Line Operators, Global Forms, and Discrete Theta Data

A gauge algebra and a value of $\tau$ do not yet define a globally complete $\mathcal N=4$ theory. One must decide which Wilson–’t Hooft lines are genuine, which are attached to surfaces, which bundles enter the path integral, and which discrete topological phase weights those bundles. For $\mathfrak{su}(N)$ these choices admit a particularly explicit finite-lattice description.

Required background. The $\mathcal N=4$ theory card supplies the local action. Genuine line spectra and discrete theta data supplies the global-theory criterion used here.

Helpful background. Electric–magnetic charge lattices and global form derives Abelian dualization with integral fluxes. Electric and magnetic one-form symmetries explains the background fields detected by line operators.

Wilson–’t Hooft charge classes

A Wilson line is labeled by an electric weight, modulo screening by dynamical adjoint fields. An ’t Hooft line is labeled by a magnetic cocharacter. For a fixed algebra $\mathfrak{su}(N)$, their unscreened center-charge classes can be written

$$(e,m)\in\mathbb Z_N\times\mathbb Z_N.$$

Two lines can both be genuine only if their Dirac pairing is trivial:

$$\langle(e,m),(e',m')\rangle =em'-me'\pmod N.$$

Equivalently, transporting one line around the other gives the phase

$$\exp\!\left[\frac{2\pi i}{N}(em'-me')\right],$$

which must be unity for a mutually local set. A complete choice is a maximal isotropic subgroup $L\subset\mathbb Z_N^2$: the pairing vanishes on $L$, and no further independent charge class can be added while preserving mutual locality. The classification of Wilson–’t Hooft operators by weight and coweight data is developed in Kapustin 2006, §§2–4.

The word “genuine” is essential. A line charged under a gauged one-form symmetry may require a topological surface ending on it. Such a line is a useful relative defect but is not an isolated operator of the four-dimensional theory.

The $(SU(N)/\mathbb Z_k)_n$ family

Let $k$ divide $N$ and set $k'=N/k$. On a spin four-manifold, the theory conventionally denoted

$$\left(SU(N)/\mathbb Z_k\right)_n, \qquad n\in\mathbb Z_k,$$

has the genuine charge subgroup

$$L_{k,n} = \left\langle (k,0),\ (n,k') \right\rangle \subset\mathbb Z_N^2.$$

The first generator is the smallest genuine purely electric class. The second is a minimally magnetic line dressed by electric charge $n$. Their pairing is $kk'=N$, hence zero modulo $N$, and $L_{k,n}$ contains $N$ charge classes. This realizes a maximal isotropic subgroup.

The limiting cases are instructive:

global theory	generators of $L$	interpretation
$SU(N)$	$(1,0)$	all center Wilson classes are genuine
$(SU(N)/\mathbb Z_N)_0$	$(0,1)$	a purely magnetic generator is genuine
$(SU(N)/\mathbb Z_N)_n$	$(n,1)$	the minimal magnetic line is dyonically dressed

The integer $n$ is a discrete theta datum. It changes the weighting of topological sectors and the genuine dyonic line, even though the perturbative adjoint-field Lagrangian is unchanged. A systematic derivation and the duality action on these theories appear in Aharony, Seiberg, and Tachikawa 2013, §§2–3.

For the smallest example, $\mathfrak{su}(2)$ has three maximal choices:

$$\begin{aligned} L_{SU(2)}&=\{(0,0),(1,0)\},\\ L_{SO(3)_+}&=\{(0,0),(0,1)\},\\ L_{SO(3)_-}&=\{(0,0),(1,1)\}. \end{aligned}$$

These theories share the same local fields but have different genuine probes.

