higher dimensional arithmetic geometry

> This entry is about traditional [[arithmetic geometry]] over [[higher local fields]]. For "[[E-∞ arithmetic geometry]]" see there.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

the study of [[arithmetic geometry]] which concentrates on arithmetic schemes of higher dimensions and uses associated higher structures such as  [[higher local fields]], higher adelic structures, (commutative) higher class field theory and hence Milnor K-theory is often called higher arithmetic geometry.

The [[zeta functions]] in higher dimensional arithmetic geometry are called _[[arithmetic zeta functions]]_.

## Related concepts

* [[perfectoid space]]

## References

* [[Alexander Beilinson]], _Residues and adeles_, Funct. Anal. and Appl. 14(1980), 34-35

* [[Ivan Fesenko]], M. Kurihara (eds.), _Invitation to Higher Local Fields_, Geometry and Topology Monographs vol 3, Warwick 2000, 304 pp. ([web](https://www.maths.nottingham.ac.uk/personal/ibf/volume.html))

* [[Ivan Fesenko]], _Analysis on arithmetic schemes. I_,  Docum. Math. Extra Volume Kato (2003) 261--284 ([web] (http://www.mathematik.uni-bielefeld.de/documenta/vol-kato/fesenko.dm.html))

* [[Ivan Fesenko]], _Adelic approch to the zeta function of arithmetic schemes in dimension two_, Moscow Math. J. 8 (2008), 273–317 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/ada.pdf))

* [[Ivan Fesenko]], _Geometric adeles and the Riemann–Roch theorem for 1-cycles on surfaces_ ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/ar.pdf))

* {#Fesenko06} [[Ivan Fesenko]], _Measure, integration and elements of harmonic analysis on generalized loop spaces_, Proc. of
St Petersburg Math. Soc. 12(2005), 179–199; English transl. in AMS Transl. Series 2, 219 (2006), 149–164 ([pdf](http://www.maths.nott.ac.uk/personal/ibf/aoh.pdf))

* [[Ivan Fesenko]],  G. Ricotta,  M. Suzuki, _Mean-periodicity and zeta functions_, Ann. L'Inst. Fourier, 62(2012), 1819-1887 ([pdf] (https://www.maths.nottingham.ac.uk/personal/ibf/m-p.pdf))

* K. Kato, Sh. Saito, Two-dimensional class field theory,
Adv. Stud. Pure Math., vol. 2, 1983, 103-152

* K. Kato and Sh. Saito, Global class field theory of arithmetic schemes,
 Contemp. Math., vol. 55 part I, 1986, 255-331

* M. Morrow, Grothendieck's trace map for arithmetic surfaces via residues and higher adeles, Algebra and Number Theory J., 6-7 (2012), 1503-1536

* A. Parshin, On the arithmetic of two dimensional schemes. I, Distributions and residues, Math. USSR Izv. 10(1976), 695-729

* A. Parshin,  Abelian coverings of arithmetic schemes, Dokl. Akad. Nauk SSSR  243(1978), no. 4, 855-858

* M. Suzuki, Two dimensional adelic analysis and cuspidal automorphic representations of $GL(2)$,  In Multiple Dirichlet Series, L-functions and automorphic forms, D. Bump, S. Friedberg, D. Goldfeld,  eds., Progress in Math. 300, pp.339-361, Birkhaeuser 2012

* [[Amnon Yekutieli]], _An explicit construction of the Grothendieck residue complex_, Asterisque 208, 1992.

See also the conference webpage

*  _[Symmetries and Correspondences:
Higher Structures in Number Theory](https://www.maths.nottingham.ac.uk/personal/ibf/files/hsnt1.html)_, Oxford, July 2014

[[!redirects higher arithmetic geometry]]


higher dimensional Chern-Simons theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There are two kinds of higher dimensional generalizations of ordinary 3-dimensional [[Chern-Simons theory]] that are often called "higher dimensional Chern-Simons theory" in the literature. Both are special cases of [[schreiber:infinity-Chern-Simons theory]].

Recall that for $\mathfrak{g}$ a [[Lie algebra]] (not necessarily abelian) with non-generate binary [[invariant polynomial]] $\langle -,-\rangle$, the corresponding [[schreiber:infinity-Chern-Simons theory]] [[QFT]] is ordinary [[Chern-Simons theory]] in dimension 3.

But also every other [[invariant polynomial]] $\langle-,-,\cdots,-\rangle$ on $\mathfrak{g}$ induces an [[schreiber:infinity-Chern-Simons theory]], now in higher dimension.

Moreover, every [[line Lie n-algebra]] $b^n \mathbb{R}$ carries a canonical invariant polynomial. The [[schreiber:infinity-Chern-Simons theory]] associated with that is often called _abelian higher dimensional CS theory_ .

(...)

More general "higher"-generalization of Chern-Simons theory to [[schreiber:infinity-Chern-Simons theory]] allow $\mathfrak{g}$ to be a (nonabelian) [[Lie 2-algebra]] or more generally a (nonabelian) [[L-infinity algebra]] or fully generally a [[L-infinity algebroid]].

## Higher abelian Chern-Simons theory.

### Definition

The definition of higher abelian Chern-Simons theory is simple _locally_ when certain global [[cohomology|cohomological]] effects can be ignored. We first give the simple local definition and then the full global definition.

Let $k \in \mathbb{N}$ be a [[natural number]], let $d = 4 k + 3$ and let $\Sigma$ be a [[compact space|compact]] [[smooth manifold]] of [[dimension]] $d$.

Then the _simple version_ of abelian $d$-dimensional Chern-Simons theory is defined as follows.

* the [[configuration space]] is the [[space]] of [[differential form]]s on $\Sigma$ of degree $2k+1$

  $$
    Conf_{simpl} = \Omega^{2k+1}(\Sigma)
    \,,
  $$

* the [[Lagrangian]] is

  $$
    L : B \mapsto B \wedge d_{dR} B
    \,,
  $$

* and the [[action functional]]

  $$
    S : \Omega^{2k+1}(\Sigma) \to \mathbb{R}
  $$

  is therefore

  $$
    S : B \mapsto \int_\Sigma B \wedge d_{dR}B
    \,.
  $$

Notice that generally for an $n$-form $B$ on a closed $(2n+1)$-dimensional manifold $\Sigma$ we have

$$
  \int_\Sigma B \wedge d_{dR} B
  =
  (-1)^{1+n + n(1+n)}
  \int_\Sigma B \wedge d_{dR} B
$$

by first using integration by parts and then switching the order of the wedge factors. Therefore this kind of action vanishes identically when $deg B$ is even. This is the reason for the above assumption that $deg B = 2k+1$ for $k \in \mathbb{N}$ and hence that the Chern-Simons theory is in dimension $4k+3$.

In the full theory instead the configuration space is

$$
  Conf = \mathbf{H}_{diff}^{2k+2}(\Sigma)
  \,,
$$

the space of [[circle n-bundle with connection|circle (2k+1)-bundles with connection]] (given by [[cocycle]]s in degree $2k+2$ [[ordinary differential cohomology]]). This contains the above simplified configuration space as the subspace of $(2k+1)$-connections whose underlying circle $(2k+1)$-bundle is trivial.

The action functional is given by

$$
  S : \hat B \mapsto \int_\Sigma \hat B \cup \hat B
  \,,
$$

where now the integral is [[fiber integration in ordinary differential cohomology]] and in the integrand we have the [[cup product in ordinary differential cohomology]] of differential cocycles.

(See for instance ([GT, section 4.1](#GT)), ([FMS, (1.28)](#FMS))).

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

We discuss how the above definition arises as a special case of the general notion of
[[schreiber:infinity-Chern-Simons theory]].

These theories are defined by

* an [[L-infinity algebroid]] $\mathfrak{a}$;

* equipped with an [[invariant polynomial]] $\langle \rangle$.

The abelian higher dimensional Chern-Simons theories in dimension $4k+3$ are the special case of this general situation where

* $\mathfrak{a} = b^{2k+1}\mathbb{R}$ is the [[line Lie n-algebra|line Lie (2k+1)-algebra]], the $(2k+1)$-fold [[delooping]] of the abelian [[Lie algebra]] $\mathbb{R}$;

* $\langle - \rangle$ is the canonical quadratic [[invariant polynomial]] on this.

(...)

See ([FRS, 4.1.4](#FRS)).

### Examples

Higher dimensional abelian Chern-Simons theories appear automatically as components of systems of higher [[supergravity]], for instance in [[11-dimensional supergravity]] (they are automatically induced by the requirement of  [[local supersymmetry]] in these higher dimensional supergravity theories).

* see at _[[M5-brane]]_ the section _[Conformal blocks and 7d Chern-Simons dual](http://ncatlab.org/nlab/show/M5-brane#7dCSDual)_.

### Properties

#### Holographic relation to $4k+2$-dimensional theory

Higher Chern-Simons theory in dimension $4k+3$ is related by a [[holographic principle]] to [[self-dual higher gauge theory]] in dimension $4k+2$ (at least in the abelian case).

* $(k=0)$: ordinary 3-dimensional [[Chern-Simons theory]] is related to a [[string]] [[sigma-model]] on its boundary;

* $(k=1)$: 7-dimensional Chern-Simons theory is related to a [[fivebrane]] model on its boundary;

* $(k=2)$: 11-dimensional Chern-Simons theory is related to a parts of a [[type II string theory]] on its bounday (or that of the space-filling 9-[[brane]], if one wishes) ([BelovMoore](#BelovMoore))

#### Background charges and square root action functionals
 {#BackgroundCharge}

The [[supergravity C-field]] is an example of a general phenomenon of
higher abelian Chern-Simons QFTs in the presence of
_background charge_. This phenomenon was originally noticed
in ([Witten](#Witten96)) and then made precise in
([HopkinsSinger 05](#HopkinsSinger05)).
The holographic dual of this phenomenon is that of
self-dual higher gauge theories, which for the supergravity $C$-field
is the 2-form theory on the [[M5-brane]] -- see there for a discussion of this example. Here we discuss this effect generally, for higher abelian Chern-Simons theory in arbitrary dimension $4k+3$.

Fix some natural number $k \in \mathbb{N}$ and an [[orientation|oriented]] [[manifold]]
([[compact topological space|compact]] [[manifold with boundary|with boundary]]) $X$ of [[dimension]] $4 k + 3$.
The gauge equivalence class of a $(2k+1)$-form gauge field $\hat G$ on $X$ is an element in the [[ordinary differential cohomology]] group $\hat H^{2k+2}(X)$.
The [[cup product]] $\hat G \cup \hat G \in \hat H^{4k+4}(X)$ of this with itself  has a natural [[higher holonomy]] over $X$, denoted
$$
  \exp(i S (-)) : \hat H^{2k+2}(X) \to U(1)
$$
$$
  \hat G \mapsto \exp(i \int_X \hat G \cup \hat G)
  \,.
$$
This is the exponentiated [[action functional]] for bare $(4k+3)$-dimensional
abelian Chern-Simons theory, as discussed above.

Observe now that
the above action functional may be regarded as a
_[[quadratic form]]_ on the [[cohomology group]]
$\hat H^{2k+2}(X)$. The corresponding [[bilinear form]] is the
("[[secondary characteristic class|secondary]]",
since $X$ is of dimension $4k+3$ instead of $4k+4$)
_[[intersection pairing]]_

$$
  \langle -,-\rangle : \hat H^{2k+2}(X) \times \hat H^{2k+2}(X) \to U(1)
$$
$$
  (\hat a_1 , \hat a_2) \mapsto \exp(i \int_X \hat a_1 \cup \hat a_2 )
  \,.
$$

But note that from $\exp(i S(-))$ we do _not_ obtain a
_[[quadratic refinement]]_
of the pairing. A quadratic refinement is, by definition, a function
$$
  q : \hat H^{2k+2}(X) \to U(1)
$$
(not necessarily homogenous of degree 2 as $\exp(i S(-))$ is), such that
the [[intersection pairing]] is reobtained from it by the polarization formula
$$
  \langle \hat a_1, \hat a_2\rangle
    =
  q(\hat a_1 + \hat a_2)
  q(\hat a_1)^{-1}
  q(\hat a_2)^{-1}
  q(0)
  \,.
$$
If we took $q := \exp(i S(-))$, then the above formula would yield not
$\langle -,-\rangle$, but the square $\langle -,-\rangle^2$, given by
(the exponentiation of) _twice_ the integral.

