Abstract
The low energy effective theory on a stack of D3-branes at a Calabi-Yau singularity is an $ \mathcal{N} $ = 1 quiver gauge theory. The AdS/CFT correspondence predicts that the strong coupling dynamics of the gauge theory is described by weakly coupled type IIB supergravity on AdS 5 × L 5 , where L 5 is a Sasaki-Einstein manifold. Recent results on Calabi-Yau algebras efficiently determine the Hilbert series of any superconformal quiver gauge theory. We use the Hilbert series to determine the volume of the horizon manifold in terms of the fields of the quiver gauge theory. One corollary of the AdS/CFT conjecture is that the volume of the horizon manifold L 5 is inversely proportional to the a-central charge of the gauge theory. By direct comparison of the volume determined from the Hilbert series and the a-central charge, this prediction is proved independently of the AdS/CFT conjecture.
Highlights
Functions ensures that the volume of the Sasaki-Einstein manifold is inversely proportional to the central charge a, Vol(L5)
We review the gauge theories associated to C3 and the conifold and show how the Hilbert series correctly determines the volume of their horizon manifolds
We have established the equivalence of a-maximization and volume minimization for AdS5× L5 compactifications where L5 is Sasaki-Einstein whenever the quiver gauge theory is known
Summary
The world-volume gauge theory on a stack of D3-branes at a Calabi-Yau singularity is often described by a quiver gauge theory. Quiver gauge theories describing the low energy effective field theory of D-branes at a Calabi-Yau singularity have a variant of the Green-Schwarz mechanism to cancel the anomalous U(1)’s. The gauge fields of the anomalous U(1)’s couple to RR-form fields giving them Stuckelberg masses [24,25,26] These massive vector fields decouple in the IR. At a conformal fixed point in the infrared, we expect the NSVZ 1-loop exact beta functions of the gauge groups SU(nv) and couplings λl to vanish. These constraints are β1/gv2 = 0 βλl = 0. The last condition implies that at a superconformal fixed point, every term in the superpotential has total R-charge 2
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