thermodynamics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

It is practically impossible to model a macroscopic [[physical system]] in terms of the microscopic kinematical and dynamical variables of all its [[particles]]. Thus one makes a hierarchical reduction in which this complexity is reduced to a small number of collective variables. The theoretical framework for such reductions for systems is [[statistical mechanics]] or statistical physics.

One special case of hierarchical reduction is the limit of large volumes #$V$, in which the number of particles (of each species) per volume, $N/V$, stays constant. This is called the **[[thermodynamic limit]]** in statistical physics. Under some standard assumptions like homogeneity (spacial and possibly directional) and stability (no transitory effects), there is a small number of collective variables characterizing the system. Such a description can be (and historically was) postulated as an independent self-consistent phenomenological theory even without going into the details of statistical mechanics; such a description is called **[[equilibrium]] thermodynamics**, which is believed to be deducible from statistical mechanics, as has been partially proved for some classes of systems. Sometimes transitional finite-time phenomena are described either statistically by studying stochastic processes or by a more elaborate hierarchical form of thermodynamics, so-called **nonequilibrium thermodynamics**.

One of the basic characteristics of a thermodynamical system is its [[temperature]], which has no analogue in fundamental non-statistical physics. Other common thermodynamical variables include pressure, volume, [[entropy]], enthalpy etc.

## Related entries

* [[temperature]], [[inverse temperature]], [[absolute zero]]

* [[Boltzmann constant]]

* [[Boltzmann distribution]]

* [[entropy]], [[relative entropy]],

* [[second law of thermodynamics]], [[generalized second law of thermodynamics|generalized second law]],

* [[KMS state]]

* [[intensive and extensive]]

* [[thermal quantum field theory]]

* [[eigenstate thermalization hypothesis]]

## References

### General

Introductions:

* John Denker, _Modern Thermodynamics_, ([web](http://www.av8n.com/physics/thermo), [pdf](http://www.av8n.com/physics/thermo-laws.pdf))

Mathematically rigorous treatments:

* Constantin Carathéodory, _Untersuchung über die Grundlagen der Thermodynamik_, Math. Annalen 67, 355-386

* Elliott H. Lieb, Jakob Yngvason, _The Physics and Mathematics of the Second Law of Thermodynamics_, Phys.Rept. 310 (1999) 1-96 ([arXiv:cond-mat/9708200](https://arxiv.org/abs/cond-mat/9708200))

* Elliott H. Lieb, Jakob Yngvason, _A Guide to Entropy and the Second Law of Thermodynamics_, Notices Amer. Math. Soc., 45, (1998) 571-581 ([arXiv:math-ph/9805005](https://arxiv.org/abs/math-ph/9805005))

See also

* Wikipedia: [thermodynamics](http://en.wikipedia.org/wiki/Thermodynamics), [fundamental thermodynamic relation](http://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation)

* Azimuth Project, _[Thermodynamics](http://www.azimuthproject.org/azimuth/show/Thermodynamics)_

### In terms of symplectic geometry (Souriau)
 {#ReferencesSouriau}

A covariant formalization of thermodynamics in terms of [[moment maps]] in [[symplectic geometry]] is due to

* [[Jean-Marie Souriau]], _Thermodynamique et géométrie_,  Lecture Notes in Math. 676 (1978), 369–397 ([scan](http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=197810025))

* {#IglesiasSouriau84} [[Patrick Iglesias-Zemmour]], [[Jean-Marie Souriau]] _Heat, cold and Geometry_, in: M. Cahen et al (eds.) Differential geometry and mathematical physics, 37-68, D. Reidel 1983 ([web](http://www.jmsouriau.com/Heat_Cold_And_Geometry_1983.htm), [pdf](http://www.jmsouriau.com/Publications/JMSouriau-PIglesias-HeatColdAndGeometry1983.pdf), [doi:978-94-009-7022-9_5](https://doi.org/10.1007/978-94-009-7022-9_5))

* [[Jean-Marie Souriau]], chapter IV "Statistical mechanics" of _Structure of dynamical systems. A symplectic view of physics_ . Translated from the French by C. H. Cushman-de Vries. Translation edited and with a preface by R. H. Cushman and G. M. Tuynman. Progress in Mathematics, 149. Birkhäuser Boston, Inc., Boston, MA, 1997

* [[Patrick Iglesias-Zemmour]], _Essai de «thermodynamique rationnelle» des milieux continus_, Annales de l'I.H.P. Physique théorique, Volume 34 (1981) no. 1,  p. 1-24 ([numdam:AIHPA_1981__34_1_1_0](http://www.numdam.org/item/AIHPA_1981__34_1_1_0))

* [[Jean-Marie Souriau]], J.-M. Mécanique statistique, groupes de Lie et cosmologie. In Colloques int. du CNRS; numéro 237; Aix-en-Provence, France, 1974; pp. 24–28, 59–113. English translation by F. Barbaresco, April, 2020. Available online: https://www.academia.edu/42630654/Statistical_Mechanics_Lie_Group_and_Cosmology_1_st_part_Symplectic_Model_of_Statistical_Mechanics(access on 20 April 2020)

Review includes

* {#Marle16} Charles-Michel Marle, _From tools in symplectic and Poisson geometry to Souriau's theories of statistical mechanics and thermodynamics_, Entropy 2016, 18(10), 370 ([arXiv:1608.00103](https://arxiv.org/abs/1608.00103))

The Souriau model of thermodynamics has been extented for "geometric science of information" (Koszul information geometry) with a general definition of [[Fisher metric]], Euler-Poincaré equation and variational definition of Souriau thermodynamics, as in:

* Frederic Barbaresco, _Koszul information geometry and Souriau geometric, temperature_, Capacity of Lie Group Thermodynamics, MDPI Entropy, n°16,  4521-4565 (2014) [pdf](http://www.mdpi.com/1099-4300/16/8/4521/pdf); _Symplectic structure of information geometry: Fisher metric and Euler-Poincaré equation of Souriau Lie group thermodynamics_, GSI'15, Springer LCNS __9389__, 529-540 (2015) [doi](https://doi.org/10.1007/978-3-319-25040-3_57)

