Schreiber Edinburgh 2025 (Rev #45, changes)

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A mini-course that I have lectured:

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• Urs Schreiber $\;\;$ (on work with atCQTS with H. Sati):

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  Complete Topological Quantization of Higher Gauge Fields

  • extended lecture notes:

    • pdf (v3, minor revisions)

    • arXiv:2512.12431 hep-th (v2)

    • live recording I: yt, II: yt, Playlist: yt

    • draft recording: yt (Part 1)

  SciPost Physics Lecture Notes 127 (2026)

  doi:10.21468/SciPostPhysLectNotes.127

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• lecture series at:

  Geometry, Higher Structures, and Physics (all talks playlist: yt)

  @ International Centre for Mathematical Sciences (ICMS)

  Bayes Centre, Edinburgh

  8-12 December 2025

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Abstract. After global completion of higher gauge fields (as appearing in higher-dimensional supergravity) by proper flux quantization in extraordinary nonabelian cohomology, the (non-perturbative, renormalized) topological quantum observables and quantum states of solitonic field histories are completely determined through a topological form of light-front quantization. We survey the logic of this construction and expand on aspects of the quantization argument.

In the instructive example of 5D Maxwell-Chern-Simons theory (the gauge sector of 5D SuGra) dimensionally reduced to 3D, a suitable choice of flux quantization in Cohomotopy (“Hypothesis h”) recovers fine detail of the traditionally renormalized (Wilson loop) quantum observables of abelian Chern-Simons theory and makes novel predictions about anyons in fractional quantum (anomalous) Hall systems. An analogous choice (“Hypothesis H”) of global completion of 11D higher Maxwell-Chern-Simons theory (the higher gauge sector of 11D SuGra) realizes various aspects of the topological sector of the conjectural “M-theory” and its M5-branes.

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Based on:

• Flux Quantization &

  The Character Map in Non-Abelian Cohomology

• Flux Quantization on Phase Space

• Quantum Observables of Quantized Fluxes

• Renormalization of Wilson Loop Observables via Extraordinary Flux Quantization in Cohomotopy

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Followup:

• Higher-dimensional Anyons via Higher Cohomotopy

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Related talk:

• @ Oberwolfach Mini-Workshop 2539b (Sep 2025)

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Script

The following is close to the blackboard presentation.

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Complete Topological Quantization of Higher Gauge Fields \linebreak ncatlab.org/schreiber/show/ICMS25

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Plan:

1. Completion of Higher Gauge Theories

2. Their Complete Topological Quantization

3. Application to Topological Quantum Materials

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Motivation:

• Global completion of 11D supergravity (“M-theory”),

• and of its M5-brane probes (“Theory X”),

• relatedly of FQH anyons (“topological quantum”).

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Part 1 – Completion of Higher Gauge Theories

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Recall Maxwell's equations in vacuum:

$\mathrm{d} F_2 = 0 ,\;\;\; \mathrm{d} \star_{1,3} F_2 = 0 \mathrlap{\,.}$

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Suprprisingly good idea to rewrite this in

pre-metric or duality symmetric form

$\left. \begin{aligned} \mathrm{d} F_2 & = 0 \\ \mathrm{d} G_2 &= 0 \end{aligned} \right\}\,, \;\;\;\;\; G_2 = \star_{1,3} F_2$

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In this vein, say that

higher Maxwell-type equations

on higher flux densities

$\vec F \;\coloneqq\; \Big( F^{(i)} \in \Omega^{deg_i}_{dR}(X^{1,d}) \Big)_{i \in I}$

are equations of the form

$\mathrm{d} \vec F \;=\; \vec P\big( \vec F \big) \,,\;\;\;\; \star_{1,d} \vec F = \vec \mu\big( \vec F \big)$

for

• $\vec P$ an $I$-tuple of graded-symemtric polynomials,

• $\vec \mu$ an $I$-tuple of linear maps

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Examples

RR-field in type IIA 10D supergravity:

$\mathrm{d} F_{2\bullet} = 0 \Big\} \,,\;\;\;\; \star_{1,9} F_{2\bullet} = F_{10-2\bullet} ,.$

self-dual higher gauge field:

$\mathrm{d} H_{2k+1} = 0 \Big\} \,,\;\;\;\;\; \star_{1,4k+1} H_{2k+1} = H_{2k+1}$

