The modular group SL(2, ℤ)

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in which reciprocity, acting as the modular S transformation, was conjectured to fix a hidden sector’s coupling, was superseded.

Status: Established — [T] — standard mathematics and field theory, written here from general knowledge, with the literature it names marked †. No program document names the group: #11 uses its generator S, as τ_c → −1/τ_c (§11.3), and cites Montonen–Olive and Seiberg as [18†] (§11.2), references from model memory that await the † literature battery. No verify/ script checks the group or its action on charges and couplings; the matrix [[0, 1], [−1, 0]] is machine-checked on the phase-space side, at d = 1, in verify/toy_coset.py (the toy-model scripts).

This page writes z for a point of the upper half-plane. The literature usually writes τ, a letter this wiki reserves for the dark-energy field; the gauge coupling is τ_c (notation).

The group [T]#

Whole-number matrices of determinant 1#

SL(2, ℤ) = { [[a, b], [c, d]] :  a, b, c, d whole numbers,  ad − bc = 1 }

The product of two such matrices is another, and so is the inverse of one, [[d, −b], [−c, a]] [T]. The condition on the determinant is what keeps the inverse whole: an integer matrix has an integer inverse only when its determinant is 1 or −1. The integer matrices of determinant ±1 are therefore exactly the changes of basis of a lattice, the linear maps that carry the grid of whole-number pairs onto itself [T]. Requiring +1 also keeps the antisymmetric pairing u₁v₂ − u₂v₁ of two lattice vectors u and v, which is their determinant and is multiplied by det M under a map M [T]. The charge lattice below carries such a pairing.

The generators S and T#

S = [[0, 1], [−1, 0]]     z → −1/z        (#11 §11.3's matrix)
T = [[1, 1], [0, 1]]      z → z + 1

Every element of SL(2, ℤ) is a product of S, T and their inverses [T]. The proof is Euclid’s algorithm for the greatest common divisor, run on the first column (a, c) of a matrix: multiplying by Tⁿ from the left replaces a by a + nc, multiplying by S from the left replaces (a, c) by (c, −a), and alternating the two shrinks the column until it is (±1, 0). A matrix with that first column and determinant 1 is ±Tⁿ for some whole number n, and −1 = S².

As maps of z, S applied twice is the identity, and so is the product S·T applied three times; every relation between S and T as maps follows from these two [T]. As matrices, S² = −1, minus the unit matrix, which the toy-model note states for the same matrix as ρ: “ρ = [[0, 1], [−1, 0]] is symplectic and squares to −1” (papers/reciprocity-toy-models.md §1). Many textbooks write S with the opposite sign, [[0, −1], [1, 0]], the inverse of #11’s matrix. Both give the map z → −1/z, but relations written with matrices can differ in sign between the two conventions [T].

How it acts on the upper half-plane [T]#

A matrix [[a, b], [c, d]] acts on a complex number z by the fractional-linear, or Möbius, map, and the imaginary part of the image follows from the determinant:

z → (az + b)/(cz + d)

Im[(az + b)/(cz + d)] = (ad − bc) · Im z / |cz + d|²

A matrix of determinant 1 therefore keeps a point with Im z > 0 in the upper half-plane [T]. Composing two maps corresponds to multiplying their matrices, and a matrix and its negative give the same map, so the group of distinct maps is PSL(2, ℤ), SL(2, ℤ) with each M identified with −M [T]. The same formula with real entries is the action of the continuous group SL(2, ℝ), the d = 1 case of the symmetry group of the Siegel domain; the modular group is its subgroup with whole-number entries.

What S and T do#

• T moves every point one unit to the right.
• S sends z = r·e^{iφ} to (1/r)·e^{i(π − φ)}: it inverts the distance from the origin and reflects the point across the imaginary axis. It carries the unit semicircle |z| = 1 onto itself and leaves one point fixed, z = i (inversion of the half-plane).

The elements of SL(2, ℤ) that leave z = i in place are the powers of S [T]. In SL(2, ℝ) the matrices that fix i form a continuous circle of rotations, and S is one of them, the quarter-turn (the upper half-plane).

