Electromagnetic Duality Anomaly

• Electromagnetic duality anomaly is the quantum breakdown of classical electric–magnetic rotation symmetry in Maxwell theory, manifesting in local, global, and lattice settings.
• It is analyzed using methods like Fujikawa’s technique and heat-kernel expansion, revealing how gravitational curvature induces helicity non-conservation.
• The anomaly has practical implications, including non-invertible duality defects, charge lattice constraints, and analogue-gravity experiments demonstrating helicity-selective vacuum pair creation.

Electromagnetic duality anomaly denotes the quantum failure, or more broadly the quantum reformulation, of the classical electric–magnetic rotation symmetry of Maxwell theory. In four-dimensional source-free electrodynamics, the classical equations admit continuous duality rotations; with a theta term, they extend to a classical $SL_2(\mathbb{R})$ action on the complex coupling $\tau$ and on the pair $(F,G)$. In the quantum literature, however, the phrase is used in several technically distinct senses: a local curved-spacetime anomaly of the duality current, a flux-quantization-induced reduction of classical $SL_2(\mathbb{R})$ to $SL_2(\mathbb{Z})$ that recent SymTFT work emphasizes is not an ’t Hooft anomaly, and a global anomaly of partition functions in nontrivial duality bundles. Recent work further recasts the classical $U(1)$ stabilizer as a non-invertible symmetry and realizes duality anomalies through defects, condensates, and analogue-gravity systems (Agullo et al., 2018, Hasan et al., 2024, Hsieh et al., 2019).

1. Classical duality, helicity, and the $SL_2(\mathbb{R})$ structure

In source-free Maxwell theory on a four-dimensional spacetime, the action

$$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$

is classically invariant, up to the standard Noether sense, under global electric–magnetic duality rotations

$$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$

Equivalently, in self-dual and anti-self-dual variables,

$$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$

so the duality acts as a $\tau$0 phase rotation on the two chiral sectors of the spin-1 field (Agullo et al., 2018).

The associated Noether charge is the optical helicity. Using a dual gauge potential $\tau$1 or, in Hamiltonian language, an “electric potential” $\tau$2, the current can be written as

$\tau$3

and the charge as

$\tau$4

with convention-dependent sign choices across the literature. In momentum space this charge measures the difference between right- and left-circularly polarized radiation, $\tau$5, so duality is simultaneously a symmetry of the equations of motion and a helicity symmetry (Agullo et al., 2018).

With a theta term and coupling $\tau$6, classical Maxwell theory admits a larger continuous duality structure. Writing

$\tau$7

and

$\tau$8

the classical group $\tau$9 acts by

$(F,G)$0

For fixed $(F,G)$1 in the upper half-plane, the stabilizer inside $(F,G)$2 is a $(F,G)$3 subgroup isomorphic to $(F,G)$4, corresponding to continuous duality rotations at fixed coupling (Hasan et al., 2024).

Usage of “electromagnetic duality anomaly”	Core statement	Representative source
Local curved-spacetime anomaly	$(F,G)$5 is sourced by gravitational curvature	(Agullo et al., 2018)
Quantization obstruction in flat Maxwell theory	classical $(F,G)$6 is reduced to $(F,G)$7 by charge/flux quantization, not by inflow	(Hasan et al., 2024)
Global anomaly in duality bundles	Maxwell partition functions acquire nontrivial phases in $(F,G)$8 monodromy backgrounds	(Hsieh et al., 2019)

2. Curved-spacetime quantum anomaly of the duality current

The most direct use of the term refers to the quantum non-conservation of the duality current in curved spacetime. In the spin-1 “Dirac-like” formulation, one quantizes the Maxwell field in terms of a first-order operator $(F,G)$9, and the renormalized divergence of the duality current is obtained from the second DeWitt coefficient $SL_2(\mathbb{R})$0 or, equivalently, from the Jacobian of a Fujikawa-type local duality rotation of the functional measure. The later papers in this line give the anomaly equation

$SL_2(\mathbb{R})$1

with

$SL_2(\mathbb{R})$2

The anomaly is described there as one-loop, local, independent of the choice of quantum vacuum state, and entirely due to the renormalization that preserves gauge invariance and general covariance rather than duality invariance (Agullo et al., 2018).

