D2-brane Chern-Simons theories: F-maximization = a-maximization

TL;DR

The paper links 3d N=2 Chern-Simons quivers, realized on N D2-branes at Calabi–Yau singularities with a Romans mass, to a 4d N=1 parent through a D-brane reduction and a massive IIA AdS4 dual. Using localization, it shows the large-N free energy scales as F(R) ∝ (nN)^{1/3} a(R)^{2/3}, establishing that F-maximization in 3d is equivalent to a-maximization in the 4d parent; this leads to an exact match with the holographic free energy from the AdS4 massive IIA solution and with the Wilson loop VEVs. The field-theory result expresses the dependence on the 4d a-function via a precise relation with Vol(Y), and the gravity analysis confirms the leading large-N scaling, providing a concrete 3d/4d duality in this class of Calabi–Yau singularities. Overall, the work extends the a-maximization paradigm to 3d CS theories, linking extremization principles across dimensions and confirming holographic predictions in massive IIA setups.

Abstract

We study a system of N D2-branes probing a generic Calabi-Yau three-fold singularity in the presence of a non-zero quantized Romans mass n. We argue that the low-energy effective N = 2 Chern-Simons quiver gauge theory flows to a superconformal fixed point in the IR, and construct the dual AdS_4 solution in massive IIA supergravity. We compute the free energy F of the gauge theory on S^3 using localization. In the large N limit we find F = c(nN)^{1/3}a^{2/3}, where c is a universal constant and a is the a-function of the "parent" four-dimensional N = 1 theory on N D3-branes probing the same Calabi-Yau singularity. It follows that maximizing F over the space of admissible R-symmetries is equivalent to maximizing a for this class of theories. Moreover, we show that the gauge theory result precisely matches the holographic free energy of the supergravity solution, and provide a similar matching of the VEV of a BPS Wilson loop operator.