Baryonic branches and resolutions of Ricci-flat Kahler cones

TL;DR

The paper develops a geometric framework for baryonic deformations of AdS/CFT duals, showing that turning on VEVs for baryonic operators corresponds to (partial) resolutions of Calabi–Yau cones and yields warped geometries that interpolate between UV AdS$_5\times Y$ and IR AdS$_5\times Z$ throat regions. Central to the approach is the Green’s-function warp factor on AC Ricci-flat Kähler manifolds and the use of Euclidean D3-branes to probe baryonic condensates, with scaling dimensions $Δ(Σ)=\dfrac{Nπ\mathrm{vol}(Σ)}{2\mathrm{vol}(Y)}$ and charges tied to harmonic 3-forms on $Y$. The Y^{p,q} theories provide explicit realizations, where partial resolutions I/II and canonical resolutions map to Higgsing patterns in the corresponding quiver gauge theories, yielding IR orbifold theories such as $\mathbb{C}^3/\mathbb{Z}_n$ and $\mathcal{N}=4$ or $\mathcal{N}=2$ limits, in agreement with the dual supergravity solutions. The results underscore a coherent picture of RG flows in non-compact Calabi–Yau backgrounds and offer practical criteria for when condensates vanish, guided by toric (pq-web) descriptions. The work lays groundwork for a full treatment of background fluxes and torsion data in the baryonic sector (to be pursued in follow-up work).

Abstract

We consider deformations of N=1 superconformal field theories that are AdS/CFT dual to Type IIB string theory on Sasaki-Einstein manifolds, characterised by non-zero vacuum expectation values for certain baryonic operators. Such baryonic branches are constructed from (partially) resolved, asymptotically conical Ricci-flat Kahler manifolds, together with a choice of point where the stack of D3-branes is placed. The complete solution then describes a renormalisation group flow between two AdS fixed points. We discuss the use of probe Euclidean D3-branes in these backgrounds as a means to compute expectation values of baryonic operators. The Y^{p,q} theories are used as illustrative examples throughout the paper. In particular, we present supergravity solutions describing flows from the Y^{p,q} theories to various different orbifold field theories in the infra-red, and successfully match this to an explicit field theory analysis.

Figures (21)

• Figure 1: Toric diagram of a $Y^{p,q}$ singularity, with internal point $(s,s)$ shown. Here $0<s<p$.
• Figure 2: On the left: pq-web with D3-branes at a toric singularity. On the right: a partially resolved geometry, with D3-branes localised at a residual singularity. If a toric divisor $D$ asymptotic to $C(\Sigma)$ intersects the point-like D3-branes, the corresponding baryonic operators do not acquire a VEV. On the other hand, toric divisors $D$ that do not intersect the D3-branes may give rise to a condensate, as denoted by the shaded region.
• Figure 3: The pq-webs for the cone $C(Y^{p,q})$ and its two small partial resolutions.
• Figure 4: pq-web of a canonical orbifold resolution of a $C(Y^{p,q})$ singularity. The quadrangle represents the compact divisor $D_5$, which is the Fano orbifold $M$. The four non-compact divisors $D_a=\{z_a=0\}$, $a=1,\dots,4$, are the total spaces of the orbifold line bundles ${\cal O}_{\mathbb{WCP}^1_{[p-s,s]}}(-p)$, ${\cal O}_{\mathbb{CP}^1}(-p-q)$, ${\cal O}_{\mathbb{WCP}^1_{[p-s,s]}}(-p)$, ${\cal O}_{\mathbb{CP}^1}(-p+q)$, respectively. Slightly more precisely, these are all Kähler quotients of $\mathbb{C}^3$ by the $U(1)$ actions with weights $(p-s,s,-p)$, $(p-s,p-s,-p-q)$, $(p-s,s,-p)$, $(s,s,-p+q)$, respectively, and with positive moment map level.
• Figure 5: On the left hand side: a $Y^{4,2}$ quiver diagram in the toric phase that we adopt in this paper. On the right hand side: a $Y^{4,2}$ quiver in a different toric phase. The two are related by Seiberg duality.