Maxwell-Chern-Simons field in minimal 5D supergravity

$\left. \begin{aligned} \mathrm{d} F_2 & = 0 \\ \mathrm{d} F_3 & = \tfrac{1}{2} F_2 \wedge F_2 \end{aligned} \right\} \,,\;\;\;\; F_3 = \star_{1,4} F_2$

C-field in 11D supergravity:

$\left. \begin{aligned} \mathrm{d} G_4 & = 0 \\ \mathrm{d} G_7 & = \tfrac{1}{2} G_4 \wedge G_4 \end{aligned} \right\} \,,\;\;\;\; G_7 = \star_{1,10} G_4 \mathrlap{\,.}$

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Next, assume a globally hyperbolic spacetime

$X^{1,d} \simeq \mathbb{R}^{1,0} \times X^d$

with Cauchy surface $X^d \xhookrightarrow{ \iota_t } \mathbb{R}^{1,0} \times X^d$

and consider the time-local solution space

$LocSol_t \;\coloneqq\; \left\{ \begin{array}{l} \text{ temporal germs of solutions } \\ \text{ of the above EoMs around }\; \{t\} \times X^d \end{array} \right\} \mathrlap{\,.}$

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Proposition.

Pullback to the Cauchy surface constitutes a bijection:

$LocSol_t \xrightarrow[\sim]{\phantom{--}\iota_t^\ast\phantom{--}} \Big\{ \vec B \coloneqq \big( B^{(i)} \in \Omega^{deg_i}_{dR}(X^d) \big)_{i \in I} \;\Big\vert\; \overset{ \text{higher Gauss law} }{ \overbrace{ \mathrm{d}\vec B = \vec P(\vec B) } } \Big\}$

so:

• dual flux densities become canonical momenta,

• Hodge duality relation disappears (absorbed in the iso),

• remaining differential relation is higher Gauss law.

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Proposition: Moreover:

The higher Gauss law is equivalently the closure (flatness)

condition of $\mathfrak{a}$-valued differential forms

$\Big\{ \vec B \;\Big\vert\; \mathrm{d}\vec B = \vec P(\vec B) \Big\} \;\; \simeq \;\; \Omega^1_{dR}\big(X^d; \mathfrak{a}\big)_{cl}$

for a characteristic $L_\infty$-algebra $\mathfrak{a}$

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Let’s recall what this means:

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connected $L_\infty$-algebras of finite type are

$\begin{array}{ccc} L_\infty Alg^{ft}_{conn} &\xhookrightarrow{\phantom{--}CE\phantom{--}}& dgcAlg^{op} \\ \left( \mathfrak{g} ,\; \begin{array}{l} [-], [-,-], \\ [-,-,-], \cdots \end{array} \right) &\mapsto& \left( \wedge^\bullet \mathfrak{g}^{\vee} ,\, \mathrm{d}_{\vert \wedge^1 \mathfrak{g}^{\vee}} = \begin{array}{l} [-]^\ast + [-,-]^\ast + \\ [-,-,-]^\ast + \cdots \end{array} \right) \end{array}$

where $\mathfrak{g}$ is

• an $\mathbb{N}$-graded vector space

• degreewise of finite dimension.

and closed $L_\infty$-valued forms are

$\Omega^1_{dR}\big(X; \mathfrak{g}\big)_{cl} \;\coloneqq\; Hom_{dgAlg}\Big( CE(\mathfrak{g}) , \Omega^\bullet_{dR}(X) \Big)$

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Example/Definition/Proposition

For $\mathcal{A}$ a connected topological space

with abelian fundamental groups (for simplicity)

and $dim_{\mathbb{R}}\big(H^\bullet(\mathcal{A},\mathbb{R})\big) \lt \infty$,

then its real Whitehead L∞-algebra $\mathfrak{l}\mathcal{A}$ is

$(\mathfrak{l}\mathcal{A})_\bullet \coloneqq \pi_\bullet(\Omega \mathcal{A}) \otimes_{\mathbb{Z}} \mathbb{R}$

with brackets uniquely determined by the condition that

$H^\bullet\big( CE(\mathfrak{l}\mathcal{A}) \big) \;\simeq\; H^\bullet\big( \mathcal{A}; \mathbb{R} \big) \mathrlap{\,.}$

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Examples

electromagnetism: $\mathfrak{a} = \mathfrak{l}(B \mathrm{U}(1)^2)$

self-dual higher gauge field: $\mathfrak{a} = \mathfrak{l} B^{2k} \mathbb{Z}$

RR-field: $\mathfrak{a} = \mathfrak{l}(B \mathrm{U})$

Maxwell-Chern-Simons field: $\mathfrak{a} = \mathfrak{l}S^2$

C-field: $\mathfrak{a} = \mathfrak{l}S^4$

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Definition/Proposition.