Integrating the anomaly between Cauchy surfaces gives

$SL_2(\mathbb{R})$3

so spacetime regions with nonzero gravitational Chern–Pontryagin density produce a net change in optical helicity. In Ricci-flat backgrounds this reduces to the Weyl Pontryagin density, and the identity

$SL_2(\mathbb{R})$4

makes explicit that parity-odd curvature sources a right/left photon asymmetry. The backgrounds explicitly identified as having potentially nonzero integrated $SL_2(\mathbb{R})$5 include rotating Kerr geometries, chiral gravitational-wave backgrounds, and dynamical strong-field processes such as collapses and compact-object mergers, whereas conformally flat Friedmann–Robertson–Walker spacetimes have vanishing Weyl tensor and hence no anomaly-induced helicity change (Rio, 2024).

The literature represented here is not uniform in its normalization statements. An earlier Fujikawa/heat-kernel presentation wrote

$SL_2(\mathbb{R})$6

again interpreting the effect as the spin-1 analogue of the chiral anomaly for fermions (Agullo et al., 2016). A different earlier renormalization analysis on spatially flat FLRW backgrounds expressed the broken Ward identity through the local Ricci combination

$SL_2(\mathbb{R})$7

and found unequal renormalized vacuum fluctuations of electric and magnetic fields even in a duality-invariant conformal vacuum (Agullo et al., 2014). The supplied literature therefore records a stable physical claim—quantum curved-spacetime non-conservation of optical helicity—together with multiple formulations of its local coefficient.

3. Flux quantization, charge lattices, and the reduction to $SL_2(\mathbb{R})$8

A distinct question concerns Maxwell theory quantized with a fixed lattice of line operators. In that setting, the classical $SL_2(\mathbb{R})$9 duality is not realized as an exact symmetry of a given quantum theory. Wilson and ’t Hooft lines carry integer electric and magnetic charges,

$SL_2(\mathbb{Z})$0

with $SL_2(\mathbb{Z})$1, so the dyonic charge lattice is $SL_2(\mathbb{Z})$2. Under an $SL_2(\mathbb{Z})$3 element

$SL_2(\mathbb{Z})$4

the charges transform as

$SL_2(\mathbb{Z})$5

and integrality is preserved. Irrational duality rotations would typically map integer charges to non-integer ones, violating the charge lattice. The exact quantum duality group of a fixed integral Maxwell theory is therefore $SL_2(\mathbb{Z})$6, not $SL_2(\mathbb{Z})$7 (Hasan et al., 2024).

In this sense, the obstruction is not an ’t Hooft anomaly. The theory remains fully consistent; the point is instead that flux and charge quantization restrict the admissible duality action. The $SL_2(\mathbb{Z})$8 and $SL_2(\mathbb{Z})$9 generators act by

$U(1)$0

$U(1)$1

and these are exact invertible symmetries because they preserve the quantized lattice. The continuous $U(1)$2 stabilizer of a fixed $U(1)$3 is therefore absent as an ordinary symmetry in the standard quantum theory not because of inflow, but because the line-operator spectrum furnishes a rigid integral global structure (Hasan et al., 2024).

This distinction is central to the modern terminology. In curved spacetime, the duality current can be anomalous in the usual quantum-field-theoretic sense. In fixed-lattice Maxwell theory, by contrast, the “breaking” of classical continuous duality is attributed to quantization data. The two phenomena are related by subject matter but not by mechanism.

4. SymTFT, discrete gauging, and non-invertible restoration of the classical $U(1)$4

The SymTFT construction of Maxwell theory provides a precise way to separate classical automorphisms from quantum realizability. The five-dimensional topological theory is a BF-like system

$U(1)$5

with topological operators on 2-cycles $U(1)$6,

$U(1)$7

obeying the Heisenberg relation

$U(1)$8

On the integrated variables $U(1)$9 and $SL_2(\mathbb{R})$0, one has $SL_2(\mathbb{R})$1, and explicit generators $SL_2(\mathbb{R})$2, $SL_2(\mathbb{R})$3, and $SL_2(\mathbb{R})$4 act as $SL_2(\mathbb{R})$5 automorphisms of this Heisenberg algebra (Hasan et al., 2024).

