daifreed 0.1.1

Compute Dai-Freed anomalies in particle physics

daifreed

A Python library to compute Dai-Freed anomalies in particle physics, implementing the results of

I. García-Etxebarria and M. Montero, Dai-Freed anomalies in particle physics, arXiv:1808.00009.

Dai-Freed anomaly cancellation is a refinement of ordinary anomaly cancellation. For a 4d fermion theory with symmetry group G, the anomaly is a homomorphism η : Ω₅^struct(BG) → U(1), and freedom requires exp(2πi η_Y) = 1 on every closed 5-manifold Y — not just mapping tori. The strategy (paper §2.2) is:

1. Compute the bordism group Ω₅^struct(BG); if it vanishes there is no anomaly.
2. Otherwise, evaluate η on generators, via the Bahri–Gilkey eta-invariant formulas for lens spaces / spherical space forms, or the closed-form cancellation conditions derived from them.

Installation

uv add daifreed
# or
pip install daifreed

or, install from the source code to add a new model or run tests:

git clone [e-mail]:chris-w-murphy/dai-freed.git
cd dai-freed/
# if you need to install `uv` run:
# curl -LsSf https://astral.sh/uv/install.sh | sh
uv sync --all-extras  # creates a venv

What it computes

Module	Contents
daifreed.eta	Exact η-invariants of lens spaces (eqs. 2098, 2401) and spherical space forms (2071/2074)
daifreed.conditions	Closed-form conditions: spin_zn (2114–2116), spinc_zn (2422–2425), local_u1, z3_triality
daifreed.bordism	Ω_d(pt) (Table 3132), Ω_d^Spin(BG) (Table 2941), and the AHSS for BSU(n)
daifreed.topological_sc	The Z₄ / topological-superconductor mod-16 constraint
daifreed.ibanez_ross	Ibañez–Ross constraints and their comparison to Dai-Freed
daifreed.models	Ready-made spectra: Standard Model, MSSM, baryon triality, proton hexality

Every η-invariant is returned as an exact sympy.Rational (the anomaly phase is exp(2πi η)). The default SymPy backend is authoritative; an optional mpmath fast path (backend="mpmath") is available and agrees with it.

Using it with your own model

The workflow is two steps: (1) encode your spectrum, (2) call the checker that matches your theory's spacetime structure and symmetry.

Step 1 — encode the spectrum. A model is a Spectrum of chiral (right-moving Weyl) Fermions; each carries a dict of integer charges, a multiplicity (identical copies), and a spin (1/2 default, or 3/2 for a gravitino):

from daifreed.spectrum import Fermion, Spectrum

model = Spectrum([
    Fermion({"q": 1, "Z5": 1}, multiplicity=2, name="psiA"),
    Fermion({"q": -1, "Z5": 4}, multiplicity=2, name="psiB"),
    Fermion({"q": 3, "Z5": 2}, name="psiC"),           # multiplicity defaults to 1
    Fermion({"q": -3, "Z5": 3}, name="psiD"),
])

Two helpers matter for realistic models:

• spec.with_symmetry("X", fn) — derive a charge from existing ones, e.g. spec.with_symmetry("X", lambda f: -2*f.charges["Y"] + 5*f.charges["B-L"]).
• spec.scaled(k) — replicate for k generations.

Step 2 — call the checker for your setup. Each returns a result that is truthy when anomaly-free and carries .residuals (what must vanish):