The total flux of a flux density $\vec F$

is its image in non-abelian de Rham cohomology

$H^1_{dR}\big( X;\, \mathfrak{a} \big) \;\simeq\; \Omega^1_{dR}\big(X; \mathfrak{a}\big)_{cl}\big/ \text{concordance}$

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Definition/Proposition.

Denoting by

• $L^{\mathbb{Q}}\mathcal{A}$ the rationalization of $\mathcal{A}$,

• $L^{\mathbb{R}}\mathcal{A}$ its extension of scalars to the reals

• $\eta^{\mathbb{R}} \colon \mathcal{A} \longrightarrow L^{\mathbb{R}} \mathcal{A}$ the comparison map,

the nonabelian character map is

$\begin{array}{ccc} \begin{array}{c} \text{charge in} \\ \text{nonabelian cohomology} \end{array} &character& \begin{array}{c} \text{total flux in} \\ \text{nonabelian dR-cohomology} \end{array} \\ H^1(X;\,\Omega \mathcal{A}) &\xrightarrow{\phantom{--} ch^{\mathcal{A}} \phantom{--}}& H^1_{dR}\big(X;\,\mathfrak{a}\big) \\ = && \simeq \\ \pi_0\, Map\big(X, \mathcal{A}\big) &\xrightarrow{\phantom{--} \eta^{\mathbb{R}} \phantom{--}}& \pi_0\, Map\big(X, L^{\mathbb{R}}\mathcal{A}\big) \end{array}$

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Flux quantization of a higher gauge theory is

choice of $\mathcal{A}$ with $\mathfrak{l}\mathcal{A}\simeq \mathfrak{a}$

Promotion of flux densities to pairs $(\vec F, \chi)$ for

$\chi$ a charge whose character is the total flux of $\vec F$:

$\begin{array}{ccccc} \Omega^1_{dR}\big( X^d; \mathfrak{a} \big)_{cl} &\xrightarrow{ [-] }& H^1_{dR}\big( X^d;\, \mathfrak{a} \big) &\xleftarrow{ ch^{\mathcal{A}} }& H^1(X^d; \Omega \mathcal{A}) \\ \vec F &\mapsto& [\vec F] = ch^{\mathcal{A}}([\chi]) &\leftarrow\!\!\!\vert& [\chi] \end{array}$

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There are either none or infinitely many admissible $\mathcal{A}$

$\Rightarrow$ $\mathcal{A}$ is a hypothesis about the correct model for given physics

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Examples

Dirac charge quantization of EM-field: $\mathcal{A} \equiv B \mathrm{U}(1)^2$

Hypothesis K on the RR-field

Hypothesis h on Maxwell-Chern-Simons field: $\mathcal{A} \equiv S^2$

Hypothesis H on C-field: $\mathcal{A} \equiv S^4$

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Finally, the full completed higher gauge field

is not just the flux densities $\vec F$, a charge $\chi$

and an equality $[\vec F] = ch[\chi]$, but involves

a gauge transformation/homotopy $\widehat{A}$ exhibiting this equality

which out the “be” the gauge potentials.

The globally completed phase space is, schematically:

$PhsSpc \;\coloneqq\; \left\{ \begin{array}{ll} \text{flux densities} & \vec F \in \Omega^1_{dR}(X^d;\mathfrak{a})_{cl} \\ \text{local charges} & \chi \in Map(X^d, \mathcal{A}) \\ \text{gauge potentials} & \widehat{A} \colon \vec F \Rightarrow \chi \end{array} \right\}$

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More in detail, this is the smooth ∞-groupoid

which as such is the homotopy pullback

of the character map from total flux to flux densities.

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Essentially. In the following we will only need

the shape of the space space,

which turns out to be given by a concrete formula.

