1-form Symmetry in 4d N=4 SYM

• 1-form symmetry in 4d N=4 SYM is a global symmetry acting on line operators, defined by the Z_N center and closely intertwined with S-duality and anomaly constraints.
• It organizes quantum phases and topologically ordered sectors via discrete dyonic charge pairings and partition function degeneracies.
• SymTFT and superspace techniques enable systematic classification of these symmetries, illuminating their role in anomaly matching and non-invertible duality defects.

A 1-form symmetry in four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) refers to a generalized global symmetry acting on line operators such as Wilson and 't Hooft lines, intimately tied to the center of the gauge group and the global structure of the theory. In $\mathcal{N}=4$ SYM, the presence of %%%%2%%%% center symmetries and their electric/magnetic duals, non-invertible defects, and their interplay with S-duality and emergent automorphism ("bonus") symmetries, are crucial for understanding the quantum phases, duality web, and topologically ordered sectors of the theory. Superspace formulations and symmetry topological field theory (SymTFT) techniques allow systematic classification and computation of supersymmetry-protected quantities as well as anomalies involving these symmetries.

1. Definition and Characterization of 1-Form Symmetry in $\mathcal{N}=4$ SYM

The 1-form symmetry in 4d SYM, specifically for gauge group $SU(N)$, is a $\mathbb{Z}_N^{(1)}$ center symmetry acting on line operators in representations with nontrivial $N$-ality. Formally, Wilson loops transform as

$$\frac{1}{\dim r}\, \mathrm{Tr}\, \Pi \longrightarrow e^{\frac{2\pi i\,m}{N} n_r}\, \frac{1}{\dim r}\, \mathrm{Tr}\, \Pi$$

where $m \in \mathbb{Z}_N$, $n_r$ is the $N$-ality of the representation $r$, and $\Pi$ is the holonomy along the contour.

This symmetry can be coupled to background $\mathbb{Z}_N$ 2-form gauge fields, with charged operators picking up phases under the associated large gauge transformations. The symmetry is nontrivial for simply-connected groups and is modified or partially gauged in non-simply connected forms (e.g., $PSU(N)$) or when a subgroup is gauged. The presence of electric and magnetic one-form symmetries allows for mixed 't Hooft anomalies and constraints on the spectrum and phase structure.

2. Role in Quantum Phases and Topological Order

Quantum phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM exhibit an accumulation line of zero-temperature topologically ordered phases, each indexed by $N$ discrete electromagnetic one-form charges $(m,n) \in \mathbb{Z}_N \times \mathbb{Z}_N$ (Cabo-Bizet, 2021). These phases arise as bound states composed of two dyonic surface operators, each supported on a two-torus at fixed points of rotational symmetry. The states are labeled by holonomy variables

$$v_a = \frac{(a-\widehat{a})\,\mathrm{mod}\,N}{N}\,(m\,\tau+n)$$

and correspond to conformal blocks (primaries) of a $SU(N)_1$ WZNW model on the torus.

A key property is the $N$-fold degeneracy of the partition function in the Cardy-like limit: $$\mathcal{I} \sim e^{-S_\text{Casimir}\,Z_{G/G}}, \qquad Z_{G/G} = \sum_{\widehat{a}=0}^{N-1} 1 = N$$ which matches the counting from the one-form symmetry. The topological order and quantum phase transitions are thus organized by discrete one-form charges.

3. Non-Invertible Defects, Duality, and the "Bonus Symmetry"

Non-invertible symmetries, a modern notion in QFT, arise as topological defects implementing symmetry operations lacking group inverses. In Maxwell theory, combining $SL(2,\mathbb{Z})$ duality and gauging subgroups of electric/magnetic 1-form symmetries creates such defects. In $\mathcal{N}=4$ SYM, corresponding "duality" and "triality" defects exist, with duality defects defined by performing $S$-duality on a half-space, especially at self-dual points ($\tau = i$), and triality defects at other fixed points (Sela, 10 Jan 2024).

In the abelian $U(1)$ theory, these non-invertible defects act on local operators as elements of the $U(1)$ outer-automorphism group of the $\mathcal{N}=4$ superconformal algebra, historically referred to as the "bonus symmetry." The bonus symmetry combines a $U(1)$ rotation of the Maxwell field strength,

$F^{\pm} \to e^{\pm i\phi} F^{\pm},$

with a $U(1)_\psi$ fermionic symmetry, producing a net action $U(1)_Y = -U(1)_\psi - 2 U(1)_F$, under which the supercurrent transforms with charge $-1$ and the gravitational stress tensor remains invariant.

In nonabelian SYM, the bonus symmetry is only approximate and manifests in the supergravity limit (large $N$, strong coupling), as the maximal compact subgroup of the enhanced $SL(2,\mathbb{R})$ duality of IIB string theory on $AdS_5 \times S^5$. At finite $N$, only duality and triality defects are exactly topological, while additional non-invertible defects appear as approximate emergent symmetries.

