The full SU(3) holonomy domain: alcoves, traces and exact response

Identify the group slice before moving its parameters

Every unitary matrix can be diagonalized by a unitary change of frame. For W∈SU(3), the eigenvalues have modulus one and product one, so a representative is diag(e^iθ₁,e^iθ₂,e^{−i(θ₁+θ₂)}). Two angular degrees of freedom remain after conjugation. The frequently displayed family diag(e^iθ,e^−iθ,1) samples only a line through that two-dimensional space.

The source's single-block theorem permits arbitrary W∈U(m). In its two-block group S(U(3)×U(2)), the product of the block determinants is one; neither block determinant must separately be one. This page deliberately restricts to SU(3) to examine its complete color-loop domain. Temporal completion, v20, §2 and §2.1, p.8.

A continuously varied scalar e^iθI₃ lies in SU(3) only at θ=2πk/3. Continuous determinant-linked central transport instead pairs the triplet with an appropriate doublet phase. Forgetting that partner changes the group being explored. The two angular controls below describe class coordinates, not two new physical time directions.

Choose one triangular domain for the conjugacy classes

Choose real eigenangle lifts with θ₁≥θ₂≥θ₃, sum zero and θ₁−θ₃≤2π. Define the gaps α=θ₁−θ₂ and β=θ₂−θ₃. Solving these three linear relations gives

This closed triangle is a Weyl alcove: permutations of eigenvalues and allowed 2π changes of their lifts have been accounted for. Its corners give I, e^−2πi/3I and e^2πi/3I. Its edges correspond to coincident unitary eigenvalues. At the third edge the first and third lifted angles differ by 2π, so their exponentials coincide even though the lifts differ.

Some lifts exceed the principal interval [−π,π]. That is harmless: eigenvalues are their exponentials. Wrapping each angle independently and then assuming their real sum remains zero is an inconsistent coordinate procedure. Determinant one constrains the sum modulo 2π; the alcove chooses compatible lifts. These coordinates refine the source's representation of loop content by spectra up to conjugation. Completed records, v21, Proposition 5.3, p.11.

Derive the three-cusped complex trace domain

Put z=TrW. The characteristic polynomial is λ³−zλ²+z̄λ−1. Thus the full complex trace labels an SU(3) class, as derived in loop tomography. On a boundary with repeated eigenvalue e^iφ, the third eigenvalue is e^−2iφ, giving the curve z(φ)=2e^iφ+e^−2iφ. Its three cusps are the traces of the central matrices.

The polynomial discriminant provides an algebraic check:

The equality follows by multiplying the squared pairwise eigenvalue differences and using θ₁+θ₂+θ₃=0 modulo 2π. Equality marks a repeated eigenvalue. The trace image fills the interior of this deltoid boundary; it is smaller than the disk |z|≤3. For example z=−3 obeys the disk bound but has positive discriminant and is not an SU(3) trace.

These are explicit algebraic consequences of the stipulated unitary determinant-one domain used within v20, p.8, not an additional source-selection theorem.

Construct two inequivalent loops with the same real trace

Consider W_A=diag(1,e^2πi/3,e^−2πi/3). Its complex trace is zero. Now choose c=(√3−1)/2 and θ=arccos c, and set W_B=diag(e^iθ,e^iθ,e^−2iθ). Its real trace is 2c+2c²−1=0 as well. Its imaginary trace is positive:

Both loops therefore have the same fundamental real trace deficit, three, but different spectra. W_A is an interior class; W_B lies on a repeated-eigenvalue boundary. They are not related merely by reversing the loop because reversing W_A still has trace zero.

Their second power traces already expose additional spectral information: ReTrW_A²=0, whereas ReTrW_B²=9−6√3≈−1.39230484541. The fourth- and sixth-order energy terms displayed in the source depend only on ReTrW and agree for this pair; the complete secular function retains more information. V20, equation (5), pp.9–10, and the next hopping-order expansion, p.12.

