It has been several years since I submitted my PhD thesis and the accompanying papers. For a long time, the code and calculations sat in a collection of standalone Cadabra notebooks — functional, verified, but not in a form that was easy to share or extend. With the help of AI tools, I have finally had the time and energy to clean everything up, unify it under a single driver, and release it publicly.

My PhD thesis, Dynamical supersymmetry enhancement of black hole horizons (arXiv:1910.01080), investigates how near the horizons of supersymmetric black holes in supergravity, the number of preserved supersymmetries doubles — the horizon conjecture — proved across IIA, massive IIA (Romans), and D=5 supergravity. The companion paper Symmetry enhancement of Killing horizons in D=6 supergravity (arXiv:1912.04249) extends the analysis to six-dimensional gauged supergravity.

The structural argument at the heart of both works rests on KSE integrability: proving that the integrability conditions of the Killing Spinor Equations are automatically satisfied on-shell. This is the Lichnerowicz-type argument that allows one to trade global analysis of the KSEs for local algebraic conditions — verified symbolically, for each theory, using Cadabra2.

The code and full reference are at github.com/drrobotk/py_integrability_sugra, released under CC BY-NC 4.0. If you use this work, please cite arXiv:1910.01080 and arXiv:1912.04249.

py_integrability_sugra

Symbolic verification of Killing Spinor Equation (KSE) integrability conditions for supergravity theories in D = 4, 5, 6, 10, 11 dimensions, using the Cadabra2 computer algebra system.

All computations are driven by a unified Python script (integrability_driver.py) that reads theory data from a structured JSON schema (theories.json). Legacy standalone scripts for each theory are preserved in legacy/.

Contents

• Mathematical Background
• Installation
• Quick Start
• Theory Reference
  • D=11 Supergravity
  • D=10 Type IIA (Massless)
  • D=10 Type IIA (Romans Mass)
  • D=10 Heterotic
  • D=5 Vector Multiplets (Ungauged)
  • D=5 Vector Multiplets (Gauged)
  • D=5 Minimal
  • D=6 N=(1,0)
  • D=4 Einstein–Maxwell
  • D=4 Minimal Gauged
• Unified Driver
• theories.json Schema
• Extending to New Theories
• Verified Results
• Known Limitations
• References

Mathematical Background

KSE Integrability

A supergravity field configuration admits a supersymmetric Killing spinor $\varepsilon$ if it satisfies the Killing Spinor Equations (KSEs):

where (\Psi_a) is the supercovariant connection — a matrix-valued 1-form built from the bosonic fields — and $\mathcal{A}$ is the dilatino or gaugino operator.

Integrability requires the curvature of $\mathcal{D}$ to annihilate any Killing spinor:

Expanding the gravitino commutator yields the gravitino integrability operator:

(the precise form depends on the theory; the last three terms are present only when a dilatino KSE exists).

The Main Theorem

The integrability conditions are consequences of the bosonic field equations and Bianchi identities. For any on-shell field configuration:

where each coefficient (C_i) is a linear combination of on-shell residuals (EOM or Bianchi) that vanish on any solution. This guarantees that one need only solve the KSEs directly.

Lichnerowicz Argument

The converse also holds: the KSEs together with a subset of the field equations imply the full set of equations of motion (via a spinorial Lichnerowicz identity). This is the structural result used to classify all supersymmetric near-horizon geometries.

Residual Notation

All equations of motion and Bianchi identities are written in residual form:

All residuals vanish on any solution to the field equations.