The boundary conditions determine which quantum Maxwell theory is realized.

Boundary variant	Boundary data	Duality behavior
$SL_2(\mathbb{R})$6-Maxwell	Dirichlet for $SL_2(\mathbb{R})$7, Neumann for $SL_2(\mathbb{R})$8	full $SL_2(\mathbb{R})$9 orbit
$$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$0-Maxwell	only $$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$1 and $$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$2 with $$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$3 end on the boundary	$$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$4 preserves integrality

Within the $$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$5-Maxwell variant, rational rescalings arise through discrete gauging of a non-anomalous subgroup $$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$6 of the $$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$7 1-form symmetry. The partition function rescales as

$$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$8

with $$S[A_\mu] \;=\; -\frac{1}{4}\int d^4x\,\sqrt{-g}\;F_{\mu\nu}F^{\mu\nu}$$9 ensuring the absence of a 1-form anomaly. In operator language, the $$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$0 automorphism with $$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$1 reproduces this rescaling. More general irrational $$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$2 operations are realized as limits of infinite sequences of rational gaugings, so the classical $$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$3 stabilizer can be “restored” only as a non-invertible symmetry (Hasan et al., 2024).

The cost of this restoration is encoded in condensates. Fusing rational gauging defects produces condensates, and for a continuous rotation $$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$4 the fusion with its opposite gives a “continuous condensate,” schematically

$$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$5

These condensates act as projectors on line operators: for a discrete condensate $$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$6, a line survives only when its electric and magnetic charges satisfy the corresponding divisibility conditions. A continuous condensate is an infinite product of such projectors, and for any fixed line operator some factor eventually projects it out. In that condensed phase, all Wilson and ’t Hooft lines are trivialized. The SymTFT therefore makes precise a striking statement: the classical $$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$7 duality can re-emerge quantum mechanically, but only as a non-invertible structure whose implementation eliminates the very line operators that diagnose the original charge lattice (Hasan et al., 2024).

5. Global duality anomalies, modular phases, and topological reformulations

A third use of the term concerns global anomalies of Maxwell partition functions in backgrounds with $$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$8 monodromy. In this formulation, the issue is not the local divergence of $$F'_{\mu\nu} \;=\; F_{\mu\nu}\cos\theta \;+\; \tilde F_{\mu\nu}\sin\theta,\qquad \tilde F'_{\mu\nu} \;=\; -F_{\mu\nu}\sin\theta \;+\; \tilde F_{\mu\nu}\cos\theta.$$9 and not the fixed-lattice reduction of $$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$0, but the failure of the partition function to be single-valued when duality twists are turned on around nontrivial cycles. On a closed oriented 4-manifold $$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$1, the Maxwell partition function transforms under $$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$2 with modular weights determined by $$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$3 and with a multiplier $$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$4 computed by a $$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$5-dimensional invertible bulk phase. The anomaly is characterized by

$$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$6

where $$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$7 is the twisted signature $$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$8-invariant and $$F_\pm \equiv \frac{1}{2}\big(F \mp i\,\tilde F\big),\qquad F_\pm \to e^{\pm i\theta}F_\pm,$$9 is an Arf invariant of a quadratic refinement on torsion cohomology (Hsieh et al., 2019).

The principal quantitative statement of that work is that the electromagnetic duality anomaly of 3+1-dimensional Maxwell theory is 56 times that of a Weyl fermion. Equivalently, in the spin–$\tau$00 bordism classification, the Maxwell class is 56 times the unit-charge Weyl-fermion class. This is presented both through $\tau$01-dimensional inflow and through a compactification argument using the $\tau$02-dimensional E-string theory, where the Higgs-branch description produces 56 Weyl fermions in four dimensions (Hsieh et al., 2019).

