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

Hisham Sati (Mathematics, Division of Science, and Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, New York University Abu Dhabi, United Arab Emirates)

Urs Schreiber (Mathematics, Division of Science, and Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, New York University Abu Dhabi, United Arab Emirates)

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