One representative of each orbit#

The set of points that a group can reach from a given z is the orbit of z. The continuous group SL(2, ℝ) has a single orbit, the whole half-plane [T]. The modular group has many, and every one of them meets the region

F = { z :  |Re z| ≤ 1/2  and  |z| ≥ 1 }

T moves any point into the strip |Re z| ≤ 1/2, and S moves a point of the strip that lies inside the unit circle to one outside it, with a larger imaginary part, since Im(−1/z) = Im z / |z|². Alternating the two brings every point into F, and no two points of the interior of F are related by the group. F, the standard fundamental domain, therefore holds one representative of each orbit, apart from identifications along its edges [T] (Serre 1973 †). The fixed point z = i lies on the edge of F.

Charge lattices [T]#

The lattice and its pairing#

In a theory with one unbroken U(1) gauge field, each particle carries an electric charge e and a magnetic charge g, and the allowed pairs q = (e, g) form a lattice (the charge lattice). The Dirac–Schwinger–Zwanziger (DSZ) pairing of two charges, ⟨q, q′⟩ = e₁g₂ − e₂g₁, which #11 §11.3 places in 2πℤ, is the determinant of the two charge vectors. Written in a basis of the lattice, the maps that carry the lattice onto itself and keep the pairing are therefore the integer matrices of determinant 1: SL(2, ℤ) [T] (dyons and the DSZ pairing).

With the charges labelled by whole numbers (n, m), n electric and m magnetic units, the generators act as follows [T]:

• S sends (n, m) to (m, −n): electric and magnetic charges change places, as the fields do in Maxwell’s exchange E → B, B → −E (duality in Maxwell’s equations). Applied twice it sends every charge to its opposite, (n, m) → (−n, −m), which is charge conjugation. The sign that S² = −1 carries is invisible on z and visible on charges.
• T sends (n, m) to (n + m, m), adding the magnetic label to the electric one. This is the relabelling of dyons that the Witten effect produces when the vacuum angle θ rises by 2π (shifting θ by 2π relabels the dyons).

Charges whose pairing vanishes can be described together by one local Lagrangian. For one U(1), a largest such set, a Lagrangian sublattice, consists of the lattice points on one line through the origin, and an element of the group can turn any such line onto the electric axis [T]. Every duality frame, every choice of which charges count as electric, is therefore related to every other by the group (a duality frame is a polarization).

Several gauge fields: Sp(2r, ℤ)#

With r unbroken U(1) factors the lattice has 2r dimensions, and the integer maps that keep the pairing form the integer symplectic group Sp(2r, ℤ), of which SL(2, ℤ) is the case r = 1 [T] (the same geometry as phase space). The couplings and vacuum angles then form a symmetric r × r complex matrix τ_c whose imaginary part is positive definite, as positive field energy requires. That makes τ_c a point of the Siegel domain of size r, the matrix version of the upper half-plane (the Siegel domain), and Sp(2r, ℤ) acts on it by the matrix form of the same map, τ_c → (Aτ_c + B)(Cτ_c + D)⁻¹ [T]. Mathematicians call Sp(2r, ℤ) the Siegel modular group. Without charges, the duality rotations of the classical fields form the continuous group Sp(2r, ℝ), which acts on the couplings as well (Gaillard and Zumino 1981 †); quantized charges restrict it to the integer matrices [T]. No program document considers the charge lattice of several gauge fields.

The complexified coupling [T]#

A gauge theory’s coupling g, here the gauge coupling and not a magnetic charge, and its vacuum angle θ, the coefficient of a term proportional to E·B (the θ term), combine into one complex number, τ_c = θ/2π + 4πi/g² in the usual normalization [T]. The notation is #11’s. Since g² > 0, τ_c lies in the upper half-plane, and the modular group acts on it by the maps above while the corresponding matrix relabels the charges [T].

