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Critical Spin(2,3) / PGT Quantization Packet v1

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corpus/programmes/closure/spin23-gravity-route/quantization-packet-v1.md
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Why this page is here: the shortest known corpus-link route from an Edition 3 source is:

  1. Leech pair-Gram geometry reaches Spin(2,3), but needs an orientation twist and affine theta shift
  2. Corpus crosswalk: what this programme inherits and what it changes
  3. Critical Spin(2,3) / PGT Quantization Packet v1

Critical Spin(2,3) / PGT Quantization Packet v1

Frozen: 22 October 2026

Scope: the fixed-spurion, constrained two-copy branch in its broken Spin(2,3) phase. The gauged-copy branch is a different theory and is not silently included.

Purpose: define one reproducible background-field convention for the critical (0^-) Poincare-gauge sector. This packet defines every Lagrangian and gauge-fixing choice needed to construct the quadratic operator. It does not claim that the complete nonlinear second-class measure has already been evaluated.

1. Geometry and orientation

Lorentzian signature is ((-+++)), (epsilon{0123}=+1), and (eta=*1). Internal indices are raised with (eta{ab}={\rm diag}(-1,1,1,1)). The independent fields after unitary breaking are an invertible coframe (e^a) and metric-compatible connection (omega^{ab}=-omega^{ba}):

\[ T^a=de^a+\omega^a{}_b\wedge e^b, \qquad R^{ab}=d\omega^{ab}+\omega^a{}_c\wedge\omega^{cb}. \]

Let (Ea\lrcorner e^b=\deltaa{}^b). Define the trace and axial one-forms

\[ t=E_a\lrcorner T^a, \qquad p=*(e_a\wedge T^a), \]

and freeze the three torsion projectors as

\[ {}^{(2)}T^a={1\over3}e^a\wedge t, \qquad {}^{(3)}T^a=-{1\over3}*(e^a\wedge p), \qquad {}^{(1)}T^a=T^a-{}^{(2)}T^a-{}^{(3)}T^a. \]

These equations, rather than a literature numbering convention, define the packet's (a1,a2,a_3). The pseudoscalar curvature is

\[ \widehat X=*(R_{ab}\wedge e^a\wedge e^b). \]

Its overall normalization is fixed by this equation. No later rescaling of (widehat X) is allowed without transforming its coupling and normal map.

2. Frozen action

With (eta{ab}=*(ea\wedge e_b)), the Lorentzian action is

\[ \begin{aligned} S_L={}&{1\over2\kappa}\int R^{ab}\wedge\eta_{ab} -{\Lambda\over\kappa}\int\eta\\ &+{1\over2\kappa}\sum_{I=1}^3a_I \int{}^{(I)}T^a\wedge *{}^{(I)}T_a +{b_X\over2}\int\widehat X^2\eta+S_{\rm con}, \end{aligned} \]

where (Lambda=-3/\ell^2). The critical surface is

\[ 2a_1+a_2=0, \qquad a_1+2a_3=0, \qquad b_X\ne0, \]

with every complementary curvature-square coefficient zero. The sign and mass calibration is performed by expanding about the torsionless vacuum and requiring

\[ {1\over2}A S_\mu S^\mu +{1\over2}B(\bar\nabla_\mu S^\mu)^2, \qquad A>0,quad B>0,quad m_{0^-}^2=A/B. \]

This calibration is part of the convention: it fixes the conversion from ((aI,bX,\kappa,\ell)) to the normalized axial field (S_\mu).

The constrained selector sector is

\[ S_{\rm con}=\int\rho(V^2+\ell^2)+\int B_AW^A, \]

with (V^A=\ell\delta^A4), (W^A=0) imposed before the broken-phase Hessian is formed. Its broken-gauge FP matrix is (ell\delta^a{}b), and its algebraic constraint Jacobian has rank six. The resulting field-independent factors are retained in the normalization of (Z), not counted as propagating determinants.

3. Background split

Use a torsionless Einstein background

\[ \bar T^a=0, \qquad \bar R^{ab}=-\ell^{-2}\bar e^a\wedge\bar e^b. \]

The coframe split is exponential,

\[ e^a{}_{\mu}=\bar e^a{}_{\nu}[\exp(f)]^\nu{}_{\mu}, \]

and the independent connection split is affine,

\[ \omega^{ab}{}_{\mu}=\bar\omega^{ab}{}_{\mu}+k^{ab}{}_{\mu}. \]

The metric fluctuation is (h{\mu\nu}=2f{(\mu\nu)}+O(f^2)). The antisymmetric coframe fluctuation (f_{[ab]}) fixes local Lorentz rotations. The selector fluctuation (v^a) fixes the four broken parent rotations.