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Part 2 – Their Complete Topological Quantization

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observables are the smooth functions on phase space

$O \,\colon\, PhsSpc \longrightarrow \mathbb{C}$

the topological observables are the locally constant functions

$O_{top} \,\colon\, PhsSpc \longrightarrow \flat \mathbb{C}$

(where $\flat \mathcal{C}$ is the discrete set underlying $\mathcal{C}$)

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Proposition

The topological observables

$PhsSpc \xrightarrow{\phantom{--}} \flat \mathbb{C}$

are naturally equivalent to the maps

$\pi_0 \, Map(X^d, \mathcal{A}) \xrightarrow{\phantom{--}} \mathcal{C}$

out of the connected components of the mapping space

from the Cauchy surface into the classifying space,

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“Topologically, fields are only seen through their charges.”

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Moreover, realistic observables are

supported on finitely many components, hence:

$TopObs \;\coloneqq\; \mathbb{C}\big[ \pi_0 \, Map(X^d, \mathcal{A}) \big] \,\simeq\, H_0\big( Map(X^d, \mathcal{A}) ;\, \mathbb{C} \big)$

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More specifically, we look at field solitons,

whose charge vanishes at infinity

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To formalize this, consider a pointed topological space

$\big( X^d_{dom}, \infty \big) \in Top^\ast$

with basepoint thought of as a point at infinity:

$X^d \,\simeq\, X^d_{dom} \setminus \{\infty\}$

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Also think of $\mathcal{A}$ as pointed by 0-charge

$\big(\mathcal{A}, 0\big) \in Top^\ast \,.$

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Then a pointed map $\chi \in Map^\ast\big(X^d_{dom}, \mathcal{A}\big)$

literally takes the value $0$ at $\infty$

$\begin{array}{ccc} X^d_{dom} &\xrightarrow{\phantom{--} \chi \phantom{--}}& \mathcal{A} \\ \uparrow && \uparrow \\ \{\infty\} &\xrightarrow{\phantom{----}}& \{0\} \end{array}$

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So the solitonic topological observabels are

$TopObs \,\coloneqq\, \mathbb{C}\Big[ Map^\ast\big(X^d_{dom}, \mathcal{A}\big) \Big]$

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Question: What is the quantum operator product $\star$ on $TopObs$?

Feynman 1948: the time-ordered ordinary product

e.g. for field operators $\phi$ and their canonical momenta:

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NB: The analogue is still true for light front quantization

with respect to ordering in the light-front parameter $x^+ \coloneqq x^0 + x^1$.

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So consider light-front quantization on spacetime if the form

$\begin{aligned} X^{1,d} &\simeq\, \mathbb{R}^{1,1} \times X^{d-1} \\ X^d & \simeq \mathbb{R}^1 \times X^{d-1} \\ X^d_{dom} & = \mathbb{R}^1_{\cup \{\infty\}} \wedge X^{d-1}_{dom} \end{aligned}$

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For topological observables which are necessarily $x^0$-independent

their light-front ordering is their $x^1$-ordering.

This is given by the Pontrjagin product/fundamental group algebra-structure on:

$\begin{aligned} TopObs & = H_0\Big( Map^\ast\big( \mathbb{R}^1_{\cup \{\infty\}} \wedge X^d_{dom} \big) \Big) \\ & \simeq H_0\Big( \Omega Map^\ast\big( X^d_{dom} \big) \Big) \\ & \simeq \mathbb{C}\Big[ \pi_1 \, Map^\ast\big( X^d_{dom} \big) \Big] \end{aligned}$

induced under pushforward in homology

$H_0(\Omega M) \otimes H_0(\Omega M) \simeq H_0(\Omega M \times \Omega M) \xrightarrow{\phantom{--} \star_\ast \phantom{--}} H_0(\Omega M)$

from concatenation of loops

$\begin{array}{ccc} \Omega M \times \Omega M &\xrightarrow{\phantom{--} \star \phantom{--}}& \Omega M \\ \big( \ell_2, \ell_1 \big) &\mapsto& \ell_2 \star \ell_1 & \colon\; x^1 \,\mapsto\, \left\{ \begin{array}{lcl} \ell_1\big(\ln(-\tfrac{1}{x^1})\big) & \vert & x^1 \geq 0 \\ \ell_2\big(\ln(+x^1)\big) & \vert & x^1 \leq 0 \end{array} \right. \end{array}$

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So where $\delta_\ell \in TopObs$ \linebreak is the $\{0,1\}$-valued observable asking:

“Is the field history of shape $[\ell]$?”

the observable $\delta_{\ell_2} \star \delta_{\ell_1}$ asks

“Is the field history \linebreak first of shape $\ell_1$ and \linebreak then of shape $\ell_2$?”

