The Siegel domain Sp(2d, ℝ)/U(d)

The Siegel domain is the space of complex symmetric d × d matrices Z = X + iY whose imaginary part Y is positive definite. For a single coordinate (d = 1) it is the upper half of the complex plane; for larger d it is the matrix version of that half-plane. Each point Z names exactly one positive phase-space metric H_Z that is compatible with a fixed symplectic form ω, and every such metric has a point, so the domain is the space of all of them. The symplectic group moves its points by fractional-linear (Möbius) maps, the matrix versions of z → (αz + β)/(γz + δ). The program uses the domain as the arena of the modular field once only ω is held rigid. The space and its parametrization are standard mathematics, and battery test T2 checks the parametrization’s identities at one sampled point.

Status: Established — standard mathematics (calc note #5 heads its section “[D — structural; math facts standard]”); the parametrization identities are machine-checked [V] at one sampled point at d = 4 (battery test T2; the verification battery) and symbolically at d = 1 (verify/toy_coset.py).

What the domain is#

The upper half-plane (d = 1)#

With one coordinate the domain is the upper half-plane, the complex numbers z = x + iy with y > 0. The real 2 × 2 matrices of unit determinant, the group SL(2, ℝ), act on it by fractional-linear maps [T]:

z → (α z + β) / (γ z + δ),        [[α, β], [γ, δ]] ∈ SL(2, ℝ)

Each such map sends the half-plane onto itself, and any point can be carried to any other. The maps that leave the point z = i fixed are the rotations [[cos θ, sin θ], [−sin θ, cos θ]], which form the circle group SO(2) ≅ U(1). The half-plane is therefore the coset space SL(2, ℝ)/SO(2): a point of the half-plane is a group element up to a rotation [T]. In two dimensions SL(2, ℝ) is the same group as Sp(2, ℝ), the linear maps of the (x, p) plane that preserve the area form ω.

The matrix version (general d)#

For d coordinates, z becomes a complex symmetric d × d matrix Z = X + iY, with X and Y real symmetric and Y positive definite (Y > 0). This is Siegel’s upper half-space [T] (Siegel 1943 †). A symplectic matrix written in d × d blocks, S = [[A, B], [C, D]], acts by the matrix form of the same map:

Z → (A Z + B)(C Z + D)⁻¹

Sp(2d, ℝ) reaches every point, and the elements that fix the base point Z = i·1 (i times the unit matrix) form the unitary group U(d), sitting inside Sp(2d, ℝ) as the symplectic maps that are also rotations. Hence the name Sp(2d, ℝ)/U(d) [T]. X and Y each have d(d + 1)/2 independent entries, ten each at d = 4, the counts calc note #5 gives for the metric and for X (§D).

Two kinds of map recur in the program’s documents:

  • the translations [[1, B], [0, 1]], with B real symmetric, act as Z → Z + B: they shift X and leave Y alone (#6 §1);
  • the quarter-turn [[0, 1], [−1, 0]] acts as Z → −Z⁻¹, which is how Born reciprocity acts on the modular field (#5 §C; #11 §6; modular inversion).

Together with the linear changes Z → AZAᵀ (A invertible), these two kinds generate the whole action [T]. One convention needs care [T]: the toy-model note lets a symplectic S act on a metric by congruence, H → SᵀHS (§2), and in that convention the map induced on Z is the fractional-linear map of Sᵀ, so Z → Z + B comes from [[1, 0], [B, 1]]. For ρ the two readings agree, because ρᵀ = −ρ and S and −S induce the same map.

The domain as a space of phase-space metrics#

The program meets the domain from the other side, as a space of metrics. The symplectic form is fixed as ω = [[0, 1], [−1, 0]] in d × d blocks, in the (x, p) basis of the lab notebook, #1 §2. A positive symmetric matrix H on phase space is compatible with ω when I = ω⁻¹H squares to −1, so that I is a complex structure (see the structures ω, η, H). For a symmetric H this is the same as H itself being a symplectic matrix [T]; model I of the toy-model note takes H to be “a positive symmetric symplectic matrix” with I² = −1 (§2). Calc note #5: “positive H compatible with ω alone form the Siegel domain Sp(2d,ℝ)/U(d), coordinatized by a complex symmetric matrix” (§C):

Z = X + iY,   Y > 0

H_Z = [[ Y + XY⁻¹X ,  XY⁻¹ ],
       [ Y⁻¹X      ,  Y⁻¹  ]]

#11 calls H_Z “the standard generalized metric” (§6). Its complex structure, in what calc note #7 calls “the verified Siegel parametrization” (§2), is

I[Z] = ω⁻¹ H_Z = [[ −Y⁻¹X     ,  −Y⁻¹  ],
                  [ Y + XY⁻¹X ,  XY⁻¹  ]]

The upper-left block of H_Z measures position directions, the lower-right block measures momentum directions, and the off-diagonal blocks XY⁻¹ and Y⁻¹X couple the two. At X = 0 the coupling vanishes and H = diag(Y, Y⁻¹): position metric Y, momentum metric Y⁻¹. That is exactly the form diag(g, g⁻¹) which the compatibility theorem allows when the neutral metric η is held rigid as well. A point with X ≠ 0 is compatible with ω but mixes the position and momentum directions.

