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Loop Corrected Supercharges from Holomorphic Anomalies

Kasia Budzik, Justin Kulp

TL;DR

This work formalizes how loop corrections to the supercharge in 4d Lagrangian SUSY gauge theories emerge from holomorphic-twist anomalies encoded by a shifted L_infty-algebra. By treating the twisted theory as a holomorphic BRST system with higher brackets, the authors derive a perturbative expansion Q = Q_free + Q_tree + ħ Q1 + …, and provide a concrete master-integral scheme to compute these corrections via triangle diagrams and Laman-graph constraints. They obtain explicit one-loop expressions for Q1 across N = 1, 2, 4 supersymmetries, including a compact superfield reformulation in N = 4 that mirrors the classical Lie algebra cohomology differential, and they discuss planar-limit simplifications and potential implications for the BPS spectrum. The results illuminate how holomorphic confinement and twice-generalized Konishi anomalies shape the quantum semi-chiral ring and offer a practical route to assess loop-corrected Q-cohomology in four-dimensional theories.

Abstract

We describe the loop corrections to supercharges in supersymmetric quantum field theories using the holomorphic twist formalism. We begin by reviewing the relation between supercharge corrections and the "twice-generalized" Konishi anomaly, which corrects the semi-chiral ring. In the holomorphic twist, these corrections appear as BRST anomalies and are computed using the higher operations of an underlying $L_\infty$ conformal algebra. We then apply this formalism to obtain the complete one-loop corrections to the supercharge of four-dimensional Lagrangian supersymmetric gauge theories, including $\mathcal{N}=4$ SYM, where it admits a remarkably compact expression in terms of superfields.

Loop Corrected Supercharges from Holomorphic Anomalies

TL;DR

This work formalizes how loop corrections to the supercharge in 4d Lagrangian SUSY gauge theories emerge from holomorphic-twist anomalies encoded by a shifted L_infty-algebra. By treating the twisted theory as a holomorphic BRST system with higher brackets, the authors derive a perturbative expansion Q = Q_free + Q_tree + ħ Q1 + …, and provide a concrete master-integral scheme to compute these corrections via triangle diagrams and Laman-graph constraints. They obtain explicit one-loop expressions for Q1 across N = 1, 2, 4 supersymmetries, including a compact superfield reformulation in N = 4 that mirrors the classical Lie algebra cohomology differential, and they discuss planar-limit simplifications and potential implications for the BPS spectrum. The results illuminate how holomorphic confinement and twice-generalized Konishi anomalies shape the quantum semi-chiral ring and offer a practical route to assess loop-corrected Q-cohomology in four-dimensional theories.

Abstract

We describe the loop corrections to supercharges in supersymmetric quantum field theories using the holomorphic twist formalism. We begin by reviewing the relation between supercharge corrections and the "twice-generalized" Konishi anomaly, which corrects the semi-chiral ring. In the holomorphic twist, these corrections appear as BRST anomalies and are computed using the higher operations of an underlying conformal algebra. We then apply this formalism to obtain the complete one-loop corrections to the supercharge of four-dimensional Lagrangian supersymmetric gauge theories, including SYM, where it admits a remarkably compact expression in terms of superfields.

Paper Structure

This paper contains 16 sections, 110 equations, 1 figure.

Table of Contents

  1. Introduction and Summary
  2. The Holomorphic Twist and Corrections to Q from Anomalies
  3. Semi-Chiral Superfields and Q-Cohomology
  4. Corrections to Q from Anomalies
  5. Warmup: The Generalized Konishi Anomaly and the Chiral Ring
  6. Anomalies as Higher Operations
  7. Computing Loop Corrections in the Holomorphic Twist
  8. Lagrangian SUSY Gauge Theories
  9. One-Loop Supercharge
  10. N=1 Super Yang-Mills
  11. N=2 Super Yang-Mills
  12. N=4 Super Yang-Mills
  13. Loop Corrections in the Planar Limit
  14. The One-Loop Master Integral
  15. One-Loop Supercharges on Derivatives
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Laman graphs are the only graphs which can appear in the computation of brackets $\{\,\cdot\,,\dots,\,\cdot\,\}_0$ in a theory with action \ref{['eq:FirstOrderAction']}. Even then, the contributions of some of these graphs to the action of $\bm{Q}$ may be ruled out by the interaction $\mathop{\mathrm{\mathcal{I}}}\nolimits$. The propagator, the one-loop, the two-loop, and 3 three-loop Laman graphs are drawn above.