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Futaki Invariants and Reflexive Polygons

Jiakang Bao, Eugene Choi, Yang-Hui He, Rak-Kyeong Seong, Shing-Tung Yau

TL;DR

This work analyzes Futaki invariants as a diagnostic for K-stability of mesonic moduli spaces arising from 4d $\mathcal{N}=1$ gauge theories on D3-branes probing toric Calabi–Yau cones with reflexive toric diagrams. By computing $F(\mathcal{X}; \zeta_R, \eta)$ and $F(\mathcal{X}; \zeta_p, \eta)$ across the 16 reflexive polygons, it demonstrates robust bounds linking Futaki invariants to geometric/topological data including $V_{min}$, Euler $\chi$, and the first Chern number $C_1$, with a key bound $F(\mathcal{X}; \zeta_p, \eta) \le \tfrac{8}{27} F(\mathcal{X}; \zeta_R, \eta)$ saturating for select orbifold cases. The analysis provides explicit connections between Hilbert-series coefficients $A_0,A_1$ and physical quantities via Reeb minimization and divisor volumes, and reveals how higher Laurent coefficients $A_2,A_3$ relate to integrated curvatures, hinting at generalized stability notions. A concrete example, the $L_{1,3,1}/\mathbb{Z}_2$ model, illustrates the methodology and the interplay between $R$-charges, divisors, and Futaki invariants. Overall, the results illuminate how topological and geometric invariants constrain stability conditions of moduli spaces, with implications for identifying IR SCFTs and guiding extensions of K-stability beyond its classical formulation.

Abstract

Futaki invariants of the classical moduli space of 4d N=1 supersymmetric gauge theories determine whether they have a conformal fixed point in the IR. We systematically compute the Futaki invariants for a large family of 4d N=1 supersymmetric gauge theories coming from D3-branes probing Calabi-Yau 3-fold singularities whose bases are Gorenstein Fano surfaces. In particular, we focus on the toric case where the Fano surfaces are given by the 16 reflexive convex polygons and the moduli spaces are given by the corresponding toric Calabi-Yau 3-folds. We study the distribution of and conjecture new bounds on the Futaki invariants with respect to various topological and geometric quantities. These include the minimum volume of the Sasaki-Einstein base manifolds as well as the Chern and Euler numbers of the toric Fano surfaces. Even though the moduli spaces for the family of theories studied are known to be K-stable, our work sheds new light on how the topological and geometric quantities restrict the Futaki invariants for a plethora of moduli spaces.

Futaki Invariants and Reflexive Polygons

TL;DR

This work analyzes Futaki invariants as a diagnostic for K-stability of mesonic moduli spaces arising from 4d gauge theories on D3-branes probing toric Calabi–Yau cones with reflexive toric diagrams. By computing and across the 16 reflexive polygons, it demonstrates robust bounds linking Futaki invariants to geometric/topological data including , Euler , and the first Chern number , with a key bound saturating for select orbifold cases. The analysis provides explicit connections between Hilbert-series coefficients and physical quantities via Reeb minimization and divisor volumes, and reveals how higher Laurent coefficients relate to integrated curvatures, hinting at generalized stability notions. A concrete example, the model, illustrates the methodology and the interplay between -charges, divisors, and Futaki invariants. Overall, the results illuminate how topological and geometric invariants constrain stability conditions of moduli spaces, with implications for identifying IR SCFTs and guiding extensions of K-stability beyond its classical formulation.

Abstract

Futaki invariants of the classical moduli space of 4d N=1 supersymmetric gauge theories determine whether they have a conformal fixed point in the IR. We systematically compute the Futaki invariants for a large family of 4d N=1 supersymmetric gauge theories coming from D3-branes probing Calabi-Yau 3-fold singularities whose bases are Gorenstein Fano surfaces. In particular, we focus on the toric case where the Fano surfaces are given by the 16 reflexive convex polygons and the moduli spaces are given by the corresponding toric Calabi-Yau 3-folds. We study the distribution of and conjecture new bounds on the Futaki invariants with respect to various topological and geometric quantities. These include the minimum volume of the Sasaki-Einstein base manifolds as well as the Chern and Euler numbers of the toric Fano surfaces. Even though the moduli spaces for the family of theories studied are known to be K-stable, our work sheds new light on how the topological and geometric quantities restrict the Futaki invariants for a plethora of moduli spaces.

Paper Structure

This paper contains 31 sections, 6 theorems, 98 equations, 19 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Nomenclature
  3. Background
  4. Toric Varieties and Reflexive Polytopes
  5. Hilbert Series and the Mesonic Moduli Space
  6. Grading and Fugacities.
  7. Plethystics.
  8. Laurent Expansion.
  9. The Sasaki-Einstein Base.
  10. Minimized Volumes and Topological Invariants
  11. Volume Function and Minimization.
  12. Topological Invariants and Volume Bounds.
  13. Divisor Volumes
  14. Hilbert Series and Volume functions for Divisors.
  15. $U(1)_R$ Charges and Divisor Volumes.
  16. ...and 16 more sections

Key Result

Corollary 15

Under a general test symmetry $\eta=\eta_{(w_1, \dots, w_k)}$ giving weights $(w_1, \dots, w_k)\in \mathbb{Z}_{\geq 0}^k$ to generators $(x_1,\dots,x_k)$ of $\mathcal{X}$, the corresponding Futaki invariant takes the form,

Figures (19)

  • Figure 1: The 16 reflexive polygons in $\mathbb{Z}^2$. The polygons are arranged in such a way that horizontally we have the number of extremal vertices in the polygons and vertically we have the normalized area of the polygons. Each reflexive polygon is gives rise to a toric Calabi-Yau 3-fold which is associated to at least one $4d$$\mathcal{N}=1$ supersymmetric gauge theory Hanany:2012hi.
  • Figure 2: The toric diagram for $L_{1, 3, 1}/\mathbb{Z}_{2}$ with orbifold action $(0,1,1,1)$Hanany:2012hi.
  • Figure 3: The quiver for the $4d$$\mathcal{N}=1$ supersymmetric gauge theory (phase a) corresponding to $L_{1, 3, 1}/\mathbb{Z}_{2}$ with orbifold action $(0,1,1,1)$Hanany:2012hi.
  • Figure 4: Futaki invariants $F(\mathcal{X}_a, \zeta_R, \eta_h)$ [$F_R$] against $F(\mathcal{X}_a, \zeta_p, \eta_h)$$[F_p]$, where $a=1, \dots, 16$ labels the 16 reflexive polygons and their corresponding toric Calabi-Yau 3-folds $\mathcal{X}_a$, and $h=1, \dots, k_a$ labels the generators $x_h$ for a given $\mathcal{X}_a$.
  • Figure 5: The minimum volume $V_{min} = V(b_i^*; Y_a)$ of the Sasaki-Einstein $5$-manifold $Y_a$ associated to the toric Calabi-Yau 3-fold $\mathcal{X}_a$ with one of the 16 reflexive polygons as its toric diagram, plotted against the Futaki invariants $F(\mathcal{X}_a; \zeta_R, \eta_h)$ for all generators $x_h$ corresponding to $\mathcal{X}_a$.
  • ...and 14 more figures

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 6
  • Definition 11
  • Definition 13
  • Corollary 15
  • Definition 16
  • Definition 17
  • ...and 5 more