* Barbaresco, F.; Gay-Balmaz, F. Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics. Entropy 2020, 22, 498. https://www.mdpi.com/1099-4300/22/5/498

* {#Amari87} Shun-ichi Amari, Chapter 2: _Differential Geometrical Theory of Statistics_ in _Differential geometry in statistical inference_, Institute of Mathematical Statistics Lecture Notes - Monograph Series 1987, 19-94 ([euclid:1215467059](https://projecteuclid.org/euclid.lnms/1215467059))

* A. Bravetti, C. S. Lopez-Monsalvo, F. Nettel, _Contact symmetries and Hamiltonian thermodynamics_, [arxiv/1409.7340](http://arxiv.org/abs/1409.7340)

### Irreversible thermodynamics

A survey of [[irreversible thermodynamics]] is in

* Ivan Vavruch, _Conceptual problems of modern irreversible thermodynamics_, Chem. Listy 96 (2002) ([pdf](http://www.chemicke-listy.cz/docs/full/2002_05_01.pdf))

For more on this see also _[[rational thermodynamics]]_.

* Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, Christopher Perry, _Fluctuating work: from quantum thermodynamical identities to a second law equality_, Phys. Rev. X 6, 041017 [doi](http://dx.doi.org/10.1103/PhysRevX.6.041017)

### Relativistic thermodynamics
 {#ReferencesRelativisticThermodynamics}

Making sense of thermodynamics when taking into account [[special relativity]] and ultimately, possibly, [[general relativity]] ([[gravity]]) is notoriously subtle (even ignoring the issue of [[Bekenstein-Hawking entropy]]).

> Shortly after the advent of the relativity theory, Planck, Hassenoerl, Einstein and others advanced separately a formulation of the thermodynamical laws in accordance with the special principle of relativity. This treatment was adopted unchanged including the first edition of this monograph. However it was shown by Ott and indepently by Arzelies, that the old formulation was not quite satisfactory, in particular because generalized forces were used instead of the true mechanical forces in the description of thermodynamical processes.

> The papers of Ott and Arzelies gave rise to many controversial discussions in the literature and at the present there is no generally accepted description of relativistic thermodynamics.

(quote from Moller, _[The theory of relativity](https://archive.org/details/theoryofrelativi029229mbp), 1952_)

A standard textbook has been

* {#Tolman34} Richard Tolman, _Relativity, Thermodynamics and Cosmology_, Oxford 1934, reprinted by Dover 1987

but Tolman's approach has been called into question, see e.g.

* Christian Fronsdal, _Relativistic thermodynamics_, 2014 ([pdf](http://fronsdal.physics.ucla.edu/system/files/Gen.Rel_.LMP_.pdf))

See also

* Nils Andersson, _General relativistic thermo-dynamics_, survey talk 2014 ([pdf](http://www.dpg-physik.de/dpg/pbh/aktuelles/pdf/Andersson.pdf))

* Sean A. Hayward, _Relativistic thermodynamics_ ([arXiv:gr-qc/9803007](https://arxiv.org/abs/gr-qc/9803007))

* Paul Frampton, Stephen D.H. Hsu, Thomas W. Kephart, David Reeb, _What is the entropy of the universe?_, Class. Quant. Grav.26:145005, 2009 ([arXiv:0801.1847](https://arxiv.org/abs/0801.1847))

### Further generalizations

Some formal generalizations of thermodynamical formalism include mixing time and temperature in formalisms with complex time-temperature like Matsubara formalism in QFT.

Mathematical analogies and generalizations include also

* [[John Baez]], [[Mike Stay]], _Algorithmic thermodynamics_, [pdf](http://math.ucr.edu/home/baez/thermo.pdf), [cafe](http://golem.ph.utexas.edu/category/2010/02/algorithmic_thermodynamics.html)

* [[M. Marcolli]], [[Ryan Thorngren]], _Thermodynamical semirings_, [arXiv/1108.2874](http://arxiv.org/abs/1108.2874)

* M. Zinsmeister, _Thermodynamic formalism and holomorphic dynamical systems_, Amer. Math. Soc. 2000.

* I. Itenberg, G. Mikhalkin, _Geometry in the tropical limit_, [arXiv/1108.3111](http://arxiv.org/abs/1108.3111)

On [[gravity]] in relation to thermodynamics:

* [[Thanu Padmanabhan]], *Gravity and/is Thermodynamics*,  Current Science, **109** (2015) 2236-2242 [[doi:10.18520/v109/i12/2236-2242](https://doi.org/10.18520/v109/i12/2236-2242)]

* [[Thanu Padmanabhan]], *Exploring the Nature of Gravity*, talk notes [[arXiv:1602.01474](https://arxiv.org/abs/1602.01474)]

[[!redirects irreversible thermodynamics]]


theta angle

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

#Contents#
* table of contents
{:toc}

## Idea

In [[Yang-Mills theory]] and specifically in its application to [[QCD]], the _theta angle_ refers to the prefactor $\theta$ in the expression of the [[action functional]] of the theory in front of the piece of [[topological Yang-Mills theory]]

$$
  \nabla \mapsto
  \frac{1}{g^2 }\int_X F_\nabla \wedge \star F_\nabla  \;+\; i \theta \int_X F_\nabla \wedge F_\nabla
$$

(see also at _[[S-duality]]_ for more).

## Phenomenology

In [[phenomenology]] the theta angle has to be very close to an integer multiple of $\pi$, see at _[[CP problem]]_. That it is indeed $\theta_{QCD}\sim 0$ instead of $\theta_{QCD} \sim \pi$ that matches experiment is argued at the end of [Crewther-DiVecchia-Veneziano-Witten 79](#CrewtherDiVecchiaVenezianoWitten79), [PO discussion](http://www.physicsoverflow.org/23722/how-do-we-know-%24-theta_-mathrm-qcd-ne-pi%24).

