Duality Groupoids, Walls, and Generalized-Symmetry Refinements

Once global forms and line spectra are retained, the natural duality structure is a groupoid: its objects are complete theories, and its invertible arrows are equivalences between possibly different objects. A duality wall realizes an arrow as a codimension-one interface. Stacking walls then turns categorical composition into a physical operation that can be checked on lines, boundary conditions, and background fields.

Required background. Line operators and global forms defines the objects. Montonen–Olive and S-duality defines the modular arrows. Duality defects, walls, and interfaces supplies the folding and composition constructions.

Helpful background. Non-invertible topological defects and fusion explains why generalized defects need not have group-like fusion.

The duality groupoid

Fix a Lie algebra and collect all admissible global forms, genuine line lattices, discrete theta data, and background counterterms. An object is

$$\mathcal T_i=(\mathfrak g_i,G_i,L_i,\eta_i;\tau_i).$$

An arrow $D:\mathcal T_i\to\mathcal T_j$ is an equivalence together with a dictionary for observables and backgrounds. It has an inverse arrow, identity arrows exist at every object, and composable arrows associate. These are precisely the axioms of a groupoid.

The modular matrix alone does not determine the arrow. One must also specify:

• its action on $L_i$ and on the global gauge group;
• the transformation of continuous and discrete theta data;
• local counterterms for background one-form fields;
• possible anomalous phases on curved manifolds; and
• the identification of operator and defect sectors.

If an arrow begins and ends at the same object, it belongs to the automorphism group $\operatorname{Aut}(\mathcal T_i)$. Thus the familiar phrase “duality group” properly refers to the stabilizer of one globally specified theory, or to an action on a family after the object changes have been made explicit.

The $\mathfrak{su}(2)$ orbit

Let

$$\begin{aligned} A&=SU(2),&L_A&=\langle(1,0)\rangle,\\ B&=SO(3)_+,&L_B&=\langle(0,1)\rangle,\\ C&=SO(3)_-,&L_C&=\langle(1,1)\rangle. \end{aligned}$$

Modulo two, the passive generators act as

arrow	$A$	$B$	$C$
$S$	$B$	$A$	$C$
$T$	$A$	$C$	$B$

For example, $S:A\to B$ and $T:B\to C$, so the composition $T\circ S:A\to C$ is defined. The same symbols acting on $\tau$ obey modular relations up to the central element $-I$, which acts as charge conjugation on $(e,m)$. Whether $-I$ is trivial depends on the full observable dictionary.

This finite example also shows why projecting every object to the common algebra $\mathfrak{su}(2)$ creates false loops. The projected picture forgets exactly the line and discrete data that distinguish source from target.

Walls as physical arrows

Place $\mathcal T_i$ on $x^3<0$ and $\mathcal T_j$ on $x^3>0$. A duality wall $W_D$ couples their boundary values so that crossing the wall applies $D$. After folding the right half-space, the wall is a boundary condition for

$$\mathcal T_i\otimes\overline{\mathcal T_j}.$$

This makes several tests available. A line approaching the wall must emerge as its dictionary image or end on a wall operator. Conserved currents must obey the appropriate gluing condition. Background one-form gauge fields must be related by the same finite-lattice transformation as the genuine lines.

For the $S$ generator, half-BPS boundary conditions are related to three-dimensional $\mathcal N=4$ theories denoted $T[G]$. Their Higgs and Coulomb symmetries couple to the two sides and are exchanged by three-dimensional mirror symmetry. Gaiotto and Witten develop this interface description and its S-duality action in Gaiotto and Witten 2009, §§3–4 and 8.

A $T$ wall is more elementary locally: it implements a theta-angle shift and carries the corresponding three-dimensional Chern–Simons contact term. Globally, that term also shifts the electric dressing of a magnetic line, so it can change the discrete theta label.

