[Paper Review] K stability and stability of chiral ring

Tristan C. Collins, Dan Xie+1 more|arXiv (Cornell University)|Jun 29, 2016

Geometry and complex manifolds29 references15 citations

TL;DR

This paper introduces a stability condition for chiral rings in 4D $σ=1$ superconformal field theories (SCFTs) using test chiral rings and generalized $a$-maximization. It proves that K-stability of the associated 3-fold singularity—ensuring a Ricci-flat conical metric—equivalently characterizes the stability of the chiral ring, providing a geometric criterion for when a chiral ring arises from an SCFT.

ABSTRACT

We define a notion of stability for chiral ring of four dimensional N=1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N=1 field theory derived from D3 branes probing a three-fold singularity X, and show that the K stability which implies the existence of Ricci-flat conic metric on X is equivalent to the stability of chiral ring of the corresponding field theory.

Motivation & Objective

• To define a stability condition for chiral rings in 4D $σ=1$ field theories that determines whether they originate from a superconformal field theory (SCFT).
• To address the ambiguity in chiral ring structure under renormalization group flow, where the UV chiral ring may differ from the IR SCFT chiral ring due to unitarity violations or irrelevant superpotentials.
• To establish a geometric field-theoretic correspondence by linking the stability of the chiral ring to K-stability of the 3-fold singularity in D3-brane compactifications.
• To provide a systematic method—generalized $a$-maximization and test chiral rings—to detect when chiral operators become free or superpotential terms become irrelevant in the IR.

Proposed method

• Define a test chiral ring as the flat limit of a 1-parameter $\mathrm{U}(1)$-action (generated by a symmetry $\eta$) on the original chiral ring $\mathcal{R}$, using the initial terms of relations under the $\eta$-weight decomposition.
• Introduce generalized $a$-maximization over test chiral rings to compute the central charge $a$ and identify the maximal $a$-value, which corresponds to the IR SCFT.
• Use the Futaki invariant from K-stability of the 3-fold singularity $X$ to compute the stability condition for the chiral ring, with the vanishing of the Futaki invariant indicating K-stability.
• Relate the stability of the chiral ring to the existence of a Ricci-flat conical metric on the 3-fold singularity $X$, via the equivalence of K-stability and chiral ring stability.
• Apply the framework to D3-brane probes of 3-fold singularities, such as $z_0^2 + z_1^2 + z_2^p + z_3^q = 0$, to derive explicit bounds on parameters $p, q$ for stability.
• Use the test configuration with charge $\eta = (0,0,1,-1/q)$ to compute the flat limit and the resulting chiral ring $z_0^2 + z_1^2 + z_2 z_3^q = 0$, which is the IR chiral ring when unstable.

Experimental results

Research questions

• RQ1 When is a given chiral ring $\mathcal{R}$ the chiral ring of a 4D $\mathcal{N}=1$ superconformal field theory?
• RQ2 How can one systematically detect when chiral operators become free or superpotential terms become irrelevant in the IR, leading to a different chiral ring?
• RQ3 What is the geometric condition on the 3-fold singularity $X$ probed by D3-branes that ensures the associated chiral ring is stable and corresponds to an SCFT?
• RQ4 Is there a one-to-one correspondence between K-stability of the singularity $X$ and the stability of the chiral ring of the corresponding field theory?
• RQ5 Can the chiral ring of the IR SCFT be reconstructed from the destabilizing test configuration of the UV chiral ring?

Key findings

• The chiral ring of a 4D $\mathcal{N}=1$ SCFT is stable if and only if it satisfies the proposed stability condition based on test chiral rings and generalized $a$-maximization.
• For the singularity $z_0^2 + z_1^2 + z_2^p + z_3^q = 0$, the chiral ring is unstable when $q < 34/11$ for $p=6$, despite satisfying the unitarity bound $5/2 < q < 10$, indicating a stronger dynamical instability.
• The IR chiral ring for $q=3$ and $p=6$ is $z_0^2 + z_1^2 + z_2 z_3^3 = 0$, which is a 3D quotient singularity and corresponds to the central fiber of the destabilizing test configuration.
• The Futaki invariant computed from the test configuration with charge $\eta = (0,0,1,-1/q)$ vanishes if and only if $q > (p^2 - 1)/(2p - 1)$, which is the K-stability condition.
• K-stability of the 3-fold singularity $X$ is equivalent to the stability of the chiral ring of the D3-brane field theory, establishing a geometric-field-theoretic duality.
• The method explains known phenomena: $a$-maximization selects the correct $R$-charge, operators violating the unitarity bound become free, and irrelevant superpotentials are dropped in the IR.