A related but broader topological reformulation appears for abelian $\tau$03-forms on compact Euclidean manifolds. There the dual effective actions agree exactly in odd spacetime dimension, while in even dimension they differ by a term proportional to the Euler number: $\tau$04 Despite this duality anomaly, the trace of the stress tensor agrees between dual descriptions, and the corresponding entanglement anomaly is identified with the duality anomaly of an edge-mode theory in two fewer dimensions (Donnelly et al., 2016).

These topological formulations sharpen an important conceptual point. In this setting, duality acts covariantly rather than strictly invariantly on the partition function. The anomaly is global, encoded by modular multipliers, bordism classes, or Euler-characteristic terms, and is naturally distinguished from both the local Pontryagin-density anomaly of curved-spacetime helicity non-conservation and the fixed-lattice reduction of $\tau$05 to $\tau$06.

6. Analogue systems, defect realizations, and related anomaly structures

Recent work has pursued experimentally accessible and defect-theoretic realizations of duality anomaly physics. In an infinitely long cylindrical waveguide that starts at rest and is accelerated into simultaneous rotation and translation, the in/out mode decomposition becomes helicity dependent. Under duality-preserving boundary conditions, the late-time positive-frequency domains $\tau$07 differ between right- and left-handed sectors, producing a spectral asymmetry and hence a nonzero vacuum expectation value of the optical-helicity charge. The central result is that the net helicity produced from the vacuum is nonzero if and only if both the angular velocity and the longitudinal velocity are nonzero; a related estimate gives

$\tau$08

with $\tau$09 photons in the illustrative regime $\tau$10, $\tau$11. The effect is presented as a laboratory analogue of the electromagnetic duality anomaly, realized through helicity-selective pair creation from the vacuum (Río, 26 May 2025).

A subsequent analogue-gravity treatment emphasizes the same mechanism in terms of frame dragging of the polarization basis. In that description, rotating, accelerating waveguides induce opposite helicity-dependent phase shifts in right- and left-handed modes, and the resulting mode-counting asymmetry is explicitly compared to the Adler–Bell–Jackiw anomaly in $\tau$12 dimensions. The paper does not require the explicit local coefficient $\tau$13 of the curved-spacetime anomaly equation; instead it tracks the anomaly through spectral asymmetry and net helicity production (Rio, 1 Sep 2025).

Beyond pure Maxwell theory, electromagnetic duality also reorganizes anomaly data in interacting systems. In four-dimensional $\tau$14 gauge theory with massless fermions, the ABJ chiral anomaly is mapped covariantly under $\tau$15 duality: in the QED frame it is proportional to $\tau$16, while in the dual frame it becomes a specific $\tau$17-determined linear combination of $\tau$18 and $\tau$19. The organizing structure is a conserved two-form current

$\tau$20

and the duality transformation is described there as covariant rather than anomalous in itself: it reshuffles the mixed anomaly between chiral and one-form symmetries (Choi et al., 17 Sep 2025).

Defect-theoretic work extends the same theme to non-invertible duality defects and their endpoints. For rational couplings, codimension-1 non-invertible duality defects in 4d Maxwell theory can end on codimension-2 twist defects around which $\tau$21. The operator spectrum of the twist defect factorizes into a generalized free-field sector and a chiral current sector governed by a right-moving compact boson with $\tau$22 current algebra, chiral central charge $\tau$23, and a defect ’t Hooft anomaly involving a chiral $\tau$24 symmetry. This realizes anomaly inflow locally on the endpoint of a non-invertible symmetry and ties electromagnetic duality anomaly to edge-mode physics of Chern–Simons type (Shao et al., 25 Sep 2025).

Taken together, these developments show that “electromagnetic duality anomaly” is not a single phenomenon but a family of quantum effects with distinct diagnostics. In curved spacetime it is a local failure of helicity conservation, usually written in terms of $\tau$25. In fixed-lattice Maxwell theory it names the loss of continuous $\tau$26 inside one quantum global variant, a loss attributed to charge quantization rather than inflow. In duality-twisted backgrounds it is a global modular anomaly with $\tau$27-dimensional inflow. And in recent non-invertible and analogue-gravity constructions, it appears as condensate-induced trivialization of line operators, chiral defect modes, or helicity-selective vacuum pair creation.