higher category theory

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### Context
#### Higher category theory
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[[!include higher category theory - contents]]
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# Contents
* table of contents
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## Idea

_Higher category theory_ is the generalization of [[category theory]] to a context where there are not only [[morphism]]s between [[object]]s, but generally [[k-morphism]]s between $(k-1)$-morphisms, for all $k \in \mathbb{N}$.

Higher category theory studies the generalization of [[∞-groupoid]]s  -- and hence, via the [[homotopy hypothesis]], of [[topological space]]s -- to that of [[directed space]]s and their _combinatorial or algebraic models_ . It is to the theory of [[∞-groupoid]]s as [[category theory]] is to the theory of [[groupoids]] (and hence of [[groups]]).

These [[geometric definition of higher category|combinatorial]] or [[algebraic definition of higher category|algebraic]] models are known as [[n-category|n-categories]] or, when $n \to \infty$, as [[∞-category|∞-categories]] or [[ω-categories]], or, in more detail, as [[(n,r)-category|(n,r)-categories]]:

* the natural number $n$ denotes the maximal dimension of _non-trivial_ cells in the model,

* while the natural number $r$ denotes the maximal dimension of the _directed_ cells.

So an ordinary [[topological space]] or [[∞-groupoid]] is an [[(∞,0)-category]]: it has cells of arbitrary dimension and all of them are reversible.

In contrast to that, a [[geometric definition of higher category|combinatorial]] or [[algebraic definition of higher category|algebraic]] model for a [[directed space]] in which the 1-dimensional paths may not all be reversible is an [[(∞,1)-category]]: it still has cells of arbitrary dimension, but only those of dimension greater than 1 are guaranteed to be reversible.

Often it is convenient in practice to consider the case where the possible dimension $n$ of non-trivial cells is finite. This is familiar from how a [[topological space]] that happens to have vanishing [[homotopy group]]s in dimension above some $n$ -- a [[homotopy n-type]] -- is modeled by an [[n-groupoid]]. A fully directed version of this is an [[n-category]], which is short for [[(n,n)-category]]: non-trivial cells up to and including dimension $n$, and all of them allowed to be non-reversible. Actually, it is possible to go on to an $(n,n+1)$-category, or $(n+1)$-[[n-poset|poset]]; you can either consider than the $n$-cells are ordered, or else consider that there are irreversible $(n+1)$-cells which are indistinguishable. (Reversible indistinguishable $(n+1)$-cells are all identities and so might as well not exist.)

For low $n$ very explicit [[algebraic definition of higher category|algebraic models]] for $n$-categories are available but in their full generality become quickly rather untractable as $n$ increases: the series starts with [[bicategory]], [[tricategory]] and [[tetracategory]]. While bicategories have found plenty of applications, already the axioms of tricategories are rather involved and their theory mainly serves to produce the statement that there is a good [[semi-strict infinity-category|semi-strictifications]] of tricategories called [[Gray-category|Gray-categories]].

Indeed, there are many _strictified_ models for higher categories: combinatorial or algebraic models that sacrifice full generality for a better concrete control. Notably there is a useful combinatorial/algebraic model for [[strict ∞-category|strict ∞-categories]] which, while falling short, already goes a long way towards describing general higher categorical structures. In fact, by [[Simpson's conjecture]] every [[∞-category]] is equivalent to one that looks like a [[strict ∞-category]] except for possibly having weak unit laws.

The challenge of describing fully general [[∞-category|∞-categories]] is to achieve a combinatorial or algebraic control of all the higher composition rules of higher cells. One can distinguish roughly two orthogonal approaches to dealing with the problem:

in the [[algebraic definition of higher category]] an algebraic machinery is set up that allows to concretely handle the explicit _choices_ of composites of cells. Such machinery usually involves [[operad]]ic tools in one way or other. The most sophisticated definitions of this kind are the closely related [[Batanin ∞-category]] and [[Trimble n-category|Trimble ∞-category]].

On the other hand, in the [[geometric definition of higher category]] a combinatorial machinery is set up that allows to guarantee _existence_ of composites of cells. In the [[simplicial model for weak omega-categories|simplicial models for weak ∞-categories]] higher categories are characterized as [[simplicial set]]s with the extra [[stuff, structure, property|property]] that certain composites exist. The issue here is to characterize these existence laws correctly.

The basic example for such "existence laws" is the _Kan-filler condition_ that characterizes simplicial sets that are [[Kan complex]]es, the models for [[(∞,0)-category|(∞,0)-categories]]. More general higher categories are obtained by relaxing the Kan condition in just the right way. For instance by simply restricting the Kan-condition to just a certain sub-set of all cells yields the definition of simplicial sets that are called [[quasi-category|quasi-categories]]. These model [[(∞,1)-category|(∞,1)-categories]].

The right further relaxation of the (weak) Kan filler condition is more involved. An approach to capture this has been given by [[Dominic Verity]]'s definition of simplicial sets that are called [[complicial set]]s and [[weak complicial set]]s.

One expects that every algebraic definition of higher categories admits a construction called a [[nerve]] that maps it to a [[simplicial set]] that would constitute the corresponding geometric model.

Another approach to handle the geometric definition of higher categories is a recursive one that uses $n$-fold simplicial sets. This is based on the notion of [[complete Segal space]], which is essentially a variation of the concept of [[quasi-category]]. Its advantage is that its definition may be applied recursively to yield the notion of [[n-fold complete Segal space]]s. These model [[(∞,n)-category|(∞,n)-categories]] for finite $n$.

Finally, a large supply of further models exists for [[(∞,1)-category|(∞,1)-categories]] in terms of [[enriched category theory]]. [[simplicial model category|Simplicially enriched model categories]] are a highly-developed toolkit for handling [[presentable (infinity,1)-category|presentable (∞,1)-categories]]. [[pretriangulated dg-category|Pretriangulated dg-enriched categories]] and [[A-infinity category|A-∞ categories]] are a comparably highly developed toolkit for handling [[stable (∞,1)-category|stable (∞,1)-categories]].

## Basic concepts

The basic concept on which higher category theory is built is the notion of **[[k-morphism]]** for all $k \in \mathbb{N}$, equipped with a notion of composition, such that **[[coherence law]]s** are satisfied.

This is what it's all about.

## Basic constructions

### Higher presheaves

* [[higher topos theory]]

### Higher universal constructions

* [[2-limit]]

* [[adjoint (∞,1)-functor|(∞,1)-adjunction]]

* [[(∞,1)-Kan extension]]

  * [[limit in a quasi-category|(∞,1)-limit]]

* [[(∞,1)-Grothendieck construction]]

## Basic theorems

* [[homotopy hypothesis]]-theorem

* [[delooping hypothesis]]-theorem

* [[periodic table]]

* [[stabilization hypothesis]]-theorem

* [[michaelshulman:exactness hypothesis]]

* [[holographic principle of higher category theory|holographic principle]]

## Idea of relation to other perspectives on higher structures

As outlined in the introduction to this article, one motivation for higher category theory comes from the relation of [[infinity-groupoid|$\infty$-groupoids]] and [[topological spaces]] via the [[homotopy hypothesis]]. One can interpret the equivalence stated by the homotopy hypothesis as saying that both $\infty$-groupoids and topological spaces are models of the *same* kind of higher structures. Concretely, the higher structures in this example may be called [[homotopy type|homotopy types]].

Similarly, generalizing the homotopy hypothesis, higher categories and [[directed space|higher directed spaces]] should model the same kind of higher structures: [[directed homotopy type|directed homotopy types]]. The difference between higher categories and directed spaces is that the latter notion (like topological spaces) usually relies on the continuum $\mathbb{R}$ in its definition---as such, we can call directed spaces a 'continuum' (or 'topological') model of directed homotopy types. In contrast, higher categories usually need not rely on the continuum but are defined in combinatorial terms (as 'sets of higher morphisms' etc.)---as such, we may call them 'combinatorial' models of higher structures.

One can argue that there is also a third pole of models of higher structures, namely 'geometric' models (here, 'geometric' refers to definitions in terms of [[stratified space|stratified manifolds]]; this is different from the earlier '[[geometric definition of higher categories|geometric composition paradigm]] in combinatorial structures'). The argument is described, for instance, on the [$n$-Category Café](https://golem.ph.utexas.edu/category/2023/03/an_invitation_to_geometric_hig.html). It underlies the idea of applying higher category theory to cobordism theory, as described in the next section. It also leads to notions of [[geometric n-category|geometric higher categories]].

## Applications

See

* [[applications of (higher) category theory]].

### Extended cobordisms

One major application of higher category theory and one of the driving forces in developing it has been [[FQFT|extended functorial quantum field theory]]. This has recently led to what may become one of the central theorems of higher category theory, the proof of the [[cobordism hypothesis]]. This roughly characterizes the [[(∞,n)-category of cobordisms]] $Bord_n$ as the free [[(∞,n)-category]] with duals on a single generator.

## Models

There are many different _models_ for bringing the abstract notion of higher category onto paper.

* [[(n × k)-category]]

* [[n-fold category]]

* [[(n,r)-category]]

  * [[Theta-space]]

  * [[∞-category]]/[[∞-category]]

  * [[(∞,n)-category]]

    * [[n-fold complete Segal space]]

  * [[(∞,2)-category]]

  * [[(∞,1)-category]]

    * [[quasi-category]]

      * [[algebraic quasi-category]]

    * [[simplicially enriched category]]

    * [[complete Segal space]]

    * [[model category]]

    * [[internal category in homotopy type theory]]

  * [[(∞,0)-category]]/[[∞-groupoid]]

    * [[Kan complex]]

      * [[algebraic Kan complex]]

      * [[simplicial T-complex]]

  * [[n-category]] = (n,n)-category

    * [[2-category]], [[(2,1)-category]]

    * [[1-category]]

    * [[0-category]]

    * [[(-1)-category]]

    * [[(-2)-category]]

  * [[n-poset]] = (n−1,n)-category

    * [[2-poset]]

  * [[n-groupoid]] = (n,0)-category

    * [[2-groupoid]], [[3-groupoid]]

* [[categorification]]/[[decategorification]]

* [[geometric definition of higher category]]

  * [[Kan complex]]

  * [[quasi-category]]

  * [[simplicial model for weak ∞-categories]]

    * [[complicial set]]

    * [[weak complicial set]]

* [[algebraic definition of higher category]]

  * [[bicategory]]

  * [[bigroupoid]]

  * [[tricategory]]

  * [[tetracategory]]

  * [[strict ∞-category]]

  * [[Batanin ∞-category]]

  * [[Trimble n-category|Trimble ∞-category]]

  * [[Grothendieck-Maltsiniotis ∞-categories]]

* [[stable homotopy theory]]

  * [[symmetric monoidal category]]

  * [[symmetric monoidal (∞,1)-category]]

  * [[stable (∞,1)-category]]

    * [[dg-category]]

    * [[A-∞ category]]

    * [[triangulated category]]

### Via 1-categorical presentations

* [[homotopical category]]

* [[model category|model category theory]]

* [[enriched category theory]]

## Related concepts

* [[higher structures]]

* [[directed homotopy type theory]]

  * [[simplicial homotopy type theory]]

  * [[opetopic type theory]]

[[!include table of category theories]]

## References

For a very gentle introduction to higher category theory, try [The Tale of n-Categories](http://math.ucr.edu/home/baez/week73.html#tale), which begins in "week73" of This Week's Finds and goes on from there ...; keep clicking the links.

For a slightly more formal but still pathetically easy introduction, try:

* [[John Baez]], [An Introduction to n-Categories](http://arxiv.org/abs/q-alg/9705009), in
7th Conference on Category Theory and Computer Science, eds.
E. Moggi and G. Rosolini, Springer Lecture Notes in
Computer Science vol. 1290, Springer, Berlin, 1997.

For a free introductory text on $n$-categories that's *full of pictures*, try this:

* [[Eugenia Cheng]] and [[Aaron Lauda]], [Higher-Dimensional Categories: An Illustrated Guidebook](http://www.cheng.staff.shef.ac.uk/guidebook/).

