[Paper Review] Advances and applications of lattice supersymmetry

Joel Giedt|ArXiv.org|Jan 5, 2007

Black Holes and Theoretical Physics78 references5 citations

TL;DR

This paper reviews lattice formulations of supersymmetric field theories that preserve exact supersymmetry without fine-tuning of counterterms, focusing on Q-exact actions and Ginsparg-Wilson fermions. It demonstrates that models like supersymmetric quantum mechanics and 2d ${\cal N}=2$ Wess-Zumino models achieve exact spectrum degeneracy and Ward identities on the lattice, enabling nonperturbative study of AdS/CFT duals and quantum gravity via lattice simulations of strongly coupled gauge theories.

ABSTRACT

I review motivations for the study of supersymmetric field theories by lattice techniques. In particular, some of the more interesting potential applications are described. These are models of quantum gravity, that rely on the AdS/CFT correspondence. The dual gauge theory is an ideal place for lattice studies to make a contribution. I also survey some of the recent advances in lattice formulations of supersymmetric theories.

Motivation & Objective

• To develop lattice formulations of supersymmetric field theories that preserve exact supersymmetry without fine-tuning of counterterms.
• To enable nonperturbative study of strongly coupled gauge theories dual to quantum gravity via the AdS/CFT correspondence.
• To explore applications of lattice supersymmetry in models of warped extra dimensions and deconstructed gauge theories.
• To assess the viability of lattice methods for studying supersymmetry breaking and spectrum degeneracy in quantum gravity models.
• To promote the use of Q-exact actions and Ginsparg-Wilson fermions as tools for simulating realistic, low-energy supersymmetric theories.

Proposed method

• Utilizes Q-exact lattice actions where the action is written as $ S = QX $, ensuring $ Q^2 = 0 $, which guarantees exact supersymmetry and absence of counterterms.
• Applies Ginsparg-Wilson fermions to 4d $ \mathcal{N}=1 $ SYM, preserving chiral symmetry and eliminating gaugino mass counterterms in the chiral limit.
• Employs lattice formulations with exact lattice symmetries that protect continuum supersymmetry in the $ a \to 0 $ limit.
• Uses Monte Carlo simulations to test spectrum degeneracy, Ward identities, and R-symmetry as evidence of unbroken supersymmetry.
• Applies deconstruction techniques to build 2d $ (4,4) $ super-QCD models from higher-dimensional theories, enabling lattice simulation of $ AdS_3/CFT_2 $.
• Combines lattice gauge theory with holographic duals, such as the Klebanov-Strassler deformed conifold, to model realistic supersymmetric quantum field theories.

Experimental results

Research questions

• RQ1 Can lattice formulations of supersymmetric field theories be constructed without fine-tuning of counterterms?
• RQ2 To what extent can exact lattice supersymmetry be preserved in nontrivial gauge theories, such as $ \mathcal{N}=1 $ SYM?
• RQ3 How can lattice methods be used to probe the nonperturbative spectrum of AdS/CFT duals, particularly in warped compactifications?
• RQ4 Can Q-exact actions in 2d $ \mathcal{N}=2 $ Wess-Zumino models reproduce continuum spectrum degeneracy and Ward identities without counterterms?
• RQ5 What is the role of lattice formulations in studying the resolution of conifold singularities and their dual gauge theories?

Key findings

• In Q-exact supersymmetric quantum mechanics, lattice simulations show Bose-Fermi mass degeneracy at finite lattice spacing, with $ m_B \approx m_F $ up to $ N=256 $, and extrapolation to $ N\to\infty $ matches the continuum Schrödinger solution.
• For the 2d $ \mathcal{N}=2 $ Wess-Zumino model, all-order perturbative and nonperturbative evidence confirms the absence of counterterms due to Q-exactness and Ward identities.
• In $ \mathcal{N}=1 $ 4d SYM with Ginsparg-Wilson fermions, the chiral limit and supersymmetric limit coincide, eliminating the need for gaugino mass counterterms.
• Numerical evidence from spectrum degeneracy, Ward identities, and R-symmetry supports the absence of fine-tuning in the Q-exact 2d model, even nonperturbatively.
• The deconstructed $ (4,4) $ super-QCD model on the lattice realizes two exact supercharges and allows for the study of $ AdS_3/CFT_2 $ duality via lattice simulations.
• The paper identifies the Klebanov-Strassler deformed conifold as a promising candidate for lattice study, with its dual gauge theory exhibiting confinement and breaking of conformal symmetry, relevant to realistic models of the electroweak sector.