Triangle Anomalies from Einstein Manifolds

TL;DR

This work derives a geometric, metric-free formula for five-dimensional Chern-Simons couplings $c_{IJK}$ in Type IIB on ${ ext{AdS}}_5 imes X$, with $c_{IJK}=rac{N^2}{2}\int_X oldsymbol{ extomega}_{igrace I}\wedge ext{iota}_{oldsymbol{k}_J}oldsymbol{ extomega}_{Kigrace}$. It demonstrates that, for Sasaki–Einstein $X$, these couplings—and hence the boundary triangle anomalies—can be computed from topological data alone, and checks them against dual quiver gauge theories for circle bundles over del Pezzo surfaces and toric SE manifolds. The paper shows that in toric cases $c_{IJK}=rac{N^2}{2}ig| ext{det}(k_I,k_J,k_K)ig|$, equivalent to the area of a toric triangle, and that the same results hold in del Pezzo geometries via Higgsing/dibaryon flows, connecting giant gravitons to dibaryon operators. The findings illuminate a unifying picture where dibaryon condensation drives a flow among Sasaki–Einstein vacua, while preserving the Chern–Simons data that matches the CFT anomalies, with implications for $a$-maximization and $Z$-minimization in AdS/CFT.

Abstract

The triangle anomalies in conformal field theory, which can be used to determine the central charge a, correspond to the Chern-Simons couplings of gauge fields in AdS under the gauge/gravity correspondence. We present a simple geometrical formula for the Chern-Simons couplings in the case of type IIB supergravity compactified on a five-dimensional Einstein manifold X. When X is a circle bundle over del Pezzo surfaces or a toric Sasaki-Einstein manifold, we show that the gravity result is in perfect agreement with the corresponding quiver gauge theory. Our analysis reveals an interesting connection with the condensation of giant gravitons or dibaryon operators which effectively induces a rolling among Sasaki-Einstein vacua.

Figures (5)

• Figure 1: Construction of $\omega_I$. The polygon designates the image of the moment map. The red blob $S$ is the support of $\mathcal{F}$ and the blue region $R_I$ is the support of $\mathcal{A}_I$.
• Figure 2: Pictorial representation of the toric formula $c_{IJK}=\frac{N^2}{2} \left| \det(k_I, k_J,k_K) \right|$.
• Figure 3: A generic toric diagram with four corners, i.e. a generic $L^{p,q|r}$, and the associated $(p,q)$-web. We have $s= p+q-r$. The integers $a$ and $b$ are such that $as-bp=q$.
• Figure 4: An example of "folded quiver." From a generic toric diagram with four nodes we can immediately compute the multiplicities of $6$ sets of bifundamental fields.
• Figure 5: Schematic depiction of the dibaryon condensation. Each edge corresponds to a three-cycle in the toric Sasaki-Einstein around which D3-branes can be wrapped. Higgsing with the corresponding dibaryon operator in the quiver CFT eliminates that edge.