The observation in ([Witten96](#Witten96)) was
that for the correct [[holographic principle|holographic]] physics, we need instead an action functional
which is indeed a genuine [[quadratic refinement]] of the [[intersection pairing]].

But since the [[differential cohomology|differential classes]] in $\hat H^{2k+2}(X)$ refine
_[[integral cohomology]]_, we cannot in general simply divide by 2 and
pass from $\exp( i \int_X \hat G \cup \hat G)$ to
$\exp( i  \int_X \frac{1}{2} \hat G \cup \hat G)$. The integrand in the
latter expression does not make sense in general in differential cohomology.
If one tried to write it out in the "obvious" local formulas one would
find that it is a functional on fields which is not
[[gauge invariance|gauge invariant]].
The analog of this fact is familiar from nonabelian $G$-[[Chern-Simons theory]]
with [[simply connected space|simply connected]] $G$, where also the theory is consistent only at
interger _levels_. The "level" here is nothing but the
underlying integral class $G \cup G$.

Therefore the only way to obtain a [[square root]] of the [[quadratic form]]
$\exp(i S(-))$ is to _shift its origin_. Here we think of
the analogy with a
quadratic form $q : x \mapsto x^2$ on the [[real numbers]] (a parabola in the plane).
Replacing this by $q^{\lambda} : x \mapsto x^2 + \lambda x$ for some real number
$\lambda$ means keeping the shape of the form, but shifting its [[minimum]] from 0 to
$-\frac{1}{2}\lambda$. If we think of this as the potential term for a scalar field
$x$ previously with rotation-symmetric dynamics about $x = 0$, then the new potential
exhibits [[spontaneous symmetry breaking]]: its ground state is now at $x = -\frac{1}{2}\lambda$
(and has [[energy]] $-\frac{1}{4}\lambda^2$ there). We may say that there is
a _background field_ or _background [[charge]]_ that pushes
the field out of its free equilibrium.

To lift this reasoning to our action quadratic form
$\exp(i S(-))$ on differential cocycles, we need a
differential class $\hat \lambda \in H^{2k+2}(X)$ such that
for every $\hat a \in H^{2k+2}(X)$ the composite class
$$
  \hat a \cup \hat a + \hat a \cup \hat \lambda
  \in
  H^{4k+4}(X)
$$
is even, hence is divisible by 2. Because then we could define a shifted
action functional

$$
  \exp(i S^\lambda(-)) : \hat a \mapsto \exp(i \int_X \frac{1}{2}(\hat a \cup \hat a + \hat a \cup \hat \lambda))
  \,,
$$

where now the fraction $\frac{1}{2}$ in the integrand does make sense. One directly
sees that if this exists, then this shifted action is indeed now a
[[quadratic refinement]] of the [[intersection pairing]]

$$
  \exp(i S^\lambda(\hat a + \hat b))
  \exp(i S^\lambda(\hat a))^{-1}
  \exp(i S^\lambda(\hat b))^{-1}
  \exp(i S^\lambda(0)
  =
  \exp(i \int_X \hat a \cup \hat b))
  \,.
$$

The condition on the existence of $\hat \lambda$ here means equivalently that the
image of the underlying integral class in cohomology with coefficients
in $\mathbb{Z}_2$ vanishes:
$$
   (a)_{\mathbb{Z}_2} \cup (a)_{\mathbb{Z}_2}
     +
   (a)_{\mathbb{Z}_2} \cup (\lambda)_{\mathbb{Z}_2}
   =
   0 \in H^{4k+4}(X, \mathbb{Z}_2)
  \,,
$$

Precisely such a class $(\lambda)_{\mathbb{Z}_2}$ does uniquely exist on
every oriented manifold. It is called the _[[Wu class]]_
$\nu_{2k+2} \in H^{2k+2}(X,\mathbb{Z}_2)$, and may be _defined_ by this condition.
Moreover, if $X$ is a [[spin structure|Spin-manifold]], then every second
[[Wu class]], $\nu_{4k}$, has a pre-image in [[integral cohomology]],
hence $\lambda$ does exist as required above

$$
  (\lambda)_{\mathbb{Z}_2}
    =
  \nu_{2k+2}
  \,.
$$

It is given by  [[polynomials]] in the [[Pontryagin classes]] of $X$. For instance
the degree-4 Wu class (for $k = 1$) is refined by the
[[first fractional Pontryagin class]]
$\frac{1}{2}p_1$

$$
  (\frac{1}{2}p_1)_{\mathbb{Z}_2} = \nu_4
  \,.
$$

This was the original observation in [Witten96, around (3.3)](#Witten96).

Notice that the [[equations of motion]] of the shifted action $\exp(i S^\lambda(\hat a))$
are no longer $F_a = 0$, but are now $F_a = - \frac{1}{2}F_\lambda$.
Comparing this to the [[Maxwell equations]], we see that $-\frac{1}{2}\hat \lambda$
here plays the role of a _background [[charge]]_ (or rather, of the
_background [[current]]_ that underlies a background charge). We therefore think of
$\exp(i S^\lambda(-))$ as the exponentiated action functional for
_higher dimensional abelian Chern-Simons theory with background charge
$-\frac{1}{2}\lambda$_.

This of course only makes sense if $X$ is such that
$\lambda$ is further divisible by 2, which we will
assume now. In ([Hopkins-Singer 05](#HopkinsSinger05)) is discussed a way to
make sense of the further division in general if one passes to a certain notion of  twisted differential cohomology. One can also adopt a different perspective and interpret the condition that $\frac{1}{2}p_1$ is further divisible by
2 precisely as a $\mathrm{String}^{2a}$-structure
([SSS3](http://ncatlab.org/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures)). This is a higher analog
of a [[Spin^c structure]].

With respect to the shifted action functional it makes sense to introduce the
shifted field

$$
  \hat G  := \hat a  + \frac{1}{2}\lambda
  \,.
$$

This is simply a re-parameterization such that the
Chern-Simons equations of motion again look homogenous, namely $FieldStrength(\hat G) = 0$.
In terms of this shifted field the action $\exp(i S^\lambda(\hat a))$
from above equivalently reads

$$
  \exp(i S^\lambda(\hat G)) =
  \exp(
    i \int_X \frac{1}{2}(\hat G \cup \hat G - (\frac{1}{2}\hat \lambda)^2)
  )
  \,.
$$

This is the form of the action functional first given as ([Witten96 (3.6)](#Witten96)) in for the case $k = 1$.

In the language of [[twisted differential c-structures]], we may summarize this sitation
as follows:

in order for the action functional of higher abelian Chern-Simons theory
to be correctly divisible, the images of the fields in $\mathbb{Z}_2$-cohomology
need to form a [[twisted Wu structure]]. Therefore the fields themselves
need to constitute a _twisted $\lambda$-structure_. For $k = 1$ this is
a [[twisted differential string structure|twisted string structure]] and explains for instance the quantization
condition on the [[supergravity C-field]] in [[11-dimensional supergravity]]. For that case see also the corresponding discussion at _[[M5-brane]]_.

## Higher-dimensional non-abelian CS theory

Chern-Simons actions for [[Lie algebra]]s $\mathfrak{g}$ but with higher-degree [[invariant polynomial]]s have in particular received attention for $\mathfrak{g} = \mathfrak{siso}$ the [[super Poincare Lie algebra]]. In this case these action functionals can be regarded as defining higher [[Chern-Simons supergravity]].

### 1d Chern-Simons theory

* [[1d Chern-Simons theory]]

### 2d Chern-Simons theory

* [[2d Chern-Simons theory]]

  * [[Poisson sigma-model]]

### 3d Chern-Simons theory

* [[3d Chern-Simons theory]]

  * [[Chern-Simons theory]]

  * [[Dijkgraaf-Witten model]]

  * [[Courant sigma-model]]

### 4d Chern-Simons theory

* [[4d Chern-Simons theory]]

  * [[Yetter model]]

### 5d Chern-Simons theory

* [[5d Chern-Simons theory]]

### 6d Chern-Simons theory

* [[6d Chern-Simons theory]]

### 7d Chern-Simons theory

* [[7d Chern-Simons theory]]

### 11d Chern-Simons theory

* [[11d Chern-Simons theory]]

### Infinite-dimensional Chern-Simons theory

* [[infinite-dimensional Chern-Simons theory]]

### AKSZ $\sigma$-models

* [[AKSZ sigma-model]]

### String field theory

* [[string field theory]]

## Related concepts

* [[Wu classes]]

* [[schreiber:infinity-Chern-Simons theory]]

  * [[Chern-Simons theory]]

  * **higher dimensional Chern-Simons theory**

    * [[Chern-Simons gravity]]

  * [[AKSZ sigma-model]]

    * [[Poisson sigma model]]

      * [[A-model]], [[B-model]]

    * [[Courant sigma-model]]

      * [[Chern-Simons theory]]

  * [[1-dimensional Chern-Simons theory]]

* [[higher dimensional WZW models]]

## References

### General

For higher dimensional non-abelian Chern-Simons theory see for instance

* Máximo Bañados, _Higher dimensional Chern-Simons theories and black holes_ ([pdf](http://worldscibooks.com/etextbook/4388/4388_chap01.pdf))

* Máximo Bañados, Luis Garay, [[Marc Henneaux]], _Existence of local degrees of freedom for higher dimensional pure Chern-Simons theories_ Phys. Rev. D 53, R593–R596 (1996)  ([pdf](http://prd.aps.org/pdf/PRD/v53/i2/pR593_1))

* Máximo Bañados, Luis Garay, [[Marc Henneaux]], _The dynamical structure of higher dimensional Chern-Simons theory_ ([arXiv:hep-th/9605159](http://arxiv.org/abs/hep-th/9605159))

* G Giachetta, L Mangiarotti, G Sardanashvily, _Noether conservation laws in higher-dimensional Chern-Simons theory_ Modern Physics Letters A Volume 18(2003) ([pdf](http://gnsardan.appfarm.ru/a84.pdf))

* Danhua Song, Mengyao Wu, Ke Wu, Jie Yang, *Higher Chern-Simons based on (2-)crossed modules*, JHEP **2023** 207 (2023) [[arXiv:2212.04667](https://arxiv.org/abs/2212.04667), doi:10.1007/JHEP07(2023)207]

### Formulation in differential cohomology

For the formulation of abelian Chern-Simons theory by fiber integration over cup products in [[ordinary differential cohomology]] see

* Enore Guadagnini, Frank Thuillier, _Deligne-Beilinson cohomology and abelian links invariants_ SIGMA 4 (2008), 078, 30 ([arXiv:0801.1445](http://arxiv.org/abs/0801.1445))
  {#GT}

* R. Floreanini, R. Percacci, _Higher dimensional abelian Chern-Simons theory_ Physics Letters B Volume 224, Issue 3, 29 (1989)

* [[Daniel Freed]], [[Gregory Moore]], [[Graeme Segal]], _The Uncertainty of Fluxes_ ([arXiv:hep-th/0605198](http://arxiv.org/abs/hep-th/0605198))
  {#FMS}

### Relation to self-dual theories

The idea of describing [[self-dual higher gauge theory]] by abelian Chern-Simons theory in one dimension higher originates in

* [[Edward Witten]], _Five-Brane Effective Action In M-Theory_ Journal of Geometry and Physics, Volume 22, Issue 2, May 1997, Pages 103-133  ([arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234))
 {#Witten96}

(there for the [[6d (2,0)-susy QFT]] on the [[fivebrane]]) and

* [[Edward Witten]], _Duality Relations Among Topological Effects In String Theory_ ([arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086))

Motivated by this the [[differential cohomology]] of self-dual fields had been discussed in

* {#HopkinsSinger05} [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_, 2005

More discussion of the general principle is in

* Dmitriy Belov, [[Greg Moore]], _Holographic Action for the Self-Dual Field_ ([arXiv:hep-th/0605038](http://arxiv.org/abs/hep-th/0605038))

The application of this to the description of
[[type II string theory]] in 10-dimensions to [[schreiber:infinity-Chern-Simons theory|11-dimensional Chern-Simons theory]] is in the followup

* Dmitriy Belov, [[Greg Moore]], _Type II Actions from 11-Dimensional Chern-Simons Theories_ ([arXiv:hep-th/0611020](http://arxiv.org/abs/hep-th/0611020))
 {#BelovMoore}

### Higher Chern-Simons (super)gravity

There are various discussions identifying or conjecturing higher dimensional Chern-Simons theories as parts of or related to [[gravity]] and [[supergravity]].