## In string theory and AdS/QCD

Within string theory, with [[Yang-Mills theory]] realized on [[intersecting D-brane models]], as on [[D4-branes]] for [[holographic QCD]], the theta angle corresponds to the [[graviphoton]] [[RR-field]] 1-form potential in the [[higher WZW term]] of the [[D4-brane]], which is ([CGNSW 96 (7.4)](#CGNSW96) [APPS97b (51)](#APPS97b), [CAIB 99, 6.1](#CAIB99))

$$
  \mathbf{L}_{D4}^{WZ}
  \;\propto\;
  C_1 \wedge \langle F \wedge F\rangle
  \,.
$$

## Related concepts

* [[theta vacuum]]

## References

### General

* {#CrewtherDiVecchiaVenezianoWitten79} [[Rodney Crewther]], [[Paolo Di Vecchia]], [[Gabriele Veneziano]], [[Edward Witten]], _Chiral estimate of the electric dipole moment of the neutron in quantum chromodynamics_, Phys. Lett. B 88 (1979) 123-127 ([cds:133382](http://cds.cern.ch/record/133382)).

* [[Davide Gaiotto]], [[Anton Kapustin]], [[Zohar Komargodski]], [[Nathan Seiberg]], _Theta, Time Reversal, and Temperature_, JHEP05(2017)091 ([arXiv:1703.00501](https://arxiv.org/abs/1703.00501))

Discussion of a similar $\theta$-angle in a 3d field theory, via [[extended TQFT]] and [[stable homotopy theory]] is in

* {#FreedKomargodskiSeiberg17} [[Daniel Freed]], [[Zohar Komargodski]], [[Nathan Seiberg]], _The Sum Over Topological Sectors and $\theta$ in the 2+1-Dimensional $\mathbb{C}\mathbb{P}^1$ $\sigma$-Model_ ([arXiv:1707.05448](https://arxiv.org/abs/1707.05448))

In relation to [[confinement]]:

* Massimo D'Elia, Francesco Negro, _Theta dependence of the deconfinement temperature in Yang-Mills theories_, Phys. Rev. Lett. 109, 2012 ([arXiv:1205.0538](https://arxiv.org/abs/1205.0538))

### In string theory
 {#ReferencesInStringTheory}

The $\theta$-angle as the [[graviphoton]] [[RR-field]]-potential $C_1$ in the [[higher WZW term]] of the [[D4-brane]]:

* {#CGNSW96} [[Martin Cederwall]], Alexander von Gussich, [[Bengt Nilsson]], Per Sundell, Anders Westerberg, around (4.6) of _The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity_, Nucl.Phys. B490 (1997) 179-201 ([arXiv:hep-th/9611159](http://arxiv.org/abs/hep-th/9611159))

* {#APPS97b} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], Section 6 of _Dual D-Brane Actions_, Nucl. Phys. B496 (1997) 215-230 ([arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133))

* {#CAIB99} C. Chryssomalakos, [[José de Azcárraga]], J. M. Izquierdo and C. Pérez Bueno, Sec. 6.1 of _The geometry of branes and extended superspaces_, Nuclear Physics B Volume 567, Issues 1–2, 14 February 2000, Pages 293–330 ([arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137))

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Theorem 7.4 of _[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]_

For more see at _[Green-Schwarz sigma model -- References -- For D-branes](Green-Schwarz+action+functional#ReferencesForDBranes)_.

Discussion explicitly in view of the [[Witten-Sakai-Sugimoto model]] for [[QCD]] on [[D4-branes]]:

*  Si-wen Li, around (3.1) of _The theta-dependent Yang-Mills theory at finite temperature in a holographic description_ ([arXiv:1907.10277](https://arxiv.org/abs/1907.10277))

[[!redirects theta angles]]


Theta category

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

#Contents#
* table of contents
{:toc}

## Idea ##

For $n \in \mathbb{N}$ the [[category]] $\Theta_n$ -- **Joyal's disk category** or **cell category** -- may be thought of as the
[[full subcategory]] of the category $Str n Cat$ of [[strict ∞-category|strict n-categories]] on those $n$-categories that are [[free construction|free]] on [[pasting diagrams]] of $n$-[[globe]]s.

For instance $\Theta_2$ contains an [[object]] that is depicted as

$$
  \array{
    & \nearrow &\Downarrow& \searrow &&
    && \nearrow && \searrow
    \\
    a && \to && b &\to & c && \Downarrow && b
    \\
    & \searrow &\Downarrow& \nearrow &&
    && \searrow && \nearrow &&
  }
  \,,
$$

being the pasting diagram of two 2-[[globes]] along a common 1-globe and of the result with a 1-globe and another 2-globe along common 0-globes.

Such pasting diagrams may be alternatively encoded in [[planar trees]], the above one corresponds to the tree:

$$
  \array{
    \nwarrow \nearrow &  & & \uparrow &&& 2
    \\
    & \nwarrow & \uparrow & \nearrow &&& 1
    \\
    && {*}
  }
  \,.
$$

Accordingly, $\Theta_n$ is also the category of planar rooted trees of level $\leq n$.

In low degree we have

* $\Theta_0 = *$ is the [[point]].

* $\Theta_1 = \Delta$ is the [[simplex category]]: the $n$-[[simplex]] $[n]$ is thought of as a linear [[quiver]] and as such the pasting diagram of $n$ 1-morphisms

  $$
    0 \to 1 \to \cdots \to n
    \,.
  $$

  Dually, this is the planar rooted tree of the form

  $$
    \array{
      \nwarrow &\uparrow & \cdots \nearrow
      \\
      &{*}
    }
  $$

  with $n$-branches.

## Definition ##

We discuss two equivalent definitions

* [Via the free strict ω-category](#ViaFreeStrictOmegaCategory)

* [Via iterated wreath product](#ViaIteratedWreathProduct)

* [Via duals of disks](#ViaDualsOfDisks)

### Via the free strict ω-category
 {#ViaFreeStrictOmegaCategory}

Let $T(1)$ denote the free [[strict ∞-category]] generated from the [[terminal object|terminal]] [[globular set]] $1$.

Notice that this terminal globular set consists of precisely one $k$-[[globe]] for each $k \in \mathbb{N}$: one point, one edge from the point to itself, one disk from the edge to itself, and so on.