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With the algebra of topological quantum observables understood,

the topological quantum states are its modules,

which here are the linear representations of the moduli fundamental group:

$\begin{aligned} TopQuStSpc & \coloneqq (TopObs, \star) Mod \\ & \equiv \Big( H_0\big( \Omega Map^\ast(X^{d-1}_{dom}, \mathcal{A}) \big), \star \Big) Mod \\ & \simeq \Big( \mathbb{C}\big[ \pi_1 Map^\ast(X^{d-1}_{dom}, \mathcal{A}) \big] \Big) Mod \\ & \simeq \big( \pi_1 Map^\ast(X^{d-1}_{dom}, \mathcal{A}) \big) Rep \end{aligned}$

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Part 3 – Application to Topological Quantum Materials

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Consider $\mathfrak{g}$-Yang-Mills theory on

$X^{1,3} = \mathbb{R}^{1,1} \times \Sigma^2$

What are its standard topological flux quantum observables?

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Proposition.

The Poisson brackets on linear E/M YM-flux observables through $\Sigma^2$ are:

$\underset{\text{electric flux}}{ \underbrace{ C^\infty\big( \Sigma^2, \mathfrak{g}_{\hbar} \big) } } \ltimes_{ad} \underset{\text{magnetic flux}}{ \underbrace{ C^\infty\big( \Sigma^2, \mathfrak{g}_{0} \big) } \mathrlap{\,,} }$

where $\mathfrak{g}_{\hbar}$ is $\mathfrak{g}$ with Lie bracket rescaled by $\hbar \in \mathbb{R}_{\gt 0}$.

Their non-perturbative C-star algebraic deformation quantization

is the $C^\ast$-convolution algebra:

$\mathbb{C}\Big[ C^\infty\big( \Sigma^2, G \big) \ltimes_{Ad} C^\infty\big( \Sigma^2, \mathfrak{g}_{0}/\Lambda \big) \Big]$

for choices of

• Lie group $G$ integrating $\mathfrak{g}$

• $ad$-invariant lattice $\Lambda \subset \mathfrak{g}$.

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Hence the traditional non-perturbative topological quantum observables

on E/M Yang-Mills fluxes through a surface $\Sigma^2$ are:

$\begin{aligned} TopObs^{trad} & = \mathbb{C}\Big[ \pi_0 C^\infty\big( \Sigma^2, G \ltimes \mathfrak{g}_{0}/\Lambda \big) \Big] \\ & \simeq \mathbb{C}\Big[ \pi_0 Map\big( \Sigma^2, G \ltimes \mathfrak{g}_{0}/\Lambda \big) \Big] \\ & \simeq \mathbb{C}\Big[ \pi_1 Map\big( \Sigma^2, B(G \ltimes \mathfrak{g}_{0}/\Lambda) \big) \Big] \end{aligned}$

Specifically for Maxwell theory, where $\mathfrak{g} = \mathfrak{u}(1)$,

if we choose $G = \mathrm{U}(1)$ and $\Lambda = \mathbb{Z} \subset \mathbb{R}$ then

$TopObs^{trad} = \mathbb{C}\Big[ \pi_1 Map\big( \Sigma^2, B \mathrm{U}(1)^2 \big) \Big]$

which coincides with our general $TopObs$ from above.

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Example.

For $\Sigma^2 = T^2$ the 2-torus,

$\begin{aligned} \pi_1 Map\big( T^2 , B \mathrm{U}(1) \big) & \simeq \pi_0 Map\big( T^2 , \mathrm{U}(1) \big) \\ & \simeq H^1(T^2; \mathbb{Z}) \\ & \simeq \mathbb{Z}_a \times \mathbb{Z}_b \end{aligned}$

Algebra of Maxwell Wilson loops on torus.