An original articles includes

* Máximo Bañados, Ricardo Troncoso, Jorge Zanelli, _Higher dimensional Chern-Simons supergravity_ Phys. Rev. D 54, 2605–2611 (1996) ([pdf](http://cdsweb.cern.ch/record/293524/files/9601003.pdf))

An introduction and survey is in

* Jorge Zanelli, _Lecture notes on Chern-Simons (super-)gravities_ ([arXiv:hep-th/0502193](http://arxiv.org/abs/hep-th/0502193))

For more references see at

* _[[Chern-Simons gravity]]_ .

### Higher Chern-Simons invariants
 {#ReferencesInvariants}

* F. Thuillier, _Deligne-Beilinson cohomology and abelian link invariants: torsion case_ J. Math. Phys. 50, 122301 (2009)

* B. Broda, _Higher-dimensional Chern-Simons theory and link invariants_ Physics Letters B Volume 280, Issues 3-4, 30 (1992) Pages 213-218

* L. Gallot, E. Pilon, F. Thuillier, _Higher dimensional abelian Chern-Simons theories and their link invariants_ ([arXiv:1207.1270](http://arxiv.org/abs/1207.1270))

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

Below (5.11) of ([Hopkins-Singer 05](#HopkinsSinger05)).

Section 4.1.4 of

* {#FRS} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher Chern-Weil Derivation of AKSZ Sigma-Models]]_

Section 4.4 of

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

### Boundary theories

Boundary [[higher dimensional WZW models]] for nonabelian [[higher dimensional Chern-Simons theory]] are discussed in

* J. Gegenberg , G. Kunstatter, _Boundary Dynamics of Higher Dimensional Chern-Simons Gravity_ ([arXiv:hep-th/0010020](http://arxiv.org/abs/hep-th/0010020))

* J. Gegenberg , G. Kunstatter, _Boundary Dynamics of Higher Dimensional AdS Spacetime_ ([arXiv:http://arxiv.org/abs/hep-th/9905228](http://arxiv.org/abs/hep-th/9905228))

[[!redirects higher Chern-Simons theory]]
[[!redirects higher Chern-Simons theories]]

[[!redirects higher dimensional Chern-Simons theories]]


higher dimensional transition system

Higher dimensional transitions systems are thought of as modelling concurrent operation of interacting [[transition system]]s. They are related to applications of [[directed homotopy theory]], and [[directed algebraic topology]].

We will give the definition that Gaucher uses (see below).  The version of Cattani and Sassone includes some extra conditions that yield a structure nearer to that of a process algebra.

A non-empty set, $\Sigma$, of labels is fixed.

##Definition:

 A weak higher dimensional transition system is a triple, $(S,\mu:L\to \Sigma,T)$, where $S$ is a set of states, $L$ a set of actions, $\mu$ is a labelling map and $T = \bigcup_{n\geq 1}T_n$, where $T_n\subset S\times L^n\times S$ is a set of $n$-transitions, such that two axioms hold, that ensure that the $n$-transitions are closed under permutations of labels and also satisfy a coherence (patching) axiom.

The case $n=1$ corresponds to [[transition system]]s.

##References

* [[Gian Luca Cattani]], [[Vladimiro Sassone]]  _[Higher Dimensional Transition Systems](http://eprints.soton.ac.uk/261954/1/hdtsLICSoff.pdf)_ In, 11th Symposium of Logics in Computer Science, LICS '96. IEEE Press, 55-62,(1996).

* [[Philippe Gaucher]], _Directed algebraic topology and higher dimensional transition system_, New-York Journal of Mathematics 16 (2010), 409 - 461, available as [arxiv:0903.4276](http://arxiv.org/abs/0903.4276)

*  [[Philippe Gaucher]], _Towards a homotopy theory of higher dimensional transition systems_, Theory and Applications of Categories, Vol. 25, No. 12, 2011, pp. 295–341, available as [arxiv:1011.0918](http://arxiv.org/abs/1011.0918)

[[!redirects higher dimensional transition systems]]

category:computer science


higher dimensional WZW model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Wess-Zumino-Witten theory
+--{: .hide}
[[!include infinity-Wess-Zumino-Witten theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Analogs of the 2d [[WZW model]] in higher dimension.

## Examples

* [[4d WZW model]]

* [[Green-Schwarz sigma-model for super p-branes]]

  * [[Hopf-Wess-Zumino term]] for [[M5-branes]]

## Related concepts

* [[definite globalization of WZW term]]

## References

* [[Dan Freed]], _Pions and generalized cohomology_, J. Differential Geom. Volume 80, Number 1 (2008), 45-77. ([arxiv:hep-th/0607134](http://arxiv.org/abs/hep-th/0607134))

* Takeo Inami, Hiroaki Kanno and Tatsuya Ueno, _Higher-dimensional WZW Model
on Kähler Manifold and Toroidal Lie Algebra_ ([arXiv:hep-th/9704010](http://arxiv.org/abs/hep-th/9704010))

Boundary higher dimensional WZW models for nonabelian [[higher dimensional Chern-Simons theory]] are discussed in

* J. Gegenberg , G. Kunstatter, _Boundary Dynamics of Higher Dimensional Chern-Simons Gravity_ ([arXiv:hep-th/0010020](http://arxiv.org/abs/hep-th/0010020))

* J. Gegenberg , G. Kunstatter, _Boundary Dynamics of Higher Dimensional AdS Spacetime_ ([arXiv:hep-th/9905228](http://arxiv.org/abs/hep-th/9905228))

Discussion of [[symmetry protected topological order]] phases of matter in [[solid state physics]] via higher WZW models is in

* Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, [[Xiao-Gang Wen]], _Symmetry protected topological orders and the group cohomology of their symmetry group_, Phys. Rev. B 87, 155114 (2013) ([arXiv:1106.4772](http://arxiv.org/abs/1106.4772))

The global formulation (cancelling [[classical anomalies]]) is discussed in

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]] _[[schreiber:The brane bouquet]]_, International Journal of Geometric Methods in Modern Physics, Vol. 12 (2015) 1550018 ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

[[!redirects higher dimensional WZW models]]
[[!redirects higher dimensional WZW theory]]
[[!redirects higher dimensional WZW theories]]

[[!redirects higher WZW sigma-model]]
[[!redirects higher WZW sigma-models]]

[[!redirects higher WZW model]]
[[!redirects higher WZW models]]

[[!redirects higher WZW term]]
[[!redirects higher WZW terms]]

[[!redirects higher WZ term]]
[[!redirects higher WZ terms]]


higher direct image

The higher direct image functors are the [[derived functor]]s of the [[pushforward]] functor.


higher doctrine

# Higher doctrines
* table of contents
{: toc}

## Doctrinal logic

The notion of logical [[theory]] in categorical logic, that is given by a 2-category of categories with additional structures, due to Lawvere and Ehresmann, may be straightforwardly generalized using the metalanguage of higher and homotopical (multi)-categories, in a way that is directly inspired by the Baez--Dolan categorification method.

This setting, combined with a higher Yoneda lemma, is very natural to treat the semantics of mathematical theories in a coordinate free fashion, characterizing a theory by its universal property. This gives unicity statements.

The main problem of the notion of higher doctrine, is to find a proper notion of syntax for the doctrines in play, that allows one to define the notion of _coherent theory_. This problem is already solved for 2-doctrines by Ehresmann's notion of sketches, and may be solved by using higher monads, or other descriptions of higher homotopical categories by [[generators and relations]]. This main problem is presently solved only in particular cases.

When combined with the _axiom of choice_ and the categorification of classical methods of mathematical logic given by [[model theory]] (following Goedel's groundbreaking work), this formalism can be used to prove general existence results, for the models of coherent theories in doctrines with good finiteness properties. An example of this general approach is given by Deligne's theorem about the existence of enough points for a coherent topos.
Remark that if one does not want to ground the categorical metalanguage on set theory, one may also formulate a doctrinal axiom of choice, once defined a precise notion of coherent theory:

+-- {: .un_axiom}
###### Doctrinal axiom of choice
Every coherent theory has at least a model.
=--

Basing mathematics on this ground, one can define the notion of
well posed mathematical problem,
generalizing Hadamard's approach to the partial differential systems of mathematical physics: a mathematical problem/theory is well posed if it has a unique solution.

Remark that showing that a mathematical problem is well posed in the above sense may be a bit harder than solving it directly by hand, but both actions are homotopically equivalent, in the sense of doctrinal logic, as you may be able to prove it, with some unworthy efforts.

This modest program of generalizing model theory methods to higher doctrines may be put at the heart of a categorical foundation of mathematics, a la Lawvere, but with a more concrete flavor, since the methods also allow to prove existence results (for objets, theories, and even proofs), under finiteness (coherence) hypothesis.

Remark that this abstract nonsense  may also be pedagogically very helpful to understand on a conceptual way the recent and numerous applications of higher categorical methods in mathematics and physics.

## Definition and challenges

Suppose given a metalanguage of homotopical higher (and even multi-) categories, that we will call simply higher categories.

+-- {: .un_defn}
###### Definition
A __doctrine__ $D$ is an $(n+1)$-category whose objects are $n$-categories equipped with some specified finitary structure.  A theory of type $D$ is an object $T$ of $D$. A model for a theory of type $D$ in another one is a morphism
$$M\colon T_1 \to T_2.$$
=--

The main challenges of higher doctrine theory are the following:

* Define a precise notion of (possibly homotopical) syntax for a given theory
(analog of a site underlying a topos, or of a sketch underlying a theory a la Ehresmann). This is what it meant by the generators and relations data.

* Basing on the above, give a precise definition of coherent theory (finiteness property for the syntax) in a given doctrine.

* Prove that, under the set theoretic axiom of choice and the formalization of higher categories through set theoretical methods (i.e., by one of the standard delooping machines), the doctrinal axiom of choice is true.

## Examples

It may be interesting to show that these methods can be used to study concrete mathematical problems:

1. A good testing example is the following: prove that the problem of proving the representability of the projective space $P^1$ (or grassmanians) is well posed, by showing that it is the unique (by Yoneda) model of a coherent theory (the theory of lines in the affine space of the big Zariski topos on rings) in a well chosen doctrine.

2. Another example is: prove that the fixed point theorem in functional analysis of Banach spaces is the solution of a well posed mathematical problem (or more specifically, that the limit of a cauchy filter in a complete uniform structure, à la Bourbaki, is an existing model of a theory associated to the uniform space, that is coherent by completeness of the uniform structure).

3. Show that compactness arguments in analysis (those saying that one can extract a convergent subsequence from a sequence on a compact topological space) may be formalized by saying that the subsequence is a model of a theory associated to the topological space, that is coherent because of its
compactness.

## References

* See the Section [[doctrine]] for historical background and refined definitions in lower dimension, and

* [[Frédéric Paugam]]: Chapter 1 of
[Towards the mathematics of quantum field theory](http://people.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf).

[[!redirects higher doctrine]]
[[!redirects higher doctrines]]
[[!redirects Higher doctrine]]
[[!redirects Higher doctrines]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[action functional]] for [[higher U(1)-gauge theory]] in the presence of background [[electric charge]] contains a charge-coupling term which is of [[schreiber:infinity-Chern-Simons theory]]-type.

## Definition

Let $X$ be a [[smooth manifold]] of [[dimension]] $d$ and let $n \in \mathbb{N}$. Then a [[higher U(1)-gauge field|degree n U(1)-gauge field]] on $X$ is a [[circle n-bundle with connection]] $\hat F : X \to \mathbf{B}^n U(1)_{conn}$.

### For smooth currents

A background [[electric current]] for this is a circle $(d-n-1)$-bundle with connection $\hat j_{el} : X \to \mathbf{B}^{d-n-1} U(1)_{conn}$.

The coupling [[action functional]] is

$$
  \exp(i S_{el}(-)) : \hat F \mapsto \exp(i \int_X \hat F \cup \hat j)
$$

given by the [[higher holonomy]]/[[fiber integration in ordinary differential cohomology]] of the [[Beilinson-Deligne cup product]] of the gauge field with the higher electric background.

