N=4 Moduli, BPS States, and Protected Sectors

The $\mathcal N=4$ moduli space is simple enough to write in one line, yet its singular strata mark where charged gauge multiplets become massless. Away from those strata, electric and magnetic charges enter the central-charge matrix, while at the conformal origin local operators are organized by superconformal shortening. This page keeps those three layers—vacuum geometry, particle BPS data, 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

In the overall-coupling convention of the preceding page, the potential vanishes precisely when

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

The $X^I$ are Hermitian. A mutually commuting set can therefore be diagonalized simultaneously and conjugated into a Cartan subalgebra $\mathfrak t$. Continuous Cartan gauge transformations act trivially on these expectation values, while the remaining Weyl group $W$ permutes equivalent Cartan representatives. Thus

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

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

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

For $SU(2)$ this reduces to the beginner’s model

$$X^I=\frac{v^I}{2}\sigma_3, \qquad \mathcal M_{\mathfrak{su}(2)}=\mathbb R^6/\mathbb Z_2, \qquad \vec v\sim-\vec v.$$

The quotient is smooth away from $\vec v=0$ and has an orbifold singularity at the origin. The singularity is not a failure of the microscopic theory: it warns that the Abelian low-energy description has omitted fields that become massless there.

At a generic point the non-Abelian algebra breaks to $\mathfrak u(1)^r$, where $r=\operatorname{rank}\mathfrak g$. The off-diagonal vector multiplet labeled by a root $\alpha$ has

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

It becomes massless on the locus $\alpha(X^I)=0$ for all six $I$. Several simultaneous root conditions produce higher-codimension strata with a larger unbroken algebra. 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, particle charge sectors on nontrivial spatial topology, and genuine lines.

On each smooth covering chart, the two-derivative Abelian metric is flat and its overall normalization is set by $\operatorname{Im}\tau$. This does not make the Coulomb-branch theory free at all orders in derivatives: integrating out massive $W$ multiplets produces interactions such as the supersymmetric completion of $F^4/|X|^4$. Their protection properties are a separate, higher-derivative statement Dine and Seiberg 1997, §3. Nor may the flat chart be continued through an enhanced-symmetry stratum after its additional massless fields have been discarded.

Electric–magnetic central charges

On the generic branch, the unbroken theory is Abelian. Allowed electric charges are drawn from weight data and allowed magnetic charges from the cocharacter lattice, with an integral Dirac pairing; the particle spectrum occupies a dynamical subset of those possibilities. The precise lattices depend on the global gauge group. When discussing line-charge classes, one further quotients by charges of dynamical particles that can screen a line. A particle charge lattice and a screened finite line lattice are therefore related inputs, not interchangeable notation.

For a rank-one factor, choose canonical scalar normalization and integral particle-charge units $(e,m)$ in which the unit electric $W$ boson has mass $g_{\rm YM}|v_{\rm can}|$. These integers are not the finite center-charge classes used later to classify screened lines. Define

$$Z_{e,m} =\sqrt{\frac{4\pi}{\operatorname{Im}\tau}}\, v_{\rm can}(e+\tau m).$$

A half-BPS state of that charge obeys $M_{\rm BPS}=|Z_{e,m}|$, or

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

Two limiting checks fix the normalization at $\theta=0$:

$$(e,m)=(1,0):\quad M_W=g_{\rm YM}|v_{\rm can}|,$$ $$(e,m)=(0,1):\quad M_{\rm mon}=\frac{4\pi}{g_{\rm YM}}|v_{\rm can}|.$$

Changing the normalization of roots, $v$, or the gauge kinetic term changes the prefactor but not the invariant pairing of the scalar expectation value with an integral electric–magnetic charge. Osborn derived the $\mathcal N=4$ topological central charges and showed why a magnetically charged short multiplet has the spin content required by the Montonen–Olive proposal Osborn 1979, pp. 321–326.

Use the passive chapter convention

$$\tau'=\frac{a\tau+b}{c\tau+d}, \qquad \begin{pmatrix}e'\\m'\end{pmatrix} = \begin{pmatrix}a&-b\\-c&d\end{pmatrix} \begin{pmatrix}e\\m\end{pmatrix}.$$

Direct substitution gives

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

Therefore $|e+\tau m|^2/\operatorname{Im}\tau$ is invariant. This proves covariance of the formula under the proposed dictionary. It does not prove that the source and target global theories are equivalent or that a stable one-particle state exists for every formal pair $(e,m)$.

Half-BPS and quarter-BPS particle sectors

The $\mathcal N=4$ particle central charge is an antisymmetric complex tensor $Z^{AB}$ of $SU(4)_R$. A unitary change of supercharge basis puts it into two $2\times2$ skew blocks with complex entries $z_1,z_2$. In a convention where the massive bound is

$$M\geq\max(|z_1|,|z_2|),$$

saturation with $|z_1|=|z_2|$ gives a half-BPS multiplet, while saturation of only the larger unequal singular value gives a quarter-BPS multiplet.

The same distinction has a geometric expression. Absorb the common coupling normalization and theta-induced electric shift into two physical $Spin(6)_R$ vectors $\vec Q_E$ and $\vec Q_M$. The larger central-charge eigenvalue obeys

$$M_{\rm BPS}^2 =|\vec Q_E|^2+|\vec Q_M|^2 +2\sqrt{|\vec Q_E|^2|\vec Q_M|^2 -(\vec Q_E\!\cdot\!\vec Q_M)^2}.$$

The square root is the area $|\vec Q_E\wedge\vec Q_M|$ of the parallelogram spanned by the two charge vectors.

