Gauge theories from wrapped and fractional branes

TL;DR

This work analyzes two gauge/gravity realizations of the same nonconformal 3D ${N}=4$, ${D}=2+1$ SYM: D4-branes wrapped on a two-cycle inside a Calabi–Yau and fractional D2/D6-branes on ${\mathbb R}^4/{\mathbb Z}_2$. Using the Maldacena–Núñez approach and probe brane techniques, the authors extract perturbative running couplings and moduli-space metrics, showing that a single geometric quantity—the cycle's stringy volume—controls the running in both cases. In both constructions the moduli space metric takes a hyperkähler Taub–NUT form, but the solutions exhibit an enhançon where the supergravity description breaks down, signaling the need for additional (instanton) degrees of freedom to capture nonperturbative physics. They unite the wrapped and fractional brane pictures through a master formula $\frac{1}{g^2_{YM}(\mu)} = \frac{V_{ST}(\Sigma_2)}{g^2_{Dp}}$, with $V_{ST}(\Sigma_2)$ encoding the cycle geometry and $B$-field, and show that the two pictures reduce to the same perturbative data in their respective limits.

Abstract

We compare two applications of the gauge/gravity correspondence to a non conformal gauge theory, based respectively on the study of D-branes wrapped on supersymmetric cycles and of fractional D-branes on orbifolds. We study two brane systems whose geometry is dual to N=4, D=2+1 super Yang-Mills theory, the first one describing D4-branes wrapped on a two-sphere inside a Calabi-Yau two-fold and the second one corresponding to a system of fractional D2/D6-branes on the orbifold R^4/Z_2. By probing both geometries we recover the exact perturbative running coupling constant and metric on the moduli space of the gauge theory. We also find a general expression for the running coupling constant of the gauge theory in terms of the "stringy volume" of the two-cycle which is involved in both types of brane systems.