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Now consider lifting these particles in 4D to strings in 5D

$X^{1,4} \simeq \mathbb{R}^{1,1} \times (\mathbb{R}^1 \times \Sigma^2)$

$X^4_{dom} = \mathbb{R}^1_{\sqcup \{\infty\}} \wedge (\mathbb{R}^1 \times \Sigma^2)_{\cup \infty} \,.$

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Under Hyothesis h, with $\mathcal{A} = S^2 \xhookrightarrow{\iota_2} B \mathrm{U}(1)$,

the topological observables are deformed to:

Proposition.

$\pi_1 Map\big( T^2, S^2 \big) \simeq \widehat{\mathbb{Z}^2}$

which is the integer Heisenberg group at level=2:

$\widehat{\mathbb{Z}^2} \coloneqq_{Set} (\mathbb{Z}_a \times \mathbb{Z}_b) \times \mathbb{Z}$

with generating elements

$\begin{aligned} W_a &\coloneqq \big((1,0), 0\big) \\ W_b &\coloneqq \big((0,1), 0\big) \\ \zeta &\coloneqq \big((0,0), 1\big) \end{aligned}$

on which the only non-trivial group commutator is

$[W_a, W_b] = \zeta^2 \mathrlap{\,.}$

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This is a nonabelian deformation of the Maxwell Wilson loops.

In fact this is exactly the algebra of observables of:

1. abelian Chern-Simons theory

2. anyons in fractional quantum Hall systems,

the latter being (vortices in a 2D electron liquid induced by)

surplus magnetic flux quanta

on top of some rational number of flux quanta per electrons.

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Let’s further lift this situation to 11D supergravity on

$X^{1,10} \coloneqq \mathbb{R}^{1,3} \times \big( \mathbb{R}^1 \times S^3 \times S^3 \big)$

with

$X^{10}_{dom} \coloneqq \mathbb{R}^3_{\sqcup \{\infty\}} \wedge \big( \mathbb{R}^1 \times S^3 \times S^3 \big)_{\cup \{\infty\}}$

globally completed according to Hypothesis H:

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Proposition.

$\pi_1 Map\big( S^3 \times S^3, S^4 \big) \simeq \widehat{\mathbb{Z}^2} \times \mathbb{Z}_{/12}$

Under the Pontrjagin theorem this says that

5-branes on this background, wrapped on $\mathbb{R}^{1,3}$

have the same anyon braiding in $S^3 \times S^3$ as FQH anyons on $T^2$

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Back to the case in 5D,

to see more transparently where the anyon braiding comes from

consider the simpler situation flux through the plane

$\begin{aligned} X^{1,4} & = \mathbb{R}^{1,1} \times (\mathbb{R}^1 \times \mathbb{R}^2) \\ X^4_{dom} & = \mathbb{R}^1_{\sqcup \{\infty\}} \wedge \mathbb{R}^3_{\cup \infty} \end{aligned}$

Then the 2-Cohomotopical quantum observables are spanned by

$\pi_1 \, \mathrm{Map}\big( S^2 ,\, S^2 \big) \simeq \pi_3(S^2) \simeq \mathbb{Z}$

This looks simple, but let’s see what these observables observe:

Pontrjagin theorem:

$\pi^2(S^3) = \Big\{ \text{framed links} \Big\} \Big/ \big(\text{cobordism}\big)$

In our situation, these links are

vacuum diagrams of flux quanta on the surface.

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Proposition.

The observables

$\begin{array}{ccc} \Big\{ \text{framed links} \Big\} &\longrightarrow& \mathbb{Z} \\ L &\mapsto& \# L \end{array}$

compute the writhe or total linking of a link.

Hence in a pure quantum state $\vert K \rangle$ this is

$\begin{aligned} \langle K \vert \mathcal{O}_L \vert K \rangle & = \exp\big( \tfrac{\pi \mathrm{i}}{K} \# L \big) \\ & = \exp\bigg( \tfrac{\pi\mathrm{i}}{K} \Big( \textstyle{\sum_{i \neq j}} lnk(L_i, L_j) + \textstyle{\sum_{i}} frm(L_i) \Big) \bigg) \end{aligned}$

These are exactly the expectation values of traditionally

renormalized Wilson loop observable of abelian Chern-Simons.

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The first line shows that a braiding phase

$\zeta \coloneqq \exp(\tfrac{\pi \mathrm{i}}{K})$

is picked up for each crossing in the link diagram

hence for each braiding of the anyon worldlines.

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For more see at ncatlab.org/schreiber/show/ICMS25

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