Four Lectures On The Gauge/Gravity Correspondence

TL;DR

These lectures survey non-conformal extensions of the gauge/gravity correspondence in four dimensions, focusing on two concrete realizations: fractional D-branes on orbifolds and D-branes wrapped on Calabi–Yau cycles. They demonstrate how supergravity backgrounds encode perturbative gauge data such as running couplings and anomalies, and in some cases non-perturbative effects like gaugino condensation, via holographic fields such as twisted scalars and the gaugino condensate proxy a(ρ). The discussion highlights singularities (e.g., enhançon) and geometric transitions that resolve them, and shows how different dual pictures (MN, KS, Vafa-type setups) fit into a unified duality web across string/M-theory frameworks. A central caveat is that decoupling is not clean in these non-conformal settings, so a complete non-perturbative duality remains challenging, with potential mixing between gauge and Kaluza-Klein/stringy degrees of freedom in the deep IR.

Abstract

We review in a pedagogical manner some of the efforts aiming to extend the gauge/gravity correspondence to non-conformal supersymmetric gauge theories in four dimensions. After giving a general overview, we discuss in detail two specific examples: fractional D-branes on orbifolds and D-branes wrapped on supersymmetric cycles of Calabi-Yau spaces. We explore in particular which gauge theory information can be extracted from the corresponding supergravity solutions, and what the remaining open problems are. We also briefly explain the connection between these and other approaches, such as fractional branes on conifolds, branes suspended between branes, M5-branes on Riemann surfaces and M-theory on G2-holonomy manifolds, and discuss the role played by geometric transitions in all that.

Figures (6)

• Figure 1: A freely moving D-brane on the most simple orbifold as it appears in the covering space. The brane $D$ and its image $D'$ are identified in the physical space. The $X$ axis represents the orbifold directions (these being all transverse to the brane), the $Y$ axis the flat longitudinal directions. To see the flat transverse directions... we would need an extra dimension! But they are there, of course.
• Figure 2: A fractional D-brane on the most simple orbifold as it appears in the covering space. The brane does not have any image in this case and it is stuck at the orbifold fixed plane.
• Figure 3: A configuration of D4-branes stretched between two parallel NS5-branes. The $x_6$ direction is compact, this meaning that the first and the third NS5-brane in the picture are identified. One can have as well D4-branes stretched from the second NS5-brane two the third. These would couple to the NS5-brane world volume fields with equal strength as the former, but with opposite sign.
• Figure 4: The ${\cal N}=2$ duality web. On the left hand side up we have D4-branes stretched between two NS5-branes. By performing a T-duality along $x_6$ we get fractional D3-branes on the orbifold $\,\hbox{${\rm C}$}_2/{ Z Z}_2$. The distance $L$ between the NS5-branes translates into the (background) value of the $B_{(2)}$ flux. Performing a T-duality along $x_7$, instead, we get D5-branes wrapped on a two-sphere inside an $A_1$ ALE space. The distance $L$ between the NS5-branes becomes now the (background) value of the volume of the two-sphere. Finally, performing a S-duality (left hand side down) we end up in M-theory with a M5-brane wrapped on a Riemann surface. All these different configurations describe the same physics at low energy: four-dimensional ${\cal N}=2$ SYM.
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