Duality Groupoids, Walls, and Generalized-Symmetry Refinements

Once global forms and line spectra are retained, the invertible part of the proposed duality structure is a groupoid: its objects are complete theories, and its arrows are equivalences between possibly different objects. Exact charge-lattice and source–target maps identify candidate arrows; for the non-Abelian $\mathcal N=4$ examples, promoting them to equivalences is conditional on the S-duality conjecture and the complete background-dependent dictionary. A duality wall realizes such an arrow as a codimension-one interface. Stacking walls turns composition into a physical operation that can be checked on lines, background fields, and wall-localized sectors. The larger collection of all interfaces is not a groupoid because it also contains non-invertible walls.

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 invertible core of the interface category

Choose a Langlands-duality orbit of root data and collect all admissible global forms, genuine line lattices, discrete theta data, and background counterterms. An object is

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

Here $\eta_i$ includes the discrete theta choice and the background contact terms needed to say what the partition function means. An arrow

$$D_1\in\operatorname{Hom}(\mathcal T_i,\mathcal T_j)$$

is an equivalence together with a dictionary for observables, defects, and backgrounds. Write $s(D_1)=\mathcal T_i$ and $t(D_1)=\mathcal T_j$ for its source and target. If

$$D_2\in\operatorname{Hom}(\mathcal T_j,\mathcal T_k),$$

then, and only then, the composite $D_2\circ D_1$ is defined. Our convention is that the rightmost arrow acts first, so

$$s(D_2\circ D_1)=\mathcal T_i, \qquad t(D_2\circ D_1)=\mathcal T_k.$$

Every object has an identity arrow, and an invertible arrow has $D_1^{-1}:\mathcal T_j\to\mathcal T_i$. Composition is associative after interfaces related by a wall-local renormalization-group equivalence are identified. Before making that identification, walls, their junctions, and junction operators are more accurately organized as a bicategory: associativity is implemented by a coherent junction isomorphism rather than literal equality. The prerequisite discussion of operator transport and junction associativity develops this structure at the wall-junction level.

The modular matrix alone does not determine an 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. A modular group may act on a family of objects without being an internal symmetry group of each member.

The su(2) global-form 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 on the three objects as

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

The symbol $S$ therefore abbreviates three source-labeled arrows, $S_A:A\to B$, $S_B:B\to A$, and $S_C:C\to C$; similarly for $T$. For example, $T_B\circ S_A:A\to C$ is defined, whereas $T_A\circ S_A$ is not, because the target of $S_A$ is $B$ but the source of $T_A$ is $A$. At the level of the displayed charge classes, $S_B\circ S_A=\operatorname{id}_A$. A physical wall composition can still retain an invertible phase, as discussed below.

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, not just on the three mod-two lattices.

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. The global-theory orbit figure keeps the three lattices and stabilizers visible while displaying the same source-labeled arrows.

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. This statement first fixes the algebra-level interface; choosing its global form and allowed line endings completes the wall. Gaiotto and Witten develop the boundary-condition and $T[G]$ construction 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 $W_{D_1}:\mathcal T_i\to\mathcal T_j$ and $W_{D_2}:\mathcal T_j\to\mathcal T_k$ parallel, with a thin slab of the same complete object $\mathcal T_j$ 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. If the two sides of the slab differ in polarization, discrete theta data, or background counterterms, the walls are not composable until an additional interface supplies that change. 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.

Duality anomalies and projective composition

In backgrounds $B$, a proposed arrow can transform the partition function as

$$Z_{\mathcal T_j}[D B] =\exp\!\bigl(2\pi i\,\mathcal A_D[M,B]\bigr) Z_{\mathcal T_i}[B].$$

If $\mathcal A_D$ is the variation of an allowed local counterterm, choosing that counterterm completes the arrow. If it is not removable, the four-dimensional theory is relative to a five-dimensional invertible inflow theory and composition is projective until that inflow data is retained. Seiberg, Tachikawa, and Yonekura formulate duality anomalies as nontrivial fibrations of background-counterterm data over the coupling space in Seiberg, Tachikawa, and Yonekura 2018, §§2–4.

An anomalous phase therefore does not automatically invalidate a wall. It does mean that the phase is part of the arrow and of every junction associator. Dropping it can make a modular relation appear to hold when the corresponding physical walls actually differ by an invertible theory.

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 of the form

$$\mathcal D\circ\overline{\mathcal D} =\mathcal C_{\Gamma^{(1)}},$$

where $\mathcal C_{\Gamma^{(1)}}$ is the codimension-one condensation defect obtained by gauging the finite one-form symmetry on the wall. On a chosen wall topology it can be expanded as a topology-dependent sum over networks of one-form-symmetry surfaces, often with a three-dimensional topological field theory as coefficient. It is not, in general, a topology-independent finite sum of group elements. Since the right-hand side is not the identity wall, $\mathcal D$ 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, spin structure, 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 the required coherence relations, including any topological-theory-valued coefficients.

Exercises

1. Check source and target. Let $D_1:A\to B$, $D_2:B\to C$, and $D_3:A\to C$. Which of $D_2\circ D_1$ and $D_3\circ D_1$ is defined?

Solution

$D_2\circ D_1$ is defined because $t(D_1)=B=s(D_2)$, and it is an arrow $A\to C$. The expression $D_3\circ D_1$ is not defined: $D_1$ ends at $B$, while $D_3$ begins at $A$.

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

Solution

$T_B$ maps $B$ to $C$, and $S_C$ fixes $C$. Therefore the ordered composite is $S_C\circ T_B:B\to C$. The rightmost wall acts first.

3. Diagnose a projective relation. Suppose two wall words have the same bulk dictionary but their partition functions differ by a background-dependent phase that no allowed four-dimensional counterterm removes. Are they the same groupoid arrow of an absolute four-dimensional theory?

Solution

No. They differ by anomaly-inflow data. They can be compared as relative walls after adjoining the appropriate five-dimensional invertible theory, but identifying them as absolute arrows would discard part of the composition law.

4. Diagnose non-invertibility. If $\mathcal D\overline{\mathcal D}=\mathcal C_{\Gamma^{(1)}}$ and the condensation defect is not the identity, can $\overline{\mathcal D}$ be an inverse?

Solution

No. An inverse must fuse to the identity wall, up to a specified invertible phase. The nontrivial condensation defect records the one-form sectors summed over by the gauging operation and is the obstruction to invertibility.

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.
• Seiberg, Nathan, Yuji Tachikawa, and Kazuya Yonekura. “Anomalies of Duality Groups and Extended Conformal Manifolds.” Progress of Theoretical and Experimental Physics 2018 (2018): 073B04. doi:10.1093/ptep/pty069.