Schreiber Perì Pantheōrías

Expository notes on provable aspects of metaphysics

leading to previously elusive

predictions on strongly coupled quantum materials.

Assorted results,
beginning with my thesis on cohesively geometric homotopy theory [Sc13 v2:‘17, SS20],
via work on 11D supergravity [Sc17] & its global completion (Hypothesis H, FSS20),
towards current research on topologically ordered quantum materials [Sc25a, Sc25b]
at the Center for Quantum and Topological Systems @ NYUAD.

Introduction

Concerning Theories of Everything, how deep can we go?

Use suitable mathematical language as metaphysical microscope!

G. Galilei, 1623:

[Physics] is written in this grand book — I mean the universe — […] in the language of mathematics […] without which it is humanly impossible to understand a single word of it; without [it], one is wandering around in a dark labyrinth.

I. Newton, 1687:

natural philosophy [has] mathematical principles

I. Kant, 1786:

I maintain that in each particular natural science there is only as much true science as there is mathematics.

D. Hilbert, 1930:

we do not master a theory in natural science until we have extracted its mathematical kernel and laid it completely bare.

E. Wigner, 1959:

the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious [which] raises the question of the uniqueness of our physical theories

Beware: Contemporary public discussion knows the meme that current high energy physics in general and string theory in particular would be “too mathematical” — while with the forefathers above we perceive just the opposite. The resolution of this apparent paradox is that even the meaning of “mathematics in the natural sciences” is being forgotten: We mean mathematical foundations with precise definitions and rigorous proofs (e.g. when speaking about “branes” or other commonly ill-defined concepts of the second superstring revolution).

The following is expository survey of some fragments
of modernized Principia Mathematica that I have found.

The Objective Logic: Category Theory of Duality

(Further reading:
$\phantom{--}$ Sc16: Progression of the notion towards physics,
$\phantom{--}$ Sc18: geometry of physics – Categories and Toposes
$\phantom{--}$ Sc25: Higher Topos Theory in Physics)

Beyond the all-familiar mathematics of quantity,

there is a mathematics of, in this order:

structure, duality, quality & effects.

That, of course, is category theory:

categories, adjunctions, modalities & monads

provide an objective logic for metaphysics.

The many faces of Categories.

• Categories are directed graphs with composition:

  vertices are called objects,

  edges are called morphisms.

• We may think of

  objects as generalized spaces

  (physical spaces, classifying spaces, …)

  morphisms as the maps between these;

• or

  objects as data types,

  morphisms as programs.

• Like groupoids are sets with gauge symmetries,

  so categories are sets with “non-invertible symmetries”.

It is a famous saying (Freyd 1964, p 1) that categories were defined in order to

• define functors

  (maps from one category to anohter, preserving the (units and) composition of morphisms)

which, in turn, are are needed to

• define natural transformations

  (deformations of one functor into another).

But this is at best half the truth:

Category Theory is the Theory of Duality.

Actual category theory comes to life with the next level of this hierarchy: adjunctions $L \dashv R$:

Pairs of functors $L \colon \mathcal{C} \leftrightarrows \mathcal{D} \colon R$ and natural transformations $\eta \colon id_{\mathcal{C}} \Rightarrow R \circ L$ & $\epsilon \colon L \circ R \Rightarrow id_{\mathcal{D}}$ satisfying the Yin-Yang identity shown on the right.

Instructive example: Duality between even numbers and odd numbers is an adjunction: here.

Primordial example: In a category with initial object $\varnothing$ and terminal object $\ast$, the endofunctors constant on these objects are adjoint: $\varnothing \;\dashv\; \ast \,.$

Advanced example: Some “stringy dualities” are adjunctions, notably double dimensional reduction, and with it topological T-duality & M/IIA duality [FSS18, BMSS19, GSS24].

Make place for physics: Gros Toposes. (cf. [Sc16, Sc17, Sc19, Sc25, GS25])

Core role of category theory in physics is to provide

categories of generalized spaces (field spaces, super-spaces, dg-spaces, moduli spaces, orbifolds,…)

by declaring how these are probed by generalized coordinate charts.