Your situation	Call
Discrete ℤₙ (n odd), Spin manifolds	spin_zn(spec, n, label="Z5")
ℤ₃ specifically (proton triality, …)	z3_triality(spec, label="Z3")
Discrete ℤₙ inside a U(1) / Spin^c	spinc_zn(spec, n, s_label="Z5", q_label="q")
Continuous U(1) (perturbative)	local_u1(spec, label="q")
ℤ₄ / topological superconductor (Spin^{ℤ₄}, mod 16)	topological_superconductor(spec, label="Z4")
Mixed ℤ₄–gauge (S¹×S⁴)	mixed_z4_gauge_anomaly(spec, "Z4", "index")
Ibañez–Ross comparison	ibanez_ross(spec, n, ...) / embeds_in_anomaly_free_u1(...)
"Can this group even have an anomaly?"	omega_spin_BG("SU(5)", 5), is_anomaly_free("Spin(10)")

from daifreed.conditions import spin_zn
r = spin_zn(model, 5, label="Z5")
r.is_free      # True -> anomaly-free  (bool(r) works too, so `if r: ...`)
r.residuals    # {'cubic (L^3(n))': 0, 'linear (K3xL^1(n))': 0}

Which checker you want is dictated by what you gauged: a plain Spin structure + a discrete ℤₙ → spin_zn; a gauged U(1) (so spacetime is Spin^c) → spinc_zn; a ℤ₄ whose ℤ₂ is fermion parity (Spin^{ℤ₄}) → topological_superconductor (put a gravitino in as spin=Fraction(3, 2)); a non-abelian gauge group → check is_anomaly_free(...) first (a vanishing bordism group means no Dai-Freed anomaly at all).

If your case isn't covered by a canned checker, drop to the engine and evaluate η directly on the relevant generator — e.g. a ℤₙ Spin fermion of charge s on the lens space L^k(n) is eta_spin_lens(s, n, k), a general spherical space form S^{2k-1}/τ(G) is eta_general(...) — then require the spectrum sum to be ≡ 0 (mod 1).

Reproducing the paper

from daifreed import models
from daifreed.bordism import is_anomaly_free
from daifreed.conditions import local_u1, z3_triality
from daifreed.topological_sc import topological_superconductor
from daifreed.eta import eta_spinc_lens
import sympy as sp

# --- The Standard Model and GUTs are Dai-Freed anomaly-free (§3.4) ----------
assert is_anomaly_free("SU(5)")     # Ω₅^Spin(BSU(5)) = 0
assert is_anomaly_free("Spin(10)")  # ... hence the SM (⊂ SU(5)) too

# X = -2Y + 5(B-L) is a local-anomaly-free U(1) (Σq = Σq³ = 0), which forces
# the number of fermions per generation to be a multiple of 16 (§4.3):
sm = models.standard_model(generations=1)
assert sm.total_multiplicity() == 16
assert local_u1(sm, "X").is_free

# --- Topological superconductor: 16 fermions ⇒ mod-16 anomaly-free ----------
assert topological_superconductor(sm, "Z4").is_free          # SM: 16 Majorana fermions
assert topological_superconductor(models.mssm(1), "Z4").is_free  # MSSM: 16+12+4=32

# --- Baryon triality has a mod-9 anomaly, needs 3 generations (§4.2) --------
assert not z3_triality(models.baryon_triality(1)).is_free     # one generation: anomalous
assert z3_triality(models.baryon_triality(3)).is_free         # three generations: OK

# --- η(L¹(n)) = -s/n mod 1  (Bahri-Gilkey, eq. 2466) ------------------------
assert eta_spinc_lens(2, 1, 5, 1) == sp.Rational(3, 5)        # = -2/5 mod 1

Testing

uv run pytest

The suite reproduces each headline claim of the paper: SM/GUT anomaly-freedom, the baryon-triality mod-9 anomaly and its 3-generation cancellation, the Standard-Model / MSSM mod-16 count, the η(L¹(n)) = -s/n identity, the agreement of the closed-form spin_zn conditions with the two bordism generators L³(n) and K3 × L¹(n), and the Ibañez–Ross counterexample (eq. 2498) that satisfies Ibañez–Ross but fails the stronger Dai-Freed conditions.

Scope

The library covers the exact η-invariant engine, the closed-form conditions, the tabulated bordism groups, and a documented Atiyah–Hirzebruch computation for BSU(n). A fully general spectral-sequence engine (arbitrary BG) is out of scope — those computations are done by hand in the paper.