Computation Pipeline

Each integrability operator is processed by a 9-step Cadabra2 algorithm:

1. KSE substitution — substitute (\Psi_a), $\mathcal{A}$, $\mathcal{N}$ by their expressions in bosonic fields
2. Field equation substitution — replace (R_{ab}), (\nabla^c H_{abc}), etc., introducing EOM residuals
3. Clifford expansion (3 passes) — join_gamma + distribute
4. Leibniz rule — product_rule expands $\nabla$ on products
5. Clifford expansion (2 passes) + unwrap
6. Second field substitution — handles new EOM terms from step 4
7. Bianchi substitution — replaces $\nabla H \cdot \Gamma$ contractions by Bianchi residuals
8. $\varepsilon \to \Gamma$ — replaces the $\varepsilon$-tensor by $\pm C\,\Gamma$ (or $+\Gamma$ in $D=11$)
9. Cleanup — expand $\nabla(e^{n\Phi})$, simplify $e^{n\Phi}\cdot e^{m\Phi}$, apply $C^2=1$, $i^2=-1$, factor out $C$

The Cadabra post_process hook fires after every algorithm step: sort_product → eliminate_kronecker → canonicalise → collect_terms.

String/M-Theory Origins

All the supergravity theories studied here arise as low-energy limits of string/M-theory compactifications. The diagram below shows the principal descent relations. D=11 supergravity sits at the top as the unique maximal theory; all others emerge by dimensional reduction or duality.

D=11 SUGRA (unique)
│
├── ×S¹ ──────────────────────► D=10 Type IIA (massless)
│                                       │
│                               + Romans mass m ──► D=10 Type IIA (massive)
│
├── ×CY₃ ─────────────────────► D=5 N=2 ungauged (M-theory on CY₃)
│                                       │
│                               + U(1) gauging ──► D=5 N=2 gauged
│
├── ×CY₃ × S¹ ───────────────► D=4 N=2 (M-theory on CY₃, or IIA on CY₃)
│
├── ×K3 ──────────────────────► D=7 N=1 (not yet in driver)
│
└── ×S¹/ℤ₂ (Hořava-Witten) ──► D=10 Heterotic E₈×E₈
                                        │
                                ×T⁴ ───► D=6 N=(1,0)
                                        │
                                ×CY₃ ──► D=4 N=1

D=11 → Type IIA

The most direct reduction: compactify D=11 on a circle $S^1$ of radius (R_{11}), truncating all Kaluza–Klein modes with masses (\sim 1/R_{11}). The metric decomposes as in (Dimensional Reduction), identifying the dilaton (e^\Phi = R_{11}^{3/2}), the RR 1-form (A_\mu) as the Kaluza–Klein gauge field, and the NS-NS 2-form (B_{\mu\nu}) plus RR 3-form (C_{\mu\nu\rho}) from the D=11 3-form potential. The $D=11$ gravitino gives both the IIA gravitino and dilatino.

D=11 → D=5 N=2 (M-theory on (CY_3))

Compactify M-theory on a Calabi–Yau threefold (CY_3) with Hodge numbers $h^{(1,1)}, h^{(2,1)}$ (see D=5 Vector Multiplets (Ungauged)). The massless spectrum contains:

• Gravity multiplet: (g_{\mu\nu}), graviphoton (A^0_\mu), gravitino
• $h^{(1,1)}-1$ vector multiplets: each contains a vector (A^I_\mu), a real scalar $X^I$ (Kähler modulus), and a gaugino
• $h^{(2,1)}+1$ hypermultiplets: complex structure moduli and universal hypermultiplet

The Kähler moduli satisfy the very-special-geometry constraint (C_{IJK}X^IX^JX^K = 1) where (C_{IJK}) are the triple intersection numbers of (CY_3). Hypermultiplets decouple from stationary solutions and are set to constants. The resulting 5d action is exactly that encoded in d5_vector_ungauged.

D=5 Ungauged → D=5 Gauged

The gauged theory is obtained by turning on a $U(1)$ subgroup of the (SU(2)R) automorphism group of the $\mathcal{N}=2$ algebra. This is equivalent to gauging a linear combination (V_I A^I\mu) of the abelian vectors with coupling $\chi$. The scalar potential (U = 9V_IV_J(X^IX^J - \frac{1}{2}Q^{IJ})) arises from the $D$-term of the gauging. The D=5 gauged theory also arises from type IIB supergravity compactified on $S^5$.

