A noncommutative Cartier formula for E1-ring spectra is proven and applied to show that p-curvature of the quantum connection computes quantum Steenrod operations for Calabi-Yau symplectic manifolds.
Blumberg, Michael A
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abstract
We describe a structure on a commutative ring (pre)cyclotomic spectrum $R$ that gives rise to a (pre)cyclotomic structure on topological Hochschild homology ($THH$) relative to its underlying commutative ring spectrum. This lets us construct $TC$ relative to $R$, denoted $TC^{R}$, and we prove some descent results relating $TC^{R}$ and $TC$. We explore several examples of this structure on familiar $\mathbb{T}$-equivariant commutative ring spectra including the periodic $\mathbb{T}$-equivariant complex cobordism spectrum $MUP_{\mathbb{T}}$ and a new (connective) equivariant version of the complex cobordism spectrum $MU$.
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Geometric norms together with inflations assemble into a functor that is an equivalence between rational ultracommutative ring spectra and certain functors on the span category of finite connected groupoids.
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Noncommutative Cartier Formulae
A noncommutative Cartier formula for E1-ring spectra is proven and applied to show that p-curvature of the quantum connection computes quantum Steenrod operations for Calabi-Yau symplectic manifolds.
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An algebraic model for rational ultracommutative rings
Geometric norms together with inflations assemble into a functor that is an equivalence between rational ultracommutative ring spectra and certain functors on the span category of finite connected groupoids.