4. BRST differential and gauge fermion

The odd ghosts are the diffeomorphism vector (c^\mu), Lorentz scalar (\lambda^{ab}=-\lambda^{ba}), and broken-parent scalar (u^a). With the sign convention (s\omega={\cal L}c\omega-D\lambda) and (se={\cal L}c e+\lambda e), use

\[ sc^\mu=c^\nu\partial_\nu c^\mu, \qquad s\lambda={\cal L}_c\lambda+\lambda^2. \]

The parent transformation of (V) fixes (su) by the Spin(2,3) Lie algebra; after unitary gauge its FP operator is the algebraic matrix (ell\delta^a{}_b). Every antighost and Nakanishi--Lautrup field is a BRST doublet, (s\bar c=b, sb=0), with analogous Lorentz and broken pairs.

Freeze

\[ F_\mu=\bar\nabla^\nu h_{\mu\nu}-{1\over2}\bar\nabla_\mu h, \qquad G_{ab}=f_{[ab]}, \qquad H_a=v_a, \]

and the gauge fermion

\[ \Psi=\int\bar\eta\left[ \bar c^\mu(F_\mu+{\alpha\over2}b_\mu) +\bar\lambda^{ab}(G_{ab}+{\beta\over2}b_{ab}) +\bar u^a(H_a+{\gamma\over2}b_a) \right]. \]

The gauge-fixed action is (S_{\rm gf+gh}=s\Psi). The reference point is (alpha=\beta=gamma=1). This formula defines all ghost and mixed vertices; discarding the connection terms in (sG) or the Lie-derivative terms in (sF) is not permitted. At the background, the propagating FP operator is

\[ (M_{\rm diff})_\mu{}^\nu =-\bar\nabla^2\delta_\mu{}^\nu-\bar R_\mu{}^\nu, \]

while the Lorentz and broken-parent FP blocks are algebraic and invertible.

5. Euclidean domain and contours

Bulk logarithmic divergences are defined on a smooth compact quotient of Euclidean (H^4), with no physical boundary. Zero modes are removed from primed determinants and treated by collective coordinates. This choice avoids importing unlisted boundary counterterms. Lorentzian observables are defined by analytic continuation back to AdS.

The Euclidean coframe traceless modes and the (SO(4)) connection are integrated over their real slices. The conformal coframe direction uses the standard steepest-descent imaginary contour. Multipliers (\rho,B_A) use imaginary Fourier contours so their integrals impose delta functionals. The positive-normalized pseudoscalar is real. Ghost and antighost variables are independent Grassmann fields.

6. Measure and constrained-system rule

Use ultralocal background (L^2) metrics for (f) and (k), with all dimensionful normalizations expressed through the subtraction scale (mu). Every irreducible decomposition or change from connection to contorsion must carry its induced Jacobian. The fundamental variables are (e,omega); therefore no connection-to-contorsion Jacobian is silently set to one.

For comparison with the canonical critical theory, the phase-space measure is required to contain

\[ \delta[\chi_A]\sqrt{\det\{\chi_A,\chi_B\}}, \]

for the complete second-class set. The published massive (0^-) bracket block proves nonvanishing rank, but the repository does not yet contain all 34 constraints and their full field-dependent bracket matrix in this packet's convention. This factor is therefore the sole non-executable input of v1. It must not be replaced by the known sub-block.

7. Renormalization and quotient

Use dimensional regularization in (d=4-2\epsilon), background-field (\overline{\rm MS}), and record the unquotiented local divergence first. The basis includes all parity-even (R^2,T^2,(DT)^2,RDT) invariants generated at the calculated derivative order. Only afterward quotient Euler and other topological densities, BRST-exact terms, and terms removable by explicitly listed local field redefinitions. Equation-of-motion removal is allowed only for on-shell observables and must be labelled as such.

For the first displayed shell, order the coupling coordinates as

\[ g=(t_1,t_2,t_3,c_C,c_R,c_X,c_4,c_5,c_6,z_q) \]

and project with

\[ Cg=(2t_1+t_2, t_1+2t_3, c_C, c_R, c_4, c_5, c_6, z_q). \]

The fixed-spurion branch omits the last coordinate; it is retained in joined ledgers only to compare the gauged-copy alternative.

8. Matching observables

Running gravitational coefficients are matched to background 1PI vertices. The pseudoscalar mass and residue are matched to the pole and derivative of the gauge-invariant (widehat X)-(widehat X) two-point function. A proposed fixed point is admissible only after the physical graviton and pseudoscalar spectral measures remain nonnegative.

9. v1 execution boundary

The action, projectors, split, gauge fermion, ghost definition, local measure, Euclidean domain, contours, subtraction scheme, counterterm ordering, normal map and observables are now frozen. The next executable operation is the quadratic Hessian and principal-symbol factorization.

The next non-executable operation is the claimed complete off-shell one-loop functional: that requires the full 34-by-34 second-class bracket operator (or a demonstrated covariant BV conversion) in these conventions.

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