D=11 → D=4 N=2 (M-theory on (CY_3), or IIA on (CY_3))

Two equivalent paths:

• M-theory on (CY_3): gives 5d N=2 as above; further reducing on $S^1$ gives 4d N=2
• Type IIA on (CY_3): directly gives 4d N=2 with $h^{(1,1)}$ vector multiplets and $h^{(2,1)}+1$ hypermultiplets

The 4d Einstein–Maxwell theory in d4_einstein_maxwell is the simplest case ($h^{(1,1)}=1$, no vector multiplets beyond the graviphoton). The gauged version d4_minimal_gauged adds an (AdS_4) vacuum stabilised by the cosmological constant $\Lambda = -3/\ell^2$.

D=11 / Hořava–Witten → D=10 Heterotic

M-theory on the orbifold (S^1/\mathbb{Z}2) (Hořava–Witten construction) gives D=10 heterotic (E_8\times E_8) supergravity at low energy. The two fixed-point boundaries of the interval each carry an (E_8) gauge multiplet. In the weak-coupling limit the interval shrinks and one recovers the perturbative heterotic string. The D=10 heterotic supergravity encoded in d10_heterotic is the effective field theory of this construction, with a single (abelianised) gauge 2-form (F{ab}).

The heterotic string also has an $SO(32)$ variant (related to Type I by S-duality); the NS-NS sector supergravity is the same in both cases.

D=10 Heterotic → D=6 N=(1,0)

Compactify heterotic string theory on the torus $T^4$. Generically this gives D=6 with N=(1,1) or N=(2,0) supersymmetry depending on the gauge bundle. With a non-trivial gauge bundle satisfying (\int_{T^4} \text{tr}F^2 = \int_{T^4} R^2) (the anomaly cancellation condition), one breaks to D=6 N=(1,0), which is the theory encoded in d6_n10. The bosonic content (metric, anti-self-dual 3-form $H$, gauge 2-form $F$, dilaton $\Phi$) matches the tensor + vector multiplet sector.

Alternatively, M-theory on $K3$ gives D=7 with 16 supercharges; further compactification or orbifolding gives D=6 N=(1,0).

D=10 Heterotic / IIA → D=4 N=1

Compactify heterotic string theory on a Calabi–Yau threefold (CY_3). With holonomy group $SU(3)\subset SO(6)$, this preserves $\mathcal{N}=1$ supersymmetry in D=4. The resulting supergravity has a Kähler potential determined by the CY moduli and is the effective theory of the heterotic landscape. The D=4 theories in d4_einstein_maxwell and d4_minimal_gauged are truncations of this richer structure to the gravity + gauge sector.

Massive IIA from Romans deformation

The Romans mass $m$ cannot be obtained as a standard Kaluza–Klein reduction of D=11 supergravity — it is an intrinsically 10-dimensional deformation. It can be thought of as a vacuum expectation value for the dual of a 0-form field strength. The theory exists consistently at the quantum level (as a type IIA string background) and its near-horizon geometries include (AdS_4\times S^6) and warped $AdS$ vacua relevant to ABJM theory.

Installation

Cadabra2

Cadabra2 is a computer algebra system specialised for field theory. It provides both a notebook GUI and a Python library (cadabra2).

macOS — Homebrew (recommended)

brew install cadabra2

The Python library lands at:

/opt/homebrew/Cellar/cadabra2/<version>/libexec/lib/python3.13/site-packages

Linux / macOS — From Source

sudo apt-get install cmake python3-dev libgmp-dev libpcre3-dev \
     libboost-all-dev libgtkmm-3.0-dev python3-matplotlib

git clone https://github.com/kpeeters/cadabra2.git
cd cadabra2 && mkdir build && cd build
cmake .. -DENABLE_JUPYTER=OFF
make -j4 && sudo make install

See cadabra.science/download.html for full instructions.

Verify

/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 -c "from cadabra2 import *; print('Cadabra2 OK')"

Python

The driver uses Cadabra’s bundled Python 3.13. No additional packages are needed.

Important: Always invoke scripts with Cadabra’s own Python binary. The cadabra2 C extension is linked against it specifically.