Composition by stacking

Put walls $W_{D_1}$ and $W_{D_2}$ parallel with a thin slab of the intermediate theory between them. At distances much larger than the separation, renormalization-group flow produces a composite wall

$$W_{D_2}\circ W_{D_1} \simeq W_{D_2\circ D_1}.$$

The equality is an infrared equivalence, not necessarily equality of microscopic wall Lagrangians. Decoupled topological sectors and invertible phases can remain. Consequently a reliable composition check includes:

1. the source and target global theory;
2. the induced map on every genuine-line class;
3. background counterterms and anomaly inflow;
4. localized wall degrees of freedom; and
5. any topological factor produced when the intermediate slab is removed.

An anomalous phase does not automatically invalidate the wall. It may mean the duality acts projectively on partition functions and requires a five-dimensional invertible inflow theory. Omitting that phase, however, gives an incomplete composition law.

From invertible arrows to non-invertible defects

An ordinary duality wall between two distinct objects is invertible as an interface: stacking the inverse returns the identity, possibly with a controlled invertible phase. Non-invertibility arises after a further operation such as gauging a finite one-form symmetry in half of spacetime or summing over global sectors.

At a self-dual value of $\tau$, compose a duality interface with a one-form gauging interface so that the endpoints define the same theory. The resulting defect $\mathcal D$ can obey a fusion rule schematically of the form

$$\mathcal D\circ\overline{\mathcal D} =\sum_{\gamma\in\Gamma^{(1)}}\mathcal U_\gamma,$$

where $\mathcal U_\gamma$ are one-form-symmetry surface sectors, with possible three-dimensional topological coefficients. Since the right-hand side is a sum rather than one identity defect, $\mathcal D$ has no inverse. General four-dimensional condensation, duality, and triality defects of this kind—including $\mathcal N=4$ examples—are constructed in Choi et al. 2023, §§2–4 and 6.

This refinement should not be attached automatically to every S-duality wall. It requires a specified gauging operation, a self-dual object or orbit, and compatible anomalies. The invertible groupoid remains the correct starting structure.

Limitations

Interfaces preserve only the supersymmetry compatible with their orientation and couplings. A half-BPS wall need not determine nonsupersymmetric defect observables. The three-dimensional wall theory can also flow to an interacting fixed point whose microscopic description is not unique.

Fusion is sensitive to global form, polarization, and background counterterms. Two walls with the same action on local bulk operators can differ by a three-dimensional invertible theory. Finally, a proposed non-invertible fusion algebra is meaningful only after its junctions satisfy associativity, including any topological-theory-valued coefficients.

Exercises

1. Compose the $\mathfrak{su}(2)$ arrows. Starting at $B$, apply $T$, then $S$. Where do you end?

Solution

$T$ maps $B$ to $C$, and $S$ fixes $C$. Therefore $S\circ T$ is an arrow $B\to C$.

2. Distinguish a group from a groupoid. Why can the full modular action on $\{A,B,C\}$ not be called the internal symmetry group of $A$?

Solution

Because $S$ sends $A$ to the distinct object $B$. Only modular words whose transformed global data return to $A$ are automorphisms of $A$.

3. Diagnose non-invertibility. If $\mathcal D\overline{\mathcal D}=1+\mathcal U$ with $\mathcal U\neq0$, can $\overline{\mathcal D}$ be an inverse?

Solution

No. An inverse would fuse to the identity alone. The additional topological sector is the obstruction and records the summed or gauged one-form sectors.

References

• Choi, Yichul, Clay Córdova, Po-Shen Hsin, Ho Tat Lam, and Shu-Heng Shao. “Non-Invertible Condensation, Duality, and Triality Defects in $3+1$ Dimensions.” Communications in Mathematical Physics 402 (2023): 489–542. doi:10.1007/s00220-023-04727-4.
• Gaiotto, Davide, and Edward Witten. “S-Duality of Boundary Conditions in $\mathcal N=4$ Super Yang–Mills Theory.” Advances in Theoretical and Mathematical Physics 13 (2009): 721–896. doi:10.4310/ATMP.2009.v13.n3.a5.