The higher algebra of the spin orientation

David Aretz

July 16, 2026

ENFFT Lausanne

MPIM Bonn

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Geometric Spin Structures
Stolz–Teichner
spin structure on V → X = irreducible Cl(V)-Clrk(V)-bimodule bundle Σ
spin structure on V → X = Morita equivalence
Σ : Cl(V) ≃ Cl(ℝrk(V))
Orientation
φ : det(V) ≅ det(ℝrk(V))
Upshot
SO-structure: trivialization of the determinant line
Spin-structure: trivialization of the Clifford algebra super 2-line
1 / 4
Categorical Whitehead Tower
Vect Spin Pic ( sAlg 2 ) Vect SO Vect O sLine trace det Cl
B Spin τ ≤2 BO BSO BO B ℤ/2 w 1
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Twisted Cohomology

E ring spectrum, V → X vector bundle.

Twisted cohomology: E∗(Th(V)) = E∗cs(V) = E∗+τV(M).

Homotopically:

X → BOn J→ Sp → Mod(E) .
  • Classifies bundle of E-modules over X.
  • Twist trivial ⟺ bundle of modules trivial. V is E-oriented.
KO-twist
Via J, any V → X determines a bundle of KO-modules τV → X.
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Synthesis
Theorem (DA)
⊔ BO n Pic ( KO ) Pic ( sAlg 2 ) J Cl ⟳ K gr
Punchline
KO-twist τV trivial (KO-oriented) ⟺ super 2 line bundle Cl(V) trivial (Spin)
Corollary
MSpin → KO map of ring spectra
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Thank you!