabling nonperturbative study of AdS/CFT duality. The method ensures gauge invariance, eliminates fermion doublers, and constrains quantum corrections, allowing successful numerical simulations that confirm holographic predictions like black hole entropy and phase transitions in thermal Yang-Mills systems.

ABSTRACT

I review recent approaches to constructing supersymmetric lattice theories focusing in particular on the concept of topological twisting. The latter technique is shown to expose a nilpotent, scalar supersymmetry which can be implemented exactly in the lattice theory. Using these ideas a lattice action for $\mathcal{N}=4$ super Yang-Mills in four dimensions can be written down which is gauge invariant, free of fermion doublers and respects one out of a total of 16 continuum supersymmetries. It is shown how these exact symmetries together with the large point group symmetry of the lattice strongly constrain the possible counterterms needed to renormalize the theory and hence determine how much residual fine tuning will be needed to restore all supersymmetries in the continuum limit. We report on progress to study these renormalization effects at one loop. We go on to give examples of applications of these supersymmetric lattice theories to explore the connections between gauge theories and gravity.

Motivation & Objective

• To construct a nonperturbative lattice formulation of ${\cal N}=4$ super Yang-Mills theory that preserves exact supersymmetry.
• To overcome the traditional problem of supersymmetry anomaly and fermion doubling in lattice field theories.
• To enable numerical studies of gauge-gravity duality using standard lattice Monte Carlo methods.
• To test holographic predictions such as black hole entropy and phase transitions in thermal Yang-Mills systems.
• To constrain quantum corrections and fine-tuning requirements via exact lattice symmetries.

Proposed method

• Apply topological twisting to the ${\cal N}=4$ SYM action, exposing a nilpotent scalar supersymmetry that can be exactly preserved on the lattice.
• Construct a lattice action that is gauge invariant, free of fermion doublers, and respects one of the 16 continuum supersymmetries.
• Use the large point group symmetry of the lattice to constrain possible counterterms in the renormalized action.
• Perform one-loop lattice perturbation theory to analyze quantum corrections and residual fine-tuning.
• Simulate the dimensionally reduced theories (e.g., thermal SYMQM and 2D SYM) using Monte Carlo methods.
• Compare lattice results with supergravity predictions for black hole entropy and phase boundaries.

Experimental results

Research questions

• RQ1 Can a lattice formulation of ${\cal N}=4$ super Yang-Mills preserve exact supersymmetry while avoiding fermion doubling?
• RQ2 To what extent do the exact symmetries of the lattice action constrain quantum corrections and fine-tuning in the continuum limit?
• RQ3 Do numerical simulations of the lattice theory reproduce holographic predictions such as black hole entropy and phase transitions?
• RQ4 What is the behavior of the spatial Polyakov line in 2D SYM, and does it match the supergravity prediction for the Gregory-LaFramme instability?
• RQ5 How well do lattice results for thermal Yang-Mills quantum mechanics agree with the Bekenstein-Hawking entropy of D0-brane black holes?

Key findings

• The lattice action for ${\cal N}=4$ SYM preserves one exact supersymmetry, is gauge invariant, and eliminates fermion doublers.
• Monte Carlo simulations of thermal Yang-Mills quantum mechanics show excellent agreement with black hole entropy predictions from supergravity, especially in the dynamical fermion case.
• The quenched theory diverges from the black hole curve at low temperatures, while the dynamical fermion version turns over to zero, confirming supersymmetry restoration.
• In 2D SYM, the phase boundary between confined and deconfined phases matches the supergravity prediction $r_\tau \sim c r_x^2$ with $c \approx 3.5$, confirming the Gregory-LaFramme instability.
• The lattice phase boundary for $SU(3)$ and $SU(4)$ agrees well with both analytic limits and supergravity, validating the numerical approach.
• One-loop perturbation theory analysis shows that the exact symmetries strongly constrain counterterms, reducing the need for fine-tuning in the continuum limit.