[[Tom Leinster]] has written about "comparative $\infty$-categoriology" (to [borrow](http://golem.ph.utexas.edu/category/2008/01/comparative_smootheology.html) a term):

* Tom Leinster, _A Survey of Definitions of n-Category_ ([arXiv](http://arxiv.org/abs/math.CT/0107188))

* Tom Leinster, _Higher Operads, Higher Categories_ ([arXiv](http://arxiv.org/abs/math/0305049))

A grand picture of the theory of higher categories is drawn in

* [[Carlos Simpson]], _[[Homotopy Theory of Higher Categories]]_ ([pdf](http://hal.archives-ouvertes.fr/docs/00/44/98/26/PDF/main.pdf))

Another collection of discussions of definitions of higher categories is given at

* [[John Baez]], [[Peter May]] [[Approaching Higher Category Theory]]

A brief useful survey of approaches to the definition of higher categories is provided by the set of slides

* [[Andre Joyal]], [[Tim Porter]], [[Peter May]], _Weak categories_ ([pdf](https://web.archive.org/web/20150326110254/http://www.ima.umn.edu/talks/workshops/SP6.7-18.04/may/PorterMay.pdf))

The theory of [[quasi-categories]] as [[(∞,1)-categories]] has reached a point where it is well developed and being applied to a wealth of problems with

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ ([arXiv](http://arxiv.org/abs/math.CT/0608040))

There's a lot more to add here, even if we restrict ourselves to very general texts.  (More specialized stuff should go under more specialized subcategories!)

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higher category theory - contents

**[[higher category theory]]**

* [[category theory]]
* [[homotopy theory]]

## Basic concepts

* [[k-morphism]], [[coherence]]
* [[looping and delooping]]
* [[stabilization|looping and suspension]]

## Basic theorems

* [[homotopy hypothesis]]-theorem
* [[delooping hypothesis]]-theorem
* [[periodic table]]
* [[stabilization hypothesis]]-theorem
* [[michaelshulman:exactness hypothesis]]
* [[holographic principle of higher category theory|holographic principle]]

## Applications

* [[applications of (higher) category theory]]
* [[higher category theory and physics]]

## Models

* [[(n,r)-category]]
  * [[Theta-space]]
  * [[∞-category]]/[[∞-category]]
  * [[(∞,n)-category]]
    * [[n-fold complete Segal space]]
  * [[(∞,2)-category]]
  * [[(∞,1)-category]]
    * [[quasi-category]]
      * [[algebraic quasi-category]]
    * [[simplicially enriched category]]
    * [[complete Segal space]]
    * [[model category]]
  * [[(∞,0)-category]]/[[∞-groupoid]]
    * [[Kan complex]]
      * [[algebraic Kan complex]]
      * [[simplicial T-complex]]
  * [[n-category]] = (n,n)-category
    * [[2-category]], [[(2,1)-category]]
    * [[1-category]]
    * [[0-category]]
    * [[(-1)-category]]
    * [[(-2)-category]]
  * [[n-poset]] = [[n-poset|(n-1,n)-category]]
    * [[poset]] = [[(0,1)-category]]
    * [[2-poset]] = [[(1,2)-category]]
  * [[n-groupoid]] = (n,0)-category
    * [[2-groupoid]], [[3-groupoid]]
* [[categorification]]/[[decategorification]]
* [[geometric definition of higher category]]
  * [[Kan complex]]
  * [[quasi-category]]
  * [[simplicial model for weak ∞-categories]]
    * [[complicial set]]
    * [[weak complicial set]]
* [[algebraic definition of higher category]]
  * [[bicategory]]
  * [[bigroupoid]]
  * [[tricategory]]
  * [[tetracategory]]
  * [[strict ∞-category]]
  * [[Batanin ∞-category]]
  * [[Trimble n-category|Trimble ∞-category]]
  * [[Grothendieck-Maltsiniotis ∞-categories]]
* [[stable homotopy theory]]
  * [[symmetric monoidal category]]
  * [[symmetric monoidal (∞,1)-category]]
  * [[stable (∞,1)-category]]
    * [[dg-category]]
    * [[A-∞ category]]
    * [[triangulated category]]

## Morphisms

* [[k-morphism]]
  * [[2-morphism]]
* [[transfor]]
  * [[natural transformation]]
  * [[modification]]

## Functors

* [[functor]]
* [[2-functor]]
  * [[pseudofunctor]]
  * [[lax functor]]
* [[(∞,1)-functor]]

## Universal constructions

* [[2-limit]]
* [[adjoint (∞,1)-functor|(∞,1)-adjunction]]
* [[(∞,1)-Kan extension]]
  * [[limit in a quasi-category|(∞,1)-limit]]
* [[(∞,1)-Grothendieck construction]]

## Extra properties and structure

* [[cosmic cube]]
  * [[k-tuply monoidal n-category]]
  * [[strict ∞-category]], [[strict ∞-groupoid]]
* [[stable (∞,1)-category]]
* [[(∞,1)-topos]]

## 1-categorical presentations

* [[homotopical category]]
* [[model category|model category theory]]
* [[enriched category theory]]


higher category theory and physics

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###Context###
#### Physics
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[[!include physicscontents]]
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#### Category theory
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[[!include category theory - contents]]
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#### Higher category theory
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[[!include higher category theory - contents]]
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This entry lists aspects of [[fundamental physics]] from the [[nPOV]]: its description in terms of [[category theory]] and [[higher category theory]]. For a more coherent exposition, starting with introduction of the very basics, see also at _[[geometry of physics]]_.

#Contents#
* table of contents goes here
{:toc}

## The basic structures {#BasicStructures}

We discuss the setting in which fundamental physics takes place.

### Dynamics in space {#Fundamentals}

Physics is _dynamics_ in _spaces_ .

[[higher topos theory|Higher topos theory]] provides the formalizations of this most fundamental aspect of physics.

* A general context for **[[space]]s** is a [[big topos|big]] [[(∞,1)-topos]] $\mathbf{H}$.

* A general context for **[[higher geometry|geometrical]] spaces** is a [[local (∞,1)-topos]].

* A general context for geometrical spaces and **processes** in these spaces is a [[cohesive (∞,1)-topos]].

An example of relevance for much of physics is the cohesive $(\infty,1)$-topos
$\mathbf{H} = $ [[∞LieGrpd]] of [[∞-Lie groupoid]]s. This contains

* [[smooth manifold]]s;

* [[orbifold]]s;

* [[diffeological space]]s;

* smooth [[path space]]s;

In its [[Cahiers topos]]-version it contains also

* [[infinitesimal space]]s;

* such as [[∞-Lie algebroid]]s.

* [[de Rham space]]s, smooth [[D-module]]s;

In its full [[derived geometry]]-version it contains also

* [[derived ∞-Lie algebroid]]s such as the [[BV-BRST complex]]es of [[gauge theories]].

Every $(\infty,1)$-topos comes with its intrinsic notion of [[cohomology]]. This encodes **kinematics** in physics. Such as [[orientation]]; [[spin structure]]s; [[string structure]]s, [[fivebrane structure]]s.

Every cohesive $(\infty,1)$-topos is in particular a [[locally ∞-connected (∞,1)-topos]]. For these their intrinsic cohomology refines to [[schreiber:differential cohomology in an (∞,1)-topos]] classifying [[connections on ∞-bundles]]. This encodes **dynamics** in physics: a connection on a principal $\infty$-bundle is a **[[gauge field]]** which exerts **[[force]]**s. Such as:

* the [[electromagnetic field]], [[Yang-Mills field]], the field of [[gravity]], of [[supergravity]], the [[Kalb-Ramond field]], the [[supergravity C-field]], the [[RR-field]].

In every such connected $(\infty,1)$-topos every [[characteristic class]] gives rise to its [[∞-Chern-Weil homomorphism]] that sends gauge fields encoded as [[connections on ∞-bundles|∞-connections]] on [[principal ∞-bundles]] to [[circle n-bundles with connection]]. Such as:

* the [[Chern-Simons circle 3-bundle]] known from the [[Green-Schwarz mechanism]] or the [[Chern-Simons circle 7-bundle]] known from [[dual heterotic string theory]].

Under the [[higher parallel transport]] of these circle $n$-bundles with connection, this assignment is the [[action functional]] for the [[schreiber:∞-Chern-Simons theory]] of the corresponding characteristic class. This includes as special cases various [[sigma-model]] [[quantum field theories]] such as:

* [[Dijkgraaf-Witten theory]], ordinary [[Chern-Simons theory]], all versions of [[AKSZ theory]], (Chern-Simons-)[[supergravity]], [[BF-theory]] coupled to [[topological Yang-Mills theory]] with cosmological constant.

These are all [[topological quantum field theories]]. But by the [[holographic principle of higher category theory]] we have that on their boundaries live non-topological theories:

* on the boundary of [[Chern-Simons theory]] sits the [[Wess-Zumino-Witten model]].

* on the boundary of the [[A-model]] sits the [[quantum mechanics]] dynamics of any classical [[symplectic manifold|symplectic]] [[phase space]] and on the boundary of the [[Poisson sigma-model]] that of a [[Poisson manifold]] phase space.

### Quantum dynamics

The [[quantum mechanics]] associated with such [[sigma-model]]s is the collection of data given by

* on each closed piece $\Sigma_{d-n}$ of worldvolume [[cobordism]] of codimension $n$ the [[n-vector space]] of [[state]]s of the system;

* on each piece with boundary $\partial \Sigma_{in} \to \Sigma \leftarrow \partial \Sigma_{out}$ a [[morphism]] between these $n$-vector spaces encoding the propagation of states;

* on each open subset $U \subset \Sigma$ the _algebra of [[observable]]s_ .

Two dual formalizations axiomatize this:

* [[AQFT]]/[[factorization algebra]]s: the assignment of algebras of observables is encoded in an [[(∞,1)-presheaf|(∞,n)-copresheaf]] of [[∞-algebra]]s on $\Sigma$ with suitable properties;

* [[FQFT]]: the assignment of spaces of states and propagators is encoded in an [[(∞,n)-functor]] on the [[(∞,n)-category of cobordisms]] (see [[cobordism hypothesis]]).

Well-understood examples of such quantum field theories include

* [[AQFT]]:

  * 2-dimensional [[conformal field theory]] by [[conformal net]]s and [[vertex operator algebra]]

  * [[topological chiral homology]]

* [[FQFT]]:

  * 2-dimensional [[conformal field theory]] (see [[FFRS-formalism]])

  * [[TCFT]]s ([[(∞,1)-category]]-formulations of 2d [[TQFT]]): the [[A-model]], the [[B-model]], their duality under [[homological mirror symmetry]]

### Quantization

By the above there is a fairly well-developed formalization of

1. background [[gauge field]]s and their [[sigma-model]] [[action functional]];

1. [[quantum field theories]].

The idea is that the former induce examples of the latter by a process called [[quantization]]. This is imagined to be given by a [[path integral]] over the action functional.

This step in full generality is not yet well understood formally. For a list of literature addressing this problem see [Literature on quantization](#LitOnQuantization).

But special aspects of quantization are quite well understood. See for instance

* [[deformation quantization]]

* [[geometric quantization]], [[multisymplectic geometry]]

* [[Hopf-algebraic renormalization]]

## The detailed structures

We look at some aspects of the [above](#BasicStructures) general abstract story in more detail.

### Gauge theory
  {#GaugeTheory}

#### Introduction
 {#GaugeTheoryIntroduction}

The discovery of [[gauge theory]]  is effectively the discovery of [[groupoid]]s in fundamental physics. The notion of [[gauge transformation]] is close to synonymous to the notion [[isomorphism]] and more generally to _[[equivalence in an (∞,1)-category]]_ .

