Pure ℝ Maxwell Theory: Duality & Symmetry

• Pure ℝ Maxwell Theory is a re-formulation of electromagnetism using the additive real group and differential forms underpinned by Hodge theory.
• The framework incorporates a classical action with Maxwell and θ terms, revealing continuous SL₂(ℝ) duality and the transition to discrete quantum symmetries.
• Higher-form symmetries and a 5D SymTFT unify surface operator algebras, clarifying how classical continuous symmetries become non-invertible in the quantum regime.

Pure $\mathbb{R}$ Maxwell theory is an augmented classical and quantum field-theoretic framework for abelian gauge fields in four dimensions, formulated with the gauge group extended from the compact $U(1)$ to the additive group of real numbers $\mathbb{R}$, and equipped with continuous symmetry and duality structures. This perspective foregrounds the geometric and Hodge-theoretic origins of Maxwell’s equations, elucidates the $SL_2(\mathbb{R})$ duality structure underlying classical electromagnetism, and clarifies the fate of symmetries—both invertible and non-invertible—upon quantization and global compactification (Sattinger, 2013, Hasan et al., 2024).

1. Hodge-Theoretic Foundation of Maxwell Theory

In four-dimensional Minkowski space-time $M^4$ with Lorentzian metric $g$ of signature $(+,-,-,-)$, electromagnetism is naturally recast in terms of differential forms and Hodge theory. The space $\Omega^p(M^4)$ of smooth $p$-forms supports the exterior derivative $d : \Omega^p \to \Omega^{p+1}$, characterized by $U(1)$0 and the graded Leibniz rule. The metric $U(1)$1 defines the Hodge star $U(1)$2 by

$U(1)$3

with explicit component expressions involving the Levi-Civita symbol. The coderivative $U(1)$4 is formally adjoint to $U(1)$5 with respect to the $U(1)$6 inner product:

$U(1)$7

The gauge potential $U(1)$8 yields the electromagnetic field strength $U(1)$9. Maxwell’s equations are succinctly recast:

$\mathbb{R}$0

where $\mathbb{R}$1 is a conserved 1-form encoding electric charge and current. This formulation is fundamentally geometric and independent of physical mechanism—applying to any field sourced by a density and current, governed by the Hodge decomposition and differential structure of $\mathbb{R}$2 (Sattinger, 2013).

2. Classical Action and Duality Structure

Maxwell theory on a closed oriented 4-manifold $\mathbb{R}$3 is defined by a gauge field $\mathbb{R}$4 and curvature $\mathbb{R}$5. The classical (Euclidean signature) action integrates both Maxwell and $\mathbb{R}$6 terms:

$\mathbb{R}$7

with the complexified coupling $\mathbb{R}$8, yielding an action

$\mathbb{R}$9

where $SL_2(\mathbb{R})$0 denote self-dual and anti-self-dual projections of $SL_2(\mathbb{R})$1 (Hasan et al., 2024). Classically, the theory possesses an $SL_2(\mathbb{R})$2 duality group acting on $SL_2(\mathbb{R})$3 by fractional linear transformations and rotating electric and magnetic fields (the "axial" U(1)).

3. Symmetry TFT and Higher-Form Symmetries

To capture the full spectrum of 1-form symmetries and dualities, the dynamical theory is realized as a boundary of a five-dimensional Symmetry Topological Field Theory (SymTFT), a BF-type theory with real two-form gauge fields $SL_2(\mathbb{R})$4:

$SL_2(\mathbb{R})$5

The equations of motion $SL_2(\mathbb{R})$6 restrict $SL_2(\mathbb{R})$7 and $SL_2(\mathbb{R})$8 to cohomology classes $SL_2(\mathbb{R})$9. Surface operators $M^4$0 and $M^4$1 satisfy a continuous Heisenberg algebra, encoding a $M^4$2 electric/magnetic pairing via intersection forms (Hasan et al., 2024).

