Free Energy vs Sasaki-Einstein Volume for Infinite Families of M2-Brane Theories

TL;DR

This paper explores the relationship between free energy and Sasaki-Einstein volume in infinite families of 3d N=2 superconformal theories on M2-branes, introducing new algorithms and formulas to analyze their properties.

Contribution

It develops a lifting algorithm from CY_3 to CY_4 and introduces a systematic method to relate toric diagram symmetries to R-charge constraints, enhancing analysis of these theories.

Findings

Explicit correspondence between free energy and volume before extremization.

A new lifting algorithm from CY_3 to CY_4 models.

F^2 expressed as a quartic function in R-charges, suggesting a universal formula.

Abstract

We investigate infinite families of 3d N=2 superconformal Chern-Simons quivers with an arbitrarily large number of gauge groups arising on M2-branes over toric CY_4's. These theories have the same matter content and superpotential of those on D3-branes probing cones over L^{a,b,a} Sasaki-Einstein manifolds. For all these infinite families, we explicitly show the correspondence between the free energy F on S^3 and the volume of the 7-dimensional base of the associated CY_4, even before extremization. Our results add to those existing in the literature, providing further support for the correspondence. We develop a lifting algorithm, based on the Type IIB realization of these theories, that takes from CY_3's to CY_4's and we use it to efficiently generate the models studied in the paper. We also introduce a procedure, based on the mapping between extremal points in the toric diagram (GLSM fields) and chiral fields in the quiver, which systematically translates symmetries of the toric diagram into constraints of the trial R-charges of the quiver, beyond those arising from marginality of the superpotential. This method can be exploited for reducing the dimension of the space of trial R-charges over which the free energy is maximized. Finally, we show that in all the infinite families we consider F^2 can be expressed, even off-shell, as a quartic function in R-charges associated to certain 5-cycles. This suggests that a quartic formula on R-charges, analogous to a similar cubic function for the central charge a in 4d, exists for all toric toric CY_4's and we present some ideas regarding its general form.