From a modern point of view, the mathematical model for a [[gauge field]] in physics is a [[cocycle]] in (nonabelian) [[differential cohomology]]: a [[principal bundle]] with [[connection on a bundle|connection]] and its higher analogs. These naturally do not form just a set, but a [[groupoid]] and generally an [[∞-groupoid]], whose morphisms are gauge transformations, and higher morphisms are gauge-of-gauge transformations. The development of [[differential cohomology]] has to a fair extent been motivated by and influenced by its application to fundamental theoretical physics in general and [[gauge theory]] in particular.

Around 1850 Maxwell realized that the [[field strength]] of the [[electromagnetic field]] is modeled by what today we call a closed [[differential form|differential 2-form]] on [[spacetime]]. In the 1930s Dirac observed that more precisely this 2-form is the [[curvature]] 2-form of a [[circle group|U(1)]]-[[principal bundle]] with [[connection on a bundle|connection]], hence that the electromagnetic field is modeled
by what today is called a degree 2-cocycle in _[[ordinary differential cohomology]]_ .

Meanwhile, in 1915, Einstein had identified also the [[field strength]] of the [[gravity|field of gravity]] as the $\mathfrak{so}(d,1)$-valued curvature 2-form of the canonical [[orthogonal group|O(d,1)]]-[[principal bundle]] [[connection on a bundle|with connection]] on a $d+1$-dimensional [[spacetime]] [[Lorentzian manifold]]. This is a cocycle in differential [[nonabelian cohomology]]: in [[Chern-Weil theory]].

In the 1950s [[Yang-Mills theory]] identified the [[field strength]] of all the [[gauge field]]s in the [[standard model of particle physics]] as the $\mathfrak{u}(n)$-valued [[curvature]] 2-forms of [[unitary group|U(n)]]-[[principal bundle]]s [[connection on a bundle|with connection]]. This is again a cocycle in differential [[nonabelian cohomology]].

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**Entities of ordinary [[gauge theory]]**

[[Lie algebra]] $\mathfrak{g}$ with [[gauge group|gauge]] [[Lie group]] $G$ --   [[connection on a bundle|connection]] with values in $\mathfrak{g}$ on $G$-[[principal bundle]] over a [[smooth manifold]] $X$.

=--

It is noteworthy that already in this mathematical formulation of experimentally well-confirmed fundamental physics the seed of higher differential cohomology is hidden: Dirac had not only identified the [[electromagnetic field]] as a [[line bundle]] with connection, but he also correctly identified (rephrased in modern language) its underlying cohomological [[Chern class]] with the (physically hypothetical but formally inevitable) [[magnetic charge]] located in [[spacetime]]. But in order to make sense of this, he had to resort to removing the support of the magnetic charge density from the spacetime manifold, because Maxwell's equations imply that at the support of any magnetic charge the 2-form representing the [[field strength]] of the [[electromagnetic field]] is in fact _not_ closed and hence in particular _not_ the curvature 2-form of an ordinary connection on
an ordinary bundle.

In ([Freed](#Freed)) this old argument was improved by refining the model for the electromagnetic field one more step:
[[Dan Freed]] notices that the charge current 3-form is itself to be regarded as a curvature, but for a connection on a [[circle n-bundle with connection|circle 2-bundle with connection]] -- also called a [[bundle gerbe]] -- , which is a cocycle in degree 3 [[ordinary differential cohomology]]. Accordingly, the [[electromagnetic field]] is fundamentally not quite a [[line bundle]], but a [[twisted bundle]] with connection, with the twist being the [[magnetic charge]] 3-cocycle. Freed shows that this perspective is inevitable for understanding the [[quantum anomaly]] of the [[action functional]] for [[electromagnetism]] is the presence of magnetic charge.

In summary, the experimentally verified models, to date, of fundamental physics are based on the notion of (twisted) $U(n)$-principal bundles with connection for the [[Yang-Mills field]] and $O(d,1)$-principal bundles with connection for the description of [[gravity]], hence on nonabelian differential cohomology in degree 2 (possibly with a degree-3 twist).

In attempts to better understand the structure of these two theories and their interrelation, theoretical physicists were led to consider variations and generalizations of them that are known as [[supergravity]] and [[string theory]]. In these theories the notion of [[gauge field]] turns out to generalize: instead of just [[Lie algebra]]s, [[Lie group]]s and connections with values in these, one finds structures called [[Lie 2-algebra]]s, [[Lie 2-group]]s and the gauge fields themselves behave like generalized connections with values in these.

+-- {: .standout}

**Entities of 2-[[nLab:gauge theory]]**

[[Lie 2-algebra]] $\mathfrak{g}$ with gauge [[Lie 2-group]] $G$ --   [[connection on a 2-bundle]] with values in $\mathfrak{g}$ on $G$-[[principal 2-bundle]]/[[gerbe]] over an [[orbifold]] $X$.

=--

Notably the [[string theory|string]] is charged under a field called the _[[Kalb-Ramond field]]_ or _$B$-field_ which is modeled by a $\mathbf{B}U(1)$-[[principal 2-bundle]] with connection, where $\mathbf{B}U(1)$ is the [[nLab:Lie 2-group]] [[delooping]] of the [[circle group]]: the circle Lie 2-group. Its [[Lie 2-algebra]]
$\mathbf{B}\mathfrak{u}(1)$ is given by the [[differential crossed module]] $[\mathfrak{u}(1) \to 0]$ which has $\mathfrak{u}(1)$ shifted up by one in homological degree.

So far all these differential cocycles were known and understood mostly as concrete constructs, without making their abstract home in [[differential cohomology]] explicit. It is the next gauge field that made [[Dan Freed|Freed]] and [[Mike Hopkins|Hopkins]] propose ([FreedHopkins](#FreedHopkins), [Freed](#Freed)) that the theory of differential cohomology is generally the formalism that models [[gauge field]]s in physics:

The superstring is charged also under what is called the [[RR-field]], a  [[gauge field]] modeled by cocycles in [[differential K-theory]]. In even degrees we may think of this as a differential cocycle whose [[curvature]] form has coefficients in the [[∞-Lie algebra]]
$\oplus_{n=0}^\infty \mathbf{B}^{2 n} \mathfrak{u}(1)$. Here $b^{2n} \mathfrak{u}(1)$ is the abelian [[L-∞-algebra|2n-Lie algebra]] whose underlying complex is concentrated in degree $2 n$ on $\mathbb{R}$.

So fully generally, one finds [[∞-Lie algebra]]s, [[∞-Lie group]]s and gauge fields behaving like connections with values in these.

+-- {: .standout}

**Entities of general [[gauge theory]]**

[[∞-Lie algebra]] $\mathfrak{g}$ with gauge [[∞-Lie group]] $G$ --  [[connection on an ∞-bundle]] with values in $\mathfrak{g}$ on $G$-[[principal ∞-bundle]] over an [[∞-Lie groupoid]] $X$.

=--

The [[curvature characteristic form]]s / [[Chern character]]s in the abelian formulation of differential cohomology take values in abelian [[∞-Lie algebra]]s and are therefore effectively nothing but differential forms with values in a complex of vector spaces, but more generally in [[∞-Chern-Weil theory]] on nonabelian [[principal ∞-bundle]]s, the [[curvature]]s forms themselves take values in general [[∞-Lie algebra]]s, such as the [[string Lie 2-algebra]], the [[supergravity Lie 3-algebra]] and the [[fivebrane Lie 6-algebra]].

Apart from generalizing the notion of gauge [[Lie group]]s to [[Lie 2-group]]s and further, structural considerations in fundamental physics also led theoretical physicists to consider models for [[spacetime]] that are more general than than the notion of a [[smooth manifold]]. In [[string theory]] [[spacetime]] is allowed to be more generally an [[orbifold]] or a generalization thereof, such as an [[orientifold]]. The natural mathematical model for these generalized spaces are [[Lie groupoid]]s or, essentially equivalently, _[[differentiable stack]]s_ .

It is noteworthy that the notions of generalized gauge groups and the generalized spacetime models encountered this way have a natural common context: all of these are examples of [[smooth ∞-groupoid]]s.

There is a natural mathematical concept that serves to describe contexts of such [[nLab:space|generalized space]]s: a [[nLab:gros topos|gros]] [[(∞,1)-topos]]. The notion of [[schreiber:differential cohomology in an (∞,1)-topos]] provides a unifying perspective on the mathematical structure encoding the generalized [[gauge field]]s and generalized [[spacetime]] models encountered in modern theoretical physics in such a general context.

#### Classes of examples {#ClassesOfExamples}

We discuss classes of examples of gauge theories that have been considered.
For all of these the configuration space is a space of
[[connections on ∞-bundles]] $\{\nabla\}$ over [[spacetime]] $X$ of sorts, which one might take to be the
defining property of a gauge theory. But there are different types of
[[action functional]]s on these configuration spaces.

##### Generalized Yang-Mills theory

In [[Yang-Mills theory]] the [[action functional]] is of the form

$$
  \nabla \mapsto \int_X \langle F_\nabla \wedge \star F_\nabla \rangle
  \,,
$$

where

* $F_\nabla$ is the [[curvature]] [[differential form]],

* "$\star$" the [[Hodge star]] operator with respect to a fixed
([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian metric]]-structure
on $X$

* and $\langle -\rangle$ some [[invariant polynomial]].

This is the original notion of _gauge theory_
and might be taken to be the strict sense of the term.

* For [[gauge group]] $G = U(1)$ the [[circle group]] this is [[electromagnetism]];

* For gauge group a general nonabelian [[Lie group]] $G$ this is [[Yang-Mills theory]] proper.

* Specifically for $G$ a discrete quotient of $SU(3) \times SU(2) \times U(1)$ this is the gauge-field part of the [[standard model of particle physics]].

* For structure group $G = \mathbf{B}U(1)$ the circle 2-group this yields the [[Kalb-Ramond field]]

* For structure group $G = \mathbf{B}^2U(1)$ the circle 3-group this yields the [[supergravity C-field]].

* For structure group the [[K-theory spectrum]] we get [[differential K-theory]] describing the [[RR-field]].

A [[nonabelian cohomology]] version of higher Yang-Mills theory -- replacing a [[connection on a bundle]] by a [[connection on a 2-bundle]] is expected to control certain 6-dimensional theories that compactify to ordinary Yang-Mills on the torus and thereby explain _[[S-duality]]_ . See there for more details on this.

##### $\infty$-Chern-Simons theory

In [[schreiber:∞-Chern-Simons theory]] the [[action functional]] is of the form

$$
  \nabla \mapsto \int_X CS(\nabla)
  \,,
$$

* where $CS(-)$ is a [[Chern-Simons element]] for an [[invariant polynomial]]
$\langle - \rangle$ on an [[∞-Lie algebroid]] $\mathfrak{a}$.

This includes ordinary [[Chern-Simons theory]] in the case that
$\mathfrak{a}$ is a [[semisimple Lie algebra]], but for general
$\mathfrak{a}$ it subsumes a wide variety of types of
[[TQFT]]s that are often counted as of different type than Chern-Simons
theory, such as [[BF-theory]] and [[AKSZ theory]].

The action functional of [[∞-Chern-Simons theory]] stands out by the fact
that it arises by [[category theory|general abstract]] construction:

* the underlying Lagrangian $CS(\nabla)$
is nothing but the [[∞-Chern-Weil homomorphism]]

  $$
    CS(-) : \mathbf{B}G_{diff} \to \mathbf{B}^n U(1)_{diff}
  $$

  induced by the given [[invariant polynomial]] $\langle - \rangle$. This sends gauge fields in the form of $G$-valued [[connections on ∞-bundles]] to the [[circle n-bundle with connection]] whose [[higher parallel transport]] is given by the Lagrangian;

* the integral over $CS(\nabla)$ is induced by postcomposition with the [[truncated|trunction]] morphism

  $$
    \mathbf{H}(\Sigma, \mathbf{B}G_{conn})
      \stackrel{CS(-)}{\to}
     \mathbf{H}(\Sigma,\mathbf{B}^n U(1)_{diff})
     \simeq
     \infty Grpd(\Pi(\Sigma), B^n U(1))
    \stackrel{\int_\Sigma}{\to}
      \tau_{n-dim \Sigma}
      \infty Grpd(\Pi(\Sigma), B^n U(1)) \simeq B^{n-dim \Sigma} U(1)
    \,.
  $$

* For $\mathfrak{a}$ a [[semisimple Lie algebra]] equipped with its [[Killing form]] [[invariant polynomial]] we have that $CS(-)$ is the ordinary [[Chern-Simons element]] and $\int_X CS(\nabla)$ the ordinary [[Chern-Simons theory]] action functional. This may be understood as  the [[higher parallel transport]] of a [[Chern-Simons circle 3-bundle]] with connection.