# Correct
/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 integrability_driver.py d11_supergravity

# Wrong — system Python cannot import cadabra2
python3 integrability_driver.py d11_supergravity

Quick Start

# List available theories
/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 integrability_driver.py --list

# Run one theory
/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 integrability_driver.py d11_supergravity
/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 integrability_driver.py d10_iia
/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 integrability_driver.py d6_n10

# Run all theories
/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 integrability_driver.py --all

Define an alias for convenience:

alias cdbpy="/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3"
cdbpy integrability_driver.py --all

Theory Reference

D=11 Supergravity

Theory: The unique maximal ($\mathcal{N}=1$) supergravity in eleven dimensions. Supersymmetry completely fixes the theory. The field content consists of the graviton (G_{MN}) (44 off-shell components), the 3-form potential (A^{(3)}_{MNP}) (84 components), and the gravitino (\psi_M) (128 fermionic components) — matching exactly the massless spectrum of type II string theory.

Fields: metric (G_{MN}), 3-form (A_{MNP}), 4-form field strength (G_{MNPQ})

Index range: $M,N,\ldots \in {0,1,\ldots,10}$

Bosonic Action

The last term is the Chern–Simons coupling, which is essential for the field equations to be consistent.

Field Equations

Einstein equation:

4-form equation (with Chern–Simons source):

Bianchi identity: $dG^{(4)} = 0$.

Supersymmetry Variations

Bosonic fields:

Gravitino:

Gravitino KSE

The KSE is the vanishing of (\delta\psi_M) on the bosonic background:

No dilatino or gaugino KSE. No chirality matrix.

Dimensional Reduction to IIA

Writing the eleven-dimensional metric as

and reducing the 3-form as (A_{MNP} \to C_{\mu\nu\rho}) (three 10d legs) and (B_{\mu\nu} = A_{\mu\nu,10}) (one leg in the 11th direction), one recovers type IIA supergravity on (\mathcal{M}_{10}).

Field Equations (residual form, Cadabra conventions)

Epsilon Identity (D=11: no $i$, no $C$)

Integrability Result (gravitino)

All coefficients multiply on-shell residuals, confirming automatic satisfaction of the integrability condition.

D=10 Type IIA (Massless)

Theory: Type IIA supergravity obtained by dimensional reduction of D=11 supergravity on (\mathcal{M}{10}\times S^1). Non-chiral theory with two Majorana (non-Weyl) spinors. The chirality matrix (\Gamma{11}) distinguishes the NS-NS sector (odd) from the RR sector (even).

Fields: metric (g_{\mu\nu}), dilaton $\Phi$, NS-NS 2-form (B_{\mu\nu}), RR 1-form (A_\mu), RR 3-form (C_{\mu\nu\rho})

Field strengths:

Index range: $\mu,\nu,\ldots \in {0,1,\ldots,9}$

Bianchi Identities

Bosonic Action (string frame)

Field Equations

Einstein equation:

Dilaton equation:

2-form (RR) equation:

3-form (NS-NS) equation:

4-form (RR) equation:

Chirality Matrix

The chirality matrix (\Gamma_{11} = C) satisfies:

Commutation with gamma matrices:

Supersymmetry Variations

Bosonic fields:

Gravitino:

Dilatino:

KSEs

Setting $m=0$ (massless case), the KSEs (\mathcal{D}\mu\varepsilon = 0) and $\mathcal{A}\varepsilon = 0$ become (using $C$ for (\Gamma{11})):

Supercovariant Connection and Integrability

The gravitino integrability condition is:

The auxiliary connection (\Phi_\mu) is:

The dilatino integrability condition is:

where the scalar $\theta$ is:

Integrability Results

Gravitino:

Dilatino:

D=10 Type IIA (Romans Mass)

Theory: Massive deformation of type IIA supergravity (Romans 1986). The mass parameter $m$ deforms the field strengths and introduces a cosmological constant-like term. The massless theory is recovered in the limit $m \to 0$.

Additional parameter: Romans mass $m$ (denoted $\kappa$ in the Cadabra code)

Modified Field Strengths

Modified Bianchi Identities

The standard Bianchi $dF = 0$ is replaced by $d\tilde{F} = mH$, which encodes the mass deformation.

