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

Hisham Sati, Urs Schreiber

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.

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

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.

Paper Structure

This paper contains 15 sections, 11 theorems, 70 equations.

Table of Contents

  1. Introduction and overview
  2. Intersecting brane charges in differential Cohomotopy
  3. Charges vanishing at infinity
  4. Configuration spaces of points
  5. Differential Cohomotopy cocycle spaces
  6. Intersecting brane charges in differential Cohomotopy
  7. Higher observables on intersecting brane configurations
  8. Weight systems on chord diagrams
  9. Horizontal chord diagrams
  10. Round chord diagrams
  11. Lie algebra weight systems
  12. On stacks of coincident strands
  13. Quantum states on chord diagrams
  14. Chord diagrams and intersecting branes
  15. Lie algebra weight systems give matrix model single trace observables

Key Result

Proposition 2.4

For $D \in \mathbb{N}$, there is a homotopy equivalence between the disjoint union of ordered unlabelled configuration spaces in $\mathbb{R}^D$ and the fiber product of unordered but labelled configuration spaces (Def. ConfigurationSpaces) as follows: where the fiber product on the right is that induced from the maps in Example MapsOfConfigurationSpacesForOrderedFiberProduct.

Theorems & Definitions (37)

  • Definition 2.1: Configuration spaces of points
  • Definition 2.2: Category of Penrose diagrams
  • Example 2.3: Maps of configuration spaces for ordered fiber product
  • Proposition 2.4: Ordered unlabeled configurations as fiber product of unordered labeled configurations
  • Proposition 2.5: Labelled configuration spaces via Cohomotopy cocycles
  • Remark 2.6: Cohomotopy charge map
  • Example 2.7: Unlabeled from labeled
  • Proposition 2.8: Configurations vanishing at the boundary
  • Proposition 2.9: Differential Cohomotopy on Penrose diagrams via configuration spaces
  • proof
  • ...and 27 more