So $T(1)$ is freely generated under composition from these cells.
As described above, this means that each element of the underlying globular set of $T(1)$ may be depicted by a [[pasting diagram]] made out of [[globes]], and such a pasting diagram itself may be considered as a [[globular set]] whose $k$-cells are _instances_ of the $k$-globes appearing in the diagram.

We now describe this formally.

The [[k-morphisms|n-cells]] of $T(1)$ may be identified with **planar [[trees]]** $\tau$ of height $n$, which by definition are functors

$$\tau: [n]^{op} \to \Delta$$

($\Delta$ is the category of [[simplex|simplices]] and $[n] \in \Delta$ is a simplex, i.e., ordered set $\{0 \lt 1 \lt \ldots \lt n\}$, regarded as a category) such that $\tau(0) = 1$. Such a $\tau$ is exhibited as a chain of morphisms in $\Delta$,

$$\tau(n) \to \tau(n-1) \to \ldots \to \tau(0) = 1,$$

and we will denote each of the maps in the chain by $i$. Thus, for each $x \in \tau(k)$, there is a fiber $i^{-1}(x)$ which is a linearly ordered set.
(Need to fill in how $\circ_j$ composition of such trees is defined.)

To each planar tree $\tau$ we associate an underlying globular set $[\tau]$, as follows. Given $\tau$, define a new tree $\tau'$ where we adjoin a new bottom and top $x_0$, $x_1$ to every fiber $i^{-1}(x)$ of $\tau$, for every $x \in \tau(k)$:

$$i_{\tau'}^{-1}(x) = \{x_0\} \cup i_{\tau}^{-1}(x) \cup \{x_1\}$$

Now define a $\tau$-**sector** to be a triple $(x, y, z)$ where $i(y) = x = i(z)$ and $y, z$ are consecutive edges of $i_{\tau'}^{-1}(x)$. A $k$-**cell** of the globular set $[\tau]$ is a $\tau$-sector $(x, y, z)$ where $x \in \tau(k)$. If $k \geq 1$, the **source** of a $k$-cell $(x, y, z)$ is the $(k-1)$-cell $(i(x), u, x)$ and the **target** is the $(k-1)$-cell $(i(x), x, v)$ where $u \lt x \lt v$ are consecutive elements in $i_{\tau'}^{-1}(i(x))$. It is trivial to check that the globular axioms are satisfied.

Now let $T([\tau])$ denote the free strict $\omega$-category generated by the globular set $[\tau]$.

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

$\Theta$ is the [[full subcategory]] of $Str \omega Cat$ on the  [[strict ∞-categories]] $T([\tau])$, as $\tau$ ranges over cells in the underlying globular set of $T(1)$.

=--

### Via iterated wreath product
 {#ViaIteratedWreathProduct}

+-- {: .num_prop #ByWreathProduct}
###### Proposition/Definition

$\Theta_n$ is the $n$-fold [[categorical wreath product]] of the [[simplex category]] with itself

$$
  \Theta_n \simeq \Delta^{\wr n}
  \,.
$$

=--

([Berger, section 3](#Berger))

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

So

$$
  \Theta_1 = \Delta
$$

$$
  \Theta_2 = \Delta \wr \Delta
$$

etc.

=--

+-- {: .num_cor}
###### Corollary

For all $n \in \mathbb{N}$ there is a canonical embedding

$$
  \sigma : \Theta_n \hookrightarrow \Theta_{n+1}
$$

given by $\sigma : a \mapsto ([1], a)$.

=--

### Via duals of disks
 {#ViaDualsOfDisks}

In analogy to how the [[simplex category]] is [[equivalence of categories|equivalent]] to the [[opposite category]] of finite strict linear [[intervals]], $\Delta \simeq \mathbb{I}^{op}$, so the $\Theta$-category is equivalent to the opposite of the category of Joyal's combinatorial finite [[disks]].

$$
  \Theta \coloneqq \mathbb{D}^{op}
  \,.
$$

(...)

## Properties

### Embedding of grids (products of the simplex category)
 {#EmbeddingOfGrids}

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

For any [[small category]] $A$ there is a canonical functor

$$
  \delta_A : \Delta \times A \to \Delta \wr A
$$

given by

$$
  \delta_A([n], a) = ([n], a^n)
  \,.
$$

=--

([Berger, def. 3.8](#Berger))

+-- {: .num_remark #ProductOfDeltaInTheta}
###### Remark

By iteration, this induces a canonical functor

$$
  \delta_n : \Delta^{\times n} \to \Theta_n
  \,.
$$

=--

### Embedding into strict $n$-categories

Write $Str n Cat$ for the category of [[strict n-categories]].

+-- {: .num_prop}
###### Proposition

There is a [[dense subcategory|dense]] [[full and faithful functor|full embedding]]

$$
  \Theta_n \hookrightarrow Str n Cat
  \,.
$$

=--

This was conjectured in ([Batanin-Street](#BataninStreet)) and shown in terms of free $n$-categories on $n$-graphs in
([Makkai-Zawadowsky, theorem 5.10](#MakkaiZawadowsky)) and ([Berger 02, prop. 2.2](#BergerCellular)). In terms of the wreath product presentation, prop. \ref{ByWreathProduct} this is ([Berger 05, theorem 3.7](#Berger)).

+-- {: .num_prop}
###### Proposition

Under this embedding an object $([k], (a_1, \cdots, a_k)) \in \Delta \wr \Delta^{\wr (n-1)}$ is identified with the $k$-fold [[horizontal composition]] of the pasting composition of the $(n-1)$-morphisms $a_i$:

$$
  ([k], (a_1, \cdots, a_k))
  =
  a_1 \cdot a_2 \cdot \cdots \cdot a_k
  \,.
$$

=--

+-- {: .num_example}
###### Example

The [[pasting diagram]]

$$
  \array{
    & \nearrow &\Downarrow& \searrow &&
    && \nearrow && \searrow
    \\
    a && \to && b &\to & c && \Downarrow && b
    \\
    & \searrow &\Downarrow& \nearrow &&
    && \searrow && \nearrow &&
  }
$$

corresponds to the objects of $\Theta_2 = \Delta \wr \Delta$ given by

$$
  ([3], (a_1, a_2, a_3))
  \,,
$$

where in turn

* $a_1 = [2]$

* $a_2 = [0]$

* $a_3 = [1]$.