* For $\mathfrak{a}$ a [[semisimple Lie algebra]] equipped with its degree 8 [[invariant polynomial]] this yields 7-dimensional, whose action functional may be understood as  the [[higher parallel transport]] of a [[Chern-Simons circle 7-bundle]] with connection.

* For $\mathfrak{a}$ a [[symplectic Lie n-algebroid]] equipped with its canonical [[invariant polynomial]] $\omega$ of degree $n+2$ $\int_X CS_\omega(-)$ is the actional functional of [[AKSZ theory]]:

  * for $n = 1$ this is the [[Poisson sigma-model]];

  * for $n = 2$ this is the [[Courant sigma-model]].

* For $\mathfrak{a}$ a [[Lie 2-algebra]] equipped with the [[Killing form]] invariant polynomial, $\int_X CS(-)$ is the action functional of [[BF-theory]] coupled to [[topological Yang-Mills theory]] with a [[cosmological constant]].

##### Gravity

The configuration spaces of [[gravity]] and [[supergravity]] may be identified with spaces of [[connections on ∞-bundles]] with gauge group a variant of the [[Poincare group]]. This parameterization of the configuration space of gravity is known as the [[first order formulation of gravity]], to be contrasted with a formulation explicitly over a space of [[pseudo-Riemannian manifold]]s.

* The [[special orthogonal group]] $S O$-component of the connection in this case is called the [[spin connection]];

* the translation group-component of $\nabla$ is called the [[vielbein]].

For the ordinary Poincare group this yields the [[Palatini action]] expression for the standard [[Einstein-Hilbert action]] of [[general relativity]].

In contrast to the $\infty$-Chern-Simons theory discussed above, the general abstract nature, if any, of the action functional for gravity remains somewhat inconclusive and subject of a plethora of speculations. If one passes from connections to their associated [[Dirac operator]]s and interprets these as parts of a [[spectral triple]] there is the [[spectral action]] functional on the space of spectral triples. This we discuss in more detail [below](SpecStandModAndGravity).

There are various higher [[group extension]]s of the Poincare group and the orthonormal group that lead accordingly to higher order variations of gravity.

* lifting $S O$-connections through the smooth Whitehead tower

  $$
    \cdots \to Fivebrane \to String \to Spin \to S O \to O
  $$

  yields, in order of appearance,

  * [[spin structure]]s and [[spin group]]-[[principal bundle]]s with [[connection on a bundle|connection]].

    This lift is necessary to cancel the [[quantum anomaly]] of
    spinning particles coupled to gravity;

  * [[string structure]]s and [[string 2-group]]-[[principal 2-bundle]]s
    with [[connection on a 2-bundle|2-connection]]

    This lift is necessary to cancel the quantum anomaly of heterotic super[[string theory|string]]s;

  * [[fivebrane structure]]s and [[fivebrane 6-group]]-[[principal ∞-bundle|principal 6-bundle]] with [[connection on an ∞-bundle|6-connection]]

  This lift is necessary to cancel the quantum anomaly of [[brane|super 5-brane]]s.

* lifting to [[super Lie group]] extensions of $SO$ yields
  action functionals for [[supergravity]]

  The [[D'Auria-Fre formulation of supergravity]] explicitly describes higher dimensional supergravity theories as gauge theories with higher super Lie structure groups

  * 11-dimensional supergravity is a gauge theory for the [[supergravity Lie 6-algebra]].

### Phenomenological models: the standard model and gravity

Theoretical physics consists of two parts: theory and models, laws and initial conditions, axioms and phenomenology.

For instance the theory called [[general relativity]] describes the classical dynamics of [[gravity]], but does not predict the value of the [[cosmological constant]]. Rather, for each choice of the latter does the theory predict a certain dynamics the large-scale universe.

The theory that describes the fundamental forces and particles except gravity is [[Yang-Mills theory]]. This, too, does not predict the fundamental particle species seen in experiments, but for a correct choice and identification of these, the theory does predict the dynamics of these particles, as observed in accelerators.

The total collection of these choices of fundamental particles that are observed in experiments is called the [[standard model of particle physics]]. It consists basically of

1. a choice of [[gauge group]] $G$, such that all observed [[gauge field]]s are components of a [[connection on a bundle|connection]] on a $G$-[[principal bundle]];

1. a choice of linear [[representation]] $\rho$ of $G$, such that all observed [[fermion]] fields are components of [[section]]s of a $\rho$-[[associated bundle]].

What precisely the "standard" model of particle physics is changes slightly over time, as new experimental insights are gained. Its particles were added item-by-item as they were discovered. More recently the mass of the particles called [[neutrino]]s, which was originally thought to be precisely 0, was measured to be very small, but non-vanishing.

The standard model as far as understood today exhibits a curious mixture of pattern and irregularity. This seems to suggest that it ought to have a more fundamental description in terms of a conceptually simpler structure out of which these patterns with their irregularities emerge. Since also the force of [[gravity]] is not presently included in the [[quantization]] of the standard model, it may seem plausible that this underlying structure is related to [[quantum gravity]].

We discuss in the following some of the proposals that have been suggested for how to formalize this situation.

#### Spectral standard model and gravity {#SpecStandModAndGravity}

A fundamental [[relativistic particle]] is technically a 1-dimensional [[sigma-model]] [[QFT]] on 1-[[dimension]]al [[cobordism]]s with target the [[spacetime]] $X$ that it propagates in. Since both [[gravity]] as well as [[Yang-Mills field]]s are encoded in [[connection on a bundle|connections]] it is plausible to assume that the only background field on $X$ that the particle couples to is a connection $\nabla_\rho$ on a $\rho$-[[associated bundle]] over a $G$-[[principal bundle]].

Here

* roughly every [[semisimple Lie algebra]] [[direct sum|summand]] in $\mathfrak{g} = Lie(G)$ is one [[gauge field]] -- a bosonic field;

* every [[irreducible representation]] that $\rho$ decomposes into is one matter particle species -- a [[spinors in Yang-Mills theory|fermion field]].

This way a single $\sigma$-model may encode a rich multiple particle content and we shall speak of a single [[superparticle]] with different _excitations_ or _modes_ .

An early proposal for a single unified connection $\nabla$ that would subsume both gravity as well as Yang-Mills forces in a phenomenologically realistic way is the [[Kaluza-Klein mechanism]]. This assumes a single [[Levi-Civita connection]] but on a [[pseudo-Riemannian manifold]] $X$ which is locally of the [[product]] form $X_4 \times F_{d}$ with $X_4$ a 4-[[dimension]]al pseudo-Riemannian manifold and $F_d$ a $d$-dimensional [[Riemannian manifold]] of very small Riemannian volume. As described in more detail at [[Kaluza-Klein mechanism]], this makes the [[isometry]] groups of $F_d$ appear as extra [[gauge group]] factors as seen on $X_4$. As also described in more detail there, while this Ansatz does reproduce the correct general form of gravity coupled to Yang-Mills forces, in its original form it does also have some phenomenologically unviable aspects.

It was observed by [[Alain Connes]] and collaborators that the Kaluza-Klein mechanism works better when used not [[internalization|internal]] to the [[category]] [[Diff]] of [[smooth manifolds]] but in context for more general [[higher geometry|geometry]]: [[noncommutative geometry]]. This is the content of the _[[Connes-Lott-Chamseddine model]]_.

This more general geometry turns out to model exactly the most general $\sigma$-model backgrounds for a 1-dimensional [[FQFT]]: because such is algebraically specified by

1. the ([[Hilbert space|Hilbert]]) space $\mathcal{H}$ of [[state]]s that it assigns to the point;

1. An [[associative algebra]] $A \hookrightarrow \mathcal{H}$ whose [[monoid|multiplication operation]] $A\otimes A \to A$ is the [[operator product expansion|operator product]] assigned to the trivalent interaction vertex

   $$
     \array{
       \bullet
       \\
       & \searrow
       \\
       && \bullet & \to
       \\
       & \nearrow
       \\
       \bullet
     }
   $$

1. A [[Dirac operator]] $D$ whose [[Dyson formula]] exponential $\exp(t D^2 + \theta D)$ is assigned to a piece of 1-dimensional cobordism of superlength $(t, \theta)$.

This data is that of [[spectral triple]], which is well known to enocode Riemannian noncommutative geometry (rather: _spectral geometry_ ). It is therefore natural to search for a Kaluza-Klein ansatz in spectral geometry that would produce the standard model context.

A very detailed such construction was given by [[Alain Connes]] (see the [references below](#FunctorialSpectral)).

It turns out that the realistic model has $K$-theory dimension $D = 4+6$.

By the result at [[(1,1)-dimensional Euclidean field theories and K-theory]] this $K$-theory dimension is precisely the intrinsic dimension of target space as seen by the [[superparticle]]. Moreover, $D = 4+6$ is precisely the dimension for which the 2-dimensional super-[[CFT]] [[sigma-model]] is critical and hence allows to lift the 1-dimensional superparticle described here to [[string theory]].

This means that Connes' [[spectral triple]] whose particle spectrum reproduces the [[standard model of particle physics]] has a chance of being the point particle limit or [[decategorification]] of the kind of [[2-spectral triple]] -- a 2-dimensional [[CFT|superconformal field theory]] -- of the kind that is considered in [[string theory]]. If so the lift of Connes' model to the corresponding element in the [[moduli space]] of 2-spectral triples called the [[landscape of string theory vacua]] might provide, via the second quantization of the latter, a (perturbative) quantization of the [[spectral action]] of the former.

## Related introductions
  {#RelatedIntroductions}

* [[higher structures]]

* [[fiber bundles in physics]]

* [[motivation for sheaves, cohomology and higher stacks]]

* [[string theory FAQ]]

* [[motivation for higher differential geometry]]

* [[applications of (higher) category theory]]

* [[motivation for cohesion]]

* [[Hilbert's sixth problem]]

* [[motives in physics]]

* [[model theory and physics]]

* [[L-infinity algebras in physics]]

## References

We list various references related to higher category  theory and fundamental physics.

A discussion of an axiomatization of basic concepts of [[gauge theory|gauge]] [[quantum field theory]] in [[cohesive homotopy type theory]] is in

* [[Urs Schreiber]], [[Mike Shulman]], _[[schreiber:Quantum gauge field theory in Cohesive homotopy type theory]]_
 {#SchreiberShulman}

For general (formal) accounts of physics see also the references at [[books and reviews in mathematical physics]], such as

* [[Frédéric Paugam]], _[Towards the mathematics of quantum field theory](http://people.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf)_

### Introductions to category theory in physics {#CatsinPhysics}

The translation between the axioms of [[functorial quantum field theory]] and [[path integral]] imagery is explained in

* [[Edward Witten]], Section 1 of: _Geometric Langlands From Six Dimensions_ ([arXiv:0905.2720](https://arxiv.org/abs/0905.2720))

In

* [[Bob Coecke]], _Introducing categories to the practicing physicist_ ([arXiv:0808.1032](http://arxiv.org/abs/0808.1032))

* [[Bob Coecke]], _Categories for the practising physicist_ ([arXiv:0905.3010](http://arxiv.org/abs/0905.3010))

the authors try to motivate and introduce some basic concepts of [[category theory]] for an audience familiar with standard physics and in particular with [[quantum mechanics]]. The article focuses towards the end on [[monoidal categories]], their description in terms of [[string diagram]]s and [[quantum mechanics in terms of dagger-compact categories]].