Boundary conditions distinguish between the "pure $M^4$3 Maxwell theory" with continuous-valued one-form symmetries and the quantum (compact) Maxwell theory with $M^4$4 1-form symmetry. Duality operations in the 5D bulk interchange these boundary conditions. This construction elucidates how higher-form symmetries, surface operators, and dualities are unified within the SymTFT framework.

4. $M^4$5 and Non-Invertible Symmetries

The continuous Heisenberg algebra of surface operators admits $M^4$6 as its group of automorphisms (Stone–von Neumann theorem). The $M^4$7 Lie algebra is generated by explicit 0-form symmetry operators acting on the algebra of surface operators:

• $M^4$8 generates shifts;
• $M^4$9 generates "magnetic" translations;
• $g$0 rescales electric and magnetic labels.

The action on the complexified coupling is $g$1 for $g$2. The classical stabilizer of $g$3 is an $g$4 axial symmetry rotating $g$5 (Hasan et al., 2024).

In the quantum theory on compact $g$6, invertible dualities are reduced to $g$7, since only integer-valued electric/magnetic fluxes are allowed. Rational rescalings $g$8 are implemented via discrete gauging of finite $g$9 one-form subgroups, while irrational transformations require an infinite composition ("infinite gauging"). Each such "gauging" introduces a non-invertible fusion algebra.

5. Condensates, Projectors, and the Fate of Line Operators

Product fusions of rational gauging defects $(+,-,-,-)$0 yield topological "condensation" operators $(+,-,-,-)$1, realized as summations over all $(+,-,-,-)$2 magnetic and $(+,-,-,-)$3 electric fluxes. In the limit of an infinite product over rationals converging to an irrational parameter, one obtains a "continuous condensate" $(+,-,-,-)$4 that acts as a projector annihilating all nontrivial Wilson and ’t Hooft lines (Hasan et al., 2024).

The classical $(+,-,-,-)$5, which stabilizes $(+,-,-,-)$6 and generates continuous electromagnetic duality rotations, is restored in the quantum theory as a genuinely non-invertible symmetry: $(+,-,-,-)$7 acts as a topological defect, and its fusion $(+,-,-,-)$8 yields the projector that trivializes all charged line operators. The presence of these condensates reflects the deep interplay between non-invertible symmetries and quantized flux sectors.

6. Implications and Analogy with Gravity

A parallel Hodge-theoretic structure applies to any force field sourced by a conserved current, including gravitational fields linearized on Minkowski spacetime. A sign-flip in coupling constants recasts the theory to describe attractive forces, yielding "Maxwell-style" field equations for weak gravity. However, the linearization of Einstein’s equations produces spin-2 fields, whereas Maxwell’s theory is fundamentally spin-1. This distinction ensures that any Maxwell-type reformulation for gravity is an approximate model valid only for weak fields; the true gravitational degrees of freedom remain governed by general relativity (Sattinger, 2013).

7. Summary Table of Key Structural Elements

Feature	$(+,-,-,-)$9 Maxwell (Quantum)	Pure $\Omega^p(M^4)$0 Maxwell (Classical)
Duality group	$\Omega^p(M^4)$1 (invertible)	$\Omega^p(M^4)$2 (continuous)
Line operator algebra	Discrete (integer charges)	Continuous ($\Omega^p(M^4)$3-charges)
1-form symmetry	$\Omega^p(M^4)$4	$\Omega^p(M^4)$5
Stabilizer of $\Omega^p(M^4)$6 (axial symmetry)	Broken to discrete (non-invertible)	$\Omega^p(M^4)$7 continuous
Topological defects/projectors	Discrete condensates	Continuous condensates

Quantum flux quantization restricts the full $\Omega^p(M^4)$8 symmetry to its discrete subgroup and renders the classical continuous $\Omega^p(M^4)$9 stabilizer non-invertible; in the presence of continuous condensates, all nontrivial Wilson and ’t Hooft line operators are projected out, embodying the reemergence of a classical symmetry as a non-invertible topological feature in the quantum regime (Hasan et al., 2024).