Half-integral levels
Yifei Zhao
TL;DR
The paper formulates and proves a set of deep equivalences that unify four perspectives on half-integral level structures for reductive groups: (i) central extensions of loop groups by $\underline{K}_2$ and $\mathbb{G}_m$ with tame commutator, (ii) factorization super central extensions of loop groups, (iii) factorization super line bundles on the affine Grassmannian, and (iv) rigidified sections of the truncated K-theory sheaf $\underline K_{[1,2]}^{\mathrm{super}}$ on $BG$ together with BD-type combinatorial data. It introduces the Zariski sheaf $\underline K_{[1,2]}^{\mathrm{super}}$, defines a super-analytic integration along curves twisted by a spin structure, and proves two main theorems: Theorem thm-classification-super linking BD data to rigidified $BG$ sections, and Theorem thm-factorization-super-central-extension-classification equating loop-group factorization data with Grassmannian line bundles and $\vartheta^{\mathrm{super}}$ data. The framework relies on the Contou-Carrère symbol to handle tame commutators, uses descent for tori and simply connected subgroups, and identifies a universal description in characteristic zero where tameness is automatic. In particular, the Pfaffian construction and the Tate/half-integral phenomena provide new geometric realizations of half-integral levels on moduli spaces of bundles over spin curves, tying algebraic K-theory to geometric representation theory and conformal blocks. Overall, the work extends Brylinski–Deligne theory to a super and half-integral setting, unifying loop-group central extensions, Grassmannian line bundles, and K-theory data through a coherent factorization framework with broad implications for spin geometry and quantization.
Abstract
We construct equivalences among four notions associated to a reductive group scheme $G$: factorization super central extensions of the loop group of $G$ by $\mathbb G_m$ subject to a condition on the commutator, factorization super line bundles on the affine Grassmannian of $G$, rigidified sections of a quotient of $2$-truncated K-theory over the Zariski classifying stack of $G$, and combinatorial data defined by Brylinski and Deligne in a conjectural extension of their classification theorem.