Conformal Manifolds and Duality Actions

A conformal manifold is a space of inequivalent CFTs connected by exactly marginal couplings. In four-dimensional $\mathcal N=1$ theories, holomorphy turns the local problem into marginal chiral couplings modulo complexified continuous global symmetries. This gives a powerful dimension count, but global existence, singular cusps, accidental currents, and duality identifications require additional analysis.

Required background. The SQCD fixed-point page supplies interacting endpoints, and linearized RG flow distinguishes marginal from exactly marginal directions. Helpful background. Exact, infrared, and emergent equivalence clarifies how dual frames can cover one manifold.

Marginal operators and broken currents

In a four-dimensional $\mathcal N=1$ SCFT, a supersymmetric marginal deformation has the form

$$\delta S =\int d^4x\,d^2\theta\, \lambda^i\mathcal O_i+\text{h.c.},$$

where $\mathcal O_i$ is a chiral primary with

$$\Delta(\mathcal O_i)=3, \qquad R(\mathcal O_i)=2.$$

Gauge kinetic terms contribute operators $\operatorname{Tr}W^\alpha W_\alpha$ and complexified gauge couplings. Superpotential monomials of R-charge two provide further candidates.

A candidate marginal operator need not be exactly marginal. If its coupling breaks a continuous flavor current $J_a$, the current multiplet can recombine with the chiral operator into a long multiplet. The corresponding direction becomes marginally irrelevant. This recombination is the physical origin of quotienting by broken global symmetries.

The local quotient

Let $V$ be the vector space of marginal chiral couplings and let a continuous global group $G$ act on it. Locally, the conformal manifold is

$$\mathcal M_c\simeq\{\lambda\in V:D^a(\lambda,\bar\lambda)=0\}/G \simeq V^{\mathrm{stable}}/G_{\mathbb C},$$

near the reference SCFT and under the usual regularity assumptions. The real functions $D^a$ begin quadratically,

$$D^a\propto \lambda^i(T^a)_i{}^j\bar\lambda_j+\cdots,$$

and encode the beta functions associated with broken currents.

On a stratum with stabilizer $H\subset G$, the expected local complex dimension is

$$\dim_{\mathbb C}\mathcal M_c =\dim_{\mathbb C}V-\dim G+\dim H,$$

provided the constraints are independent and no accidental currents appear. The stabilizer term is essential at symmetry-enhanced loci.

This count is local. It does not prove that every stable orbit extends to a finite coupling, that the metric is complete, or that distinct patches are globally connected. The broken-current criterion and local quotient construction are developed in Green, Komargodski, Seiberg, Tachikawa, and Wecht 2010, §§2–4.

Beta functions in Lagrangian variables

For a superpotential

$$W=\sum_i\lambda^i\mathcal O_i,$$

holomorphy makes its beta functions proportional to wavefunction anomalous dimensions. For a monomial $\mathcal O_i=\prod_A\Phi_A^{n_{iA}}$,

$$\beta_{\lambda^i} =\lambda^i\left[ -3+\sum_A n_{iA}\left(1+\frac12\gamma_A\right) \right].$$

The gauge beta function supplies another constraint, for example the NSVZ numerator

$$3T(G)-\sum_A T(r_A)(1-\gamma_A)=0.$$

If the number of couplings exceeds the number of independent anomalous-dimension constraints, the remaining combinations can be exactly marginal. Redundancies among beta functions often reflect nonanomalous flavor currents; the quotient formulation makes that structure intrinsic.

Solving these equations at one perturbative order suggests local directions but does not establish exact marginality unless symmetry and holomorphy control higher orders or an all-order argument is available.

Example: the N=4 locus in N=1 language

Write $\mathcal N=4$ SYM as an $\mathcal N=1$ vector multiplet plus three adjoint chirals $\Phi_1,\Phi_2,\Phi_3$ with

$$W=h\,\operatorname{Tr}\Phi_1[\Phi_2,\Phi_3].$$

The complex gauge coupling $\tau$ and $h$ are both classically marginal. Extended supersymmetry relates their beta functions, leaving a complex one-dimensional locus with the full $\mathcal N=4$ symmetry. In an appropriate normalization, $h$ is tied to the gauge coupling; changing conventions changes the written relation but not the one-complex-dimensional family.

Viewed purely with $\mathcal N=1$ supersymmetry, additional cubic couplings can produce Leigh–Strassler deformations. Counting them requires quotienting by the complexified flavor transformations of the three adjoints and checking which symmetries survive. The familiar $\mathcal N=4$ line is a sublocus, not automatically the entire $\mathcal N=1$ conformal manifold; the original all-order beta-function analysis is Leigh and Strassler 1995, pp. 95–136.

The coupling $\tau$ is further identified by electric–magnetic duality. Globally, the physical family is a quotient by a discrete group such as $SL(2,\mathbb Z)$ or a subgroup fixed by global form and line operators. Weak-coupling cusps can represent different duality frames of the same CFT family.