=--

+-- {: .num_example #Grids}
###### Example

Composing with the functor $\delta_n$ from remark \ref{ProductOfDeltaInTheta} we obtain an embedding of $n$-fold simplices into strict $n$-categories

$$
  \Delta^{\times n} \stackrel{\delta_n}{\to}
  \Theta_n \hookrightarrow Str n Cat
  \,.
$$

Under this embedding an object $([k_1], [k_2], \cdots, [k_n])$ is sent to
the $n$-category which looks like (a globular version of) a $k_1 \times k_2 \times \cdots \times k_n$ **grid** of $n$-cells.

=--

Write

$$
  Str n Cat_{gaunt} \hookrightarrow Str n Cat
$$

for the inclusion of the [gaunt](http://ncatlab.org/nlab/show/%28infinity%2Cn%29-category#GauntStrictNCategories) strict $n$-categories into all [[strict n-categories]].

+-- {: .num_prop}
###### Proposition

$\Theta_n$ is the smallest [[full subcategory]] of $Str n Cat_{gaunt}$ containing the grids, the [[image]] of $\delta_n : \Delta^{\times n} \to Str n Cat$, example \ref{Grids}, and closed under [[retracts]].

=--

([B-SP, prop. 10.5](#BarwickSchommerPries))

### Groupoidal version

The groupoidal version $\tilde \Theta$ of $\Theta$ is a [[test category]] ([Ara](#Ara)).

## Examples

In $\Theta_0$ write $O_0$ for the unique object. Then write in $\Theta_n$

$$
  O_n := [1](O_{n-1})
  \,.
$$

This is the strict [[n-category]] free on a single $n$-[[globe]].

## Related concepts

* [[globular theory]]

* [[globular set]]

* [[globular operad]]

* [[simplex category]], [[globe category]], [[cube category]]

* A local [[model structure on simplicial presheaves]] on the Theta categories is called [[Theta space]]s and models [[(n,r)-category|(n,r)-categories]].

* A [[Cisinski model structure]] on bare [[presheaves]] on $\Theta_n$, modelling [[(∞,n)-categories]] is the [[model structure on cellular sets]].

## References

The $\Theta$-categories were introduced in

* [[Andre Joyal]], _Disks, duality and Theta-categories_ ([[JoyalThetaCategories.pdf:file]])
 {#Joyal}

A discussion with lots of pictures is in [chapter 7](http://cheng.staff.shef.ac.uk/guidebook/guidebook-new.pdf#page=131) of

* [[Eugenia Cheng]], [[Aaron Lauda]], _Higher-dimensional categories: an illustrated guidebook_ ([pdf](http://eugeniacheng.com/wp-content/uploads/2017/02/cheng-lauda-guidebook.pdf))

More discussion is in

* David Oury, _On the duality between trees and disks_, TAC vol. 24 ([pdf](http://www.tac.mta.ca/tac/volumes/24/16/24-16.pdf))

The following paper proves that $\Theta$ is a test category

* [[Denis-Charles Cisinski]], [[Georges Maltsiniotis]], _La catégorie $\Theta$ de Joyal est une catégorie test_ , JPAA **215** no.5 (2011) pp.962-982. ([pdf](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/theta.pdf), in French)

Discussion of embedding of $\Theta$ into strict $n$-categories is in

* [[Michael Batanin]], [[Ross Street]], _The universal property of the multitude of trees_, J. Pure Appl. Alg. 154 (2000), 3–13.
 {#BataninStreet}

* [[Michael Makkai]], M. Zawadowsky, _Duality for simple $\omega$-categories and disks_, Theory Appl. Categories 8 (2001), 114–243
 {#MakkaiZawadowsky}

* [[Clemens Berger]], _A cellular nerve for higher categories_, Adv. Math. 169 (2002), 118–175.
 {#BergerCellular}

The characterization in terms of $n$-fold [[categorical wreath products]] is in

* [[Clemens Berger]], _Iterated wreath product of the simplex category and iterated loop spaces_ ([arXiv:math/0512575](http://arxiv.org/abs/math/0512575)),
  {#Berger}

see also section 3 of

* [[Charles Rezk]], _A cartesian presentation of weak $n$-categories_ ([arXiv:0901.3602](http://arxiv.org/abs/0901.3602))

there leading over to the notion of [[Theta space]].

The groupoidal version $\tilde \Theta$ is discussed in

* [[Dimitri Ara]], _The groupoidal analogue $\tilde \Theta$ to Joyal's category $\Theta$ is a test category_ ([arXiv:1012.4319](http://arxiv.org/abs/1012.4319))

The relation of $\Theta_n$ to [[configuration spaces]] of points in the [[Euclidean space]] $\mathbb{R}^n$ is discussed in

* [[David Ayala]], [[Richard Hepworth]], _Configurations spaces and $\Theta_n$_ ([arXiv:1202.2806](http://de.arxiv.org/abs/1202.2806))
 {#AyalaHepworth}

Related discussion in the context of [[(infinity,n)-categories]] is also in

* [[Clark Barwick]], [[Chris Schommer-Pries]], _On the Unicity of the Homotopy Theory of Higher Categories_ ([arXiv:1112.0040](http://arxiv.org/abs/1112.0040), [slides](http://prezi.com/w0ykkhh5mxak/the-uniqueness-of-the-homotopy-theory-of-higher-categories/))
 {#BarwickSchommerPries}

[[!redirects ? category]]
[[!redirects Theta-category]]
[[!redirects ? categories]]
[[!redirects Theta-categories]]
[[!redirects disk category]]
[[!redirects disk categories]]

[[!redirects cell category]]


Theta characteristic

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

#Contents#
* table of contents
{:toc}

## Idea

For $X$ a [[space]] equipped with a notion of [[dimension]] $dim X \in \mathbb{N}$ and a notion of [[Kähler differential forms]],
a _$\Theta$-characteristic of $X$ is a choice of [[square root]] of the [[canonical characteristic class]] of $X$. See there for more details.