Modular transport of the lattice

Use

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

Because the matrix has determinant one,

$$e'm''-m'e'' =em''-me''\pmod N,$$

so mutual locality is preserved. Define $\rho(M)(e,m)^T=(e',m')^T$ by the displayed charge matrix. The transformed subgroup $L'=\rho(M)L$ defines the target theory. For the generators

$$S:\tau\mapsto-\frac1\tau,\quad(e,m)\mapsto(m,-e),$$ $$T:\tau\mapsto\tau+1,\quad(e,m)\mapsto(e-m,m).$$

The sign in the $T$ action is the passive convention of this chapter. In the common active convention the line is transformed in the opposite direction and the electric shift has the opposite sign.

For $\mathfrak{su}(2)$, reduction modulo two gives

$$S:\quad SU(2)\longleftrightarrow SO(3)_+, \qquad SO(3)_-\longmapsto SO(3)_-,$$

and

$$T:\quad SU(2)\longmapsto SU(2), \qquad SO(3)_+\longleftrightarrow SO(3)_-.$$

Thus $S$ is not an internal symmetry of the $SU(2)$ theory with its standard Wilson spectrum: it points to the $SO(3)_+$ theory. A modular word is a loop only if its final line subgroup and discrete data agree with its initial values.

One-form symmetry and background dependence

Genuine lines are charged objects for one-form global symmetries. Gauging an electric subgroup changes the global gauge group and introduces magnetic sectors; a discrete theta term can produce a mixed electric–magnetic anomaly. The modern background-field formulation of these operations is given in Gaiotto et al. 2015, §§4–6.

This background dependence is physically measurable. Partition functions on manifolds with nontrivial two-cycles can be resolved by ’t Hooft flux. Surface operators can detect whether a line is genuine, and interfaces must say how one-form backgrounds are transmitted. Local correlators of adjoint operators on $\mathbb R^4$ generally miss this information.

Scope and refinements

The finite $\mathbb Z_N^2$ description records charges after adjoint screening; it does not replace the full weight and coweight labels of individual line operators. Operator ordering, monopole bubbling, spin, and supersymmetric dressing add further data.

The formula $L_{k,n}$ is stated for spin four-manifolds, appropriate to a theory with fermions. On non-spin manifolds, line spins, quadratic refinements, and theta periodicities can change the classification. Background two-form fields can also reveal anomalies invisible in the bare subgroup $L$.

For a non-simply-laced algebra, electric and magnetic root systems differ in length and the Langlands-dual algebra need not equal the original algebra. The naive $SL(2,\mathbb Z)$ action must then be replaced by the appropriate subgroup and lacing-number transformation.

Exercises

1. Check isotropy. Prove that any two elements of $L_{k,n}$ have vanishing Dirac pairing modulo $N$.

Solution

Write the two elements as $(ak+bn,bk')$ and $(a'k+b'n,b'k')$. Their pairing is $(ab'-ba')kk'=(ab'-ba')N$, which vanishes modulo $N$.

2. Recover the three $\mathfrak{su}(2)$ theories. Evaluate $L_{k,n}$ for $(k,n)=(1,0),(2,0),(2,1)$.

Solution

For $(1,0)$ the generator is $(1,0)$, giving $SU(2)$. For $(2,0)$ the nonzero generator is $(0,1)$, giving $SO(3)_+$. For $(2,1)$ it is $(1,1)$, giving $SO(3)_-$.

3. Follow a duality word. Starting from $SU(2)$, apply $ST$ from right to left in the passive convention.

Solution

$T$ leaves $L_{SU(2)}$ fixed. $S$ then maps it to $L_{SO(3)_+}$. Thus the word is an arrow from $SU(2)$ to $SO(3)_+$, not a loop at $SU(2)$.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 08 (2013): 115. doi:10.1007/JHEP08(2013)115.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. doi:10.1007/JHEP02(2015)172.
• Kapustin, Anton. “Wilson–’t Hooft Operators in Four-Dimensional Gauge Theories and S-Duality.” Physical Review D 74 (2006): 025005. doi:10.1103/PhysRevD.74.025005.