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

TL;DR

This paper develops a differential Cohomotopy theory on Penrose spacetimes, linking brane charge quantization to intersecting brane moduli and revealing rich quantum effects via configuration spaces and chord diagrams.

Contribution

It introduces a differential refinement of Cohomotopy theory that connects brane charges to configuration spaces and chord diagrams, providing a new mathematical framework for M-theory phenomena.

Findings

Brane charge quantization implies intersecting brane moduli as configuration spaces.

Higher observables are described by weight systems on chord diagrams.

The framework captures effects like brane stacks, matrix models, and gauge/gravity duality.

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/(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 Dp/D(p+2)-brane intersections: condensation to stacks of coincident branes and their Chan-Paton factors, BMN matrix model and fuzzy funnel states, M2-brane 3-algebras, the Hanany-Witten rules, AdS3-gravity observables, supersymmetric indices of Coulomb branches 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 charge-quantized in Cohomotopy theory.