In [[complex analytic geometry]] and at least if the Theta characteristic is [[polarized variety|principally polarizing]] then its [[holomorphic sections]] are called [[theta functions]]. In particular for line bundles over the [[Jacobian variety]] of a [[Riemann surface]] they are called _[[Riemann theta functions]]_.

## Examples

### Over Riemann surfaces
 {#OverRiemannSurfaces}

+-- {: .num_prop }
###### Proposition

For $\Sigma$ a [[Riemann surface]], the choices of [[square roots]] of the [[canonical bundle]] correspond to the choice of [[spin structures]].

For $X$ of [[genus of a surface|genus]] $g$, there are $2^{2g}$ many choices of square roots of the canonical bundle.

=--

([Atiyah, prop. 3.2](#Atiyah)).

+-- {: .num_remark }
###### Remark

The first statement remains true in higher dimensions over [[Kähler manifolds]], see at  _[Spin structure -- On Kähler manifolds](spin%20structure#OverAKahlerManifold)_.

=--

+-- {: .num_prop }
###### Proposition

The function that sends a square root line bundle to the [[dimension]] of its space of [[holomorphic sections]] $mod \;2$ is a [[quadratic refinement]] of the [[intersection pairing]] on $H^1(X, \mathbb{Z}_2)$.

=--

This is due to ([Atiyah, theorem 2](#Atiyah)). A motivational survey in broader context of [[quadratic refinements]] of the [[intersection pairing]] in higher dimensions is in ([Hopkins-Singer 02, section 2.1](#HopkinsSinger02)).

### As metaplectic and Spin structure over (Kähler-)polarized varieties
 {#AsMetaplecticCorretionOfKählerPolarizations}

In the context of [[geometric quantization]] a [[metaplectic structure]] on a [[polarization]] is a [[square root]] of a certain line bundle. In the special case of [[Kähler polarization]] this is a square root precisely of the [[canonical line bundle]] of the underlying [[complex manifold]] and hence is a $\Theta$-characteristic. Also, equivalently this is a [[Spin structure]], see at _[spin structure -- Over a Kähler manifold](http://ncatlab.org/nlab/show/spin%20structure#OverAKahlerManifold)_. For more on this see at _[geometric quantization -- Quantum states as index of Dolbeault-Dirac operator](geometric+quantization#IndexOfDolbeaultDiracOperator)_.

Notice that generalizing from [[complex analytic geometry]] to [[algebraic geometry]] over other bases, then the analog of a [[Kähler polarization]] is a _[[polarized variety]]_. Hence a choice of [[Theta characteristic]] on a [[polarized variety]] is the analog of a metaplectically corrected Kähler manifold.

### Over intermediate Jacobians
 {#OverIntermediateJacobians}

A special square root of the canonical bundle on [[intermediate Jacobians]] in dimension $2k+1$ thought of as [[moduli spaces]] of (flat) [[circle n-bundles with connection|circle (2k+1)-bundles with connection]] has a unique [[section]] the [[partition function]] of abelian [[self-dual higher gauge theory]] (see there for details).  ([Witten 96](#Witten96), [Hopkins-Singer 02](#HopkinsSinger02)).

## Related concepts

[[!include square roots of line bundles - table]]

## References

The spaces of choices of $\Theta$-characteristics over [[Riemannian manifolds]] were originally discussed in

* {#Atiyah} [[Michael Atiyah]], _Riemann surfaces and spin structures_, Annales Scientifiques de l'École Normale Supérieure, (1971), Quatrième Série 4: 47–62, ISSN 0012-9593, MR0286136

See also

* M. Bertola, _Riemann surfaces and Theta Functions_, August 2010 ([pdf](http://www.mathstat.concordia.ca/faculty/bertola/ThetaCourse/ThetaCourse.pdf))

* Gavril Farkas, _Theta characteristics and their moduli_ (2012) ([arXiv:1201.2557](http://arxiv.org/abs/1201.2557))

The relation of Theta characteristics on [[intermediate Jacobians]] to [[self-dual higher gauge theory]] was first recognized in

* {#Witten96} [[Edward Witten]], _Five-Brane Effective Action In M-Theory_, J.Geom.Phys.22:103-133,1997 ([arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234))

and the argument there was made rigorous in

* {#HopkinsSinger02} [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_

Related arguments revolving around [[characteristic element of a bilinear form|characteristic elements]] for the [[intersection pairing]] appear in

* {#PoonenRains11} Bjorn Poonen, Eric Rains, _Self cup products and the theta characteristic torsor_ ([arXiv:1104.2105](http://arxiv.org/abs/1104.2105))

[[!redirects ? characteristic]]
[[!redirects ∞-characteristic]]
[[!redirects Theta-characteristic]]

[[!redirects ? characteristics]]
[[!redirects ∞-characteristics]]
[[!redirects Theta characteristics]]
[[!redirects Theta-characteristics]]

[[!redirects theta characteristic]]
[[!redirects theta characteristics]]
[[!redirects theta-characteristic]]
[[!redirects theta-characteristics]]


theta divisor

#Contents#
* table of contents
{:toc}

## Idea

A certain [[divisor (algebraic geometry)|divisor]] definable on any [[abelian variety]].

On [[Jacobian varieties]] it induces a [[principally polarized variety]] structure.