Why the identities hold [T]#

A standard factorization shows at once why H_Z has the properties the battery tests:

H_Z = Lᵀ · diag(Y, Y⁻¹) · L,        L = [[ 1 , 0 ],
                                          [ X , 1 ]]

Both L and diag(Y, Y⁻¹) are symplectic, so H_Z is symplectic and I[Z]² = −1. diag(Y, Y⁻¹) is positive definite because Y is, so H_Z is positive definite. And det H_Z = det Y · det Y⁻¹ = 1. Taking square roots of the diagonal factor writes H_Z = EᵀE with E symplectic, which is one choice of what calc note #7 calls “the symplectic vielbein E, the coset representative with H = EᵀE” (§3). The factorization is textbook algebra; the program’s documents do not use it.

How the program uses it#

The arena of the modular field#

Under the corrected postulate P2′, only ω is rigid. The metric H may then be any ω-compatible positive metric, that is, any point of the Siegel domain, and its coordinates X and Y become fields on spacetime. #11 §6: “With only ω rigid, H ranges over Sp(2d,ℝ)/U(d): Z = X + iY (symmetric; Y > 0), H_Z the standard generalized metric (I[Z]² = −1 machine-verified [V]).” The physical readings, Im Z = g the metric and Re Z = X the momentum graviton, are the program’s claims and are set out on the modular field Z = X + iY. Under the original P2, with η rigid too, only the slice X = 0 would remain.

Its symmetries as field symmetries#

  • Translations. “The Möbius action of the rigid Sp(2d,ℝ) contains the subgroup [[1, B], [0, 1]], B symmetric, acting as Z → Z + B: X → X + B, Y → Y” (#6 §1, a section tagged [D — rigorous]). Constant shifts of X are therefore exact symmetries of any action built from (ω, H): X is “axion-like”, never auxiliary (shift symmetry).
  • Inversion. Reciprocity acts as Z → −Z⁻¹ with fixed point Z = i (modular inversion).

The target metric#

A field with values in the domain needs a kinetic term, and the natural one comes from the domain’s invariant metric [T] (Siegel 1943 †):

ds² = Tr( Y⁻¹ dZ  Y⁻¹ dZ̄ )

It is unchanged by every map of the Sp(2d, ℝ) action [T]. For d = 1 it is the hyperbolic metric (dx² + dy²)/y² of the upper half-plane. Calc note #5 counts its positivity among the good signs for a healthy kinetic term: “the Siegel target metric Tr(Y⁻¹dZ Y⁻¹dZ̄) is positive-definite” (§D; see the field content of the broken vacuum). That positivity belongs to positive (Euclidean) H. With a Lorentzian metric, calc note #6 records “the stabilizer U(1, d−1) and the invariant target metric indefinite” (§3); the extension to spacetime signature is open item 3 (Open: Lorentzian dressing).

The toy model’s H-sector (d = 1)#

Model I of the toy-model note takes “H, a positive symmetric symplectic matrix, H(Z) = [[Y + X²/Y, X/Y], [X/Y, 1/Y]] for Z = X + iY in the upper half-plane, with I ≡ ω⁻¹H, I² = −1” (§2): the d = 1 case of H_Z. verify/toy_coset.py checks the d = 1 parametrization symbolically, for symbolic X and Y (the toy-model scripts): PASS H(Z) is symplectic and I = w^-1 H squares to -1. The model’s vacuum manifold is “the upper half-plane times this circle” of real structures, and the two half-plane directions are two of its three Goldstone directions (§2). The half-plane sector is non-compact: “The non-compact half-plane sector is a hyperbolic model, where symmetry breaking in 2D does occur” (§4; coset and hyperbolic sigma models; model I: action and symmetries). The handoff’s queue had proposed starting the toy “on the Siegel disk”, the bounded form of the domain described next (HANDOFF.md §8 item 4).

The boundary of the domain#

The domain is unbounded, but the Cayley transform W = (Z − i)(Z + i)⁻¹ maps it onto a bounded domain, the “Siegel disk” of complex symmetric matrices W with 1 − W̄W positive definite; the point Z = i goes to the center W = 0 [T]. For d = 1 this is the familiar map of the upper half-plane onto the unit disk.