• If $\vec Q_E$ and $\vec Q_M$ are parallel, the area vanishes, the two skew singular values coincide, and saturation is half-BPS. Rank-one dyons lie in this class because all charges couple to the same Cartan direction.
• If the vectors are not parallel, the singular values differ and saturation is generically quarter-BPS. Such sectors first require rank greater than one; their field-theory Bogomolny equations and nonparallel charge vectors are exhibited in Tong 1999, §§1–2.

This is again a bound and a shortening criterion, not a census of states. Existence requires solving the field equations or defining the quantum state, and stability requires comparing with all allowed decay products. On a marginal-stability wall, constituent central charges align so that a shortened one-particle state can meet a multiparticle threshold without violating any BPS inequality.

Local protected operators

Particle shortening applies after moving onto the moduli space and choosing a massive charge sector. Local-operator shortening is instead a radial-quantization statement at the conformal vacuum. The two uses of “BPS” share the positive-superalgebra logic but classify different Hilbert spaces.

For an $SU(N)$ example, choose $Z=X^1+iX^2$. For $p\geq2$, subject to finite-rank trace identities, the highest-weight component

$$\operatorname{Tr}Z^p$$

belongs to a half-BPS superconformal multiplet whose scalar primary has

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

An $SU(4)_R$-covariant expression introduces a null polarization $Y^I$, with $Y\cdot Y=0$, and writes

$$\mathcal O_p(x,Y) =Y^{I_1}\cdots Y^{I_p} \operatorname{Tr}\!\left( X^{(I_1}\cdots X^{I_p)} \right)_{\rm traceless}.$$

For a general gauge algebra, choose algebraically independent invariant polynomials $P_{d_i}$ of degrees $d_i$; the corresponding primitive half-BPS operators are $P_{d_i}(Z)$. In the fundamental-matrix realization of $SU(N)$ these invariants can be represented by traces, but finite-rank trace identities remove some naive operators, and products of lower-degree invariants create multi-trace operators with the same total quantum numbers. A normalized primary is therefore a basis vector in the full finite-rank mixing space, not always a bare single trace.

The $p=2$ operator is the bottom of the stress-tensor multiplet. Its descendants include the stress tensor, supersymmetry currents, $SU(4)_R$ currents, and the exactly marginal operators. Shortening fixes $\Delta=2$, but its four-point function still contains a nontrivial function of conformal cross-ratios. Superconformal Ward identities determine the allowed tensor structures and organize short, semishort, and long exchanges without fixing all dynamical OPE data 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

The half-BPS $[0,p,0]$ series is absolutely protected. More general quarter-BPS representations require care: some are absolutely protected, while semishort multiplets at a long-multiplet threshold can recombine. The full conclusion requires all Lorentz and $SU(4)_R$ Dynkin labels, the shortening type, and possible recombination partners, as classified in Córdova, Dumitrescu, and Intriligator 2019, §2.2.4. Calling an operator “BPS at zero coupling” is insufficient because a free-field expression can mix with descendants or join a long multiplet once interactions are turned on.

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.

The three charge notions can be kept straight with a simple question:

object	charge information being classified
massive particle	a state in the Coulomb-branch Hilbert space, including existence and stability
genuine line	an unscreened defect charge compatible with the chosen global theory
local BPS operator	a superconformal representation at the conformal vacuum; no electromagnetic line charge is implied

Common pitfalls

Reading an orbifold singularity as a singular microscopic theory. The Abelian effective theory is singular because fields integrated out as massive become massless. The full non-Abelian $\mathcal N=4$ theory remains the correct local description.

Treating a lattice vector as a state. Integral charges tell us which quantum numbers are allowed. The BPS algebra fixes the mass if a shortened state exists, but neither fact proves existence or stability.

Equating a protected dimension with a protected correlator. Shortening fixes selected representation data. Four-point functions can still contain coupling-dependent long-multiplet dimensions and OPE coefficients.

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. Distinguish half-BPS from quarter-BPS charges

Evaluate the higher-rank BPS formula when (a) $\vec Q_M=\lambda\vec Q_E$ and (b) $\vec Q_E\cdot\vec Q_M=0$ with both vectors nonzero. Which shortening fraction is possible in each case?

Solution

In case (a), the wedge area vanishes, so

$$M_{\rm BPS}^2=|\vec Q_E|^2+|\vec Q_M|^2.$$

The two central-charge singular values coincide, and saturation is half-BPS. In case (b), the square root is $|\vec Q_E||\vec Q_M|$, so

$$M_{\rm BPS}=|\vec Q_E|+|\vec Q_M|.$$

The charge vectors are nonparallel, the singular values are unequal, and saturation is generically quarter-BPS. In neither case does the algebra prove that a corresponding stable state exists.

4. Diagnose a protection 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

• Córdova, Clay, Thomas T. Dumitrescu, and Kenneth Intriligator. “Multiplets of Superconformal Symmetry in Diverse Dimensions.” Journal of High Energy Physics 03 (2019): 163. doi:10.1007/JHEP03(2019)163. Open PDF.
• Dine, Michael, and Nathan Seiberg. “Comments on Higher Derivative Operators in Some SUSY Field Theories.” Physics Letters B 409 (1997): 239–244. doi:10.1016/S0370-2693(97)00899-X. Open PDF.
• 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.
• Tong, David. “A Note on 1/4-BPS States.” Physics Letters B 460 (1999): 295–301. doi:10.1016/S0370-2693(99)00794-7. Open PDF.