• T: τ_c → τ_c + 1 raises θ by 2π. By the Witten effect it maps the set of dyons onto itself, relabelled, so it is a symmetry of any theory in which θ is an angle of period 2π [T] (when θ is physical).
• S: τ_c → −1/τ_c exchanges the electric and magnetic descriptions. At θ = 0 it sends τ_c = 4πi/g² to ig²/4π, the same form with g replaced by 4π/g, and its fixed point τ_c = i is the self-dual coupling, at which g = 4π/g [T]. Whether S is a symmetry is the question of S-duality. #11 records Montonen–Olive duality as “exact in maximally supersymmetric Yang–Mills” (§11.2 [T]). Harvey’s lectures, as the round-3 referee quoted them after checking the source, say that it has been “put to non-trivial tests in finite N = 4 Super Yang-Mills theory and special finite N = 2 theories”, and that for the non-supersymmetric theory “there is no reason to think that the duality of the spectrum should be maintained by quantum corrections” (review/round3/referee/audit-notes.md Step 1).

Where both generators are symmetries, so is the whole group: the theory at τ_c and the theory at (aτ_c + b)/(cτ_c + d) are the same theory with its charges relabelled, and the distinct theories correspond to the points of F [T]. This is the form in which S-duality is usually stated for N = 4 super Yang–Mills theory (SYM), the four-dimensional gauge theory with the most supersymmetry possible without gravity; Sen (1994 †) gave evidence for it from bound states of monopoles. In a theory without supersymmetry T can still be exact, while S is not expected to be; the v3 worklog, summarizing Harvey and Osborn, records that this “is an expectation, not a no-go” (review/v3-worklog.md, ‘Round-2 pending list’, item R1).

Tori and string theory [T]#

The group’s name comes from the theory of elliptic functions [T]. Identifying the points of the complex plane that differ by whole-number combinations of 1 and z turns the plane into a torus whose shape is fixed by z, its modulus. Two values of z give the same torus exactly when an element of SL(2, ℤ) carries one to the other, so the shapes of tori correspond to the points of F [T].

String theory uses the group in both ways, on the shape of a torus and on a coupling, and both uses appear in the program’s record:

• Modular invariance. At one loop a closed string sweeps out a torus, and the amplitude cannot depend on which of the equivalent values of the modulus labels it, so it must be unchanged by SL(2, ℤ) [T]. The toy-model worklog lists a derivation of the section condition of double field theory, its strong constraint, “from modular invariance plus amplitude poles” (Betz–Blumenhagen–Lüst–Rennecke, live-searched; notes/toy-lagrangian.md ‘Chunk 4 (in progress, 2026-10-19): S5’; where the condition comes from).
• S- and U-duality. In one of the ten-dimensional superstring theories, S-duality is an SL(2, ℤ) acting on a complex coupling whose imaginary part is the inverse of the string coupling [T]. Hull and Townsend (1995 †) proposed that, for the type II superstrings on a torus, it combines with T-duality into a larger discrete group, U-duality. #11 §11.3 takes “T- and S-duality cohabiting inside U-duality in string theory [T]” as the model for its conjecture (T-, S- and U-duality in string theory).

How the program uses it#

One matrix, two kinds of group#

#11 §11.3: “S-duality acts on the complexified coupling by τ_c → −1/τ_c, generated by the same [[0,1],[−1,0]] rotation as Born’s ρ and our Z → −Z⁻¹: the third appearance of one matrix in this program.” The section is tagged [D — structural; math T] (one matrix, three appearances).

The three appearances share the matrix, but the symmetry group is discrete on the charge side and continuous in the other two settings [T]:

In the toy model the continuous group is explicit: verify/toy_coset.py checks that the action is invariant under a generic element of Sp(2, ℝ), and hence under ρ (section ‘S1’). #11’s refinement of P2′ places ρ inside a continuous family, the rotations that fix the self-dual point Z = i (the fixed point): “with only ω rigid, ρ sits inside a continuous U(1), and the polarization is a Goldstone angle rather than a discrete choice [D][V]” (§4; the polarization as a Goldstone angle). On the charge side no continuous family passes through S, since in SL(2, ℤ) the elements that fix τ_c = i are the powers of S [T]. The checkpoint does not comment on the difference.