## References

* Wikipedia, _[Theta divisor](http://en.wikipedia.org/wiki/Theta_divisor)_

[[!redirects theta divisors]]

[[!redirects Theta divisor]]
[[!redirects Theta divisors]]

[[!redirects Θ divisor]]
[[!redirects Θ divisors]]


theta function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Theta functions
+--{: .hide}
[[!include theta functions - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Geometric quantization
+--{: .hide}
[[!include geometric quantization - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

Generally, a _theta function_ ($\theta$-function, $\Theta$-function) is a [[holomorphic section]] of a ([[polarized variety|principally polarizing]]) [[holomorphic line bundle]] over a [[complex torus]] / [[abelian variety]].
(e.g. [Polishchuk 03, section 17](#Polishchuk03)) and in particular over a [[Jacobian variety]] ([Beauville](#Beauville)) such as [[prequantum line bundles]] for (abelian) [[gauge theory]]. The line bundle being principally polarizing means that its space of holomorphic sections is 1-dimensional, hence that it determines the $\theta$-function up to a global complex scale factor. Typically these line bundles themselves are [[Theta characteristics]].
Expressed in [[coordinates]] $\mathbf{z}$ on the [[covering]] $\mathbb{C}^g$ of the [[complex torus]] $\mathbb{C}^g/\mathbb{Z}^g$, a $\theta$-function appears as an actual function $\mathbf{z} \mapsto \theta(\mathbf{z})$ satisfying certain transformation properties, and this is how theta functions are considered.

Those theta functions encoding sections of line bundles on a [[Jacobian variety]] $J(\Sigma)$ of a [[Riemann surface]] $\Sigma$ ([[determinant line bundles]], [Freed 87, pages 30-31](#Freed87)) typically vary in a controlled way with the [[complex structure]] modulus $\mathbf{\tau}$ of $\Sigma$ and are hence really functions also of this variable $(\mathbf{z},\mathbf{\tau}) \mapsto \theta(\mathbf{z}, \mathbf{\tau})$ with certain transformation properties. These are the _[[Riemann theta functions]]_. They are the expressions in local coordinates of the covariantly constant sections of the [[Hitchin connection]] on the [[moduli space of Riemann surfaces]] $\mathcal{M}_\Sigma$ ([Hitchin 90, remark 4.12](#Hitchin90)).  In the special case that $\Sigma$ is complex 1-dimensional of [[genus]] $g = 1$ (hence a complex [[elliptic curve]]) then such a function $(z,\tau) \mapsto \theta(z,\tau)$ of two variables with the pertinent transformation properties is a _[[Jacobi theta function]]_. Notice that in their dependency not only on $\tau$ but also on $z$ these are properly called _[[Jacobi forms]]_. Finally notice that these line bundles on [[Jacobian varieties]] have non-abelian generalizations to line bundles on [[moduli stacks of vector bundles]] of [[rank]] higher than one, whose sections may then be thought of as _generalized theta functions_ ([Beauville-Laszlo 93](#BeauvilleLaszlo93)).

Specifically in the context of [[number theory]]/[[arithmetic geometry]], by _the_ theta function one usually means the _[[Jacobi theta function]]_ (see there for more) for $z = 0$. While this is the historically first and archetypical function from which all modern generalizations derive their name, notice that at fixed $z$ as a function in $\tau$ the "theta function" is not actually a section of a line bundle anymore. The generalization in number theory of the Jacobi theta function that does again have a dependence on a twisting is the _[[Dirichlet theta function]]_ depending on a [[Dirichlet character]] (which by [[Artin reciprocity]] corresponds to a [[Galois representation]]).

Certain integrals of theta functions yield [[zeta functions]], see also at _[[function field analogy]]_.

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

### In quantization
 {#IdeaInQuantization}

Theta functions are naturally thought of as being the [[space of states (in geometric quantization)|states]] in the [[geometric quantization]] of the given complex space, the given holomorphic line bundle being the [[prequantum line bundle]] and the condition of holomorphicity of the section being the [[polarization]] condition. See for instance ([Tyurin 02](#Tyurin02)). In this context they play a proming role specifically in the quantization of [[higher dimensional Chern-Simons theory]] and of [[self-dual higher gauge theory]]. See there for more.

Specifically the fact that in [[geometric quantization|geometric]] [[quantization of Chern-Simons theory]] in the abelian case, and the [[holographic principle|holographically]] dual [[partition functions]] of the [[WZW model]] the choice of polarization is induced from the choice of [[complex structure]] $\mathbf{\tau}$ on a given [[Riemann surface]] $\Sigma$ and for each such choice there is then a section/[[partition function]] depending on a coordinate $\mathbf{z}$ in the  [[Jacobian]] $J(\Sigma)$ is reflected in the double coordinate dependence of the theta function:

$$
  \theta(\mathbf{z},\mathbf{\tau})
  =
  \theta\left(gauge\;field\;configuration\;on\;\Sigma\;, \; complex\;structure\;on\;\Sigma\right)
  \,.
$$

See ([AlvaresGaumé-Moore-Vafa 86, p. 4](#AlvaresGaumeMooreVafa86), [Freed 87, section 4](#Freed87), [Falceto-Gawedzki 94](#FalcetoGawedzki94), [Bunke-Olbrich 94, around def. 4.5](#BunkeOlbrich94), [Gelca-Uribe 10a](#GelcaUribe10a), [Gelca-Uribe 10b](#GelcaUribe10b), [Gelca-Hamilton 12](#GelcaHamilton12), [Gelca-Hamilton 14](#GelcaHamilton14), [Gelca 14](#Gelca14)).

Since from the point of view of [[Chern-Simons theory]] this is a [[wavefunction]], one might rather want to write $\Psi(\mathbf{z},\mathbf{\tau})$.

For nonabelian CS/WZW theory the same story goes through and one may the elements of the corresponding [[conformal blocks]] "generalized theta functions" ([Beauville-Laszlo 93](#BeauvilleLaszlo93)).

## Definition
 {#Definition}

Consider a [[complex torus]] $T \simeq V/\Gamma$ for given [[finite group]] $\Gamma$.

