BPS Wilson, 't Hooft, and Dyonic Line Observables

A BPS line operator is specified by more than an electric–magnetic charge. One must give the global gauge theory, the line trajectory and framing, its singular boundary condition or representation, scalar couplings, preserved supercharges, screening equivalence, monopole-bubbling sectors, and the normalization of its expectation value. S-duality acts on this complete object, not on a charge pair in isolation.

Required background. Use electric–magnetic charge lattices and global form together with framed BPS states.

Helpful background. Genuine lines and screening fix the global line set before supersymmetry is imposed.

Supersymmetric Wilson lines

In four-dimensional $N=4$ Yang–Mills theory, a Euclidean Wilson line in representation $R$ can couple to both the connection and six adjoint scalars:

$$W_R[C,n] =\operatorname{Tr}_R\, \mathcal P\exp\!\oint_C ds\, \left(iA_\mu\dot x^\mu+|\dot x|n_I\Phi^I\right),$$

where $n_In_I=1$. Supersymmetry requires the local projector

$$\left(i\dot x^\mu\gamma_\mu +|\dot x|n_I\Gamma^I\right)\varepsilon=0.$$

A straight line with constant $n_I$ is half-BPS; a circle preserves a conformally related set of supercharges. Changing the scalar coupling can destroy the projector even though the gauge Wilson line remains well defined.

The representation must define a genuine line of the chosen global gauge group. Dynamical matter can screen weights, so the conserved label is a class in the electric line lattice rather than an arbitrary highest weight.

‘t Hooft singularities and dyonic charges

An ‘t Hooft line is defined by a magnetic singularity along its worldline. On a small two-sphere linking the line,

$$\frac1{2\pi}\int_{S^2}F=B,$$

where $B$ is a cocharacter of the gauge group, modulo Weyl transformations. In a BPS configuration an adjoint scalar has a correlated $1/r$ singularity; its sign selects the preserved supercharges. The magnetic weight cannot be stated from the Lie algebra alone because the cocharacter lattice changes with global form.

A Wilson–‘t Hooft line carries a pair of weights modulo the simultaneous Weyl action,

$$(e,m)\in \left(\Lambda_{\mathrm w}\oplus\Lambda_{\mathrm{cw}}\right)/W.$$

Equivalently, fixing $m$ breaks the gauge group along the singularity to its stabilizer $G_m$, and the electric datum is an irreducible representation of $G_m$; a weight $e$ together with the simultaneous Weyl quotient encodes that representation. Global form, dynamical screening, and the chosen genuine-line set further restrict these labels. Two pairs obey the Dirac constraint

$$\langle(e,m),(e',m')\rangle =\langle e,m'\rangle-\langle e',m\rangle\in\mathbb Z.$$

The allowed maximal mutually local set, discrete theta angle, and screening quotient specify the global theory Kapustin 2006, §§2–4.

Monopole bubbling

The singular magnetic charge $B$ measured very near the line need not equal the charge $v$ measured farther away. Smooth monopoles can shrink into the defect and partially screen it:

$$B\longrightarrow v, \qquad B-v\in\text{positive coroot cone}.$$

An exact expectation value therefore sums bubbling sectors,

$$\langle L_{B,e}\rangle =\sum_{v} \int da\, Z_{\mathrm{bulk}}(a) Z_{\mathrm{mono}}(B,v;a) \mathcal L_{v,e}(a).$$

The factor $Z_{\mathrm{mono}}$ is an equivariant integral over a singular-monopole moduli space or an equivalent defect quantum mechanics. Keeping only the unscreened sector $v=B$ generally gives the wrong duality transformation and OPE Gomis, Okuda, and Pestun 2012, §§5–7.

Framing, divergences, and the protected observable

A line has ultraviolet divergences localized on its support. A perimeter term can be removed by a line counterterm, while cusps and intersections carry additional anomalous data. Framing can affect phases and the definition of magnetic or Chern–Simons-linked lines. The renormalized line must therefore include a declared scheme.

Different protected quantities can be attached to the same defect:

• its expectation value on a specified supersymmetric background;
• a framed BPS index of states bound to the line;
• its OPE or fusion coefficients with other lines;
• a difference-operator action on a partition function or block;
• defect local-operator correlation functions.

Exactness of one does not make the full defect conformal data exact.

Duality action

For Abelianized charges, an $SL(2,\mathbb Z)$ transformation acts schematically as

$$\begin{pmatrix}e\\m\end{pmatrix} \longmapsto \begin{pmatrix}a&b\\c&d\end{pmatrix} \begin{pmatrix}e\\m\end{pmatrix}.$$

In a non-Abelian theory, the transformed line can live in a theory with Langlands-dual group and different global form. Its bubbling factors, discrete theta data, and genuine-line lattice must transform as well. A Wilson line becoming an ‘t Hooft line is a shorthand for this full map.

Checks and failure modes

1. Verify the supersymmetry projector and scalar singularity.
2. Confirm that $(e,m)$ belongs to the genuine line lattice of the global theory.
3. Quotient charges screened by dynamical fields.
4. Enumerate every allowed bubbling charge $v$ and its multiplicity factor.
5. Fix framing, perimeter subtraction, and background counterterms.
6. Test weak-coupling, semiclassical, or Abelian limits.
7. Apply duality to the full line specification and compare protected observables.

Ignoring any of these steps can turn two distinct lines into the same symbol or make a correct duality map appear to fail.

Exercises

Why must the magnetic label of an ‘t Hooft line be a cocharacter?

Solution

The transition function around the Dirac string is a homomorphism $U(1)\to G$. Such homomorphisms form the cocharacter lattice. Requiring only a Lie-algebra element would not ensure that the transition function closes in the chosen global gauge group.

References

• Gomis, J., T. Okuda, and V. Pestun. “Exact Results for ‘t Hooft Loops in Gauge Theories on $S^4$.” Journal of High Energy Physics 2012, no. 5 (2012): 141. DOI; Open PDF.
• Kapustin, A. “Wilson–‘t Hooft Operators in Four-Dimensional Gauge Theories and S-Duality.” Physical Review D 74 (2006): 025005. DOI; Open PDF.