### For $\delta$-distributed charges

The object $\hat j_{el}$ above models the [[electric current]] of a smooth density of charged electric [[brane|(n-1)-branes]]. If we think of the current form $j_{el}$ as being a [[delta distribution]] on the [[worldvolume]] $\Sigma \to X$ of a single charged  [[brane|(n-1)-branes]], then (one may thing of this via [[Poincare duality]]) the electric charge coupld action functional becomes the [[higher holonomy]] of the [[higher U(1)-gauge field]] over $\Sigma$

$$
  \exp(i S_{el}(-)) : \hat F \mapsto hol_\Sigma(\hat F)
  \,.
$$

If, moreover, we restrict attention to gauge field configurations whose underlying [[circle n-bundle]] is trivial, which are given by globally defined [[differential form|n-forms]] $A$ (with $d A = F$), then this is

$$
  \cdots = \exp(i \int_\Sigma A)
  \,.
$$

In the form of this simple special case the higher electric background charge coupling is often presented in [[physics]] texts.

## References

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


higher gauge field

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An ordinary [[gauge field]] (such as the [[electromagnetic field]] or the fields that induce the [[nuclear force]]) is a [[field (physics)|field (in the sense of physics)]] which is locally represented by a [[differential 1-form]] (the "[[gauge potential]]") and whose [[field strength]] is locally a [[differential 2-form]]. For instance, in the case of the [[electromagnetic field]] this [[differential 2-form]] is the _[[Faraday tensor]]_.

Roughly speaking, a _higher gauge field_ is similarly a field which is locally represented by [[differential forms]] of higher degree.

An explanation as to why an ordinary [[gauge field]] has a [[gauge potential]] given locally by a [[differential 1-form]] $A$ is that the [[trajectory]] of a [[charged particle]] is a 1-dimensional [[curve]] in [[spacetime]] $X$, its _[[worldline]]_, hence a [[smooth function]] $\gamma \colon \Sigma_1 \to X$, and the canonical way to produce an [[action functional]] on the [[mapping space]] of such curves is the [[integration of differential forms|integration of 1-forms]] over curves:

$$
  \exp(\tfrac{i}{\hbar} S_{gauge})
  \;\colon\;
  \gamma
  \mapsto
  P \exp\left(
    \int_{\Sigma_1} \gamma^\ast A
  \right)
  \,.
$$

This is the _[[parallel transport]]_ map or the _[[holonomy]]_ map, if $\Sigma_1$ is a [[closed manifold|closed manifold]]. The contribution to the [[Euler-Lagrange equation]] of the particle obtained from the [[variational calculus|variation]] of this [[action functional]] is the _[[Lorentz force]]_ which is exerted by the [[background field|background]] [[gauge field]] on the particle.

When one generalizes in this picture from 0-dimensional [[particles]] with 1-dimensional [[worldlines]] to $p$-dimensional particles (often called "[[p-branes]]") with $(p+1)$-dimensional [[worldvolumes]] $\gamma_{p+1} \colon \Sigma_{p+1} \to X$, then one needs, locally, a [[differential n-form|differential (p+1)-form]] $A_{p+1}$ on [[spacetime]] $X$

$$
  \exp(\tfrac{i}{\hbar} S_{higher\,gauge})
  \;\colon\;
  \gamma
  \mapsto
  P \exp\left(
    \int_{\Sigma_{p+1}} \gamma_{p+1}^\ast A_{p+1}
  \right)
  \,.
$$

The [[field strength]] or _[[flux]]_ of such a higher gauge field is, accordingly, locally the $(p+2)$-form $F_{p+2}$.

The archetypical example of such a higher gauge field is the (hypothetical) _[[Kalb-Ramond field]]_ or _[[B-field]]_ (a precursor of the [[axion]] field under [[KK-compactification]]) to which the charged 1-brane, the "[[string]]", couples. This is locally a [[differential 2-form]] $B_2$, and the gauge-coupling term in the [[action functional]] for the string is accordingly, locally, of the form

$$
  \exp(\tfrac{i}{\hbar} S_{stringy\,gauge})
  \;\colon\;
  \gamma
  \mapsto
  P \exp\left(
    \int_{\Sigma_{2}} \gamma_2^\ast B_2
  \right)
  \,.
$$

This continues: next one may consider "2-branes", i.e. _[[membranes]]_, and these will couple to a 3-form gauge field. For instance, the membrane which gives the name to _[[M-theory]]_ (the [[M2-brane]]) couples to a 3-form field called the _[[supergravity C-field]]_.

But there is an important further aspect to higher gauge fields which makes this simple picture of higher degree differential forms drastically more rich:

Where an ordinary [[gauge field]] has [[gauge transformations]] $A_1 \mapsto A'_1$ given locally by smooth functions (0-forms) $\lambda_0$ via the [[de Rham differential]] $d_{dR}$

$$
  A'_1 = A_1 + d_{dR} \lambda_0
$$

so a higher gauge field has [[higher gauge transformations]] given locally by $p$-forms $\lambda_p$:

$$
  A'_{p+1} = A_{p+1} + d_{dR} \lambda_{p}
  \,.
$$

But for $p \geq 0$ then a crucial new effect appears: these gauge transformations, being higher differential forms themselves, have "[[higher gauge transformation|gauge-of-gauge transformations]]" between them, given by lower degree forms.

This phenomenon implies that higher gauge fields have a rich _global_ ("topological") structure, witnessed by the higher analog of their [[instanton sectors]]. Namely, while a higher gauge field to which a [[p-brane]] may couple is _locally_ given by a $(p+1)$-form $A_{p+1}$, as one moves across [[coordinate charts]] this form gauge transforms by a $p$-form, which then itself, as one passes along two charts, transforms by a $(p-1)$-form, and so on.

The global structure for higher gauge fields obtained by carrying out this globalization via [[higher gauge transformations]] is the higher analog of that of a [[fiber bundle]] with [[connection on a bundle]] in [[higher differential geometry]]. This is sometimes known as a _[[gerbe]]_ or, more generally, a _[[principal infinity-bundle]]_.

In fact the situation that there is just one [[gauge potential]] of degree $(p+1)$ with [[field strength]] of degree $(p+2)$ is just the simplest case, the "ordinary" case.  More abstractly one says that such higher gauge fields are [[cocycles]] in _[[ordinary differential cohomology]]_.

More generally it may happen in [[higher gauge theory]] that the gauge potential is a [[formal linear combination]] of differential forms in various degrees.

The canonical example of this phenomenon is the [[RR-field]] in [[string theory]]. This has, locally, a gauge potential which is a differential form in every even degree, or every odd degree. If one is careful about the [[higher gauge transformations]] in this situation to find the correct global structure (the "[[instanton sector]]") of the higher gauge field, then one finds that this now is a [[cocycle]] in a [[differential cohomology|differential]] _[[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]]_, namely, in what is called [[differential K-theory|differential]] [[topological K-theory]]. This may be understood as a higher and generalized form of the famous [[Dirac charge quantization]] condition for the [[electromagnetic field]], see [Freed 00](#Freed00). A lot of the fine detail of the [[anomaly cancellation]] in [[type II string theory]] depends on being careful about the global nature of this [[K-theory|K-theoretic]] higher gauge [[RR-field]] ([Distler-Freed-Moore 09](#DistlerFreedMoore09))

## Examples
 {#Examples}

Among ordinary [[gauge theories]] there is

1. "metric" gauge theory:

   1. [[Maxwell theory]],

   1. [[Yang-Mills theory]]

1. [[topological field theory|topological]] gauge theory:

   * [[Chern-Simons theory]]

(as well as mixed cases, such as the [[ABJM model]]).

While the [[field (physics)|fields]] of these theories are [[gauge fields]] of the same kind in all cases (modeled by [[connection on a bundle|connections]] on [[principal bundles]]), the nature of these theories is different, not the least because the first class depends on (couples to) a [[background field|background]] ([[pseudo-Riemannian metric|pseudo]]) [[Riemannian metric]] ([[gravity]]) while the second does not (is a [[topological field theory]]).

Moreover, even though [[Maxwell theory]] ([[electromagnetism]]) may be understood as the special case of [[Yang-Mills theory]] for [[gauge group]] specialized to the [[circle group]] $U(1)$, this makes a crucial difference for the nature of these theories and their higher generalizations.

Some example of higher gauge fields...

1. ...of higher Maxwell type:

   the fields appearing in higher-dimensional [[supergravity]] theories:

   * [[Kalb-Ramond field]] ([[B-field]])

   * [[RR-field]]

   * [[supergravity C-field]]

1. ...of higher Chern-Simons type:

   * [[AKSZ sigma-models]]

   * [[string field theory|string field]]

### Higher gauge theories of Maxwell type
 {#HigherGaugeTheoryOfMaxwellType}

Consider

* $D = 1 + d \in \mathbb{N}_{\geq 2}$ the [[spacetime]] dimension,

* $X^D$ a $D$-dimensional [[spacetime]] [[smooth manifold|manifold]],

* with [[Hodge star]]-operator on [[differential forms]]

  $$
    \star \,\colon\,
    \Omega^p_{dR}(X^D) \to \Omega^{D-p}_{dR}(X^D)
  $$

  satisfying (see [there](Hodge+star+operator#eq:HodgeSquareOnRiemannian))

  $$
    \star \, \star = - (-1)^{p(D-p)}
    \,,
  $$

* $I \,\in\, Set$ an [[index set]] of flux species

  (subsuming both electric fluxes and magnetic fluxes, both now allowed of further different species),

* $\big( deg_i \,\in\, \mathbb{N}_{\geq 1} \big)_{i \in I}$ an [[indexed set]] of degrees,

* $\vec F \,\equiv\,\big( F^{(i)} \,\in\, \Omega^{deg_i}_{dR}(X^D) \big)_{i \in I}$ an [[indexed set]] of [[flux densities]]

* $\vec P = \big( P^{(i)}  \big)_{i \in I}$ an [[indexed set]] of graded-symmetric [[polynomials]] in $I$ [[variables]],

* $\vec \mu$ an [[invertible matrix|invertible]] $I \times I$-[[matrix]]

then the corresponding **higher Maxwell-type equations** on these [[differential forms]]/[[flux densities]] are, in [[duality-symmetric higher gauge theory|duality-symmetric form]] (cf. [SS23](#SS23)):

1. the system of [[Bianchi identity]] [[differential equations]] (cf. at *[[Gauss law]]*)

   \[
     \label{HigherBianchiIdentities}
     \mathrm{d}\,\vec F
     \,=\,
     \vec P\big(
       \vec F
     \big)
   \]

1. the self-[[Hodge duality]] condition

   \[
     \label{HigherSelfDuality}
     \star \, \vec F
     \,=\,
     \vec \mu\big(\vec F\big)
     \,.
   \]

The full field content of a higher gauge theory with these flux densities is to involve *[[gauge potentials]]* for these flux densities. A systematic way to get hold of these is to choose a [[flux quantization law]] for the given flux densities, defining a ([[nonabelian cohomology|nonabelian]]) [[generalized cohomology theory]]. Then a full higher gauge field is a [[cocycle]] in the corresponding ([[nonabelian differential cohomology|nonabelian]]) [[differential cohomology]].