A bounded domain has a distinguished part of its boundary, its Shilov boundary: the smallest closed part of the boundary on which every function that is holomorphic inside and continuous up to the edge reaches its largest absolute value [T]. For the Siegel disk it is the set of symmetric unitary matrices, which is the Lagrangian Grassmannian Λ(d) [T] (Hua 1963 †; the construction is on that page). #11 §6: “Λ(d) is the Shilov boundary of the Siegel domain [T]”. In the upper-half-space picture the real symmetric matrices, the limit Y → 0, make up a dense part of that boundary [T]. The program places the dark-energy field there (τ as the Maslov phase of the boundary).

Battery test T2#

What it computes. verify/battery.py, the battery of HANDOFF.md Appendix A, works at d = 4 with numpy’s random seed 0. It draws one random symmetric X and one positive-definite Y, builds H_Z and I[Z] = ω⁻¹H_Z, and checks four properties. The gate prints:

PASS  T2 I[Z]^2 = -1
PASS  T2 H_Z symmetric
PASS  T2 det H_Z = 1  [det=1.000000000000]
PASS  T2 H_Z positive definite

HANDOFF.md §5 describes the test as “T2 Siegel parametrization (I[Z]² = −1, H ≻ 0, det H = 1 off-vacuum)”: the sampled point has X ≠ 0, away from the vacuum slice.

What it shows, and what it does not.

  • It checks one sampled member of the family. #11 §2 describes the battery as covering “the Siegel parametrization (I[Z]² = −1, H > 0, det H = 1 across the whole family)”, and #11 §5 says det H = 1 “machine-verified — persists across the whole modular family of §6”. That the identities hold for every Z is the algebra above ([T]; [D] in the program’s tagging), not a machine result. The symbolic check in verify/toy_coset.py holds for all X and Y at once, but only at d = 1.
  • It confirms a matrix identity, not the program’s reading of Y as the spacetime metric (verification confirms arithmetic, not the model).

Round 3’s independent check. The round-3 blind audit re-derived the compatibility theorem with its own code and tested an “omega-only family (Siegel, X != 0)” in 2000 random trials at each of d = 3 and d = 4: “omega residual <= 3e-14 but eta residual >= 0.52 (d=3), 1.87 (d=4)” (review/round3/referee/audit-notes.md Step 3). Every sampled member is compatible with ω to rounding error, and every one misses compatibility with η by a finite margin. The Siegel family is strictly larger than the compatibility theorem’s solution set, which is its X = 0 slice (#5 §C). The audit’s scripts sit in review/round3/referee/ and are not part of the gate.

Standard references#

  • C. L. Siegel, “Symplectic geometry”, American Journal of Mathematics (1943) † — the upper half-space, the symplectic group’s action on it and its invariant metric.
  • L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains (English translation 1963) † — the bounded realizations of these domains and their Shilov boundaries.

The program’s documents cite no reference for these facts: #5 marks them “[standard]”, and #11 tags the boundary statement [T].

Used by#

Sources#

  • notes/debt1a-geometric-core.md §C — the Siegel domain of ω-compatible metrics and H_Z; “[D — structural; math facts standard]”; the boundary fact “[standard]”.
  • notes/debt1a-geometric-core.md §D — the positive Siegel target metric.
  • notes/debt1a-iii-tuning-test.md §2, §3 — I[Z] in “the verified Siegel parametrization”; the symplectic vielbein H = EᵀE.
  • verify/battery.py T2 — the four identities at one sampled point (d = 4, seed 0).
  • HANDOFF.md §5, §8 — T2 described as an off-vacuum check; the toy queued “on the Siegel disk”.
  • notes/debt1a-ii-X-classification.md §1, §3 — the translations Z → Z + B; the Lorentzian stabilizer U(1, d−1) and the indefinite target metric.
  • papers/broken-born-checkpoint-v2.md §6 — “the standard generalized metric”; Λ(d) as the Shilov boundary [T].
  • papers/broken-born-checkpoint-v2.md §2, §5 — the battery’s description of T2; det H = 1 “across the whole modular family”.
  • papers/reciprocity-toy-models.md §2, §4 — the d = 1 case in model I; the hyperbolic half-plane sector.
  • verify/toy_coset.py — the symbolic d = 1 anchor “H(Z) is symplectic and I = w^-1 H squares to -1”.
  • review/round3/referee/audit-notes.md Step 3 — the ω-only family, checked by independent code.
  • notes/broken-born-theory.md §2 — the (x, p) block basis in which ω is fixed.
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