4. SymTFT and Systematic Classification of 1-Form Symmetries

The Symmetry Topological Field Theory (SymTFT) formalism provides a systematic way to encode and classify 1-form symmetries in 4d theories (Duan et al., 13 Oct 2024). In this framework, the five-dimensional bulk theory is described by the BF model,

$$S_{BF} = \frac{N}{2\pi} \int_{M_5} (\tilde{B} \wedge dB)$$

with $\tilde{B}$ and $B$ two-form gauge fields. The canonical quantization on a 4-manifold $M_4$ yields a Hilbert space $\mathcal{H}(M_4)$ whose basis states correspond to boundary holonomies of $B$ and $\tilde{B}$. The action of $SL(2,\mathbb{Z})$ maps between distinct topological boundary conditions, corresponding to different global forms of the gauge group (e.g., $SU(N)$, $PSU(N)$), and encodes operations such as full or partial gauging of the 1-form symmetry.

For instance, the $S$-transformation acts as a Fourier transform between "position" and "momentum" bases,

$$V_S |(t,s)\rangle = \frac{1}{N^3} \sum_{t',s'} \omega^{t \cdot s' + s \cdot t'} |(t',s')\rangle,$$

where $\omega = e^{2\pi i/N}$, and the intersection pairing $K_{ij}$ governs the non-commutative algebra of the background holonomies.

5. Interplay with Supersymmetry-Protected Indices and Anomalies

In supersymmetric gauge theories, the SymTFT framework extends to compute SUSY-protected indices such as the Witten index on $T^4$, lens space index, and Donaldson-Witten or Vafa-Witten partition functions. These observables are encoded by dynamical boundary states $|\chi_{\text{SUSY}}\rangle$, and their values in a fixed background $b$ are given by contractions,

$$Z_{\text{SUSY}}[b] = \langle b | \chi_{\text{SUSY}} \rangle$$

with SL(2,$\mathbb{Z}$) transformations relating partition functions for theories with different global forms, via universal Fourier factors.

Partially gauged one-form symmetries can give rise to mixed 't Hooft anomalies, with the partition function in backgrounds $[c + \Delta c, d]$ acquiring phase factors,

$Z[c+\Delta c,d] = \omega^{-K(d,e_l)} Z[c,d],$

signaling non-invariance except with compensating counterterms. These anomalies impose constraints on infrared dynamics and confining vacua, and are manifest in the transformation properties of topologically twisted partition functions.

6. Manifestation in Superspace Formulation and Construction of Invariants

Superspace and harmonic superspace approaches encode the symmetry structure of $\mathcal{N}=4$ SYM transparently, as in the bi-harmonic superspace formalism (Buchbinder et al., 2020). By splitting the $SU(4)$ $R$-symmetry into $SU(2) \times SU(2) \times U(1)$, introducing harmonics $u_i^{(\pm)}, v_A^{(\pm)}$, and using bridge superfields to relate central and analytic gauges, one obtains manifestly $\mathcal{N}=4$ supersymmetric invariants. The gauge connections $V^{++}$, $V^{--}$ satisfy zero-curvature harmonic constraints,

$$D^{++} V^{--} - D^{--} V^{++} + [V^{++}, V^{--}] = 0,$$

which are sensitive to global symmetries, including higher-form and center symmetries. Reduction to $\mathcal{N}=2$ superspace preserves the information about hidden symmetry structure encoded in the bi-harmonic coordinates.

7. Algebraic Interpretations and VOAs

Vertex operator algebra (VOA) constructions associated with 4d $\mathcal{N}=4$ SYM provide an alternative algebraic viewpoint on the protected sector and symmetry gradings (Buican et al., 2020). Exact graded vector space isomorphisms (GVSI) between the chiral algebras of $\mathcal{N}=4$ SYM and their dual Argyres-Douglas models preserve the crucial $U(1)_r$ grading, which parallels the charge assignment under 1-form symmetries. For example, the mapping between affine currents and stress tensors,

$y(J^-) = T,$

and dimension formula

$h_{(3,2)} = 3 h_{su(2)} + f,$

repackages the symmetry data under "distorted symmetry breaking," showing how 1-form symmetry information survives in reorganized dual descriptions.

---

The paper of 1-form symmetry in four-dimensional $\mathcal{N}=4$ SYM encompasses its role in global structure, dualities, topological order, non-invertible defects, anomaly constraints, and algebraic interpretations. The amalgamation of geometric, group-theoretic, and quantum field-theoretic methods yields a unified understanding of these symmetries, strongly influencing the classification of quantum phases, duality webs, and protected spectra in supersymmetric gauge theories.