More details on the use of string diagrams in [[dagger-categories]] for the description of quantum mechanics is is

* [[Bob Coecke]], _Kindergarten quantum mechanics_ ([arXiv:quant-ph/0510032](http://arxiv.org/abs/quant-ph/0510032))

* [[Bob Coecke]], _Quantum Picturalism_ ([arXiv:0908.1787](http://arxiv.org/abs/0908.1787))

A similar introduction to the relation between quantum mechanics and [[monoidal categories]], but more from the perspective of [[FQFT]] is in

* [[John Baez]], _Quantum quandaries_ ([arXiv:quant-ph/0404040](http://arxiv.org/abs/quant-ph/0404040))

### Introductions to higher category theory and physics

A historical introduction to some aspects of [[n-categories]] (for low $n$) in physics can be found here:

* [[John Baez]] and [[Aaron Lauda]], _A prehistory of $n$-categorical physics_ ([pdf](http://math.ucr.edu/home/baez/history.pdf)), to appear in _Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World_, ed. Hans Halvorson.

A historical introduction to applications in physics of more [[homotopy theory|homotopy theoretic]] [[higher category theory]] (revolving around [[BV-BRST formalism]]) is in:

* [[Jim Stasheff]], _[[A Survey of Cohomological Physics]]_

The following book-to-be aims to give picture of the present state of the art of describing the category-theoretic structure of the universe, as far as fundamental physics is concerned

* [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_

Amplifying the ambition towards [[Hilbert's sixth problem]]:

* {#MartinsBiezuner20} [[Yuri Ximenes Martins]], [[Rodney Josué Biezuner]], _Towards An Approach to Hilbert's Sixth Problem: A Brief Review_, 2020 ([hal:02909681](https://hal.archives-ouvertes.fr/hal-02909681), [pdf](https://hal.archives-ouvertes.fr/hal-02909681/document))

### Monographs on cohomology and higher gauge theory

An unusually comprehensive collection of detailed discussion of bundles, higher bundles, cohomology, characteristic classes, etc. with an eye on applications in physics is

* [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurco]], Martin Schottenloher,  _Basic bundle theory and K-cohomology invariants_ , Springer Lecture Notes in Physics 726, 2008 ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf))

### On formal quantum field theory

There are two axiomatizations of [[QFT]]: [[FQFT]] and [[AQFT]].

* [FQFT references](#FQFTReferences)

* [AQFT references](#AQFTReferences)

#### FQFT {#FQFTReferences}

Kontsevich's 1994 ICM article, is one of the seminal papers of the 1990s. This paper invented the categorical side of mirror symmetry (homological mirror symmetry), discovered D-[[brane]]s (before physicists realized their role --- and directly inspiring many physicists) and the fact that they naturally form [[dg-category|dg-categories]] and $A_\infty$-[[A-infinity-category|category]], and thus led to a deluge of papers involving [[category theory]] and [[higher category theory]] in close relation with mathematical physics.

In particular the [[FQFT|Moore--Segal paper]] should be seen in the light of this development. On a similar note, roughly contemporary with Moore--Segal (which is [arXiv:math/0609042](http://www.arxiv.org/abs/math/0609042) though developed earlier) are the works of [[Kevin Costello]] on [[TCFT]] ([arXiv:math/0412149](http://www.arxiv.org/abs/math/0412149)) and Kontsevich--Soibelman ([arXiv:math/0606241](http://www.arxiv.org/abs/math/0606241) --- some of the results were lectured on in various places by Kontsevich in 2003 and in particular helped inspire Costello) proving a much stronger result, which is the [[TCFT]] (or equivalently differential graded or $A_\infty$-) version of the classification of open-closed 2d [[TQFT]]s. These papers were directly motivated by [[homological mirror symmetry]] and topological [[string theory]], and have greatly influenced work in areas such as [[string topology]] which you mention and the [[cobordism hypothesis]] ([[On the Classification of Topological Field Theories|Hopkins-Lurie]] started from Costello's paper and abstracted the argument, before the general argument in n-dimensions came around).

> (in parts taken from [here](http://golem.ph.utexas.edu/category/2009/07/a_prehistory_of_ncategorical_p_1.html#c025075))

For the moment see the references at [[FQFT]] for more.

##### On functorial spectral geometry {#FunctorialSpectral}

Literature on the relation between [[spectral triple]]s and 1-dimensional super-QFT is to appear shortly. Spectral background for the standard model have been considered here:

The first proposal for such compactifications, which already came very close to the standard model, is decades old by now.

* [[Alain Connes]],  [[John Lott]], _Particle Models And Noncommutative Geometry_ Nucl.Phys.Proc.Suppl.18B:29-47,1991 ([spires](http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+author+Connes+and+author+Lott&FORMAT=WWW&SEQUENCE=))

This is reviewed in

* Daniel Kastler, Thomas Schucker, _The Standard Model a la Connes-Lott_ [arXiv:hep-th/9412185](http://arxiv.org/abs/hep-th/9412185) .

As far as I can tell, at that time the Yang-Mills terms were stilll included by hand. The main point of the spectral approach was to realize that it could nicely explain the Higgs boson and its Yukawa coupling terms to the fermions. It did (and does) so by realizing the Higgs boson as an internal part of an ordinary minimally coupled connection 1-form - an internal gauge boson.

A few years later Connes apparently realized that also the gravitational and gauge kinetic Yang-Mills terms had an inherent operator-theoretic formulation, namely the [[spectral action]] principle.

This is indicated in

* [[Alain Connes]], _Gravity coupled with matter and foundation of non-commutative geometry_ [arXiv:hep-th/9603053](http://arxiv.org/abs/hep-th/9603053)

and fully formulated in

* [[Ali Chamseddine]], [[Alain Connes]], _The Spectral Action Principle_ [arXiv:hep-th/9606001](http://arxiv.org/abs/hep-th/9606001),

both of which start by presenting the general spectral idea and then working out how to realize the standard model in detail.

A useful discussion of the details of the realization of the standard model in this approach is

* C. P. Martin, Jose M. Gracia-Bondia, Joseph C. Varilly, _The Standard Model as a noncommutative geometry: the low energy regime_ ([arXiv:hep-th/9605001](http://arxiv.org/abs/hep-th/9605001)) .

This is in particular designed to take the ordinary physicst by the hand and introduce him or her gently to the operator-theoretic spectral description by motivating these by the structure of the standard model.

So already over ten years ago people had a pretty good idea that and how the standard model action has an elegant descritption as a spectral action.

The only problem was: this description was wrong. But only in the sense in which  ideas in physics tend to be wrong - not entirely wrong but not quite right.

Namely, instead of containing the fermionic particle content of the standard model, these spectral models produced four copies of every fermion in the standard model. A slight overkill.

By a remarkable synchronicity, it seems that there was no progress on this aspect for about ten years, and now two preprints appear almost simultaneously, presenting the solution:

* [[John Barrett]], _A Lorentzian version of the non-commutative geometry of the standard model of particle physics_ [arXiv:hep-th/0608221](http://arxiv.org/abs/hep-th/0608221).

* [[Alain Connes]], _Noncommutative Geometry and the standard model with neutrino mixing_ [arXiv:hep-th/0608226](http://arxiv.org/abs/hep-th/0608226)

And the modification needed to get this solution is rather tiny, just a small change in the real structure $J$ of the spectral triple from ten years ago.

A survey is also at

* [[Urs Schreiber]], _Connes on spectral geometry of the standard model IV_ ([blog entry](http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_3.html))

#### AQFT {#AQFTReferences}

For the moment see the references at [[AQFT]] for more.

### On the quantization procedure {#LitOnQuantization}

First sketches of the idea that [[path integral]] [[quantization]] may have a formalization in terms of [[higher category theory]] appeared in

* [[Dan Freed]], _[[Higher Algebraic Structures and Quantization]]_ ([arXiv:hep-th/9212115](http://arxiv.org/abs/hep-th/9212115))

and its companion

* [[Dan Freed]], _Quantum groups and path integrals_ ([arXiv:q-alg/9501025](http://arxiv.org/abs/q-alg/9501025))

More formal aspects along these lines appear in

* [[Simon Willerton]] _The twisted Drinfeld double of a finite group via gerbes and finite groupoids_ ([arXiv:math/0503266](http://arxiv.org/abs/math.QA/0503266))

An indication of a full formalization of what that may mean for discrete QFTs such as [[Dijkgraaf-Witten theory]] is in

* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]] _[[Topological Quantum Field Theories from Compact Lie Groups]]_

based on work in

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_

### On gauge fields and differential cohomology

The observation that generally [[differential cohomology]] is the correct formalism for describing [[gauge field]]s in physics originates around

* [[Dan Freed]], [[Mike Hopkins]], _On Ramond-Ramond fields and K-theory_
Journal of High Energy Physics 0005 (2000) 044 ([pdf](http://web.me.com/teichner/Math/Smooth_Cohomology_files/Freed-Hopkins.pdf))
{#FreedHopkins}

Motivated by this, the theory of differential cohomology itself was developed in

* [[Mike Hopkins]], I. Singer _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_
{#HopkinsSinger}

and shown to encode subtle [[quantum anomaly]]-cancellation phenomena in gauge theory. A systematic formal description of the nature of these quantum anomalies and further discussion of applications on differential cohomology in physics is in

* [[Dan Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_
{#Freed}

This article starts with an introduction on basic [[electromagnetism]] and points out that already there, in the presence of [[magnetic charge]], a careful analysis of [[quantum anomalies]] shows that there are higher gauge theoretic effects even in this familiar theory.

### On topos-theoretic formulations of physics
 {#RefsOnToposTheoreticPhysics}

#### Gros toposes

The observation that any formalization of physics ought to take place in a suitable [[topos]] has been promoted early on by [[Bill Lawvere]]. Lawvere started out as a student in [[continuum mechanics]] and his search for a formalization of physics made him end up being one of the most general abstract thinkers in [[category theory]].

Lawvere thought about [[classical physics]] governed, as it is since Newton, by the theory of [[differential equation]]s. He proposed that [[differential geometry]] is something that takes place inside a [[smooth topos]] -- a [[topos]] with a suitable notion of [[infinitesimal object]]s -- an approach now known as [[synthetic differential geometry]].

In

* [[Bill Lawvere]], _Categorical dynamics_, lecture (1967)

which is recalled in

* [[Bill Lawvere]], _Toposes of laws of motion_ , transcript of a talk in Montreal, Sept. 1997 ([pdf](http://www.acsu.buffalo.edu/~wlawvere/ToposMotion.pdf))

he sketched an outline both of synthetic differenial geometry in general and its application to the formulation of the [[differential equation]]s in classical dynamics.

In this spirit he to search for [[category theory|category theoretic]] formalizations of fundamental ( _metaphysical_  in the good sense of the word) ontological concepts. A collection of such formalizations is indicated in

* [[Bill Lawvere]], _Taking categories seriously_, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. ([pdf](http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf))

On [page 14](http://www.tac.mta.ca/tac/reprints/articles/8/tr8.pdf#page=14) there appears the hint of the general abstract definition of a [[locally connected topos]] as a general abstract [[gros topos]] context for geometric spaces. This idea was then refined to the concept of a [[cohesive topos]] in

* [[Bill Lawvere]], _Axiomatic cohesion_ Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

In this spirit a generalization of formalization of physics in [[(∞,1)-topos]] is discussed in

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

* [[Urs Schreiber]], _[[twisted smooth cohomology in string theory]]_, Lectures at _[ESI program on K-theory and quantum fields](http://maths-old.anu.edu.au/esi/2012/)_ (2012)
 {#SchreiberLectures}

* [[Urs Schreiber]], [[Mike Shulman]], _[[schreiber:Quantum gauge field theory in Cohesive homotopy type theory]]_ (2012)

#### Petit toposes

Independent of these developments is another proposal for a role of topos theory in [[quantum mechanics]] initiated by [[Chris Isham]]. The motivation here is the observation that the [[Kochen-Specker theorem]] -- which may be used to formalize the way in which [[quantum mechanics]] fails to secretly a theory of classical mechanics -- has a natural interpretation in terms of the [[internal logic]] of the [[presheaf topos]] on the category of commutative sub-algebras of a given bounded [[operator algebra]].