Bosonic Action (string frame)

Modified Field Equations

Einstein equation (mass term adds (-\frac{1}{4}e^{2\Phi}m^2 g_{\mu\nu})):

Dilaton equation (mass term adds $+\frac{5}{4}e^{2\Phi}m^2$):

3-form equation (mass term adds (-mF_{\mu\nu})):

Modified KSEs

The mass parameter $m$ appears in both the gravitino and dilatino KSEs:

The Cadabra code uses $\kappa$ for the Romans mass $m$. Setting $\kappa = 0$ recovers the massless IIA theory.

Integrability Results

Same structure as massless IIA. The mass term $m^2$ enters the Einstein and dilaton residuals (E_{ab}) and $F\Phi$, which vanish on-shell. The integrability output is formally identical to massless IIA with $F$, $G$ replaced by $\tilde{F}$, $\tilde{G}$.

D=10 Heterotic

Theory: Heterotic string supergravity (NS-NS sector). Three KSEs: gravitino (\Psi_a), dilatino $\mathcal{A}$, gaugino $\mathcal{N}$. No chirality matrix. The (E_8\times E_8) or $SO(32)$ gauge group enters through the non-abelian 2-form (F_{ab}).

Fields: metric (g_{ab}), dilaton $\Phi$, NS-NS 3-form (H_{abc}), gauge 2-form (F_{ab})

Index range: $a,b,\ldots \in {0,1,\ldots,9}$

Bosonic Action

Field Equations and Bianchi Identities

Note: the dilaton equation $F\Phi = 0$ is implied by the other equations and Bianchi identities via:

KSEs

Integrability Conditions

The integrability conditions of the three KSEs express as commutator identities:

All right-hand sides vanish on-shell, confirming integrability.

Epsilon Identities (D=10)

Integrability Results

Gravitino:

Dilatino:

Gaugino:

D=5 Vector Multiplets (Ungauged)

Theory: $\mathcal{N}=2$ supergravity coupled to $k$ abelian vector multiplets, obtained from M-theory compactification on a Calabi–Yau threefold (CY_3) with Hodge numbers (h_{(1,1)}, h_{(2,1)}) and intersection numbers (C_{IJK}).

Fields: metric (g_{\mu\nu}), gauge fields (A^I_\mu), scalars $X^I$, scalar metric (Q_{IJ})

Index range: $\mu,\nu \in {0,…,4}$; (I,J,K \in {1,…,h_{(1,1)}})

Very Special Geometry

The scalars $X^I$ parametrise a very special real manifold $\mathcal{N}$ defined by the cubic constraint:

The constants (C_{IJK}) are the triple intersection numbers of (CY_3). The scalar metric is:

The dual coordinate is (X_I = \frac{1}{6}C_{IJK}X^JX^K), and the constraint becomes (X^I X_I = 1). Key identities:

Bosonic Action

The five-dimensional action obtained from M-theory reduction on (CY_3):

The last term is the five-dimensional Chern–Simons coupling, which descends from the eleven-dimensional Chern–Simons term. Supersymmetry forces the same metric (Q_{IJ}) to appear in both the scalar and vector kinetic terms.

Field Equations

Einstein equation:

Maxwell equations:

Scalar field equations:

Bianchi identity:

Decomposition $F^I = FX^I + G^I$

It is useful to decompose the gauge field strengths into components parallel and perpendicular to (X_I):

In terms of $F$ and $G^I$, the Einstein equation splits as:

The Maxwell equation splits as:

The Bianchi identity splits as:

KSEs

The full KSEs expressed in terms of $F^I$:

After the $F^I = FX^I + G^I$ decomposition:

The spinor $\varepsilon$ is a Dirac spinor of $Spin(4,1)$.

Supercovariant Connection

The supercovariant curvature (\mathcal{R}_{\mu\nu}), defined by the commutator relation

involves the auxiliary connection:

Integrability Conditions

Gravitino integrability condition:

Scalar/gaugino integrability condition:

where the auxiliary connection (\theta_{IJ}) is:

Integrability Result (Cadabra output, before very-special-geometry simplification)

The raw driver output contains:

The VSG (very-special-geometry) remainder terms involve (Q_{IJ}\nabla X^I\nabla X^J), (F^IF^JQ_{IJ}), (X_IX_JF^IF^J), and (C_{IJK}F^IF^JX^K). These cancel via the calabi/back/vanish substitution dictionaries (not yet encoded in theories.json).

