F-Maximization and Accidental Symmetries

In a three-dimensional $N=2$ SCFT, the superconformal $U(1)_R$ current can mix with Abelian flavor currents. F-maximization characterizes the exact current as a local maximum of $\mathcal F=-\log|Z_{S^3}|=-\operatorname{Re}\log Z_{S^3}$. The theorem is an infrared statement in the complete space of conserved currents. A localized ultraviolet matrix model computes that maximum only when it exposes every mixing current—including accidental ones—and is continued on a well-defined 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 exact extremum

Choose a reference R-current $R_0$ and a basis $F_a$ of nonredundant Abelian flavor currents that commute with the supercharges. The trial current is

$$R(t)=R_0+\sum_a t_a F_a.$$

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

$$\Delta_i(t)=R_0(\Phi_i)+\sum_a t_aq_i^a.$$

Every superpotential monomial present at the fixed point must have trial R-charge two. Currents broken by those couplings are not mixing directions. Topological currents of Abelian gauge factors are ordinary flavor currents in three dimensions and must be included unless a monopole interaction breaks them; monopole superpotentials impose further linear constraints.

On the round sphere, R-current mixing is implemented by an imaginary flavor mass. Let $\mu_a=r_{S^3}m_a$ source $F_a$, where $r_{S^3}$ is the sphere radius. In a chiral determinant the combination $q_i^a\mu_a$ used on the matrix-model page is continued by $\mu_a\mapsto\mu_a+i t_a$. This continuation must preserve the pole prescription inherited from the physical real-mass integral. With fixed background contact terms, define

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

At a unitary fixed point, in the complete space of genuine mixing currents,

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

The matrix $\tau_{ab}$ is the positive coefficient of the separated-point flavor-current two-point function. The Hessian is therefore negative definite after currents that vanish or are linearly dependent have been removed. Ward identities, one-point functions, and R–flavor contact terms enter the proof; the scheme-independent statement here is that the real first derivative vanishes and the displayed real Hessian is negative Jafferis 2012, §4; Closset et al. 2012, §4, Eqs. (4.9)–(4.10).

This result is local: a negative Hessian does not establish a unique global maximum across poles, disconnected convergence chambers, or singular boundaries of the trial domain.

Free chiral and XYZ benchmarks

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

$$\mathcal F''\!\left(\frac12\right)=-\frac{\pi^2}{2}<0, \qquad \mathcal F\!\left(\frac12\right)=\frac12\log2.$$

This example checks the sign of the extremization. The value $\Delta=1/2$ saturates the unitarity bound for a gauge-invariant scalar chiral primary; saturation implies a free multiplet Minwalla 1998, §§2–3.

As a second check, a symmetric Wess–Zumino model with $W=XYZ$ obeys $\Delta_X+\Delta_Y+\Delta_Z=2$. Permutation symmetry fixes a stationary point at $\Delta_X=\Delta_Y=\Delta_Z=2/3$. Writing $f(\Delta)=-\ell(1-\Delta)$, its curvature is

$$f''\!\left(\frac23\right) =\frac{\pi}{\sqrt3}-\frac{4\pi^2}{9}<0.$$

Consequently the Hessian is negative on the two-dimensional plane $\delta\Delta_X+\delta\Delta_Y+\delta\Delta_Z=0$.

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. For a magnetic cocharacter $\mathfrak m$, a common zero-mode convention gives

$$\begin{aligned} \Delta_0(T_{\mathfrak m};t) ={}&-\frac12\sum_{\alpha\in\operatorname{roots}(G)}|\alpha(\mathfrak m)| \\ &+\frac12\sum_i[1-\Delta_i(t)] \sum_{\rho_i\in R_i}|\rho_i(\mathfrak m)|. \end{aligned}$$

Mixing with matter flavor currents is already encoded in $\Delta_i(t)$. Mixing with a topological current adds its charge, proportional to the corresponding component of $\mathfrak m$. The allowed cocharacters depend on the global gauge group. The non-Abelian root-and-weight formula follows by applying the fermion charge formula in Cremonesi 2015, §2.1, eq. (2.7) to gauginos and chiral fermions. In a Chern–Simons theory a bare monopole is generally electrically charged and must be dressed before applying a gauge-invariant unitarity bound Aharony, Narayan, and Sharma 2015, §1, pp. 3–4. The dressed operator’s matter contribution, global charges, and possible fermion zero modes must all be included. The radial-quantization zero-mode construction is given in Borokhov, Kapustin, and Wu 2002, §§2–3.

Decoupled operators and accidental currents

If a naive extremum assigns a gauge-invariant chiral operator $\Delta<1/2$, that extremum cannot describe the proposed unitary fixed point. Possibilities include an incorrect fixed-point hypothesis, a missing interaction or dual description, or a free operator at $\Delta=1/2$ with an emergent $U(1)$ current absent from the ultraviolet trial basis.

The exact infrared theorem still applies after an accidental current appears, but a UV localization formula generally does not contain a source for that emergent current. Jafferis therefore states the extremization result under the assumption that no accidental symmetry mixes with the R-current Jafferis 2012, §1, and UV localization data do not in general determine the missing mixing parameter Closset et al. 2012, discussion after Eq. (4.10). There is no universal operation that repairs every matrix model by subtracting a composite determinant.

A controlled analysis instead has the following decision points:

1. enumerate elementary, composite, and monopole chiral operators in the candidate region;
2. identify every gauge-invariant operator at or below $\Delta=1/2$ and test whether the candidate fixed point is consistent;
3. look for independent evidence of infrared factorization or a dual frame in which the free multiplet and its current are manifest;
4. only in such a description, extremize the interacting sector in its complete current space and then recheck every operator.