This observation led Isham to propose a fundamental role of these presheaf toposes in [[quantum mechanics]].

* Andread Doering, C.J. Isham, _A Topos Foundation for Theories of Physics_ I-IV ([arXiv](http://arxiv.org/abs/quant-ph/0703060))

This idea was further refined in

* [[Chris Heunen]], [[Nicolaas Landsman]], [[Bas Spitters]], _A topos for algebraic quantum theory_ ([arXiv:0709.4364](http://arxiv.org/abs/0709.4364))

  **Abstract**

  The aim of this paper is to relate [[AQFT|algebraic quantum mechanics]] to [[topos theory]], so as to construct new foundations for [[quantum logic]] and quantum spaces. Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a [[C-star-algebra]] of [[observable]]s $A$ induces a [[topos]] $T(A)$ in which the amalgamation of all of its commutative subalgebras comprises a single commutative $C^*$-algebra. According to the [[constructive mathematics|constructive]] [[Gelfand duality]] theorem of Banaschewski and Mulvey, the latter has an internal spectrum $S(A)$ in $T(A)$, which in our approach plays the role of a quantum [[phase space]] of the system. Thus we associate a [[locale]] (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a [[Heyting algebra]]) to a $C^*$-algebra (which is the noncommutative notion of a space). In this setting, [[state]]s on $A$ become [[probability measure]]s (more precisely, valuations) on $S(A)$, and self-adjoint elements of $A$ define continuous functions (more precisely, locale maps) from $S(A)$ to Scott's interval domain. Noting that open subsets of $S(A)$ correspond to propositions about the system, the pairing map that assigns a (generalized) [[truth value]] to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by $A$ is essentially turned into a classical theory, internal to the topos $T(A)$.

A refinement of this from [[quantum mechanics]] to [[AQFT]]-[[quantum field theory]] is in

* [[Joost Nuiten]], _[[schreiber:bachelor thesis Nuiten|Boohrification of local nets of observables]]_ .

See _[[Bohr topos]]_ for more.

### In phenomenology and experimental physics
 {#InPhenomenology}

In the context of [[phenomenology]], [[higher category theory]] is currently applied mostly for [[2-categories]] in the context of 2-dimensional [[TQFT]] (modelling phenomena in [[solid state physics]]) and [[CFT]] (modelling critical surface phenomena in [[statistical physics]]).

Specifically 2d [[extended quantum field theory]] has been applied to the solid state physics modeled by the [[Levin-Wen model]], see

* [[Alexei Kitaev]], [[Liang Kong]], _Levin-Wen models and tensor categories ([pdf])_

For more discussion of 2d [[CFT]] in terms of higher category theory see the references there and at [[FRS formalism]].

In $d = 3$ the [[quantum hall effect]] and related effects are described to some extent by [[Chern-Simons theory]], and Chern-Simons theory comes with a rich higher categorical framework. Notably there is now a fully [[extended TQFT]] description of Chern-Simons theory down to the point, hence in terms of [[(infinity,n)-functors|(infinity,3)-functors]], see [here](Topological+Quantum+Field+Theories+from+Compact+Lie+Groups#3dCSFullyExtended).

Closely related to this, several approaches at realizing [[quantum computing]] rely on [[TQFT]] methods and are treated with methods from higher category theory. For instance the notion of a [[blob n-category]] orginates in investigations of quantum computing.

Indeed, if one counts computing as "experimental" or "phenomenology" -- at least as related to the tangible world -- then category theory is ubiquituous, see at _[[computational trinitarianism]]_ for the general idea and further pointers. For instance the _[[relation between type theory and category theory]]_ (see there) which underlies the formal understanding of modern [[computer science]], is an [[equivalence of 2-categories]] in  [[2-category theory]]. More recently, computing in the guise of _[[homotopy type theory]]_ is closely related to [[(infinity,1)-category theory]]. This carries in it the prospect of serving as the very [[foundations]] of [[mathematics]] and [[computer science]]. And hence, in effect, also of theoretical physics (see [Schreiber-Shulman 2012](#SchreiberShulman) above).

[[!redirects n-categorical physics]]

[[!redirects Physics]]

[[!redirects theoretical physics]]


higher central extension

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{:toc}

## Idea

In [[higher algebra]] a _higher central extension_ of some algebraic object should be a higher analog of a [[central extension]], hence a [[higher extension]] by something in the higher analog of the [[center]]. What exactly "higher center" should mean is subject to some choices, see for instance _[[center of an ∞-group]]_.

But there is an easily identified sub-class of higher extensions that should definitely count as _higher central extensions_:

For $G$ a [[group]], an ordinary [[central extension]]

$$
  1 \to A \longrightarrow \widehat G \longrightarrow G \to 1
$$

may equivalently be thought of, via the [[homotopy theory]] of [[∞-groups]], as part of the [[long homotopy fiber sequence]]

$$
  \array{
    A &\longrightarrow& \widehat G
    \\
    && \Big\downarrow
    \\
    && G &\overset{c_2}{\longrightarrow}& B A
  }
$$

where $B A$ is the [[delooping]] [[2-group]] of $A$, and where $c_2 \;\colon\; G \longrightarrow B A$ is the [[cocycle]] which classifies the central extension.

Equivalently, under further [[delooping]] this is the [[long homotopy fiber sequence]] of the form

$$
  \array{
    B A &\longrightarrow& B \widehat G
    \\
    && \Big\downarrow
    \\
    && B G &\overset{c_2}{\longrightarrow}& B^2 A
  }
$$

which makes explicitl that $c_2$ is a _2-cocycle_.

Phrased this way it is clear that a concept of _higher central extension_ of an [[∞-group]] $G$ should subsume at least those [[long homotopy fiber sequence]] of the form

$$
  \array{
    B^{p+1} A &\longrightarrow& B \widehat G
    \\
    && \Big\downarrow
    \\
    && B G &\overset{c_{p+2}}{\longrightarrow}& B^{p+2} A
  }
$$

where now $c_{p+2}$ is the classifying $(p+2)$-cocyle.

This concept makese sense not just for [[∞-groups]] in any [[(∞,1)-topos]], but also for instance for ([[nLab:super L-∞ algebra|super]]) [[L-∞ algebras]].

## Examples

* [[higher central ∞-group extension]]

* [[L-∞ algebra extension]]

## Related concepts

* [[universal higher central extension]]

* [[infinity-Lie algebra cohomology]]

## References

* {#NSS12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], section 4.3 of  _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal infinity-bundles - General theory]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015) ([arXiv:1207.0248](https://arxiv.org/abs/1207.0248))

[[!redirects higher central extensions]]


higher chromatic Chern character

[[!redirects higher chromatic Chern characters]]

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#Contents#
* table of contents
{:toc}

## Idea

The traditional [[Chern character]] on [[complex K-theory]] happens to be a map from [[complex oriented cohomology theory]] of [[chromatic level]] 1 to one of chromatic level 0 ([[ordinary cohomology]]).

More generally, one may ask for _higher chromatic Chern characters_ that map a [[generalized cohomology theory]] of some [[chromatic level]] to another of some other level.

This has been considered for

* maps from [[elliptic cohomology]]/[[tmf]] to [[K-theory]], see at _[[elliptic Chern character]]_

* maps from [[Morava E-theory]] at level $(n+1)$ to Morava E-theory at level $n$, tensored with some ring, see at _[[E-theoretic Chern character]]_.

## Related concepts

* [[transchromatic character]]

## References

A survey is for instance in the introduction of

* [[Takeshi Torii]], _Milnor operations and the generalized Chern character_, Geom. Topol. Monogr. 10 (2007) 383-421 ([arXiv:0903.4708](http://arxiv.org/abs/0903.4708))

* [[Takeshi Torii]], _HKR characters, p-divisible groups and the generalized Chern character_, ([pdf](http://www.ams.org/journals/tran/2010-362-11/S0002-9947-2010-05194-3/S0002-9947-2010-05194-3.pdf))

[[!redirects higher chromatic Chern characters]]

[[!redirects higher chromatic Chern characters]]

[[!redirects higher Chern character]]
[[!redirects higher Chern characters]]


higher complex analytic geometry

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#Contents#
* table of contents
{:toc}

## Idea

The [[higher geometry]] version of [[complex analytic geometry]], for instance (petit) with [[geometry (for structured ((∞,1)-toposes)]] being the [[site]] of [[Stein manifolds]] or (gros) [[complex analytic ∞-groupoids]].

## Related concepts

* [[GAGA]]

* [[higher differential geometry]]

* [[higher structures]]

## References

* [[Jacob Lurie]], section 4.4. of _[[Structured Spaces]]_

* [[Jacob Lurie]], sections 11 and 12 of _[[Closed Immersions]]_

* {#Larusson01} Finnur Larusson, _Excision for simplicial sheaves on the Stein site and Gromov's Oka principle_ ([arXiv:math/0101103](http://arxiv.org/abs/math/0101103))

* {#HopkinsQuick12} [[Michael Hopkins]], [[Gereon Quick]], _Hodge filtered complex bordism_ ([arXiv:1212.2173](http://arxiv.org/abs/1212.2173))

* _[[schreiber:Differential cohomology is Cohesive homotopy theory]]_

  talk notes on how _[[cohesion]]_ is a faithful axiomatization of generalized [[differential cohomology hexagon|differential cohomology]] including differential [[nLab:moduli stack]] constructions such as that of  [[nLab:intermediate Jacobians]] in higher complex analytic geometry.

* [[Mauro Porta]], [[Tony Yue Yu]], _Higher analytic stacks and GAGA theorems_, [arXiv:1412.5166](http://arxiv.org/abs/1412.5166).

[[!redirects complex analytic higher geometry]]


higher Courant groupoid

#Contents#
* table of contents
{:toc}

## Idea

The [[Lie integration]] of [[higher Courant Lie algebroids]].

## Related concepts

[[!include higher Atiyah groupoid - table]]

[[!include slice automorphism groups in higher prequantum geometry - table]]

[[!redirects higher Courant groupoids]]

[[!redirects Courant 2-groupoid]]
[[!redirects Courant 2-groupoids]]

[[!redirects Courant Lie 2-groupoid]]
[[!redirects Courant Lie 2-groupoids]]


higher Courant Lie algebroids

#Contents#
* table of contents
{:toc}

## Idea

The further refinement to [[higher geometry]] of the notion of _[[Courant Lie 2-algebroids]]_.

## Related concepts

[[!include higher Atiyah groupoid - table]]

## References

A proposal for [[standard Courant algebroid|standard]] higher Courant Lie algebroids is in

* [[Marco Zambon]], _$L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids_, J. Symplectic Geometry, ([arXiv:1003.1004](http://arxiv.org/abs/1003.1004))


higher curvature correction

[[!redirects higher curvature corrections]]

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#Contents#
* table of contents
{:toc}

## Idea

In the context of [[gravity]] ([[general relativity]]), _higher curvature corrections_ are modifications of the [[Einstein-Hilbert action]] that include not just the linear appearance of the [[scalar curvature]] $R$ but higher scalar powers of the [[Riemann curvature tensor]].

When viewing Einstein-Hilbert [[gravity]] as an [[effective field theory]] valid at low [[energy]]/long [[wavelengths]], then higher curvature corrections are precisely the terms that may appear at higher energy in pure gravity. Notably the effective field theories induced by [[string theory]] come with infinite towers of higher curvature corrections.

In the context of [[cosmology]], higher curvature corrections are a candidate for the [[inflaton field]], see at _[[Starobinsky model of cosmic inflation]]_.

A [[spacetime]] that extremizes the [[Einstein-Hilbert action]] for given [[cosmological constant]] and _arbitrary_ higher curvature correction is called a _[[universal spacetime]]_.