D=5 Vector Multiplets (Gauged)

Theory: Gauged extension of the D=5 vector-multiplet theory, obtained by gauging a $U(1)$ subgroup of the $SU(2)$ R-symmetry group with gauge coupling $\chi$ and coupling constants (V_I). Equivalently obtained from type IIB compactification on $S^5$.

Additional parameter: gauging parameter $\chi$, coupling constants (V_I)

Modified Action

where the scalar potential is:

with (V_I) constants and $Q^{IJ}$ the inverse of (Q_{IJ}).

Modified Einstein Equation

Modified Scalar Field Equations

The scalar equations gain additional $\chi^2$ terms:

Modified KSEs

The gravitino KSE gains additional gauge terms:

The gaugino/scalar KSE gains a mass-like term:

Modified Supercovariant Connection

The auxiliary connection (\Phi_{I\mu}) gains an additional term:

The (\theta_{IJ}) connection gains:

Integrability Result

Same structure as ungauged, with the gauging parameter $\chi$ appearing inside (E_{ab}) (cosmological term). The Cadabra output contains (V_I\nabla X^I), (V_IV_JX^IX^J) terms that vanish via the very-special-geometry constraint.

D=5 Minimal

Theory: Minimal $\mathcal{N}=1$ supergravity with a single U(1) gauge field and Chern–Simons coupling. This is the $k=1$, (C_{111}=1) limit of the vector-multiplet theory with $X^1 = 1$ (no moduli).

Fields: metric (g_{ab}), Maxwell 2-form (F_{ab})

Index range: $a,b,\ldots \in {0,1,2,3,4}$

Bosonic Action

Gravitino KSE

Field Equations

Epsilon Identity (Chern–Simons)

Integrability Result (gravitino)

D=6 N=(1,0)

Theory: Chiral $\mathcal{N}=(1,0)$ supergravity coupled to a tensor multiplet. Three KSEs: gravitino (\Psi_a), dilatino $\mathcal{A}$, gaugino $\mathcal{N}$. Rich dilaton structure with fractional exponential weights $e^{\pm\Phi/4}$, $e^{\pm\Phi/2}$.

Fields: metric (g_{ab}), anti-self-dual 3-form (H_{abc}), gauge 2-form (F_{ab} = dA), dilaton $\Phi$, gauge coupling $g$

Index range: $a,b,\ldots \in {0,1,2,3,4,5}$

KSEs (dilaton frame)

Auxiliary connections:

Integrability Operators

Field Equations

Cleanup Rules (critical for gaugino)

The dilaton derivative expansions are:

These must be applied before any product simplifications, or the gaugino result will contain unsimplified $\nabla(e^{n\Phi})$ terms.

Integrability Results

Gravitino:

Dilatino:

Gaugino:

D=4 Einstein–Maxwell

Theory: Pure $\mathcal{N}=2$ supergravity with U(1) gauge field. No dilaton, no cosmological constant.

Fields: metric (g_{ab}), Maxwell 2-form (F_{ab})

Index range: $a,b,\ldots \in {0,1,2,3}$

Gravitino KSE

Field Equations

Bianchi Identities

Epsilon Identity

Integrability Result (gravitino)

D=4 Minimal Gauged

Theory: $\mathcal{N}=2$ Einstein–Maxwell with cosmological constant $\Lambda = -3/\ell^2$ (AdS(_4) vacuum).