If one has established

$$\mathcal T_{\mathrm{IR}} =\mathcal T_{\mathrm{int}}\times n\ \text{free chirals},$$

then, up to declared contact phases and normalization,

$$\mathcal F_{\mathrm{IR}} =\mathcal F_{\mathrm{int}}+\frac n2\log2.$$

The free fields have $\Delta=1/2$ and are not varied with an incomplete UV mixing parameter. Merely clamping a composite’s dimension while leaving the same functional unchanged is inconsistent. Introducing a flipping field is a deformation of the theory unless a duality or flow argument proves that it exposes the same infrared sector. Model-dependent correction proposals can be useful diagnostics, but their hypotheses must be stated rather than promoted to a general theorem Niarchos 2011, §§2–3.

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. A zero or pole of the analytically continued $Z(t)$ makes the logarithm or its derivatives singular and marks the boundary of the smooth extremization problem.

For a numerical extremization, report:

• the exact integrand, contour, precision, and tail treatment;
• linear superpotential constraints, the convergence chamber, and the trial domain;
• the stationary point and all eigenvalues of its Hessian;
• convergence under increased precision, integration range, and admissible contour deformations;
• the smallest dimensions of ordinary and monopole chiral operators;
• any free factors removed and the independent description used to expose them;
• uncertainties from quadrature, differentiation, and flux or residue truncation.

A nearly zero Hessian eigenvalue can signal a redundant current, a missed accidental symmetry, or inadequate numerical precision. A conformal-manifold modulus is not itself an R-current mixing direction. Neither phenomenon is evidence for maximization by itself.

Exercises

1. 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 $-\pi^2/2$.

2. For the $XYZ$ model, prove that the symmetric stationary point is a strict maximum on the superpotential constraint surface.

Solution

The constraint is $\Delta_X+\Delta_Y+\Delta_Z=2$. Permutation symmetry makes $\Delta_i=2/3$ stationary. Differentiating $\ell'(z)=-\pi z\cot(\pi z)$ gives

$$f''\!\left(\frac23\right) =-\ell''\!\left(\frac13\right) =\frac{\pi}{\sqrt3}-\frac{4\pi^2}{9}<0.$$

For any nonzero allowed variation with $\delta_X+\delta_Y+\delta_Z=0$,

$$\delta^2\mathcal F =f''\!\left(\frac23\right) \left(\delta_X^2+\delta_Y^2+\delta_Z^2\right)<0.$$

Thus the Hessian restricted to the two independent mixing directions is negative definite.

3. Consider $U(N_c)$ SQCD at zero Chern–Simons level with $N_f$ fundamental–antifundamental pairs, all assigned trial R-charge $r$. For the minimal cocharacter $(1,0,\ldots,0)$, use the zero-mode formula to find the bare monopole dimension and its unitarity constraint.

Solution

The roots connecting the first Cartan entry to the other $N_c-1$ entries give the vector contribution $-(N_c-1)$. Each flavor pair has one fundamental and one antifundamental weight of absolute magnetic charge one, so it contributes $1-r$. Hence

$$\Delta(T_\pm)=N_f(1-r)-(N_c-1).$$

The gauge-invariant chiral bound requires

$$N_f(1-r)-(N_c-1)\ge\frac12, \qquad r\le1-\frac{N_c-\tfrac12}{N_f}.$$

Violation does not license setting the monopole dimension to $1/2$ by hand. It says that the naive trial extremum or its assumed infrared description is incomplete; one must search for a free monopole sector, an emergent current, or a different fixed point and then repeat the operator scan.

References

• Aharony, Ofer, Prithvi Narayan, and Tarun Sharma. “On Monopole Operators in Supersymmetric Chern–Simons-Matter Theories.” Journal of High Energy Physics 2015, no. 5 (2015): 117. DOI; Open PDF.
• Borokhov, Vadim, Anton Kapustin, and Xinkai Wu. “Topological Disorder Operators in Three-Dimensional Conformal Field Theory.” Journal of High Energy Physics 2002, no. 11 (2002): 049. doi:10.1088/1126-6708/2002/11/049. Open PDF.
• Closset, Cyril, Thomas T. Dumitrescu, Guido Festuccia, Zohar Komargodski, and Nathan Seiberg. “Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories.” Journal of High Energy Physics 2012, no. 10 (2012): 053. doi:10.1007/JHEP10(2012)053. Open PDF.
• Cremonesi, Stefano. “The Hilbert Series of 3d $\mathcal N=2$ Yang–Mills Theories with Vectorlike Matter.” Journal of Physics A: Mathematical and Theoretical 48, no. 45 (2015): 455401. DOI; Open PDF.
• Jafferis, Daniel L. “The Exact Superconformal R-Symmetry Extremizes $Z$.” Journal of High Energy Physics 2012, no. 5 (2012): 159. doi:10.1007/JHEP05(2012)159. Open PDF.
• Minwalla, Shiraz. “Restrictions Imposed by Superconformal Invariance on Quantum Field Theories.” Advances in Theoretical and Mathematical Physics 2, no. 4 (1998): 781–846. doi:10.4310/ATMP.1998.v2.n4.a4. Open PDF.
• Niarchos, Vasilis. “Comments on F-Maximization and R-Symmetry in 3D SCFTs.” Journal of Physics A: Mathematical and Theoretical 44, no. 30 (2011): 305404. doi:10.1088/1751-8113/44/30/305404. Open PDF.