Futaki Invariants and Reflexive Polygons

TL;DR

This paper systematically computes Futaki invariants for a large class of 4d N=1 supersymmetric gauge theories derived from D3-branes on Calabi-Yau 3-folds with reflexive polygons, revealing new bounds and relationships with geometric quantities.

Contribution

It provides a comprehensive analysis of Futaki invariants for theories associated with reflexive polygons, offering new bounds and insights into their geometric and topological constraints.

Findings

Distribution and bounds of Futaki invariants are characterized.

Futaki invariants relate to volume, Chern, and Euler numbers.

The study enhances understanding of stability conditions in moduli spaces.

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.