### Higher gauge theories of Chern-Simons type
 {#OfChernSimonsType}

See at *[[schreiber:infinity-Chern-Simons theory]]*

## Related entries

* [[differential nonabelian cohomology]]

* [[Wilson surface]]

* [[local prequantum field theory]]

* [[higher category theory and physics]]

* [[adjusted Weil algebra]]

## References

### General

Introduction and exposition:

* {#BaezHuerta10} [[John Baez]], [[John Huerta]], _An invitation to higher gauge theory_, General Relativity and Gravitation 43 (2011), 2335-2392 ([arXiv:1003.4485](https://arxiv.org/abs/1003.4485))

* [[Luigi Alfonsi]], §2 in: *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]

* [[Leron Borsten]], [[Mehran Jalali Farahani]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Jiří Nárožný]], [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], *Higher Gauge Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.05275](https://arxiv.org/abs/2401.05275)]

* PhysicsForums-Insights _[Why higher category theory in physics?](https://www.physicsforums.com/insights/higher-category-theory-physics/)_

* _[[geometry of physics]]_

* _[[twisted smooth cohomology in string theory]]_

* _[Examples of prequantum field theories II: Higher gauge fields](https://www.physicsforums.com/insights/examples-prequantum-field-theories-ii-higher-gauge-fields/)

For technical introduction to the [[RR-field]] as a higher gauge field (for more see at *[[Dirac charge quantization]]*):

* {#Freed00} [[Daniel Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129–194 ([arxiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220))

* {#DistlerFreedMoore09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

Deriving [[higher gauge symmetries]] through the [[variational bicomplex]]:

* [[Igor Khavkine]], *Topology, rigid cosymmetries and linearization instability in higher gauge theories*, Annales Henri Poincaré **16** (2015) 255–288, [[arXiv:1303.2406](https://arxiv.org/abs/1303.2406), [doi:10.1007/s00023-014-0321-9](https://doi.org/10.1007/s00023-014-0321-9)]

The aspect of [[flux quantization laws]]:

* {#SS23} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Flux Quantization on Phase Space]]* [[arXiv:2312.12517](https://arxiv.org/abs/2312.12517)]

Foundations of higher [[prequantum field theory]]:

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_

For foundations of higher gauge theory formalized in [[homotopy type theory]] see

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

On higher gauge theory in [[condensed matter physics]]:

* [[Chenchang Zhu]], Tian Lan, [[Xiao-Gang Wen]], _Topological non-linear σ-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems_ [[arXiv:1808.09394](https://arxiv.org/abs/1808.09394)]

* J.P. Ang, Abhishodh Prakash, _Higher categorical groups and the classification of topological defects and textures_, ([ arXiv:1810.12965](https://arxiv.org/abs/1810.12965))

Specifically on [[topological phases of matter]] via higher  [[lattice gauge theory]]:

* [[Joe Huxford]], *The higher lattice gauge theory model for topological phases of matter*, PhD thesis, Oxford (2022) [[uuid:e789bf29-179b-4b79-9168-605bf8c035ba](https://ora.ox.ac.uk/objects/uuid:e789bf29-179b-4b79-9168-605bf8c035ba), [pdf](https://ora.ox.ac.uk/objects/uuid:e789bf29-179b-4b79-9168-605bf8c035ba/files/ddj52w5202)]

* [[Joe Huxford]], [[Steven H. Simon]], *Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview* [[arXiv:2202.08294](https://arxiv.org/abs/2202.08294)]

Further on higher [[lattice gauge theory]]:

* [[Juan Orendain]], [[José A. Zapata]], *Higher homotopy and lattice gauge fields* [[arXiv:2311.02363](https://arxiv.org/abs/2311.02363)]

[[!include higher gauge theory of the Green-Schwarz mechanism -- references]]

[[!redirects higher gauge fields]]

[[!redirects higher gauge theory]]
[[!redirects higher gauge theories]]

[[!redirects higher gauge symmetry]]
[[!redirects higher gauge symmetries]]


higher gauge theory of the Green-Schwarz mechanism -- references

### Higher gauge theory of the Green-Schwarz mechanism
  {#HigherGaugeTheoryOfTheGreenSchwarzMechanismReferences}

Discussion of [[higher gauge theory]] modeling the [[Green-Schwarz mechanisms]] for [[anomaly cancellation]] in [[heterotic string theory]], on [[M5-branes]], and in related systems in terms of some kind of [[nonabelian differential cohomology]] (ordered by arXiv time-stamp):

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], pp. 13 in: *$L_\infty$-algebra connections and applications to String- and Chern-Simons $n$-transport*, in *Quantum Field Theory*, Birkhäuser (2009) 303-424 \[arXiv:0801.3480, doi:10.1007/978-3-7643-8736-5_17\]

* {#SatiSchreiberStasheff12} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Twisted Differential String and Fivebrane Structures]]_, Comm. Math. Phys. **315** (2012) 169-213 ([arXiv:0910.4001](https://arxiv.org/abs/0910.4001), [doi:10.1007/s00220-012-1510-3](https://link.springer.com/article/10.1007/s00220-012-1510-3))

  > (via [[adjusted Weil algebras]], see there for more)

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], §3.7, §3.8 in: *Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory*, Adv. Theor. Math. Phys. **18** (2014) 229-321 \[arXiv:1201.5277, [euclid:atmp/1414414836](https://projecteuclid.org/journals/advances-in-theoretical-and-mathematical-physics/volume-18/issue-2/Multiple-M5-branes-String-2-connections-and-7d-nonabelian-Chern/atmp/1414414836.full)\]

* {#FSS15} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:The moduli 3-stack of the C-field|The $E_8$ moduli 3-stack of the C-field in M-theory]]*, Comm. Math. Phys. **333** 1 (2015) 117-151 \[arXiv:1202.2455, [doi:10.1007/s00220-014-2228-1](https://doi.org/10.1007/s00220-014-2228-1)\]

* [[Clay Cordova]], [[Thomas Dumitrescu]], [[Kenneth Intriligator]], _Exploring 2-Group Global Symmetries_,  J. High Energ. Phys. **2019** 184 (2019) ([arXiv:1802.04790](https://arxiv.org/abs/1802.04790), doi:10.1007/JHEP02(2019)184)

* {#BeniniCordovaHsin19} [[Francesco Benini]], [[Clay Cordova]], [[Po-Shen Hsin]], *On 2-Group Global Symmetries and Their Anomalies*,  J. High Energ. Phys. **2019** 118 (2019) ([arXiv:1803.09336](https://arxiv.org/abs/1803.09336), doi:10.1007/JHEP03(2019)118)

* {#FSS22} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation]]*, Reviews in Mathematical Physics **34** 05 (2022) 2250013 \[doi:10.1142/S0129055X22500131, [arXiv:2008.08544](https://arxiv.org/abs/2008.08544)\]

* [[Clay Cordova]], [[Thomas T. Dumitrescu]], [[Kenneth Intriligator]], *2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories*,  J. High Energ. Phys. 2021, 252 (2021) ([arXiv:2009.00138](https://arxiv.org/abs/2009.00138), doi:10.1007/JHEP04(2021)252)

* [[Michele Del Zotto]], [[Kantaro Ohmori]], _2-Group Symmetries of 6D Little String Theories and T-Duality_, Annales Henri Poincaré **22** (2021) 2451–2474  \[arXiv:2009.03489, doi:10.1007/s00023-021-01018-3\]

* {#SS20} [[Hisham Sati]], [[Urs Schreiber]], *The character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes* \[arXiv:2011.06533\]

* [[Hisham Sati]], [[Urs Schreiber]], §2.9 in: *[[schreiber:M/F-Theory as Mf-Theory]]*, Reviews in Mathematical Physics **35** 10 (2023) \[doi:10.1142/S0129055X23500289, [arXiv:2103.01877](https://arxiv.org/abs/2103.01877)\]

* [[Yasunori Lee]], [[Kantaro Ohmori]], [[Yuji Tachikawa]], *Matching higher symmetries across Intriligator-Seiberg duality*,  J. High Energ. Phys. **2021** 114 (2021) \[arXiv:2108.05369, doi:10.1007/JHEP10(2021)114\]

* Monica Jinwoo Kang, Sungkyung Kang, *Central extensions of higher groups: Green-Schwarz mechanism and 2-connections* \[arXiv:2311.14666\]


higher gauge theory)..


higher gauge transformation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### Equality and Equivalence
+-- {: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[gauge theory]] two configurations $\phi_1, \phi_2$ of [[gauge fields]] may be different and still be [[equivalence|equivalent]]: there may be a _[[gauge transformation]]_ $\lambda \colon \phi_1 \to \phi_2$ between them.

In _[[higher gauge theory]]_ also gauge transformations themseves may be different but still equivalent: if there is a **gauge-of-gauge transformation** $\rho \colon \lambda_1 \to \lambda_2$ between them.

These higher order gauge transformations are maybe best known in the [[physics]] literature in terms of their [[infinity-Lie theory|infinitesimal approximation]], the [[BRST complex]]: here the gauge transformations correspond to _ghost fields_ and the _gauge-of-gauge transformations_ to _ghost-of-ghost fields_.

## Examples

A basic example of a gauge field that has higher order gauge transformations is the [[B-field]]. But also [[magnetic current]], if described properly, exhibits higher gauge transformations, see at _[[Dirac charge quantization]]_.

For more see at _[[geometry of physics]]_.

## Related concepts

* [[gauge]]

* [[higher gauge theory]]

* [[higher homotopy]]

## References

Survey of [[higher gauge theory]]:

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

[[!redirects higher gauge transformations]]
[[!redirects gauge-of-gauge transformation]]
[[!redirects gauge-of-gauge transformations]]

[[!redirects gauge-of-gauge equivalence]]
[[!redirects gauge-of-gauge equivalences]]

[[!redirects gauge-of-gauge symmetry]]
[[!redirects gauge-of-gauge symmetries]]

[[!redirects infinitesimal gauge-of-gauge equivalence]]
[[!redirects infinitesimal gauge-of-gauge equivalences]]

[[!redirects infinitesimal gauge-of-gauge symmetry]]
[[!redirects infinitesimal gauge-of-gauge symmetries]]


higher generation by subgroups

# Higher generation by subgroups
* table of contents
{: toc}

## Idea

(This is partially an account of the paper by Abels and Holz (1993) listed in the references below. It is augmented with some simple examples and discussion.)

For a [[group]] $G$ and a family, $\mathcal{H}$, of [[subgroups]] of $G$, consider the [[nerve]] of the covering of the set of elements of $G$ by the [[cosets]] $H g$, for $H\in \mathcal{H}$.  Although this is a Čech nerve (and thus we have an established notation for nerves as in [[Čech method]]), we will denote it $N(\mathfrak{H})$ as in the source.

*  The group $G$ is generated by the union of the subgroups $H$ in $\mathcal{H}$ if and only if $N(\mathfrak{H})$ is connected.

*  The group $G$ is the coproduct of the subgroups, amalgamated along their intersections, if and only if $N(\mathfrak{H})$ is simply connected.

The point of the article is to explore the consequences of the  higher connectivity of $N(\mathfrak{H})$, corresponding to 'higher generation by the subgroups', but, from the nPOV, perhaps the real questions are to ask in what way does the homotopy type of $N(\mathfrak{H})$ influence the properties of $G$.

The techniques used include the bar resolution construction and homotopy colimits. There are applications to Tits systems and to [[buildings]].

## The nerve of a family of subgroups

We start with a group, $G$, and a family, $\mathcal{H} = \{H_i\mid i\in I\}$ of subgroups of $G$.  Each subgroup, $H$, determines a family of right cosets, $H_g$, which cover the _set_, $G$.  Of course, these partition $G$, so there are no non-trivial intersections between them.  If we use _all_ the right cosets, $H_ig$, for _all_ the $H_i$ in $\mathcal{H}$, then, of course, we expect to get non-trivial intersections.

Let $\mathfrak{H} = \coprod_{i\in I}H_i\backslash G= \{ H_i g\mid H_i\in \mathcal{H}\},$
where the $g$ is more as an indicator of right cosets than strictly speaking  an index. This is the family of all right cosets of subgroups in $\mathcal{H}$.  This covers $G$ and we write $N(\mathfrak{H})$ for the corresponding simplicial complex, which is the _nerve_ of this covering.

More formally:

+-- {: .un_defn}
###### Definition

Let $G$ be a group and $\mathcal{H}$ a family of subgroups of $G$.  Let $\mathfrak{H}$ denote the corresponding covering family of right cosets, $H_ig$, $H_i \in \mathcal{H}$. (We will write $\mathfrak{H} = \mathfrak{H}(G,\mathcal{H})$ or even $\mathfrak{H} = (G,\mathcal{H})$, as a shorthand as well.) The _nerve_ of $\mathfrak{H}$ is the [[simplicial complex]], $N(\mathfrak{H})$, whose vertices are the cosets, $H_ig$, $i \in I$, and where a non-empty finite family, $\{H_ig_i\}_{i\in J}$, is a simplex if it has non-empty intersection.
=--

+-- {: .un_example}
###### Examples

1. If $\mathcal{H}$ consists just of one subgroup, $H$, then $\mathfrak{H}$ is just the set of cosets, $H\backslash G$ and $N(\mathfrak{H})$ is 0-dimensional, consisting just of 0-simplices / vertices.