## Examples

### For 11d Supergravity

A systematic analysis of the possible [[supersymmetry|supersymmetric]] [[higher curvature corrections]] of [[D=11 supergravity]] makes the [[I8]]-[[characteristic class|characteristic 8-form]]

\[
  I_8(R)
  \;\coloneqq\;
  \tfrac{1}{48}
  \Big(
    p_2(R) \;-\; \big( \tfrac{1}{2} p_1(R)\big)^2
  \Big)
  \;\;
  \in
  \Omega^8
\]

(a [[differential form]] built from the [[Riemann curvature]], expressing a polynomial in the first and second [[Pontryagin classes]], see at _[[I8]]_) appear as the higher curvature correction at order $\ell^6$, where $\ell$ is the [[Planck length]] in 11d ([Souères-Tsimpis 17, Section 4](#SoueresTsimpis17)).

At this order, the [[equation of motion]] for the [[supergravity C-field]] [[flux]] $G_4$ and its dual $G_7$ is ([Souères-Tsimpis 17, (4.3)](#SoueresTsimpis17))

\[
  \label{11dSugraBIAtOrder6}
  d G_7(\ell)
  \;=\;
  -\tfrac{1}{2}
  G_4(\ell) \wedge G_4(\ell)
  +
  \ell^6 I_8(R)
  \,,
\]

where the flux forms themselves appear in their higher order corrected form as [[power series]] in the [[Planck length]]

$$
  G_4(\ell)
  \;=\;
  G_4 + \ell^6 G_4^{(1)} + \cdots
$$

$$
  G_7(\ell)
  \;=\;
  G_7 + \ell^6 G_7^{(1)} + \cdots
$$

([Souères-Tsimpis 17, (4.4)](#SoueresTsimpis17))

Beware that this is not the lowest order [[higher curvature correction]]: there is already one at $\mathcal{O}(\ell^3)$, given by $\ell^3 G_4 \wedge \tfrac{1}{2}p_1(R)$ ([Souères-Tsimpis 17, Section 3.2](#SoueresTsimpis17)). Hence the full correction at $\mathcal{O}(\ell^3)$ should be the further modification of (eq:11dSugraBIAtOrder6) to (cf. [Tsimpis 2004, p. 8](#Tsimpis04)):

\[
  \label{11dSugraBIAtOrder6}
  d G_7(\ell)
  \;=\;
  -\tfrac{1}{2}
  G_4(\ell) \wedge
  \big(
    G_4(\ell)
    -
    \tfrac{1}{2} p_1(R)
  \big)
  +
  \ell^6 I_8(R)
  \,.
\]

## Related concepts

* [[universal spacetime]]

* [[non-perturbative effect]]

## References

### General

Discussion of quadratic curvature currections includes (see also at _[[Starobinsky model of cosmic inflation]]_)

* [[Luis Alvarez-Gaume]], [[Alex Kehagias]], [[Costas Kounnas]], [[Dieter Luest]], Antonio Riotto, _Aspects of Quadratic Gravity_ ([arXiv:1505.07657](http://arxiv.org/abs/1505.07657))

Discussion of [[causal locality]] in the presence of higher curvature corrections includes

* Xian O. Camanho, Jose D. Edelstein, [[Juan Maldacena]], Alexander Zhiboedov, _Causality Constraints on Corrections to the Graviton Three-Point Coupling_ ([arXiv:](http://arxiv.org/abs/1407.5597))

* Giuseppe D'Appollonio, [[Paolo Di Vecchia]], [[Rodolfo Russo]], [[Gabriele Veneziano]], _Regge behavior saves String Theory from causality violations_ ([arXiv:1502.01254](http://arxiv.org/abs/1502.01254))

Discussion in the context of corrections to [[black hole entropy]]:

* [[Thomas Mohaupt]], _Strings, higher curvature corrections, and black holes_ ([arXiv:hep-th/0512048](http://arxiv.org/abs/hep-th/0512048))

Discussion of [[renormalization]] for gravity with higher curvature corrections:

* Nobuyoshi Ohta, *One-loop divergences in higher-derivative gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.02583](https://arxiv.org/abs/2210.02583)]

Discussion of higher curvature corrections in [[cosmology]] and [[cosmic inflation]] (for more see at [[Starobinsky model of cosmic inflation]]):

* Gustavo Arciniega, Jose D. Edelstein, Luisa G. Jaime, _Towards purely geometric inflation and late time acceleration_ ([arXiv:1810.08166](https://arxiv.org/abs/1810.08166))

* Gustavo Arciniega, Pablo Bueno, Pablo A. Cano, Jose D. Edelstein, Robie A. Hennigar, Luisa G. Jaimem, _Geometric Inflation_ ([arXiv:1812.11187](https://arxiv.org/abs/1812.11187))

Relation to [[cosmic censorship hypothesis]]:

* Akash K Mishra, Sumanta Chakraborty, _Strong Cosmic Censorship in higher curvature gravity_, Phys. Rev. D 101, 064041 (2020) ([arXiv:1911.09855](https://arxiv.org/abs/1911.09855))

For [[supergravity]]:

* Mehmet Ozkan, Yi Pang, [[Ergin Sezgin]], *Higher Derivative Supergravities in Diverse Dimensions* [[arXiv:2401.08945](https://arxiv.org/abs/2401.08945)]

### Higher curvature corrections to $D=4$ supergravity

Discussion of [[higher curvature corrections]] to [[D=4 supergravity]]:

Relation to [[quintessence]]:

* [[Fotis Farakos]], _Quintessence from higher curvature supergravity_ ([arXiv:2003.09366](https://arxiv.org/abs/2003.09366))

### Higher curvature corrections to heterotic supergravity

On higher curvature corrections to [[heterotic supergravity]]:

* Eric Lescano, Carmen Núñez, Jesús A. Rodríguez, *Supersymmetry, T-duality and Heterotic $\alpha'$-corrections* ([arXiv:2104.09545](https://arxiv.org/abs/2104.09545))

* Hao-Yuan Chang, [[Ergin Sezgin]], Yoshiaki Tanii, *Dimensional reduction of higher derivative heterotic supergravity* ([arXiv:2110.13163](https://arxiv.org/abs/2110.13163))

[[!include higher curvature corrections to D=11 supergravity -- references]]

[[!include DBI-action higher curvature corrections -- references]]

[[!redirects higher order curvature corrections]]


higher curvature corrections to D=11 supergravity -- references

### Higher curvature corrections to $D=11$ supergravity
 {#ReferencesOnHigherCurvatureCorrectionsIn11dSupergravity}

Discussion of [[higher curvature corrections]] to [[11-dimensional supergravity]] (i.e. in [[M-theory]]):

* [[Arkady Tseytlin]], _$R^4$ terms in 11 dimensions and conformal anomaly of (2,0) theory_, Nucl. Phys. B584: 233-250, 2000 ([arXiv:hep-th/0005072](https://arxiv.org/abs/hep-th/0005072))

#### Via 11d superspace cohomology

Via 11d [[superspace]]-cohomology (or mostly):

* [[Kasper Peeters]], Pierre Vanhove, Anders Westerberg, _Supersymmetric $R^4$ actions and quantum corrections to superspace torsion constraints_ ([arXiv:hep-th/0010182](https://arxiv.org/abs/hep-th/0010182))

* H. Lu, [[Christopher Pope]], [[Kellogg Stelle]], [[Paul Townsend]], _Supersymmetric Deformations of $G_2$ Manifolds from Higher-Order Corrections to String and M-Theory_, JHEP 0410:019, 2004 ([arXiv:hep-th/0312002](https://arxiv.org/abs/hep-th/0312002))

  (specifically for [[M-theory on G2-manifolds]])

* H. Lu, [[Christopher Pope]], [[Kellogg Stelle]], [[Paul Townsend]], _String and M-theory Deformations of Manifolds with Special Holonomy_, JHEP 0507:075, 2005 ([arXiv:hep-th/0410176](https://arxiv.org/abs/hep-th/0410176))

  (specifically for [[M-theory on G2-manifolds]])

* {#Tsimpis04} [[Dimitrios Tsimpis]], *11D supergravity at $\mathcal{O}(\ell^3)$*, JHEP0410:046 (2004) \[arXiv:hep-th/0407271, doi:10.1088/1126-6708/2004/10/046\]

* {#Howe04} [[Paul Howe]], _$R^4$ terms in supergravity and M-theory_ ([arXiv:hep-th/0408177](https://arxiv.org/abs/hep-th/0408177))

* {#CGNT04} [[Martin Cederwall]], [[Ulf Gran]], [[Bengt Nilsson]], [[Dimitrios Tsimpis]], _Supersymmetric Corrections to Eleven-Dimensional Supergravity_, JHEP0505:052, 2005 ([arXiv:hep-th/0409107](https://arxiv.org/abs/hep-th/0409107))

* Yoshifumi Hyakutake, Sachiko Ogushi, _$R^4$ Corrections to Eleven Dimensional Supergravity via Supersymmetry_, Phys.Rev. D74 (2006) 025022 ([arXiv:hep-th/0508204](https://arxiv.org/abs/hep-th/0508204))

* Yoshifumi Hyakutake, Sachiko Ogushi, _Higher Derivative Corrections to Eleven Dimensional Supergravity via Local Supersymmetry_, JHEP0602:068, 2006 ([arXiv:hep-th/0601092](https://arxiv.org/abs/hep-th/0601092))

* [[Anirban Basu]], _Constraining gravitational interactions in the M theory effective action_, Classical and Quantum Gravity, Volume 31, Number 16, 2014 ([arXiv:1308.2564](https://arxiv.org/abs/1308.2564))

* {#SoueresTsimpis17} [[Bertrand Souères]], [[Dimitrios Tsimpis]], *The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity*, Phys. Rev. D **95** 026013 (2017) \[doi:10.1103/PhysRevD.95.026013, [arXiv:1612.02021](https://arxiv.org/abs/1612.02021)\]

* [[Bertrand Souères]], _Supergravities in Superspace_, Lyon 2018 ([tel:01998725](https://tel.archives-ouvertes.fr/tel-01998725), [pdf](https://tel.archives-ouvertes.fr/tel-01998725/document))

#### Via superparticle scattering in 11d

Via analysis of would-be [[superparticle]] [[scattering amplitudes]] on [[D=11 supergravity]] backgrounds:

* [[Kasper Peeters]], [[Jan Plefka]], [[Steffen Stern]], _Higher-derivative gauge field terms in the M-theory action_, JHEP 0508 (2005) 095 ([arXiv:hep-th/0507178](https://arxiv.org/abs/hep-th/0507178))

#### Via lifting 10d stringy corrections

From lifting [[string length scale|alpha']]-corrections in [[type IIA string theory]] through the [[duality between M-theory and type IIA string theory]]:

* [[James Liu]], [[Ruben Minasian]], _Higher-derivative couplings in string theory: dualities and the B-field_, Nucl. Phys. B 2013 ([arXiv:1304.3137](https://arxiv.org/abs/1304.3137))

Via [[type IIB supergravity]]:

* [[James Liu]], [[Ruben Minasian]], Raffaele Savelli, Andreas Schachner, *Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles* $[$[arXiv:2205.11530](https://arxiv.org/abs/2205.11530)$]$

#### Via the ABJM M2-brane model

From the [[ABJM  model]] for the [[M2-brane]]:

* Damon J. Binder, [[Shai Chester]], Silviu S. Pufu, _Absence of $D^4 R^4$ in M-Theory From ABJM_ ([arXiv:1808.10554](https://arxiv.org/abs/1808.10554))

#### See also

See also

* [[Mohammad Garousi]], _Minimal gauge invariant couplings at order $\ell^6_p$ in M-theory_ ([arXiv:2102.00639](https://arxiv.org/abs/2102.00639))

Discussion in view of the [[Starobinsky model of cosmic inflation]] is in

* [[Katrin Becker]], [[Melanie Becker]], _Supersymmetry Breaking, M-Theory and Fluxes_, JHEP 0107:038,2001 ([arXiv:hep-th/0107044](https://arxiv.org/abs/hep-th/0107044))

* Kazuho Hiraga, Yoshifumi Hyakutake, _Inflationary Cosmology via Quantum Corrections in M-theory_ ([arXiv:1809.04724](https://arxiv.org/abs/1809.04724))

* Kazuho Hiraga, Yoshifumi Hyakutake, _Scalar Cosmological Perturbations in M-theory with Higher Derivative Corrections_ ([arxiv:1910.12483](https://arxiv.org/abs/1910.12483))

and in view of [[de Sitter spacetime]] vacua:

* Johan Blåbäck, [[Ulf Danielsson]], [[Giuseppe Dibitetto]], Suvendu Giri, _Constructing stable de Sitter in M-theory from higher curvature corrections_ ([arXiv:1902.04053](https://arxiv.org/abs/1902.04053))


higher differential geometry

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#Contents#
* table of contents
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## Idea
 {#Idea}

_Higher differential geometry_ is the incarnation of [[differential geometry]] in [[higher geometry]]. Hence it is concerned with [[n-groupoid]]-versions of [[smooth spaces]] for _higher_ $n$, where the traditional theory is contained in the case $n = 0$. For $n = 1$ these higher structures are _[[Lie groupoids]]_, _[[differentiable stacks]]_, their infinitesimal approximation by [[Lie algebroids]] and the generalization to _[[smooth stacks]]_. For higher $n$ this includes ([[deloopings]] of) [[Lie 2-groups]], [[Lie 3-groups]].

