N=4 Moduli, BPS States, and Protected Sectors

The $\mathcal N=4$ vacuum space is simple enough to write in one line, yet its singular strata support the charged states that make duality nontrivial. Supersymmetry then protects selected masses and operator dimensions, not the entire interacting theory. This page keeps those three layers—vacuum geometry, charged BPS states, and local protected multiplets—distinct.

Required background. The $\mathcal N=4$ theory card fixes scalar and coupling conventions. BPS bounds and shortening supplies the relation between central charges and shortened representations.

Helpful background. Kähler and hyperkähler quotients gives a broader geometric setting for supersymmetric vacuum quotients.

The commuting-scalar quotient

The potential vanishes precisely when

$$[X^I,X^J]=0\qquad\text{for all }I,J=1,\ldots,6.$$

Every commuting set can be conjugated into a Cartan subalgebra $\mathfrak t$. The residual gauge transformations are the Weyl group $W$, so

$$\mathcal M_{\rm vac} =\frac{\mathbb R^6\otimes\mathfrak t}{W}.$$

For $\mathfrak{su}(N)$, write six-vectors $\vec x_a\in\mathbb R^6$ with $\sum_a\vec x_a=0$. Then

$$\mathcal M_{\mathfrak{su}(N)} = \left\{(\vec x_1,\ldots,\vec x_N): \sum_a\vec x_a=0\right\}\big/S_N.$$

At a generic point the non-Abelian algebra breaks to $\mathfrak u(1)^r$, where $r=\operatorname{rank}\mathfrak g$. A root field $W_\alpha$ has

$$M_\alpha^2=\sum_I\lvert\alpha(X^I)\rvert^2.$$

It becomes massless on the locus $\alpha(X^I)=0$ for all $I$. Intersections of such loci are higher-codimension strata with larger unbroken algebras. The central quotient of the gauge group acts trivially on adjoint scalars, so connected global forms with the same algebra have the same local quotient; they differ in allowed bundles and charged lines.

The sixteen supercharges protect the two-derivative flat metric on the smooth part of this orbifold. That statement does not remove higher-derivative interactions among the Abelian fields, nor does it turn a singular stratum into a weakly coupled chart.

Electric–magnetic central charges

On the generic branch, charges take values in integral electric and magnetic lattices. The magnetic lattice is a cocharacter lattice; the electric lattice is a character lattice, with both reduced by screening when one discusses line-charge classes. Which lattice occurs depends on the global gauge group.

For a rank-one factor, choose canonical scalar normalization and charge units in which a unit electric $W$ boson has mass $g_{\rm YM}\lvert v_{\rm can}\rvert$. The half-BPS mass formula can then be written

$$M_{\rm BPS}^{2} = 4\pi\lvert v_{\rm can}\rvert^2 \,\frac{\lvert e+\tau m\rvert^2}{\operatorname{Im}\tau}.$$

The numerical coefficient changes if roots or scalar kinetic terms are normalized differently; the invariant content is the complex central charge and its integral charge lattice. The appearance of magnetically charged spin-one multiplets in the maximally supersymmetric algebra was an early basis for the Montonen–Olive conjecture Osborn 1979, pp. 321–326.

Under

$$\tau'=\frac{a\tau+b}{c\tau+d}, \qquad (e',m')=(ae-bm,-ce+dm),$$

one has

$$e'+\tau'm'=\frac{e+\tau m}{c\tau+d}, \qquad \operatorname{Im}\tau' =\frac{\operatorname{Im}\tau}{\lvert c\tau+d\rvert^2}.$$

Therefore $\lvert e+\tau m\rvert^2/\operatorname{Im}\tau$ is invariant. This calculation checks a proposed charge dictionary; it does not establish that a stable state exists for every formal pair $(e,m)$.

