Classical geometry and gauge duals for fractional branes on ALE orbifolds

TL;DR

The paper develops the classical geometry and gauge-theory interpretation for fractional D3-branes on ADE-like ALE orbifolds $\mathbb{C}^2/\Gamma$, showing that twisted scalars encode the perturbative running of the world-volume $\mathcal{N}=2$ gauge couplings via $\tau_I(z)$ and that IR singularities are regulated by an enhançon mechanism. A explicit supergravity solution is constructed, featuring logarithmic running of twisted fields $\gamma_I(z)$ and a non-constant RR five-form flux $\Phi_5(\rho)$ sourced by twisted fields and brane charges, with the warp factor $H$ solving a Poisson equation on the ALE space. The analysis connects gravity to the gauge theory through open/closed string duality, demonstrating that the probe-brane kinetic terms reproduce the perturbative prepotential and that instanton effects may arise from fractional D-instantons. The work highlights the limitations of extending the gravity description to below the enhançon scales for generic orbifolds and discusses the (non-)existence of a cascade in multi-factor $\mathcal{N}=2$ theories, contributing to the broader understanding of non-conformal gauge/gravity dualities in string theory.

Abstract

We investigate the classical geometry corresponding to a collection of fractional D3 branes in the orbifold limit of an ALE space. We discuss its interpretation in terms of the world-volume gauge theory on the branes, which is in general a non conformal N=2 Yang-Mills theory with matter. The twisted fields reproduce the perturbative behaviour of the gauge theory. We regulate the IR singularities for both twisted and untwisted fields by means of an enhancon mechanism qualitatively consistent with the gauge theory expectations. The five-form flux decreases logarithmically towards the IR with a coefficient dictated by the gauge theory beta-functions.

Figures (4)

• Figure 1: The dual interpretation of a fractional brane configuration corresponding, on the one hand, to a particular corner of the Coulomb phase moduli space in the gauge theory and, on the other hand, to a single brane probe in the classical supergravity background
• Figure 2: For the $\mathbb{C}^2/\mathbb{Z}_2$ orbifold, below the enhançon $\Lambda_1$, where the fields $b_I$ assume non-BPS values (indicated by a filled circle) one can, by unit shifts to the values indicated by an empty circle, reach a BPS regime.
• Figure 3: For the $\mathbb{C}^2/\mathbb{Z}_3$ orbifold, immediately below the first enhançon $\Lambda_1$, the non-BPS values (filled circles) cannot be turned by unit shifts (leading, e.g., to the values indicated by the empty circles) all into positive values.
• Figure 4: Level curves of the function $H(x,y)$. See the text for the meaning of the special points indicated in the figure.