J. Bao

E. Choi

Y. He

R. Seong

S. Yau

We compute Futaki invariants for gauge theories from D3-branes that probe toric Calabi-Yau singularities arising from reflexive polytopes. 

Futaki invariants form a bridge between algebraic geometry and supersymmetric field theory. Here, we explore 4d N = 1 gauge theories from D3-branes probing Calabi–Yau 3-fold singularities with Gorenstein Fano bases. Studying all 16 toric reflexive polygons, we uncover correlations between Futaki invariants, Sasaki–Einstein volumes and topological quantities such as Chern and Euler numbers, revealing new geometric bounds.

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