Fields: metric (g_{ab}), Maxwell 2-form (F_{ab}), gauge potential (A_a), AdS radius $\ell$

Index range: $a,b,\ldots \in {0,1,2,3}$

Gravitino KSE

Field Equations

Integrability Result (gravitino)

The cosmological constant contributions cancel upon using the Einstein equation:

Unified Driver

The driver integrability_driver.py reads all theory data from theories.json and dispatches to the appropriate pipeline for each computation. Three pipeline variants are supported:

• Standard — the 9-step pipeline (substitute, Clifford, Leibniz, unwrap, field equations, Bianchi, epsilon identities, final Clifford, cleanup). Used for all minimal and non-matter-coupled theories.
• VSG gravitino — standard pipeline followed by very-special-geometry simplification (calabi, back, vanish rules) that resolves (Q_{IJ}), (C_{IJK}), (X^I) algebraic relations. Used for D=5 vector multiplet gravitino computations.
• VSG scalar — a custom pipeline with 3 KSE passes, intermediate calabi simplification, and extensive VSG post-processing. Used for D=5 vector multiplet scalar/gaugino computations.

Pipeline selection is automatic via the pipeline_per_computation field in theories.json.

Command-line Usage

/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 integrability_driver.py [theory_id]
/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 integrability_driver.py [theory_id] --op gravitino
/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 integrability_driver.py --list
/opt/homebrew/Cellar/cadabra2/2.5.14/libexec/bin/python3 integrability_driver.py --all

Use --op to run a single computation (e.g. --op gravitino or --op scalar).

Available Theory IDs

Python API

from integrability_driver import run_theory, THEORIES, setup_kernel

results = run_theory('d11_supergravity')
results = run_theory('d5_vector_gauged', operator_filter='scalar')
print(list(THEORIES.keys()))

theories.json Schema

All substitution strings use Cadabra LaTeX notation (backslash-escaped).

Required Fields

{
  "id": "d11_supergravity",
  "description": "...",
  "dimension": 11,
  "index_range": "0..10",
  "kse": "{\\Psi_{a} -> ...}",
  "field_equations": "{R_{a b} -> ..., \\nabla^{d}{G_{a b c d}} -> ...}",
  "bianchi_identities": "{\\nabla_{a}{G_{b c d e}}\\Gamma^{b c d e} -> ...}",
  "integrability_operators": {
    "gravitino": "-\\frac{1}{2}R_{a b}\\Gamma^{b} + ..."
  }
}

Optional Fields

Internal Indices Schema

"internal_indices": {
  "symbols": "{I,J,K,L,M,N,O,P,Q,R}",
  "position": "spinor, position=fixed, position=independent",
  "depends": ["F^{I}_{a b}", "X^{I}", "X_{I}"],
  "symmetric": ["Q_{I J}", "C_{I J K}"],
  "tableau_symmetry": [{"symbol": "F^{I}_{a b}", "spec": "shape={1,1}, indices={1,2}"}],
  "noncommuting_with_gamma": ["{\\Psi_{a}, A^{I}, \\Gamma_{#}}"],
  "commuting": ["{X_{I}, V_{I}, C_{I J K}}"]
}

Extending to New Theories

Step 1 — Add a JSON entry to theories.json

Minimum required: id, description, dimension, index_range, kse, field_equations, bianchi_identities, integrability_operators.

Step 2 — Critical rules

Spaced indices in chirality commutation (Cadabra parses abc as a single symbol — always use spaces):

// CORRECT
"anti_commuting_with_C": ["\\Gamma^{a}", "\\Gamma^{a b c}", "\\Gamma^{a b c d e}"]

// WRONG — rule never fires
"anti_commuting_with_C": ["\\Gamma^{a}", "\\Gamma^{abc}", "\\Gamma^{abcde}"]

Fully distributed operator strings (no outer parentheses):

// CORRECT
"gravitino": "... - \\nabla_{a}{A} - \\Psi_{a}A + A\\Psi_{a} + \\Phi_{a}A"

// WRONG — causes parsing differences vs standalone
"gravitino": "... - (\\nabla_{a}{A} + \\Psi_{a}A - A\\Psi_{a}) + \\Phi_{a}A"

Dilaton derivative cleanup rules — include every power of $e^{n\Phi}$ that appears in the KSE:

"cleanup_rules": [
  "\\nabla_{a}{\\exp{\\Phi}} -> \\exp{\\Phi}\\nabla_{a}{\\Phi}",
  "\\nabla_{a}{\\exp{\\frac{1}{2}\\Phi}} -> \\frac{1}{2}\\exp{\\frac{1}{2}\\Phi}\\nabla_{a}{\\Phi}",
  "\\nabla_{a}{\\exp{\\frac{1}{4}\\Phi}} -> \\frac{1}{4}\\exp{\\frac{1}{4}\\Phi}\\nabla_{a}{\\Phi}",
  "\\nabla_{a}{\\exp{-\\frac{1}{4}\\Phi}} -> -\\frac{1}{4}\\exp{-\\frac{1}{4}\\Phi}\\nabla_{a}{\\Phi}"
]

Use extra_field_sub: true when the KSE contains (A_a) (gauge potential). This applies the Maxwell field equation a second time after product_rule generates new (\nabla_a A_b) terms.

Use power notation X**2 — Cadabra parses X^{2} as $X$ with index 2.

Step 3 — Verification Checklist

• Driver result matches standalone script exactly
• All multi-index Gamma strings in chirality_commutation have spaces between every index
• Operator strings are fully distributed
• All $e^{n\Phi}$ powers in KSE have corresponding ∇(exp(...)) → cleanup rules
• If matter indices appear, add internal_indices block

Step 4 — Debugging

Add display(ex) after each pipeline step in both driver and standalone, then compare step by step. The first divergence reveals the root cause:

substitute(ex, kse)
substitute(ex, field)
substitute(ex, kse)
print("After step 1:")
display(ex)  # compare with standalone here

Verified Results

Known Limitations

Romans mass / massive IIA: The d10_iia theory in theories.json uses the Romans mass parameter $\kappa$. Setting $\kappa = 0$ in the KSEs gives the massless theory; the integrability result is the same in both cases because mass terms enter only through (E_{ab}) and $F\Phi$ residuals which vanish on-shell.

Index alphabet: The driver uses 16 spacetime indices {a,...,p}. Theories with $D > 16$ would need an extended alphabet.

Cadabra version: Tested against Cadabra2 2.5.14 (Homebrew, macOS Apple Silicon). The Python API is stable but join_gamma behaviour may differ across minor versions.

Repository Structure

py_integrability_sugra/
├── README.md                     # This file
├── theories.json                 # Unified theory schema
├── integrability_driver.py       # Unified computation driver
├── docs/
│   └── project_notes.md          # Research notes and debugging guide
└── legacy/                       # Original standalone scripts (one per theory)
    ├── integrability_d4.py
    ├── integrability_d5_minimal.py
    ├── integrability_d5_vector_gauged.py
    ├── integrability_d6.py
    ├── integrability_d10_iia.py
    ├── integrability_d11.py
    └── integrability_heterotic.py

References

• Cadabra2: P. Peeters, Introducing Cadabra: A symbolic computer algebra system for field theory problems, hep-th/0701238
• D=11 SUGRA: E. Cremmer, B. Julia, J. Scherk, Supergravity theory in eleven dimensions, Phys. Lett. B 76 (1978) 409
• Type IIA / IIB: M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory Vol. 2, Cambridge University Press
• Massive IIA: L.J. Romans, Massive N=2a supergravity in ten dimensions, Phys. Lett. B 169 (1986) 374
• D=5 N=2 ungauged: B. de Wit, H. Nicolai, d=11 supergravity with local SU(8) invariance; also A. Cadavid, A. Ceresole, R. D’Auria, S. Ferrara, Eleven-dimensional supergravity compactified on Calabi-Yau threefolds, Phys. Lett. B 357 (1995) 76
• D=5 gauged: M. Günaydin, G. Sierra, P.K. Townsend, Gauging the d=5 Maxwell-Einstein supergravity theories, Nucl. Phys. B 242 (1984) 244
• Heterotic: P. Candelas et al., Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46
• KSE integrability: U. Gran, J. Gutowski, G. Papadopoulos, various JHEP papers on classification of supersymmetric solutions
• Spinor conventions: Appendix conventions for $Spin(9,1)$ and $Spin(4,1)$ Clifford algebras