David Roberts scratch 2021

Here are some thoughts by Urs Schreiber

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Let me lay out the picture that I have at the moment, also so that you see where you fit in:

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Hypothesis H, full quantum version:

For

• a $G$-orbi $\mathbb{R}^{10,1\vert \mathbf{32}}$-fold $\mathcal{X}$ (spacetime)

• with geometric torsion-free $\mathrm{Spin}(10,1)$-structure (i.e. field of super-gravity)

• further lifted to tangential $\mathrm{Sp}(2)$-structure $\tau$ (anomaly cancellation),

• consider the cocycle space $\mathbf{\pi}_G^\tau(\mathcal{X})$ of $\tau$-twisted equivariant differential 4-Cohomotopy (i.e. the C-field configuration G-space)

• and then the $G$-equivariant $E$-cohomology cocycle space of its loop space $\mathbf{E}^\bullet_G\big( \Omega \mathbf{\pi}_G^\tau(\mathcal{X}) \big)$ (quantum states)

• regarded as a $G$-equivariant spectrum (e.g. with $E = H\mathbb{C}$ and $\mathcal{X}$ a $D6 \perp D8$ spacetime as in SS19 this has homotopy groups the weight systems on horizontal chord diagrams)

  • hence as a spectral Mackey functor

  • hence as a parameterized spectrum (over $G$-sets) equipped with consistent orientation for pull/push transformations

  • hence as a cohomological QFT-via-secondary-integral-transforms in the sense of Sec. 7 of Quantization via Linear homotopy types.

Unwinding the definitions, this QFT encodes

• quantum operations between quantum states of C-field configurations

• associated with $H$-fixed loci of the original spacetime orbifold $\mathcal{X}$.

• But since these loci are conical black brane loci (as in HSS 18) in the spacetime,

• we see that we have a “classically controlled quantum system” where the classical control parameter says where in a holographic system we are:

  • in the bulk ($1$-fixed locus)

  • or on a brane ($H$-fixed locus)

    • and if so on which of possibly several intersecting ones (choice of subgroup $H \subset G$).

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Now take this mathematical picture and compare to the current folklore in string theory. Within the large hand-waving error bars of the latter, this could match well, I think.