Electric–magnetic duality and charge lattices on The Broken-Born program

The modular group SL(2, ℤ)

Mon, 01 Jan 0001 00:00:00 +0000

The modular group SL(2, ℤ) is the group of 2 × 2 matrices with whole-number entries and determinant 1. Each such matrix [[a, b], [c, d]] moves the points z of the upper half of the complex plane by the fractional-linear map z → (az + b)/(cz + d), and two of the matrices generate all the others: S, which sends z to −1/z, and T, which sends z to z + 1. Because the matrices and their inverses have whole-number entries, they carry a two-dimensional lattice, a regular grid of points, onto itself, and in gauge theory the group acts on the lattice of electric and magnetic charges and on the complexified coupling τ_c, which combines the coupling g with the vacuum angle θ. T, the shift of θ by 2π, relabels the charges and leaves the set of particles unchanged. S exchanges electric and magnetic charges and, at θ = 0, turns g into 4π/g; as a symmetry of a quantum theory it is S-duality, which the checkpoint records as exact in maximally supersymmetric Yang–Mills theory and which is not expected to be exact without supersymmetry. The program meets the group through the matrix of S, which the checkpoint notes is also the matrix of Born reciprocity and of the program’s modular inversion Z → −Z⁻¹; on phase space, though, the symmetry group that contains that matrix is continuous, not this discrete one. Whether the two settings are parts of one enlarged “modular” structure is an open conjecture, and an earlier use, in which reciprocity, acting as the modular S transformation, was conjectured to fix a hidden sector’s coupling, was superseded.

S-duality and Montonen–Olive

Mon, 01 Jan 0001 00:00:00 +0000

S-duality is the proposed equivalence between a gauge theory at coupling g and a theory at a coupling of order 1/g in which electrically charged particles and magnetic monopoles exchange roles, so that a strongly coupled theory has a weakly coupled description. It grows out of a symmetry of Maxwell’s equations in empty space, which are unchanged when the electric and magnetic fields are exchanged. Montonen and Olive proposed the quantum version in 1977. The checkpoint records it as exact in maximally supersymmetric Yang–Mills theory, one of the few theories in which it has passed non-trivial tests, and Seiberg’s related dualities supply the “magnetic quarks” of a dual description. In string theory S-duality sits with T-duality inside a larger U-duality. The program uses S-duality as a structural comparison, because it acts on the coupling through the same matrix as Born reciprocity, and it carries a standing caution: outside special supersymmetric theories the duality is an expectation, not an established symmetry.

The dual superconductor

Mon, 01 Jan 0001 00:00:00 +0000

The dual superconductor is the leading picture of quark confinement, the observed fact that quarks are never found in isolation. In an ordinary superconductor a condensate of paired electrons expels magnetic field and squeezes whatever magnetic flux remains into thin tubes. In the dual picture, proposed by ’t Hooft and Mandelstam, the vacuum of the strong interaction holds a condensate of magnetic monopoles, which squeezes the color-electric field between quarks into a string whose energy grows with its length, so that pulling two quarks apart never frees either of them. It is one of the three pieces of established physics that absorbed the program’s founding supposition 11, a magnetic counterpart for every electric particle; the program adds no claim of its own here.

The magnetic mirror-family

Mon, 01 Jan 0001 00:00:00 +0000

The magnetic mirror-family is the idea that every electrically charged particle, quarks included, has a magnetic counterpart. It entered the program as P.R.’s founding supposition 11, and the checkpoint records it as established physics: Montonen–Olive duality, Seiberg’s magnetic quarks and the dual-superconductor picture of quark confinement already contain it. Dirac’s condition accounts for why the counterparts are not seen: a weak electric coupling forces a strong magnetic one, so magnetic particles are heavy and tightly bound. The checkpoint’s figure for the magnetic coupling of our electromagnetism, about 34, is not the hidden-sector coupling α_d ≈ 34 that died with the soliton candidate. The program’s own addition is the observation that choosing which particles count as electric is a choice of polarization, which has its own page.

Obituary: conjugate monopoles explain the null result

Mon, 01 Jan 0001 00:00:00 +0000

This was the claim that magnetic monopoles exist but belong to the conjugate, momentum-polarized half of physics, which would explain why no experiment has found one. It was the program’s founding supposition 5. The checkpoint’s audit of its monopole claims split the supposition in two and killed this explanatory half on two counts: the symmetry that exchanges electric and magnetic charge is electric–magnetic duality, not the position–momentum reciprocity the program is built on, and inflation already accounts for the absence of monopoles without new physics. The claim is dead and stays dead. The other half, that such monopoles may exist and would gravitate, survives on a separate page, and so does the prediction that no monopole of our sector will be found.

Conjugate monopoles as ontology

Mon, 01 Jan 0001 00:00:00 +0000

Conjugate monopoles are hypothetical magnetic monopoles, or other topological solitons, belonging to the program’s momentum-polarized matter sector rather than to ours. The checkpoint keeps one statement about them: if they exist, they gravitate, because every kind of matter in the program couples to the same geometry, so their existence could not be without consequence. Their use as the dark matter itself, the superheavy soliton candidate, was retired on 2026-10-19 after its abundance arithmetic failed review. What remains is a statement about what such objects would do, not a claim that they exist.

The symplectic charge lattice

Mon, 01 Jan 0001 00:00:00 +0000

The symplectic charge lattice is the program’s observation that the set of allowed electric and magnetic charges has the same geometry as phase space. The Dirac–Schwinger–Zwanziger (DSZ) pairing of two charges is an antisymmetric whole-number form, like the pairing of position with momentum; charges that can all be described together by one Lagrangian form a “Lagrangian” half of the lattice; and deciding which particles count as electric is therefore a choice of polarization, as the program’s vacuum chooses which coordinates count as position. One quarter-turn matrix generates Born reciprocity, the program’s modular inversion and S-duality. The checkpoint counts this as a structural unification [D — structural; math T]: the mathematics is established, the identification of the two choices is the program’s, and no calculation depends on it. Whether the two polarizations are one physical structure is open problem 8.

Open: one modular structure for both polarizations (OP8)

Mon, 01 Jan 0001 00:00:00 +0000

This open problem asks whether two choices the program treats in parallel are parts of one structure: the vacuum’s choice of which phase-space directions count as position (postulate P5), and the choice of which charges count as electric, which is a polarization of the charge lattice. Both choices are made with the same quarter-turn matrix, and the checkpoint conjectures that they are components of one larger “modular” structure, on the model of string theory, where T-duality and S-duality are both contained in a larger U-duality. It is open problem 8 of the checkpoint, tagged [O]. The checkpoint does not say what the larger structure would be, no calculation toward it exists in the repository, and it is not in the current work queue. The toy models that met the program’s first acceptance criterion did not address the related demand that the order parameter also select the electric charges.

Obituary: the magnetic identification (popsci-only)

Mon, 01 Jan 0001 00:00:00 +0000

The magnetic identification is a claim made only in the popular article “The Universe’s Other Half” (#12): that the program’s momentum-polarized sector is the magnetic side of electromagnetism, dark because Dirac’s condition makes its coupling strong, “about 34, against our 1/137”, with only topological knots surviving from it as free relics; those knots are the retired soliton candidate. No paper or note of the program makes this claim. The technical record supports only the observation that choosing electric charges and choosing position are the same matrix operation, together with an open conjecture that they are one structure. The claim conflicts with a dead claim, with the strict separation of the two sectors’ forces and with the program’s prediction of a permanent null in direct detection, so the wiki records it as dead.