F-Maximization and Accidental Symmetries

In a three-dimensional $N=2$ SCFT, the superconformal $U(1)_R$ current can mix with every Abelian flavor current. F-maximization determines that mixing from the localized round-three-sphere partition function: after counterterm phases are separated, the exact R-current locally maximizes $\mathcal F=-\log|Z_{S^3}|$. The method is exact only when the trial-current space includes accidental symmetries and the matrix integral remains on the declared contour.

Required background. Use the theory’s monopole operators and quantum Coulomb branch to list all chiral operators, and derive the input from the sphere matrix model.

Helpful background. Four-dimensional a-maximization with accidental symmetries is structurally similar, while linearized RG flow clarifies which mixing directions are physical.

Trial R-currents and the sphere functional

Choose a reference R-current $R_0$ and a basis $F_I$ of Abelian flavor currents that commute with the preserved supercharge. The most general trial current is

$$R(t)=R_0+\sum_I t_I F_I.$$

For a chiral field $\Phi_i$ with flavor charges $q_i^I$, its trial dimension is

$$\Delta_i(t)=R_0(\Phi_i)+\sum_I t_Iq_i^I.$$

Every superpotential monomial must have trial R-charge two. Gauge topological currents are ordinary flavor currents in three dimensions and must be included when allowed; monopole superpotentials impose additional linear constraints.

Insert $\Delta_i(t)$ into the localized matrix integral, keeping the physical real Cartan cycle and fixed background contact terms. Define

$$\mathcal F(t)=-\log\left|Z_{S^3}(t)\right|.$$

At a unitary fixed point without an omitted accidental current,

$$\left.\frac{\partial\mathcal F}{\partial t_I}\right|_{t=t_*}=0, \qquad \left.\frac{\partial^2\mathcal F}{\partial t_I\partial t_J}\right|_{t=t_*} =-\frac{\pi^2}{2}\tau_{IJ}.$$

The matrix $\tau_{IJ}$ is the positive flavor-current two-point coefficient, so the Hessian is negative definite on genuine mixing directions. This both proves local maximization and supplies an independent normalization check Jafferis 2012, §§3–4; Closset et al. 2012, §4.

Free chiral benchmark

For one free chiral multiplet, the round-sphere answer is

$$Z(\Delta)=e^{\ell(1-\Delta)}, \qquad \mathcal F(\Delta)=-\ell(1-\Delta),$$

with $\ell'(z)=-\pi z\cot(\pi z)$. Therefore

$$\mathcal F'(\Delta)=\ell'(1-\Delta),$$

which vanishes at $\Delta=1/2$. A second derivative gives

$$\mathcal F''\!\left(\frac12\right)=-\frac{\pi^2}{2}<0,$$

and

$$\mathcal F\!\left(\frac12\right)=\frac12\log2.$$

This example checks the sign of the extremization and saturates the scalar chiral-primary unitarity bound $\Delta\ge1/2$.

As a second elementary check, a symmetric Wess–Zumino model with $W=XYZ$ obeys $\Delta_X+\Delta_Y+\Delta_Z=2$. Permutation symmetry then fixes the stationary point at $\Delta_X=\Delta_Y=\Delta_Z=2/3$; the Hessian on the two independent flavor directions must be negative.

Gauge theories and monopole coordinates

For a gauge theory, $Z_{S^3}(t)$ is an integral, not a product of independent free determinants. The correct order of operations is

$$\text{choose }t \longrightarrow \text{evaluate the fixed-contour integral} \longrightarrow \text{take }-\log|Z| \longrightarrow \text{differentiate or fit}.$$

Extremizing the integrand before integration is generally wrong. The large-Cartan asymptotics must be checked throughout the trial domain because changing $\Delta_i$ can destroy absolute convergence or move poles across a deformed cycle.

Bare monopole operators can be the first operators to approach the unitarity bound. Their trial dimensions receive one-loop contributions from charged fermion zero modes. In a common normalization,

$$\Delta(T_{\mathfrak m}) =-\frac12\sum_{\alpha\in\mathfrak g}|\alpha(\mathfrak m)| +\frac12\sum_i(1-\Delta_i) \sum_{\rho_i\in R_i}|\rho_i(\mathfrak m)|,$$

before adding flavor-charge mixing. The allowed magnetic charges $\mathfrak m$ depend on the global gauge group. Checking only polynomial gauge invariants can therefore miss an accidental symmetry.

Decoupled operators and accidental currents

If a gauge-invariant chiral operator would have $\Delta<1/2$, the proposed interacting fixed point cannot be the whole answer. At the bound the operator becomes a free chiral multiplet and generates an accidental $U(1)$ current that was absent from the ultraviolet trial basis.

A controlled treatment has four steps:

1. enumerate elementary, composite, and monopole chiral operators in the candidate region;
2. identify every operator at or below $\Delta=1/2$;
3. separate the free multiplet—often by an equivalent flipping-field description—and add its emergent current to the mixing problem;
4. extremize the interacting factor again and verify all remaining operators.

One must not simply clamp the offending trial dimension to $1/2$ while leaving the rest of the functional unchanged. The decoupled sector and its accidental current alter the space over which extremization is performed.

Complex phases, contours, and uncertainty

Quantized background Chern–Simons terms multiply $Z$ by phases. F-maximization uses $-\log|Z|$, not an arbitrary branch of the complex logarithm. The phase remains useful for contact terms but is not part of the maximized real functional.

For a numerical extremization, report:

• the exact integrand, contour, precision, and tail treatment;
• linear superpotential constraints and the trial domain;
• the stationary point and all eigenvalues of its Hessian;
• convergence under increased precision and integration range;
• the smallest dimensions of ordinary and monopole chiral operators;
• any free factors removed and the description used to expose them.

A nearly zero Hessian eigenvalue can signal a redundant current, a conformal-manifold direction not described by R-mixing, a missed accidental symmetry, or inadequate numerical precision. It is not evidence for maximization by itself.

Exercises

Show directly that the free-chiral stationary point is a maximum.

Solution

From $\ell'(z)=-\pi z\cot(\pi z)$, one finds $\ell'(1/2)=0$ and $\ell''(1/2)=\pi^2/2$. Since $\mathcal F''(\Delta)=-\ell''(1-\Delta)$, the Hessian at $\Delta=1/2$ is $-\pi^2/2$.

References

• Closset, C., T. T. Dumitrescu, G. Festuccia, Z. Komargodski, and N. Seiberg. “Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories.” Journal of High Energy Physics 2012, no. 10 (2012): 053. DOI; Open PDF.
• Jafferis, D. L. “The Exact Superconformal R-Symmetry Extremizes $Z$.” Journal of High Energy Physics 2012, no. 5 (2012): 159. DOI; Open PDF.