At higher rank the central charge is an antisymmetric $SU(4)_R$ tensor with two complex skew-eigenvalues. A half-BPS state saturates the stronger condition in which the relevant electric and magnetic scalar-charge vectors are aligned. Generic dyonic configurations can preserve one quarter of the supercharges. Marginal-stability walls concern whether a shortened one-particle state can decay into other shortened states; algebraic saturation alone does not decide stability.

Local protected operators

Choose a complex scalar $Z=X^1+iX^2$. The single-trace operators schematically written

$$\mathcal O_p=\operatorname{Tr}Z^p+\text{multi-trace mixing}$$

sit in half-BPS multiplets with

$$\Delta=p,\qquad [q,p,q]_{SU(4)_R}=[0,p,0].$$

At finite rank the independent operators obey trace relations, and operators with the same quantum numbers mix. The representation label and protected dimension remain meaningful even when a naive single-trace basis does not.

The case $p=2$ is the bottom of the stress-tensor multiplet. Its descendants include the stress tensor, supersymmetry currents, and $SU(4)_R$ currents. Two- and three-point functions of suitably normalized half-BPS operators are highly constrained, while their four-point functions retain a nontrivial function of conformal cross-ratios. Superconformal Ward identities organize the short and long exchanges Dolan and Osborn 2002, §§3–7.

This yields a useful hierarchy:

datum	protection status
half-BPS scaling dimension	fixed by the algebra
stress-tensor anomaly coefficients	fixed by the local theory card
selected BPS indices	invariant under continuous deformations away from walls
half-BPS four-point function	constrained, but contains dynamical coupling dependence
long-multiplet dimensions and OPE coefficients	generally coupling dependent

Some quarter-BPS multiplets are absolutely protected, whereas semishort multiplets at a unitarity threshold can pair into a long multiplet. Calling an operator “BPS at zero coupling” is therefore insufficient: one must identify its interacting superconformal representation and check whether recombination is allowed.

Particle BPS data are not line data

A massive BPS particle on the Coulomb branch and a genuine line operator are related but not interchangeable. A particle worldline can end a line carrying the particle’s charge; consequently dynamical particles screen line charges. The spectrum of genuine lines is the quotient that remains after screening and mutual-locality constraints.

Likewise, a local half-BPS operator does not determine the global form of the gauge group. Local operator correlators on $\mathbb R^4$ can agree between theories whose Wilson–’t Hooft spectra differ. S-duality tests must therefore say which protected sector is being compared.

Exercises

1. Singular strata for $SU(3)$. In the eigenvalue description, identify the conditions for an $SU(2)$ enhancement and for full $SU(3)$ enhancement.

Solution

An $SU(2)$ enhancement occurs when two six-vectors coincide, for example $\vec x_1=\vec x_2$, making the corresponding root bosons massless. Full $SU(3)$ enhancement requires all three to coincide. The traceless condition then sets every $\vec x_a$ to zero.

2. Verify modular invariance. Use the two displayed transformation identities to prove invariance of the rank-one BPS factor.

Solution

The transformed numerator is divided by $\lvert c\tau+d\rvert^2$, and the transformed denominator is divided by the same factor. Their ratio is unchanged.

3. Diagnose an overclaim. Does $\Delta(\mathcal O_2)=2$ imply that its connected four-point function is independent of $\tau$?

Solution

No. Shortening fixes the operator dimension, and Ward identities restrict the tensorial and cross-ratio dependence, but a dynamical function remains. Its operator-product expansion contains long multiplets with coupling-dependent dimensions and coefficients.

References

• Dolan, Francis A., and Hugh Osborn. “Superconformal Symmetry, Correlation Functions and the Operator Product Expansion.” Nuclear Physics B 629 (2002): 3–73. doi:10.1016/S0550-3213(02)00096-2.
• Osborn, Hugh. “Topological Charges for $\mathcal N=4$ Supersymmetric Gauge Theories and Monopoles of Spin 1.” Physics Letters B 83 (1979): 321–326. doi:10.1016/0370-2693(79)91118-3.