Example: quartic SQCD at Nf = 2Nc

At the SQCD fixed point with $N_f=2N_c$,

$$R(Q)=R(\widetilde Q)=\frac12, \qquad R(M)=1.$$

Quartic meson operators therefore have R-charge two. A flavor-invariant deformation can be written schematically as

$$W_{\mathrm{quartic}} =\kappa\,(Q\widetilde Q)(Q\widetilde Q).$$

Together with the gauge coupling, $\kappa$ gives candidate marginal directions. Whether a particular tensor contraction is exactly marginal depends on the flavor symmetries it breaks and the independent beta-function constraints. In the magnetic self-rank description, the same deformation can be represented by a mass term for the singlet meson and, after integrating it out, an inverse quartic magnetic coupling. This produces a nontrivial duality action on the coupling coordinate.

The example illustrates the difference between local and global statements: R-charge identifies marginal candidates; quotient and beta functions find local exactly marginal directions; duality relates distant coordinate patches.

Singular loci and cusps

A conformal manifold can have:

• orbifold points, where a discrete duality stabilizes a coupling;
• symmetry-enhanced strata, where additional conserved currents appear and the quotient stabilizer grows;
• weak-coupling cusps, at infinite Zamolodchikov distance or a boundary in a chosen coordinate;
• degeneration loci, where extra operators become free or a different effective description is needed.

At an enhanced-symmetry point, a marginal direction can recombine with the new current and cease to be exactly marginal. Dimension can therefore jump between strata. A coordinate singularity in $\tau$ may be removed by a duality transformation, while a genuine new massless sector changes the local CFT data.

Zamolodchikov metric and duality action

Exactly marginal operators define a Hermitian metric through their two-point functions,

$$\langle\mathcal O_i(x)\overline{\mathcal O}_{\bar j}(0)\rangle =\frac{g_{i\bar j}}{|x|^6}.$$

After removing redundant directions, $g_{i\bar j}$ is the Zamolodchikov metric on $\mathcal M_c$. A duality acts by an isometry together with an operator-basis transformation. Contact terms define a connection on the operator bundle, so transporting an operator around a duality loop can produce nontrivial holonomy. A complementary weak-coupling treatment with spacetime-dependent gauge couplings and one-loop $SL(2,\mathbb R)$ covariance is given in Osborn 2003, pp. 174–182.

Local quotient data do not fix this metric globally. Localization or conformal perturbation theory may compute protected parts in special theories, with counterterm ambiguities treated explicitly.

A reliable workflow

1. List every marginal chiral primary, including gauge kinetic operators.
2. Determine the faithful continuous global symmetry acting on the couplings.
3. Remove descendants, equations-of-motion operators, and redundant directions.
4. Solve the moment-map or beta-function constraints and quotient by $G_{\mathbb C}$.
5. Include accidental currents and recompute the local dimension on each stratum.
6. Identify discrete dualities, cusps, and stabilizers.
7. Distinguish a local candidate patch from a globally established conformal manifold.

Common pitfalls

Counting R-charge-two operators directly. Broken currents pair with some of them and make those directions marginally irrelevant.

Subtracting the full global-group dimension at a fixed point. A stabilizer acts trivially on the chosen couplings and must be added back.

Inferring a global manifold from local beta functions. Obstructions, accidental symmetries, and duality identifications can appear away from the reference point.

Exercises

Suppose an SCFT has four marginal chiral couplings transforming under a two-dimensional continuous global group. At a generic stable point the stabilizer is trivial; at a special locus it has dimension one.

1. Give the expected local complex dimension on the generic stratum.
2. Give it on the special stratum, assuming the constraint rank changes accordingly.
3. Explain why the result does not prove both strata extend globally.

Solution

Generically,

$$\dim_{\mathbb C}\mathcal M_c=4-2=2.$$

At the special locus,

$$\dim_{\mathbb C}\mathcal M_c=4-2+1=3,$$

provided the local quotient is regular in the orbit-type sense. The jump signals an enhanced stabilizer and fewer broken-current constraints. It is only a tangent-space count near that locus; global obstructions, identifications, or the disappearance of the candidate operators can prevent extension to a three-dimensional component.

References

• Green, Daniel, Zohar Komargodski, Nathan Seiberg, Yuji Tachikawa, and Brian Wecht. “Exactly Marginal Deformations and Global Symmetries.” Journal of High Energy Physics 06 (2010): 106. arXiv:1005.3546.
• Leigh, Robert G., and Matthew J. Strassler. “Exactly Marginal Operators and Duality in Four-Dimensional $\mathcal N=1$ Supersymmetric Gauge Theory.” Nuclear Physics B 447 (1995): 95–136. arXiv:hep-th/9503121.
• Osborn, Hugh. “Local Couplings and $SL(2,\mathbb R)$ Invariance for Gauge Theories at One Loop.” Physics Letters B 561 (2003): 174–182. arXiv:hep-th/0302119.