Fully generally, higher differential geometry hence replaces [[smooth manifolds]] (and possibly variants such as [[supermanifolds]], [[formal manifolds]], [[dg-manifolds]] etc.) by _[[∞-stacks]]_ ([[(∞,1)-sheaves]]) on the [[site]] of all such. Technically this means that higher differential geometry is the study of an _[[(∞,1)-topos]]_ into which standard [[differential geometry]] faithfully embeds.
This then allows to speak of [[smooth ∞-groups]], [[Lie ∞-algebroids]].

If the ambient [[(∞,1)-topos]] is not [[n-localic (∞,1)-topos|1-localic]] (for instance over a genuine site of [[dg-manifolds]]) then one also speaks of _[[derived differential geometry]]_.

See at _[[motivation for higher differential geometry]]_ for motivation.

The standard variants of [[differential geometry]] have their higher analogs, for instance [[symplectic geometry]] generalizes to [[higher symplectic geometry]] and [[prequantum geometry]] to [[higher prequantum geometry]].

## Formalizations

One axiomatization is [[cohesion]] and [[differential cohesion]].

## Examples

* [[Smooth∞Grpd]], [[SuperFormalSmooth∞Grpd]].

* [[Lie 2-groupoid]]

* [[double Lie algebroid]], [[Lie algebroid-groupoid]], [[double Lie groupoid]]

## Related concepts

* [[higher structure]]

* [[higher Lie theory]]

* [[higher prequantum geometry]]

* [[higher complex analytic geometry]]

* [[L-infinity algebras in physics]]

## References
 {#References}

Exposition is in

* [[Urs Schreiber]], _[[schreiber:Higher Structures|Higher Structures in Mathematics and Physics]]_, talk at [Oberwolfach Workshop 1651a](https://www.mfo.de/occasion/1651a/www_view), 2016 Dec. 18-23

Details:

* _[[geometry of physics]]_, [[geometry of physics -- categories and toposes|categories and toposes]], ...

The most classical aspect of higher differential geometry is the theory of [[orbifolds]], [[Lie groupoids]] and [[Lie algebroids]] and their application in [[foliation theory]]. Original reference here include

* [[Charles Ehresmann]], _Catégories topologiques et catégories différentiables_, Colloque de Géometrie Differentielle Globale (Bruxelles, 1958), 137--150, Centre Belge Rech. Math., Louvain, 1959;

* [[Ieke Moerdijk]], [[Dorette Pronk]], _Orbifolds, sheaves and groupoids_, K-theory 12 3-21 (1997) ([pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/pronk.pdf)),  _Orbifolds as Groupoids: an Introduction_ ([arXiv:math.DG/0203100](http://arxiv.org/abs/math.DG/0203100))

Textbook accounts:

* [[Ieke Moerdijk]], [[Janez Mrcun]] _Introduction to Foliations and Lie Groupoids_, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003) ([doi:10.1017/CBO9780511615450](https://doi.org/10.1017/CBO9780511615450))

* [[Kirill Mackenzie]], _General Theory of Lie Groupoids and Lie Algebroids,_ Cambridge University Press, 2005, xxxviii + 501 pages
([website](http://kchmackenzie.staff.shef.ac.uk/gt.html))

* [[Kirill Mackenzie]], _Lie groupoids and Lie algebroids in differential geometry_, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp ([MathSciNet](http://www.ams.org/mathscinet-getitem?mr=896907))

For properly appreciating the [[homotopy theory]] of Lie groupoids and for passage to more general higher differential geometry, it is crucial to understand Lie groupoids as [[smooth stacks]] which are [[geometric stack|geometric]]:  _[[differentiable stacks]]_. Each of the following references provides introduction to this point of view:

* [[Jochen Heinloth]], _Some notes on differentiable stacks_ ([pdf](http://www.uni-due.de/~hm0002/stacks.pdf))

* [[Kai Behrend]], [[Ping Xu]], _Differentiable Stacks and Gerbes_ ([arXiv:0605.5694](http://front.math.ucdavis.edu/0605.5694)).

* Metzler, _Topological and smooth stacks_ ([arXiv:math/0306176](http://arxiv.org/abs/math/0306176))

As a warmup for these considerations it may be useful to first look at [[smooth spaces]] given by just [[sheaves]] on the site of [[smooth manifolds]], see at

* [[Urs Schreiber]], _[[geometry of physics]] -- [[geometry of physics -- smooth spaces|smooth spaces]]_

Passing from here to more general [[smooth groupoids]], to [[smooth 2-groupoids]] and then eventually to [[smooth ∞-groupoids]] involves [[(∞,1)-topos theory]] proper, with some tools as discussed at _[[model structure on simplicial presheaves]]_ over the [[site]] of [[smooth manifolds]], or equivalently just over its [[dense subsite]] of [[Cartesian spaces]].

For motivation for this step see also

* [[Urs Schreiber]], _[[twisted smooth cohomology in string theory]]_, lectures at _[ESI program on quantum fields and K-theory](http://maths-old.anu.edu.au/esi/2012/)_, 2012

Introductory exposition includes the introductory sections of

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech Cocycles for Differential characteristic Classes]]_, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _A higher stacky perspective on Chern-Simons theory_, in Damien Calaque et al. (eds.) _Mathematical Aspects of Quantum Field Theories_ Mathematical Physics Studies, Springer 2014 ([arXiv:1301.2580](http://arxiv.org/abs/1301.2580))

and sections 1.2.4 ([[geometry of physics -- smooth homotopy types]]) as well as section 1.2.5 ([[geometry of physics -- principal bundles]]) in the Introduction section of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

This goes on to discuss [[differential cohomology]] and the [[differential cohomology diagram]] formulated in [[stable homotopy type|stable]] objects in [[smooth ∞-groupoids]] (hence in [[sheaves of spectra]] on the [[site]] of [[smooth manifolds]]/[[Cartesian spaces]]) in higher differential geometry, see

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

On [[derived differential geometry]] and [[higher Lie theory]] cast in ([[differential cohesion|differential]]) [[cohesive (infinity,1)-topos|cohesive $(\infty,1)$-topos]] theory:

* [[Luigi Alfonsi]], [[Charles A. S. Young]], *Towards non-perturbative BV-theory via derived differential cohesive geometry* [[arXiv:2307.15106](https://arxiv.org/abs/2307.15106)]

exposition

* [[Luigi Alfonsi]]: *Derived $n$-plectic geometry: towards non-perturbative BV-BFV quantisation and M-theory*, [talk at](M-Theory+and+Mathematics#Alfonsi2023) *[[M-Theory and Mathematics]]* (Jan 2023) [slides: [[Alfonsi-MTheoryAndMath2023.pdf:file]], video: [YT](https://www.youtube.com/watch?v=O7pQkyqCiA4)]

Review:

* [[Urs Schreiber]], *[[schreiber:Higher Topos Theory in Physics|Higher Topos Theory in Physics]]* [[arXiv:2311.11026](https://arxiv.org/abs/2311.11026)]

* [[Luigi Alfonsi]], *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]


higher dimensional algebra

In the groupoid survey listed below,  the term _higher dimensional algebra_ was introduced in the provocative form: _n-dimensional phenomena require for their description n-dimensional algebra_. Since then the term  has come to be used in various senses, and  to include most forms of research into higher order categories, often  of a lax nature. A more specific definition is to say that Higher Dimensional Algebra  means _the study of systems of partial algebraic structures whose domains of definition are given by geometric conditions_.

Of course the partial composition of paths or of functions was used if not formalised early on. The earliest formal use  of partial operations in algebra  was that of a [[groupoid]] as defined by Brandt in 1926, in connection with the laws for the classification of quaternary quadratic forms, generalising Gauss' work on binary quadratic forms. Curiously, groupoids were  not given as an example in Eilenberg and Mac Lane's fundamental paper on category theory, although they were well known by the Chicago algebraists of the time. The general theory of such partial algebra was developed by Philip Higgins.

The next easiest to understand example in dimension 2 is possibly that of  maps $a: I^2 \to X$ where $X$ is a topological space, called _squares in $X$_. Such a map determines  four paths $\partial^\epsilon _i a: I \to X$ for $i=1,2, \epsilon=\pm$ given by $\partial^- _1 a(t)= a(0,t), \partial^+ _1 a(t)=a(1,t), \partial^- _2 a(t)= a(t,0), \partial^+ _2 a(t)=a(t,1)$. Such squares in $X$  have 2 obvious partial compositions $\circ_1,\circ_2$ where for example $a\circ_1 b$ is defined if and only if $\partial^+ _1 a= \partial^- _1 b$.

The problem is to obtain a _strict homotopy double groupoid_ from such squares. Brown and Higgins realised in 1974 that this could be achieved  fairly easily  in a relative situation, i.e. if we are given a triple $X_*=(X,A,C)$ where $C \subseteq A \subseteq X$,  and then consider maps $I^2 \to X$ which take the edges into $A$ and the vertices into $C$, and then form $\rho X_*$ of homotopy classes of such maps rel vertices. It is not quite trivial to prove that the partial compositions of such squares are inherited by $\rho X_*$ to make it a double groupoid, and that, quite importantly, there is an extra structure of _connections_. This extra structure makes the category of such objects equivalent to the category of [[crossed module]]s but of groupoids, rather than just groups.

Under this equivalence, the double groupoid $\rho X_*$ becomes the crossed module $\Pi X_* $ consisting of the family of relative homotopy groups $\pi_2(X,A,c)$, $c \in C$, with the boundary to the [[fundamental groupoid]] $\pi_1(A,C)$ and the operations of this groupoid. Hence a van Kampen type theorem for $\rho$ yields a van Kampen type theorem for $\Pi$ and so  previously unobtainable determinations of some nonabelian second relative homotopy groups.

Notice that the compositions in $\pi_2(X,A,c)$ require a choice of direction, and so is unaesthetic, whereas $\rho$ is a _symmetric_ construction. Also $\rho$ allows for convenient multiple compositions appropriate to _algebraic inverses to subdivision_, which are essential for the proof of the 2-dimensional van Kampen type theorem.

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References

*P.J. Higgins, Algebras with a scheme of operators,  Math. Nachr. 27 1963 115--132.

*R. Brown and P.J. Higgins, On the connection between the second relative homotopy groups of some related spaces,  Proc. London Math.  Soc. (3) 36 (1978) 193-212.

* R. Brown, From groups to groupoids: a brief survey, _Bull.
London Math.  Soc._ 19 (1987) 113-134.