Compare the loops with the full finite Hamiltonian

Set m=3, Δ=1 and t=0.5 in the supplied singlet model. Its exact energy is E=1−z, where the physical secular root lies above all free mobile eigenvalues λ_ak=2tcos[(θ_a+2πk)/4]. The identity ground energy is −0.451605962956. Independent diagonalization of the complete 36-dimensional Hamiltonian gives:

The shifts differ by about 0.000621889332 despite equal ReTr. Their gaps differ more strongly because the largest mobile eigenvalue also depends on the full spectrum. These are dimensionless results in units of Δ, calculated from v20, Hamiltonian (2) and Theorem 2.1, pp.8–10.

The formal fourth-order prediction for either shift is 0.125. At t/Δ=0.5 the source's weak-hopping remainder theorem does not apply; its hypothesis is |t|/Δ<1/4. The example uses the exact Hamiltonian to expose finite-hopping structure, not a perturbative error bound outside its domain.

To reproduce the pair in the upgraded singlet explorer, choose rank three and the full diagonal family. Use angle inputs (0,2π/3) and then (θ,θ), with θ≈1.19606189408616, at hopping ratio 0.5. The tool constructs the third angle as the negative sum.

Separate coordinate degeneracy from spectral protection

At an alcove edge two holonomy eigenvalues coincide, and at a central corner all three coincide. Recovering individual eigenvalues from noisy polynomial coefficients becomes poorly conditioned near such points. Several eigenvectors can also be chosen inside the same eigenspace, so a numerical diagonalizer may return changing bases while the matrix itself changes smoothly.

None of that closes the supplied singlet model's ground-state gap. For m≥2 its gap is z−λ_*≥Δ/(4m), with z the physical root. At identity the stronger lower bound is Δ/4 for nonzero t. The ground state remains unique, including at the central loops and eigenvalue-degeneracy boundaries. V20, Theorem 2.1 and identity specialization, pp.9–10.

Use invariant outputs to compare computations: ordered eigenvalue multisets, complex traces, exact energy and gap. A jump in an arbitrarily chosen eigenvector phase is not a physical transition. Conversely, an apparent nonpositive auxiliary gap should trigger a check of the root branch, sign convention or Hamiltonian assembly rather than be explained away as ordinary frame freedom.

Use the spectral sum across the whole domain

The robust secular function is the positive spectral sum R(z)=(1/12)Σ_a,k(z−λ_ak)⁻¹ for m=3. It decreases strictly from infinity above the largest pole to zero at infinity. Bisection on [λ_*+Δ/12,λ_*+Δ] locates the physical root. At t=0, use E=0 and gap=Δ directly.

The rational form in cosθ is convenient for algebra, but apparent cancellations occur at special twists. The source specifies that they are interpreted through the spectral sum. Clearing denominators and selecting a polynomial root without checking z>λ_* can pick another band or an extraneous pole-related solution. V20, equation (5) and its proof, pp.9–10.

A useful numerical cross-check builds the four-position matrix with three identity links and final forward link W, includes the spectator dual and penalizes root nonsinglets as well as nonroot positions. Its smallest eigenvalue and first excitation must agree with the scalar construction. A model that penalizes positions alone is a different Hamiltonian, even if its link diagram looks identical.

An alcove scan is not a physical distribution of loops

A uniform grid in α,β is a clear way to display a response surface. It is not a Haar-random sample of SU(3). Diagonalization carries an orbit Jacobian proportional to Π_a<b|e^iθa−e^iθb|²: each off-diagonal infinitesimal conjugation is multiplied by the corresponding eigenvalue difference. The weight vanishes on repeated-eigenvalue boundaries. A chosen gauge action would modify the distribution further.

The source specifies normalized Haar measure for its compact configuration group, then separately supplies Hamiltonian terms. V20, opening of §2, p.8. A visualization should therefore identify whether points are a coordinate grid, Haar samples or samples from a particular dynamical measure. Counting colored pixels is not a calculation of a quantum expectation value.

The full domain is valuable because it makes omissions visible: a one-angle slice misses central corners, generic complex traces and unequal-response pairs. It still does not supply autonomous link dynamics, electric normalization, matter coupling or continuum scaling. Those physical tasks remain separate from a complete parametrization of the frozen SU(3) variable.

Continue reading

• Calculate the full two-angle SU(3) family
• Reconstruct a loop from complex trace data
• The exact secular root and gap
• Normalization and physical matching
• Test the restricted finite model