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# Advances in Theoretical and Mathematical Physics

## Volume 26 (2022)

### Number 4

### Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams

Pages: 957 – 1051

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n4.a4

#### Authors

#### Abstract

We introduce a differential refinement of Cohomotopy cohomology theory, defined on Penrose diagram spacetimes, whose cocycle spaces are unordered configuration spaces of points. First we prove that brane charge quantization in this differential $4$-Cohomotopy theory implies intersecting $p \perp (p+2)$-brane moduli given by ordered configurations of points in the transversal $3$-space. Then we show that the higher (co-)observables on these brane moduli, conceived as the (co-)homology of the Cohomotopy cocycle space, are given by weight systems on horizontal chord diagrams and reflect a multitude of effects expected in the microscopic quantum theory of $\mathrm{D}_p \perp \mathrm{D}(p+2)$-brane intersections: condensation to stacks of coincident branes and their Chan–Paton factors, fuzzy funnel states and M2-brane 3-algebras, $\mathrm{AdS}_3$-gravity observables and supersymmetric indices of Coulomb branches, M2/M5-brane bound states in the BMN matrix model and the Hanany-Witten rules, as well as gauge/gravity duality between all these. We discuss this in the context of the hypothesis that the M-theory C-field is chargequantized in Cohomotopy theory.

Published 22 February 2023