2. If $\mathcal{H} = \{H_1,H_2\}$, (and $H_1$ and $H_2$ are not equal!), then any right $H_1$ coset, $H_1g$, will intersect some of the right $H_2$-cosets, for instance, $H_1g\cap H_2g$ always contains $g$.  The nerve, $N(\mathfrak{H})$, is a bipartite graph, considered as a simplicial complex. (The number of edges will depend on the sizes of $H_1$, $H_2$ and $H_1\cap H_2$, etc.) It is just a graphical way of  illustrating the intersections of the cosets, a sort of intersection diagram.

As a specific very simple example, consider:

*  $S_3 \equiv ( a,b : a^3=b^2=(ab)^2=1)$, (so $a$ denotes, say, the 3-cycle $(1 2 3)$ and $b$, a transposition $(1 2)$).

     *  Take $H_1 = \langle a \rangle = \{1, (1 2 3), (1 3 2)\}$, yielding two cosets $H_1$ and $H_1b$.

     *  Similarly take $H_2 = \langle b\rangle = \{1, (1 2)\}$ giving cosets $H_2$, $H_2a$ and $H_2a^2$.

The covering of $S_3$ is then $\mathfrak{H} =  \{H_1,H_1b,H_2,H_2a,H_2a^2\}$, and the nerve is a complete [[bipartite graph]] on $2+3$-vertices.
=--

## $n$-generating families

+-- {: .un_defn}
###### Definition

A family, $\mathcal{H}$, of subgroups of $G$ is called __$n$-generating__ if the nerve, $N(\mathfrak{H})$, of the corresponding coset covering is [[n-connected space|(n-1)-connected]], i.e., $\pi_i N(\mathfrak{H}) = 0$ for $i\lt n$.
=--

Rephrasing and extending comments made earlier, we have
+--{: .un_proposition}
######Proposition
There are isomorphisms:

   (a) $\pi_0N(\mathfrak{H}) \cong G/\langle\bigcup H_j\rangle;$

   (b) $\pi_1N(\mathfrak{H}) \cong Ker(\underset{\cap}{\sqcup} \mathcal{H}\to G).$
=--

+-- {: .un_example}
###### Examples

1.  Keeping in the case where $\mathcal{H} = \{H_1,H_2\}$, so two subgroups of $G$ then we have that $\underset{\cap}{\sqcup} \mathcal{H}\to G$ is an isomorphism if and only if $N( \mathfrak{H})$ is a [[tree]].  This gives one of the basic types of a [[graph of groups]]. In more generality, if $\mathcal{H} = \{H_1,\ldots ,H_n\}$ then there is a [[complex of groups]] associated with $N( \mathfrak{H})$.

1.  A more complex family of  examples of the above situation occurs in algebraic K-theory. This has the general linear group, $G\ell _n(R)$, of invertible $n\times n$ matrices together with a family of subgroups corresponding to lower triangular matrices, .... but with some subtleties involved.

     *  Let  $R$ be an associative ring, and now let $\sigma$ be a partial order  on $\{1,\ldots,n\}$. If $i$ is less that $j$ in the partial order $\sigma$, it is convenient to write $i\stackrel{\sigma}{\lt} j$. (Note that this means that some of the elements may only be related to themselves and hence are really not playing a role in such a $\sigma$.) We will write $PO(n)$ for the set of partial orders of $\{1,\ldots,n\}$.

      * We say an $n\times n$ matrix, $A = (a_{ij})$ is _$\sigma$-triangular_ if, when $i\stackrel{\sigma}{\leq} j$ is false, $a_{ij}=0$, and all diagonal entries, $a_{ii}$ are $1$.

      *   We let $T_n^\sigma(R)$ be the subgroup of $G\ell_n(R)$ formed by the $\sigma$-triangular matrices and then look at all such subgroups for all $n$, considering the stable general linear group $G\ell(R)$ as the colimit of the nested sequence of all the $G\ell_n(R)$, take $G = G\ell(R)$. Considering the family,  $\mathcal{H}$, of all the $T_n^\sigma(R)$, form the corresponding nerve, $N(\mathfrak{H})$.  This space has the same homotopy type as the Volodin model for [[algebraic K-theory]], since it is the Čech nerve of the covering $\mathfrak{H}$, whilst the Vietoris nerve of that covering is the [[Volodin model]].

=--
## References

* [[H. Abels]] and S. Holz, _Higher generation by subgroups_, J. Alg, 160, (1993), 311--341.


higher geometric quantization

> under construction, for a more coherent account see ([hpqg](#FiorenzaSchreiber)).

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometric quantization
+--{: .hide}
[[!include geometric quantization - contents]]
=--
#### Higher geometry
+-- {: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Higher geometric quantization is meant to complete this table:

| [[classical mechanics]] | --[[quantization]]$\to$ | [[quantum mechanics]] |
|--|--|--|
| [[symplectic geometry]] | --[[geometric quantization]]$\to$ | [[quantum field theory]] |
| [[higher symplectic geometry]] | --higher geometric quantization$\to$ | [[extended quantum field theory]] |

Being a concept in [[higher geometry]], higher geometric quantization is formulated naturally in [[(∞,1)-topos theory]]. More precisely, since it involves not just _[[cohomology]]_ but _[[differential cohomology]]_, it is formulated in _[[cohesive (∞,1)-topos|cohesive (∞,1)-topos theory]]_ ([[cohesive homotopy type theory]]).

In this context, write $\mathbf{B}^n \mathbb{G}_{conn} \in \mathbf{H}$ for the cohesive [[moduli ∞-stack]] of [[circle n-bundles with connection]], in the ambient [[cohesive (∞,1)-topos]] $\mathbf{H}$. Then for $X \in \mathbf{H}$ any object to be thought of as the [[moduli ∞-stack]] of fields or as the [[target space]] for a [[sigma-model]], a morphism

$$
  \mathbf{c}_{conn} : X \to \mathbf{B}^n \mathbb{G}_{conn}
$$

modulates a [[circle n-bundle with connection]] on $X$. We regard this as a _extended [[action functional]]_ in that for $\Sigma_{k} \in \mathbf{H}$ of [[cohomological dimension]] $k \leq n$ and sufficiently compact so that [[fiber integration in ordinary differential cohomology]] $\exp(2 \pi i \int_{\Sigma}_k(-))$ applies, the [[transgression]] of $\mathbf{c}_{conn}$ to low [[codimension]] reproduces the traditional ingredients

| $k = $ | [[transgression]] of $\mathbf{c}_{conn}$ to $[\Sigma_{n-1},X]$ | meaning in [[geometric quantization]] |
|--|--|--|
| $n$ | $\exp(2 \pi i S(-)) : [\Sigma_n, X] \stackrel{[\Sigma_n, \mathbf{c}_{conn}]}{\to} [\Sigma_n, \mathbf{B}^n \mathbb{G}_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_n}(-))}{\to}  \mathbb{G} $ |  [[action functional]] |
| $n-1$ | $\exp(2 \pi i S(-)) : [\Sigma_{n-1}, X] \stackrel{[\Sigma_{n-1}, \mathbf{c}_{conn}]}{\to} [\Sigma_{n-1}, \mathbf{B}^n \mathbb{G}_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_{n-1}}(-))}{\to}  \mathbf{B}\mathbb{G}_{conn}  \,$  | ordinary (off-shell) [[prequantum circle bundle]] |

The idea is to consider the higher geometric quantization not just of the low codimension transgressions, but of all transgressions of $\mathbf{c}_{conn}$.
The basic constructions that higher geometric quantization is concerned with are indicated in the following table. All of them have also a fundamental interpretation in [[twisted cohomology|twisted]] [[cohomology]] (independent of any interpretation in the context of [[quantization]]) this is indicated in the right column of the table:

| higher [[geometric quantization]]  | [[cohesive homotopy type theory]] |[[twisted cohomology]] |
|--|--|--|
| [[n-plectic ∞-groupoid]]  | $X \stackrel{\omega}{\to} \Omega^{n+1}_{cl}(-,\mathbb{G})$ | [[twisted cohomology|twisting]] [[cocycle]] in [[de Rham cohomology]] |
| [[symplectomorphism group]] | $\mathbf{Aut}_{/\Omega^{n+1}(-,\mathbb{G})}(\omega) = \left\{ \array{ X &&\stackrel{\simeq}{\to}&& X \\ & {}_{\mathllap{\omega}}\searrow && \swarrow_{\mathrlap{\omega}} \\ && \Omega^{n+1}_{cl}(-,\mathbb{G})  } \right\}$ |   |
| [[prequantum circle n-bundle]] | $\array{ && \mathbf{B}^n \mathbb{G}_{conn} \\ & {}^{\mathllap{\mathbf{c}_{conn}}}\nearrow & \downarrow^{\mathrlap{curv}} \\ X &\stackrel{\omega}{\to}& \Omega^{n+1}(-,\mathbb{G})}$ | [[twisted cohomology|twisting]] [[cocycle]] in [[ordinary differential cohomology|differential cohomology]] |
| [[Planck's constant]] $\hbar$ | $\tfrac{1}{\hbar}\mathbf{c}_{conn} : X \to \mathbf{B}^n \mathbb{G}_{conn}$ | divisibility of twisting class  |
| [[quantomorphism group]] $\supset$ [[Heisenberg group]] | $\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) = \left\{ \array{ X &&\stackrel{\simeq}{\to}&& X \\ & {}_{\mathllap{\mathbf{c}_{conn}}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{\mathbf{c}_{conn}}} \\ && \mathbf{B}^n \mathbb{G}_{conn} }  \right\}$ | twist [[automorphism ∞-group]] |
| [[Hamiltonian]] [[quantum operator (in geometric quantization)|quantum observables]] with [[Poisson bracket]] | $Lie(\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}))$  | [[infinitesimal cohesion|infinitesimal]] twist automorphisms |
| [[Hamiltonian actions]] of a [[smooth ∞-group]] $G$ / dual [[moment maps]]| $ \mu : \mathbf{B}G \to \mathbf{B}\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn})$ | $G$-[[∞-action]] on the twisting |
| [[gauge reduction]] | $\mathbf{c}_{conn}//G \,:\, X//G \to \mathbf{B}^n \mathbb{G}_{conn}$ | $G$-[[∞-quotient]] of the twisting  |
| [[Hamiltonian symplectomorphisms]] |  [[∞-image]] of $\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) \to \mathbf{Aut}_{/\Omega^{n+1}_{cl}(-,\mathbb{G})}(\omega)$  | twists in de Rham cohomology that lift to differential cohomology |
| [[∞-representation]] of [[n-group]] $\mathbf{B}^{n-1}\mathbb{G}$ on $V_n$| $\array{ V_n &\to& V_n//\mathbf{B}^{n-1}\mathbb{G} \\ && \downarrow^{\mathbf{p}} \\ && \mathbf{B}^n \mathbb{G} }$ | [[local coefficient bundle]] |
|[[prequantum space of states]] | $\mathbf{\Gamma}_X(E) := [\mathbf{c},\mathbf{p}]_{/\mathbf{B}^n \mathbb{G}} = \left\{ \array{ X &&\stackrel{\sigma}{\to}&& V//\mathbf{B}^{n-1}\mathbb{G} \\ & {}_{\mathllap{\mathbf{c}}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{p}}} \\ && \mathbf{B}^n \mathbb{G} } \right\} $ | [[cocycles]] in $[\mathbf{c}]$-[[twisted cohomology|twisted V-cohomology]] |
| [[prequantum operator]] | $\widehat{(-)} : \mathbf{\Gamma}_X(E) \times \mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) \to \mathbf{\Gamma}_X(E)$ | [[∞-action]] of twist automorphisms on twisted cocycles |
| [[trace]] to higher [[dimension]] | $\array{ [S^1, V_n//\mathbf{B}^{n-1}\mathbb{G}_{conn}] &\stackrel{tr\,hol_{S^1}}{\to}& V_{n-1}//\mathbf{B}^{n-2}\mathbb{G}_{conn}  \\  \downarrow^{\mathrlap{\mathbf{p}^{V_n}_{conn}}} && \downarrow^{\mathrlap{\mathbf{p}^{V_{n-1}}_{conn}}} \\ \mathbf{B}^n \mathbb{G}_{conn} &\stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to}& \mathbf{B}^{n-1} \mathbb{G}_{conn} }$  | [[fiber integration in ordinary differential cohomology]] adjoined with one in nonabelian differential cohomology |

## Examples

### Ordinary symplectic manifolds

* [[prequantum circle bundle]]

  $X \to \mathbf{B} U(1)_{conn}$

* [[local coefficient bundle]]

  $$
    \array{
      \mathbb{C} &\to& \mathbb{C}//U(1)
      \\
      && \downarrow
      \\
      && \mathbf{B} U(1)
    }
  $$

### Of non-integral 2-forms

While only integral [[presymplectic forms]] have a [[prequantization]] to a prequantum [[circle bundle with connection]], hence to a $(\mathbb{Z} \to \mathbb{R})$-[[principal 2-bundle]], a general 2-form has a higher prequantization given by a [[connection on a 2-bundle]] on a [[principal 2-bundle]] with structure-[[2-group]] that coming from the [[crossed module]] $(\Gamma \hookrightarrow \mathbb{R})$, where $\Gamma$ is the [[discrete group]] of [[periods]] of the 2-form.