Reciprocity as S on the half-plane#

#11 §6: “ρ: Z → −Z⁻¹ (Möbius; d = 1 check), fixed point Z = i”. At d = 1 the Siegel domain of the modular field is the upper half-plane, and the map is the action of S, z → −1/z, with the same fixed point (reciprocity as modular inversion). It is machine-checked symbolically at d = 1; verify/toy_coset.py prints PASS rho acts on Z as Z -> -1/Z with fixed point Z = i.

For general d the matrix with unit blocks, [[0, 1], [−1, 0]], has whole-number entries and is symplectic, so it lies in the Siegel modular group Sp(2d, ℤ) as well as in Sp(2d, ℝ) [T]. The other maps the documents use are continuous. Calc note #6 places the translations Z → Z + B, with B any real symmetric matrix, in “the Möbius action of the rigid Sp(2d,ℝ)” (§1; shift symmetry); only the translations with whole-number B belong to the Siegel modular group [T].

The coupling conjecture of the soliton manuscript#

The second version of the soliton manuscript #14 made S act on a coupling. Its §2, preserved verbatim in review/skeptic-review-console.html, conjectured “a distinct gauge structure whose complexified coupling is fixed by the reciprocity map, acting as the modular S transformation, to the image of ours - alpha_d = 1/(4 alpha) ~ 34”. The handoff describes it as “explicitly the load-bearing conjecture; without it α_d is a free strong coupling” (HANDOFF.md §2). Round 2 named the derivation the manuscript’s weakest link, “matrix coincidence ≠ physical identity” / “numerology” (HANDOFF.md §6; review rounds 1 and 2).

The worklog’s finding F2 then showed that the strong coupling did not need the conjecture, and Paul decided on 2026-10-19 that “S-duality is no longer what fixes α_d” (review/v3-worklog.md, ‘Findings neither referee raised’). Version 3 kept the shared matrix “only as motivation” (#14 §2), and the manuscript was withdrawn after round 3. The conjecture is superseded (Superseded: reciprocity acting as modular S), and the hidden monopoles’ coupling α_d ≈ 34 is dead (Obituary: the coupling α_d ≈ 34).

The conjectured enlarged structure (OP8)#

#11 §11.3 ends with a conjecture: “Conjecture [O]: the phase-space polarization (P5) and the charge-lattice polarization are components of one enlarged modular structure — the BBT-native analog of T- and S-duality cohabiting inside U-duality in string theory [T].” BBT is the Broken-Born program’s name for itself. The conjecture is item 8 of #11 §14, “Charge-lattice polarization: formalize §11.3’s unification [O]”.

On this page’s reading, the modular structure on the charge side is the duality group acting on charges and on τ_c, SL(2, ℤ) for one gauge field, and on the phase-space side it is the continuous Sp(2d, ℝ) acting on the Siegel domain (the two polarizations). The checkpoint does not say what the enlarged structure would be. The item, the earlier form it replaced and the related demand on the order parameter are on Open: one modular structure for both polarizations (OP8).

The word “modular” in the documents#

In the program’s documents “modular” mostly describes the Siegel domain and its Möbius maps under the continuous group: “the modular field” (#11 §6), “the whole modular family of §6” (#11 §5), “modular geometry” (calc note #5 §C) and “modular (Siegel-domain) symmetry” (calc note #6 §3). In mathematics the modular group is SL(2, ℤ), or PSL(2, ℤ), and the Siegel modular group is its matrix version Sp(2d, ℤ), acting on the Siegel domain by the same formula; the names refer to the integer groups [T]. The duality group itself enters the documents only through S, as in #11 §11.3’s τ_c → −1/τ_c and #14 v2’s “modular S transformation”. No document names SL(2, ℤ) or its generator T.

What is open#

• The enlarged structure. The conjecture of #11 §11.3 is open [O], as item 8 of #11 §14, and it is not in the current queue (root CLAUDE.md, ‘Since the handoff’). By the item’s own wording, a formalization of §11.3’s unification would close it; no document works toward one (Open: one modular structure for both polarizations (OP8); open problems and the queue).
• S without supersymmetry. Outside the supersymmetric theories in which it has been tested, an exact S is not expected, and the worklog records that this is “an expectation, not a no-go” (item R1). #11 states exactness only for maximally supersymmetric Yang–Mills (§11.2) and uses S for its action on τ_c (§11.3) (the exactness burden).

