[Paper Review] Supersymmetric gradient flow in N=1 SYM

Daisuke Kadoh, N. Ukita|arXiv (Cornell University)|Dec 6, 2018

Black Holes and Theoretical Physics4 citations

TL;DR

This paper proposes a supersymmetric gradient flow in four-dimensional N=1 super Yang-Mills theory using component fields in the Wess-Zumino gauge. By introducing a modified gauge-fixing term, the authors derive a flow equation that commutes with supersymmetry up to a gauge transformation, ensuring compatibility with supersymmetry and preserving the structure of the theory on the lattice.

ABSTRACT

The gradient flow equation is derived in four-dimensional N=1 supersymmetric Yang-Mills theory in terms of the component field of the Wess-Zumino gauge. We show that the flow-time derivative and supersymmetry transformation that is naively extended to 4+1 dimensions by replacing the four-dimensional fields with the corresponding flowed fields commute with each other up to a gauge transformation. In this sense, the obtained flow is supersymmetric in the Wess-Zumino gauge. We also discuss more about the symmetry of the flow equation.

Motivation & Objective

• To construct a supersymmetric gradient flow in N=1 SYM that is manifestly compatible with supersymmetry in the Wess-Zumino gauge.
• To resolve the issue of broken ordinary gauge invariance in previous formulations of the SYM gradient flow when reduced to component fields.
• To enable consistent lattice simulations and analytic comparisons by formulating the flow in terms of component fields.
• To ensure the flow equation remains invariant under supersymmetry up to a gauge transformation, preserving the physical consistency of the theory.

Proposed method

• Derives the gradient flow equation in N=1 SYM using the Yang-Mills action as the flow functional, extending it to 4+1 dimensions via flow-time evolution.
• Introduces a new gauge-fixing term that is less restrictive than in prior work, preserving ordinary gauge symmetry during the flow.
• Expresses the flow equation in terms of component fields (gauge field, gaugino, auxiliary field) in the Wess-Zumino gauge.
• Demonstrates that the commutator of the flow-time derivative and the supersymmetry transformation vanishes up to a gauge transformation.
• Uses the superfield formalism in Euclidean space to derive the flow equation in vector superfield form before reducing to component fields.
• Applies the extended gauge transformation to maintain consistency with the super-Yang-Mills structure during the flow.

Experimental results

Research questions

• RQ1 Can a supersymmetric gradient flow be consistently formulated in the Wess-Zumino gauge using component fields?
• RQ2 Does the flow equation commute with supersymmetry up to a gauge transformation, ensuring supersymmetry compatibility?
• RQ3 How can the flow preserve ordinary gauge invariance while maintaining supersymmetry in the component field formulation?
• RQ4 What is the role of the gauge-fixing term in preserving both gauge symmetry and supersymmetry during the flow?
• RQ5 Can the resulting flow be used for lattice simulations and analytic calculations in N=1 SYM without extra fermion renormalizations?

Key findings

• The proposed gradient flow equation in the Wess-Zumino gauge commutes with supersymmetry up to a gauge transformation, confirming its supersymmetry compatibility.
• The flow preserves ordinary gauge invariance due to the introduction of a milder gauge-fixing condition compared to previous approaches.
• The flow equation is derived explicitly in component fields, enabling direct use in lattice field theory and analytic calculations.
• The commutator of the flow-time derivative and the supersymmetry transformation vanishes modulo a gauge transformation, ensuring consistency with supersymmetry.
• The method avoids extra renormalizations for the gaugino, as the flow is intrinsically supersymmetric, unlike non-SUSY flows.
• The flow equation is invariant under extended gauge transformations, preserving the full structure of the super-Yang-Mills theory.