This is discussed further at _[[prequantization of non-integral 2-forms]]_.

### Of 2-plectic $\infty$-groupoids

#### In codimension 2

* [[prequantum circle n-bundle|prequantum circle 2-bundle]]

* [[local coefficient bundle]]

  $$
    \array{
      \mathbf{B}U(n) &\to& \mathbf{B}PU(n)
      \\
      && \downarrow^{\mathrlap{\mathbf{dd}_n}}
      \\
      && \mathbf{B}^2 U(1)
    }
  $$

* [[space of states]]: [[twisted bundles]]

#### In codimension 1

**Proposition** There is a lift of coefficient bundles to [[loop space]]

$$
  \array{
    [S^1,(\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}] &\stackrel{tr hol_{S^1}}{\to}& \mathbb{C}//U(1)_{conn}
    \\
    \downarrow^{\mathrlap{\mathbf{p}^{\mathbf{B}U}}}
    &&
    \downarrow^{\mathrlap{\mathbf{p}^{\mathbb{C}}}}
    \\
    [S^1,\mathbf{B}^2 U(1)_{conn}]
    &\stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to}& \mathbf{B}U(1)_{conn}
  }
$$

where on the left we have [[loop space objects]] formed in $\mathbf{H}$ and on the bottom we have [[fiber integration in ordinary differential cohomology]].

Forming the pasting composite with this sends 2-states and 2-operators in codimension 2 to ordinary states and operators in codimension 1.

In particular it sends  [[twisted bundles]]  to sections of a line bundle.

For $X$ a [[D-brane]] and $\mathbf{c}_{conn}$ the [[B-field]], this
reproduces [[Freed-Witten anomaly cancellation]] mechanism.

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

[[schreiber:∞-Chern-Simons theory]]

$G$ a [[smooth ∞-group]],

$$
  \mathbf{c}_{conn}
  :
  \mathbf{B}G_{conn}
  \to
  \mathbf{B}^n U(1)_{conn}
$$

a universal differential characteristic map.

The following examples are of this form.

#### Extended $(4k+3)d$ abelian Chern-Simons theory

[[higher dimensional Chern-Simons theory]]

[[prequantum circle n-bundle|prequantum circle (4k+3)-bundle]]
from [[Beilinson-Deligne cup product]]

$$
  \mathbf{B}^{2k+1}U(1)_{conn}
  \stackrel{(-)\cup (-)}{\to}
  \mathbf{B}^{4k+3}U(1)_{conn}
$$

The quantomorphism $\infty$-group of this should be

$$
  \mathbb{Z}_2 \simeq Aut(U(1))
  \,.
$$

For there is, up to equivalence, a unique autoequivalence

$$
  \mathbf{B}^{2k+1}U(1)_{conn} \stackrel{\simeq}{\to}
  \mathbf{B}^{2k+1}U(1)_{conn}
  \,,
$$

the one induced by the nontrivial automorphism of $U(1)$. Since the cup-product is strictly invariant under this, this extends to

$$
  \array{
     \mathbf{B}^{2k+1}U(1)_{conn} &&\stackrel{\simeq}{\to}&&
     \mathbf{B}^{2k+1}U(1)_{conn}
     \\
     & {}_{\mathllap{(-)\cup(-)}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{(-)\cup(-)}}
     \\
     && \mathbf{B}^{4k+3}U(1)_\conn
  }
  \,.
$$

But for any further nontrivial such autoequivalence in the slice we would need in particular a gauge transformation parameterized by $(2k+1)$-forms over test manifolds from $C \wedge d C$ to itself. But the only closed $2k$-forms that we can produce naturally from $C$ are multiples of $C \wedge C$. But these all vanish since $C$ is of odd degree $2k+1$.

For $k = 1$ the total space of the [[prequantum circle n-bundle|prequantum circle 3-bundle]] of $U(1)$-Chern-Simons theory over the point is the [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack|moduli 2-stack]] of [[T-Duality and Differential K-Theory|differential T-duality structures]].

#### Extended 3d $\mathrm{Spin}$-Chern-Simons theory

* [[prequantum circle n-bundle|prequantum circle 3-bundle]]:

  [[differential string structure|differential first fractional Pontryagin class]]

  $\tfrac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}Spin_{conn} \to \mathbf{B}^3 U(1)_{conn}$

So [[Planck's constant]] here is  $\hbar = 2$ (relative to the naive multiple).

The total space of the prequantum 3-bundle is

$$
  \array{
    \mathbf{B}String_{conn'} &\to& \Omega^{1 \leq \bullet \leq 3} &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbf{B}Spin_{conn} &\stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{B}^3 U(1)_{conn} &\to& \mathbf{B}^3 U(1)
  }
$$

as it appears in [[schreiber:The moduli 3-stack of the C-field]].

But the quantomorphism group of this will be small, as the [[Chern-Simons form]] is far from being gauge invariant.

See the discussion at
_[Chern-Simons theory -- Geometric quantization -- In higher codimension](Chern-Simons%20theory#GeometricQuantHigher)_.

#### Extended 3d $G \times G$-Chern-Simons theory

However, when we consider $G \times G$ CS theory given by

$$
  \mathbf{B}(G \times G)_{conn}
  \stackrel{\mathbf{c}^1_{conn}- \mathbf{c}^2_{conn}}{\to}
  \mathbf{B}^3 U(1)_{conn}
$$

then diagonal gauge transformations $\mathbf{B}(G \times G)_{conn}
 \to \mathbf{B}(G \times G)_{conn}$ have interesting extensions to quantomorphisms, because for $g : U \to G$ the given gauge transformation at stage of definition $U$, the Chern-Simons form transforms by an exact term

$$
  CS(A_1^g,A_2^g)
  =
  CS(A_1,A_2)
  +
  d \langle A_1 - A_2, g^* \theta\rangle
  \,.
$$

#### Extended 7d $\mathrm{String}$-Chern-Simons theory

* [[prequantum circle n-bundle|prequantum circle 7-bundle]]:

  [[differential fivebrane structure|differential second fractional Pontryagin class]]

  $\tfrac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn}$

So [[Planck's constant]] here is  $\hbar = 6$ (relative to the naive multiple).

The total space of the prequantum 7-bundle is

$$
  \array{
    \mathbf{B}Fivebrane_{conn'} &\to& \Omega^{1 \leq \bullet \leq 7} &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbf{B}String_{conn} &\stackrel{\tfrac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} &\to& \mathbf{B}^7 U(1)
  }
$$

### $\infty$ Wess-Zumino-Witten theory

* [[schreiber:∞-Wess-Zumino-Witten theory]]

Differentially twisted looping of $\infty$-Chern-Simons theory

$$
  \Omega \mathbf{c} : G \to \mathbf{B}^{n-1}\mathbb{G}
$$

#### Ordinary $G$-WZW model

* [[WZW model]]

$$
  \tilde\Omega \tfrac{1}{2}\mathbf{p}_1 :  G \to \mathbf{B}^2 U(1)_{conn}
$$

studied in ([Rogers PhD, section 4.2](#RogersPhD)).

#### $String$-WZW model

(...)

#### $Fivebrane$-WZW model

(...)

## Related concepts

[[!include Isbell duality - table]]

## References

2-geometric quantization over [[smooth manifolds]] is discussed in section 6 and section 7 of

* {#RogersPhD} [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))

with further indications in

* {#Rogers} [[Chris Rogers]], _Higher geometric quantization_, at _Higher Structures 2011_ ([[RogersGottingen11.pdf:file]])

The special case of geometric quantization over infinitesimal [[action groupoids]] can be described in terms of [[BRST complexes]]. For references on this see _[Geometric quantization -- References -- Geometric BRST quantization](http://ncatlab.org/nlab/show/geometric+quantization#ReferencesBRST)_.

Higher geometric quantization in a [[cohesive (∞,1)-topos]] over [[smooth ∞-groupoids]] is discussed in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Chris Rogers]], _[[schreiber:infinity-geometric prequantization]]_

and the examples of higher Chern-Simons theories in

* {#FiorenzaSchreiber} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]] _[[schreiber:Higher geometric prequantum theory]]_

Review:

* [[Luigi Alfonsi]], §3 in: *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]

[[!redirects higher geometric prequantization]]

[[!redirects extended geometric quantization]]

[[!redirects higher geometric quantisation]]
[[!redirects higher geometric quantisations]]


higher geometry

{:bluebox: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

**higher geometry** $\leftarrow$ [[Isbell duality]] $\to$ [[higher algebra]]

***

#Contents#
* table of contents
{:toc}

## Idea

_Higher geometry_ or _homotopical geometry_ is the study of concepts of [[space]] and [[geometry]] in the context of [[higher category theory]] and [[homotopy theory]].

+-- {: bluebox}
###### Slogan

higher geometry = [[geometry]] + [[homotopy theory]]/[[higher category theory]]

=--

Higher geometry subsumes notably the theory of [[orbifolds]] and [[geometric stacks]], as well as the theory of more general [[stacks]] such as [[moduli stacks]], and generalizes all this to [[∞-stacks]] and [[derived stacks]].
This way higher geometry includes what is called _[[derived geometry]]_ and it subsumes at least parts of (derived) [[noncommutative geometry]]. Many other phenomena are naturally part of higher geometry, see the list of [Examples](#Examples) below.

In any given instance of higher geometry, one starts with a notion of "local models" for the geometry. An _affine space_ will then be a formal dual of such a local model, and a general [[space]] will be formed by "gluing" these affine spaces in some appropriate way. There are two ways of formalizing this idea, coming from [[Alexander Grothendieck]]'s two definitions of [[scheme]] in [[algebraic geometry]] via [[locally ringed spaces]] and [[functors of points]]. Both are built on [[(∞,1)-topos theory]]: in one direction, a [[petit topos|petit]] [[(∞,1)-topos]] (with some additional [[structured (∞,1)-topos|structure]]) encodes a [[space]] itself; in another direction, a [[space]] is an object of a [[gros topos|gros]] [[(∞,1)-topos]] of [[∞-stacks]] on some [[(∞,1)-site]]. We discuss these axiomatizations below in _[Formalization](#Formalizations)_.

These approaches do not apply to [[noncommutative algebraic geometry]], which requires a different approach to deal with a more complicated notion of gluing; we discuss this below in _[Noncommutative algebraic geometry](#NoncommutativeAlgebraicGeometry)_.

## Formalizations
 {#Formalizations}

We discuss two different (but closely related) formalizations of these ideas.

In

* _[Gros (∞,1)-topos](#GrosHigherToposes)_

we discuss the approach of considering one big [[(∞,1)-topos]] $\mathbf{H}$ (with "big"/[[gros topos|gros]] being formalized for instance by [[cohesion]]) such that (some of) its objects are to be regarded as higher geometric spaces.

In

* _[Petit (∞,1)-toposes](#PetitHigherToposes)_

we discuss the approach of encoding a would-be higher geometric space $X$ by a [[structured (∞,1)-topos]] to be thought of as the [[petit topos|petit]] [[(∞,1)-topos]] of [[(∞,1)-sheaves]] $(Sh_\infty(X), \mathcal{O}_X)$ of $X$, canonically equipped with a [[structure sheaf]] $\mathcal{O}_X$.