Such categories are called τόποι – where physics takes place.

For example, each chart $C$ becomes a generalized space

by declaring $Plt(-,\, C) \coloneqq Hom_{Charts}(-,C)$.

The Yoneda lemma says that this is consistent

in that also $Plt(-\, C) \;\simeq\; Hom_{\mathbf{H}}(-,C)$.

The generalized charts $C$ (implicitly) used in physics are the following (cf. again Sc25, …):

We next look at the progression of dualities in the resulting

topos of super formal smooth $\infty$-groupoids.

Modality: From \nothing emerges the super-point

The Method [Sc16]:

Theorem. The topos of generalized spaces probed

by charts $\mathbb{R}^n \times \mathbb{D}^m_k \times \mathbb{R}^{0 \vert q} \;(\times \Delta^r \times \cdots)$

exhibits the following progression of modalities out of \nothing [Sc13 v2:‘17, SS20]:

This exhibits the super-point as emerging from \nothing,

via a progression through the qualia of

1. cohesion — differential topology$\;$ $\esh$ $\dashv$ $\flat$ $\dashv$ $\sharp$

2. elasticity — differential geometry$\;$$Re$ $\dashv$ $\Im$ $\dashv$ &

3. solidity — super-geometry$\;$ $\rightrightarrows$ $\dashv$ $\rightsquigarrow$ $\dashv$ $Rh$

From the super-point emerges 11D super-gravity

There are also other universal constructions.

Next we ask for emerging universal central extensions.

The following shows how, furthermore [S17, Sc19]:

From the super-point then emerges super-spacetime [HS18]

from super-spacetime emerges the super-branes [FSS15]

culminating in the M-branes embodied by the 4-Sphere [FSS17].

This progression discovers D=11 supergravity:

Asking the 4-cohomotopy cocycle $\mu_{M2/M5}$ to globalize over an 11D super-spacetime $X$

is equivalent to $X$ solving the 11D Sugra EOMs [GSS24 Thm. 3.1]

On M5 emerges strongly coupled quantum physics

The emergence of the 4-sphere suggests that this is the correct

classifying space for flux-quantization [SS25]

of D=11 supergravity – “Hypothesis H” (cf. SS24)

$\Rightarrow$ emblematic non-perturbative effects [SS25a, SS25b]:

Modal Logic of Quantum Effects and Measurement

By unwinding the available monadology we also find

accurate quantum language which knows about

quantum measurement, possible/many worlds and wavefunction collapse

[SS23, cf. Sc25]

For instance, the deferred measurement principle

receives formalization and general proof in this language (p. 81)

Outlook

The grand open problem of contemporary theoretical physics

is non-perturbative quantum systems, a huge realm

of which familiar perturbation theory

sees only the infinitesimal neighbourhood of a point.

This problem haunts

1. particle physics, where confined QCD…

2. and CMT, where strongly coupled/correlated systems…

… are being “understood” only by computer experiment.

The problem with non-perturbative is that, for the most part,

even the kind of theory has remained unclear: language is missing.

The main attack on the problem has an ironic history

(cf. Polyakov gauge-string duality & holographic QCD):

• posit that in the strong-coupling regime

  the DoFs are flux tube strings,

• find that quantumly these live in higher dimensions,

• understand the original problem as engineered

  on branes inside higher dim bulk spacetime,

• irony: also these strings are only understood perturbatively
  ($\Rightarrow$ need for large-N limit in AdS/CFT)

• BUT strings (as opposed to QFT) come with hints for

  their non-perturbative completion: working title “M-theory”

This way, M-theory (re-)appears as the eventual solution

to the Millennium Problem of making sense of non-pert. QFT!

Still, the language problem remains: How to even speak M-theory?

Whence Amati & Witten’s look towards “mathematical structures of 21st/22nd century”

But previously missing language seems

to be just what we are seeing above.

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