Standard references#

Added by the wiki from model memory (†); the program’s documents cite none of them:

• J.-P. Serre, A Course in Arithmetic (1973) †, the chapter on modular forms — the modular group, its generators S and T, and the fundamental domain.
• E. Witten (1979) † — the electric charge of a monopole in a theory with vacuum angle θ, θ/2π of the unit of charge per unit of magnetic charge: the physical content of T.
• M. K. Gaillard and B. Zumino (1981) † — duality rotations of several abelian gauge fields, the continuous group Sp(2r, ℝ).
• A. Sen (1994) † — evidence for the SL(2, ℤ) duality of N = 4 super Yang–Mills theory from bound states of monopoles.
• C. M. Hull and P. K. Townsend (1995) † — U-duality.

As the documents cite them: C. Montonen and D. Olive (1977) and N. Seiberg (1995), #11 [18]†; J. A. Harvey, hep-th/9603086, #14 [8], quoted through the round-3 audit; A. Betz, R. Blumenhagen, D. Lüst and F. Rennecke (2014), arXiv:1402.1686, the toy-model note’s [12], live-searched in the worklog.

Used by#

• The symplectic charge lattice — SL(2, ℤ) as the pairing-preserving maps of a one-U(1) charge lattice, and S as the third appearance of [[0,1],[−1,0]].
• Reciprocity as modular inversion Z → −Z⁻¹ — at d = 1 the inversion is the action of S on the upper half-plane.
• Open: one modular structure for both polarizations (OP8) — the modular structure of the charge side.
• Obituary: the coupling α_d ≈ 34 — #14 v2’s “modular S transformation” on the hidden coupling, superseded.
• P3: a reciprocity-invariant action — the recurring matrix.

Sources#

• papers/broken-born-checkpoint-v2.md §11.3 — S on τ_c and the recurring matrix [D — structural; math T]; the conjecture [O] and its U-duality model [T]; §11.2 — Montonen–Olive “exact in maximally supersymmetric Yang–Mills” [T]; §14 item 8 — the open item; §6 — ρ: Z → −Z⁻¹ with fixed point Z = i, and “the modular field”; §5 — “the whole modular family of §6”; §4 — ρ inside a continuous U(1); ref [18]† — Montonen–Olive and Seiberg.
• CLAUDE.md ‘Since the handoff’ — the current queue, which does not include item 8.
• papers/reciprocity-toy-models.md §1 — ρ = [[0, 1], [−1, 0]] “squares to −1”.
• verify/toy_coset.py ‘Anchors’, ‘S1’, ‘S2: vacua’ — ρ as the quarter turn; the action invariant under Sp(2, ℝ) and hence under ρ; ρ acting as Z → −1/Z with fixed point Z = i.
• review/round3/referee/audit-notes.md Step 1 — Harvey’s wording on where the duality has been tested, checked against the source.
• review/v3-worklog.md ‘Round-2 pending list’ (item R1), ‘Findings neither referee raised’ (F2) — “an expectation, not a no-go”; S no longer fixes α_d.
• HANDOFF.md §2 — “reciprocity-acts-as-modular-S on the coupling” as the load-bearing conjecture; §6 — round 2’s “matrix coincidence ≠ physical identity” / “numerology”.
• notes/knot-dark-matter-standalone.md §2 — the shared matrix used “only as motivation” (withdrawn manuscript).
• review/skeptic-review-console.html — #14 v2’s “modular S transformation”, verbatim.
• notes/debt1a-geometric-core.md §C — “modular geometry”.
• notes/debt1a-ii-X-classification.md §1, §3 — the translations Z → Z + B in “the Möbius action of the rigid Sp(2d,ℝ)”; “modular (Siegel-domain) symmetry”.
• notes/toy-lagrangian.md ‘Chunk 4 (in progress, 2026-10-19): S5’ — the strong constraint derived “from modular invariance plus amplitude poles”, live-searched.