### Gros (∞,1)-toposes
 {#GrosHigherToposes}

Let $\mathcal{G}$ be an [[(∞,1)-site]] whose objects are to be viewed as "local models" or "test spaces" for a geometry. Within the context of this geometry, we make the following definitions:

An **affine space** is a formal dual of an object of $\mathcal{G}$, so that the (∞,1)-category of affine spaces is the [[opposite (∞,1)-category|opposite]] of $\mathcal{G}$. A **stack** is an [[∞-stack]] on $\mathcal{G}$, so that the (∞,1)-category of stacks is the [[gros topos|gros]] [[(∞,1)-sheaf (∞,1)-topos]] on $\mathcal{G}$. Finally, a **space** is a [[stack]] $X$ that has a cover by a family of affine spaces $(f_i : U_i \to X)_i$, where each $f_i$ belongs to some nice class of morphisms (e.g. [[open immersions]], [[etale morphisms]] or [[smooth morphisms]]).

When the underlying (∞,1)-category of $\mathcal{G}$ is the (∞,1)-category of [[commutative algebras in a symmetric monoidal (∞,1)-category]], this is known as _[[homotopical algebraic geometry]]_. When it is the (∞,1)-category of [[algebras over a Lawvere theory]], this is discussed at [[derived geometry]].

Following [[Bill Lawvere]], one may ask for a set of [[axioms]] on the [[(∞,1)-sheaf (∞,1)-topos]] $Sh_\infty(\mathcal{G})$ that ensure that it is appropriate to view [[(∞,1)-sheaves]] on $\mathcal{G}$ as generalized geometric spaces. One such set of axioms is _[[cohesion]]_.

### Petit (∞,1)-toposes
 {#PetitHigherToposes}

As above, let $\mathcal{G}$ be an [[(∞,1)-category]] whose objects will be viewed as "local models" for the kind of geometry to be developed. Following [[Jacob Lurie]] (based on the theory of geometry via [[ringed toposes]] by [[Alexander Grothendieck]] and [[Monique Hakim]]), a $\mathcal{G}$-[[structured (∞,1)-topos]] is the data of an [[(∞,1)-topos]] together with a [[structure sheaf]] valued in $\mathcal{G}$. Given an appropriate choice of $\mathcal{G}$, one gets the following hierarchy of generalized spaces this way:

* [[geometry (for structured (∞,1)-toposes)|test spaces]] $\hookrightarrow$
  [[generalized scheme|spaces locally equivalent to test spaces]]
  $\hookrightarrow$
  [[structured (∞,1)-topos|concrete spaces with structure sheaves taking values in test spaces]]
  $\hookrightarrow$
  [[∞-stack|spaces probeable by test spaces]].

technically modeled by:

* [[geometry (for structured (∞,1)-toposes)]] $\mathcal{G}$
  $\hookrightarrow$

  [[generalized scheme]]s
  $\hookrightarrow$
  formal duals to $\mathcal{G}$-[[structured (∞,1)-topos]]es
  $\hookrightarrow$
  [[(∞,1)-topos]] of [[∞-stack]]s on $\mathcal{G}$.

A plethora of proposals for formalizations of higher geometry find their home in this pattern, for instance most of the concepts listed at [[generalized smooth space]].

## Relation between these approaches

Given a gros [[cohesive (∞,1)-topos]] $\mathbf{H}$ and an object $X \in \mathbf{H}$, one may in turn assign to $X$ a petit [[structured (∞,1)-topos]] $Sh_{\mathbf{H}}(X)$ of [[internal sheaves]] over $X$. See at _[[differential cohesion]]_ for how this works. This connects the "gros" perspective back to the "petit" perspective.

Conversely, given a [[structured (∞,1)-topos]] one may consider its associated [[functor of points]], which will be an object in the [[gros topos|gros]] [[(∞,1)-topos]].

## Examples
 {#Examples}

### List of examples

* [[homotopical algebraic geometry]]

  * [[derived algebraic geometry]]

  * [[étale (∞,1)-site]], [[dg-geometry]], [[Hochschild cohomology]] of [[dg-algebra]]s

  * [[schematic homotopy type]]

* [[derived noncommutative geometry]]

  * [[noncommutative geometry]]

* [[higher differential geometry]]

  * [[motivation for higher differential geometry]]

  * [[differential geometry]], [[differential topology]]

  * [[derived smooth manifold]]

  * [[smooth ∞-groupoid]], [[∞-Lie algebroid]]

* [[higher symplectic geometry]]

* [[higher complex analytic geometry]]

* [[higher prequantum geometry]]

* [[higher Klein geometry]]

* [[higher Cartan geometry]]

### Noncommutative algebraic geometry
 {#NoncommutativeAlgebraicGeometry}

The above frameworks for higher geometry are not suitable for describing [[noncommutative algebraic geometry]], because of the more complicated notions of [[localization]], gluing and [[descent]] in the latter setting. Indeed, noncommutative spaces are supposed to be obtained from affine ones (formal duals of [[associative algebras]] or [[dg-algebras]]) by gluing along _[[bimodules]]_. A good setting for such gluing is that of [[pretriangulated dg-categories]] (or [[stable (∞,1)-categories]]). Thus in [[derived noncommutative algebraic geometry]], a noncommutative space is defined to be a [[stable (∞,1)-category]].

### Connes-style noncommutative geometry
 {#ConnesStyleNoncommutativeGeometry}

The process of forming [[groupoid convolution algebras]] is a [[2-functor]] from suitable [[topological stack|topological]] and [[differentiable stacks]] to [[C*-algebras]] with [[Hilbert bimodules]] between them. Much of [[Connes]]-style [[noncommutative geometry]] turns out to deal with objects in the image of this functor, and to the extent that it does, Connes-style noncommutative geometry may be regarded as being a way of speaking about higher geometry, specifically the [[higher differential geometry]] of [[differentiable stacks]].

## Related concepts

* [[higher structures]]

* [[higher differential geometry]]

[[!include Isbell duality - table]]

For relation to [[physics]] see

* [[higher category theory and physics]]

* [[geometry of physics]]

## References
{#Lurie}

Both approaches to higher geometry are described, in the special case of [[derived algebraic geometry]], in

* [[Jacob Lurie]], _[[Derived Algebraic Geometry]]_, Ph.D. thesis.

The gros topos approach is described, in the case of [[homotopical algebraic geometry]], in

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical algebraic geometry II: geometric stacks and applications_, 2004, [arXiv:math/0404373](http://arxiv.org/abs/math/0404373).

A general exposition of the petit topos approach is proposed in

* {#Lurie} [[Jacob Lurie]], _[[Structured Spaces]]_ .

In

* {#Lawvere} [[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))

an axiomatization of generalized geometry is proposed in terms of 1-[[category theory]]. The evident generalization of this to [[(∞,1)-category theory]] provides an axiomatization for higher geometry. This is discussed at

* [[cohesive (∞,1)-topos]].

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]], §1 in: *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]

[[!redirects higher geometries]]


higher geometry - contents

**[[higher geometry]]** / **[[derived geometry]]**

**Ingredients**

* [[higher topos theory]]

* [[higher algebra]]

**Concepts**

* **geometric [[big and little toposes|little]] [[(∞,1)-topos]]es**

  * [[structured (∞,1)-topos]]

  * [[geometry (for structured (∞,1)-toposes)]]

  * [[generalized scheme]]

* **geometric [[big and little toposes|big]] [[(∞,1)-topos]]es**

  * [[cohesive (∞,1)-topos]]

* [[function algebras on ∞-stacks]]

  * [[geometric ∞-stacks]]

**Constructions**

* [[loop space object]], [[free loop space object]]

* [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] / [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|of a locally ∞-connected (∞,1)-topos]]

**Examples**

* [[derived algebraic geometry]]

  * [[étale (∞,1)-site]], [[Hochschild cohomology]] of [[dg-algebra]]s

  * [[dg-geometry]]

    * [[dg-scheme]]

  * [[schematic homotopy type]]

* [[derived noncommutative geometry]]

  * [[noncommutative geometry]]

* derived smooth geometry

  * [[differential geometry]], [[differential topology]]

  * [[derived smooth manifold]], [[dg-manifold]]

  * [[smooth ∞-groupoid]], [[∞-Lie algebroid]]

* [[higher symplectic geometry]]

* [[higher Klein geometry]]

* [[higher Cartan geometry]]

**Theorems**

* [[Isbell duality]]

* [[Hochschild cohomology|Jones' theorem]], [[Hochschild cohomology|Deligne-Kontsevich conjecture]]

* [[Tannaka duality for geometric stacks]]


higher group character

A $k$-character, $k \in \mathbb{N}$, of a group $G$ is a certain function
$$
  \chi^{(k)} : G^k \to \mathbb{C}
$$
which descends to a function of _$k$-conjugacy classes_.

A $k$-conjugacy class in $G^k$ is a minimal subset of $G^k$ which is closed under $k$-fold conjugation in the following sense:

$$
  [g_1, \cdots, g_k]
  :=
  \{
    (g'_1, \cdots, g'_k) \in G^k |
    \exists (h_1, \cdots h_k) \in G^k :
    g'_1 = h_1 g_1 h_2^{-1} \vee
    g'_2 = h_2 g_2 h_3^{-1} \vee \cdots
    g'_k = h_k g_2 h_1^{-1}
  \}
  \,.
$$

The quotient map which sends elements in $G^k$ to their $k$-conjugacy class is called (at least for $k=2$) the [[weak Cayley table]] of $G$.

#Remarks#

* Evidently, on any $k$-conjugacy class of $G$ we canonically obtain $k$ different commuting actions of $G$.

#Reference#

For a brief review and a collection of many relevant references see

* Kenneth W. Johnson, Sandro Mattarei and Surinder K. Sehgal, _Weak Cayley tables_, [Journal of the London Mathematical Society 2000 61(2):395-411](http://jlms.oxfordjournals.org/cgi/content/abstract/61/2/395) ([pdf](http://journals.cambridge.org/download.php?file=%2FJLM%2FJLM61_02%2FS0024610799008571a.pdf&code=e119cd0ef145cd79d298f403f4d38dc7))

#Functorial reformulation#

There is a succinct functorial reformulation of $k$-conjugacy classes. This is described at [[schreiber:higher group characters]].

[[!redirects higher group characters]]


higher group characters > history

< [[higher group characters]]
[[!redirects higher group characters -- history]]


Higher Groups in Homotopy Type Theory

This page compiles pointers related to:

* [[Ulrik Buchholtz]], [[Floris van Doorn]], [[Egbert Rijke]]:

  \linebreak

  **Higher Groups in Homotopy Type Theory**

  \linebreak

   LICS '18:

   Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

   (2018) 205-214

  [arXiv:1802.04315](https://arxiv.org/abs/1802.04315)

  [doi:10.1145/3209108.3209150](https://doi.org/10.1145/3209108.3209150)

on [[infinity-groups|$\infty$-groups]] in [[homotopy type theory]].

> **Abtract.** We present a development of the theory of [[higher groups]], including [[infinity-groups]] and [[connective spectra|connective]] [[spectra]], in [[homotopy type theory]]. An [[infinity-group|infinity group]] is simply the [[loops]] in a [[pointed]], [[connected]] [[type]], where the [[group]] structure comes from the structure inherent in the [[identity types]] of [[Martin-Löf type theory]]. We investigate ordinary [[groups]] from this viewpoint, as well as [[n-group|higher dimensional groups]] and groups that can be [[delooped]] more than once. A major result is the stabilization theorem, which states that if an [[n-type]] can be [[delooped]] $n+2$ times, then it is an [[infinite loop space|infinite loop type]]. Most of the results have been formalized in the [[Lean]] [[proof assistant]].

category: reference


higher homotopy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[homotopy]] of [[homotopies]], existing in higher [[homotopy type|homotopy $n$-types]] of higher $n$.

## Related concepts

* [[homotopy theory]]

* [[identity type]]

* [[higher structures]]

  * [[higher category theory]], [[higher topos theory]]

  * [[higher geometry]]

  * [[higher observational type theory]]

  * [[higher gauge transformation]]

  * [[higher differential geometry]]

  * [[higher Lie theory]]

  * [[higher prequantum geometry]]

  * [[higher complex analytic geometry]]

  * [[higher algebra]], [[homotopical algebra]]

  * [[higher linear algebra]]

  * [[higher category theory and physics]]

[[!redirects higher homotopies]]

[[!redirects homotopy of homotopies]]
[[!redirects homotopies of homotopies]]

[[!redirects 2-homotopy]]
[[!redirects 2-homotopies]]