Say that a _system of multipliers_ is a system of invertible [[holomorphic functions]]

$$
  e_\gamma \colon V \longrightarrow \mathbb{C}^\times \hookrightarrow \mathbb{C}
$$

satisfying the cocycle condition

$$
  e_{\gamma + \delta}(z) = e_\gamma(z + \delta) e_\delta(z)
  \,.
$$

Then a _theta function_ is a [[holomorphic function]]

$$
  \theta \colon V \longrightarrow \mathbb{C}
$$

for which there is a system of multipliers $\{e_\gamma\}$ satisfying the [[functional equation]] which says that for each $z \in V$ and $\gamma \in \Gamma \hookrightarrow V$ we have

$$
  \theta(z + \gamma) = e_\gamma(z) \theta(z)
  \,.
$$

e.g. ([Beauville, above prop. 2.2](#Beauville)), also ([Beauville, section 3.4](#Beauville))

## Examples

* [[Jacobi theta function]]

* [[Riemann theta function]]

* [[Ramanujan theta function]]

* [[Dedekind eta function]]

## Related concepts

* the [[Verlinde formula]] computes the [[dimension]] of spaces of non-abelian theta functions

* [[special functions]]

* [[cubical structure on a line bundle]]

* [[mock theta function]]

* [[elliptic function]]

[[!include square roots of line bundles - table]]

## References

Introductions to the traditional notion include

* D.H. Bailey et al, _The Miracle of Theta Functions_ ([web](http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/node12.html))

* M. Bertola, _Riemann surfaces and theta  functions_ ([pdf](http://www.mathstat.concordia.ca/faculty/bertola/ThetaCourse/ThetaCourse.pdf))

Modern textbook accounts include

* {#Mumford83} [[David Mumford]], _Tata Lectures on Theta_, Birkhäuser 1983

* {#Polishchuk03} [[Alexander Polishchuk]], section 17 of _Abelian varieties, Theta functions and the Fourier transform_, Cambridge University Press (2003) ([review pdf](http://math1.unice.fr/~beauvill/pubs/poli.pdf))

Further discussion with an emphasis of the origin of theta functions in [[geometric quantization]] of [[Chern-Simons theory]] is in

* {#Beauville} [[Arnaud Beauville]], _Theta functions, old and new_, Open Problems and Surveys of Contemporary Mathematics SMM6, pp. 99–131 ([pdf](http://math.unice.fr/~beauvill/pubs/thetaon.pdf))

* {#Tyurin02} [[Andrei Tyurin]], _Quantization, Classical and quantum field theory and theta functions_, AMS 2003 ([arXiv:math/0210466v1](http://arxiv.org/abs/math/0210466))

* {#Nohara} Yuichi Nohara, _Independence of polarization in geometric quantization_ ([pdf](http://geoquant.mi.ras.ru/nohara.pdf))

* Gerard Lion, [[Michele Vergne]], _The Weil representation, Maslov index and theta series_

Specifically the theta functions appearing in [[2d CFT]] as [[conformal blocks]] and as spaces of sections of [[prequantum line bundles]] in [[quantization of Chern-Simons theory]] are discussed for instance in

* {#AlvaresGaumeMooreVafa86} [[Luis Alvarez-Gaumé]], [[Gregory Moore]], [[Cumrun Vafa]], _Theta functions, modular invariance, and strings_, Communications in Mathematical Physics Volume 106, Number 1 (1986), 1-4 ([Euclid](http://projecteuclid.org/euclid.cmp/1104115581))

* {#AlvaresGaumeBostMooreNelsonVafa87} [[Luis Alvarez-Gaumé]], [[Jean-Benoit Bost]], [[Gregory Moore]], Philip Nelson, [[Cumrun Vafa]], _Bosonization on higher genus Riemann surfaces_, Communications in Mathematical Physics, Volume 112, Number 3 (1987), 503-552 ([Euclid](http://projecteuclid.org/euclid.cmp/1104159982))

* {#Freed87} [[Daniel Freed]], around p. 30-31 of _On determinant line bundles_, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987,  (revised [pdf](http://www.math.utexas.edu/~dafr/Index/determinants.pdf), [dg-ga/9505002](http://arxiv.org/abs/dg-ga/9505002))

* {#FalcetoGawedzki94} Fernando Falceto, [[Krzysztof Gawędzki]], _Chern-Simons states at genus one_, Comm. Math. Phys. Volume 159, Number 3 (1994), 549-579. ([Euclid](https://projecteuclid.org/euclid.cmp/1104254732))

* Kazuhiro Hikami, _Mock (False) Theta Functions as Quantum Invariants_ ([arXiv:math-ph/0506073](http://arxiv.org/abs/math-ph/0506073))

* Kazuhiro Hikami, _Quantum Invariants, Modular Forms, and Lattice Points II_, J. Math. Phys. 47, 102301-32pages (2006) ([arXiv:math/0604091](http://arxiv.org/abs/math/0604091))

* {#GelcaUribe10a} [[Razvan Gelca]], [[Alejandro Uribe]], _From classical theta functions to topological quantum field theory_  ([arXiv:1006.3252](http://arxiv.org/abs/1006.3252), [slides pdf](http://www.math.ttu.edu/~rgelca/berk.pdf))

* {#GelcaUribe10b} [[Razvan Gelca]], [[Alejandro Uribe]], _Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$_ ([arXiv:1007.2010](http://arxiv.org/abs/1007.2010))

* {#GelcaHamilton12} [[Razvan Gelca]], [[Alastair Hamilton]], _Classical theta functions from a quantum group perspective_ ([arXiv:1209.1135](http://arxiv.org/abs/1209.1135))

* {#GelcaHamilton14} [[Razvan Gelca]], [[Alastair Hamilton]], _The topological quantum field theory of Riemann's theta functions_ ([arXiv:1406.4269](http://arxiv.org/abs/1406.4269))

* {#Gelca14} [[Razvan Gelca]], _Theta Functions and Knots_,  World Scientific 2014 ([prologue pdf](http://www.worldscientific.com/doi/suppl/10.1142/8872/suppl_file/8872_chap01.pdf), [publisher page](http://www.worldscientific.com/worldscibooks/10.1142/8872))

* {#Andersen11} Johan Martens [[Jørgen Andersen]], notes by Søren Jørgensen, p. 53 of _Topological quantum field theories and moduli spaces_, 2011 ([pdf](http://maths.fuglede.dk/noter/tqftms.pdf))

and more generally the [[partition functions]] of connection-twisted Dirac operators on even-dimensional locally symmetric spaces is discussed in

* {#BunkeOlbrich94}  [[Ulrich Bunke]], [[Martin Olbrich]], _Theta and zeta functions for locally symmetric spaces of rank one_ ([arXiv:dg-ga/9407013](http://arxiv.org/abs/dg-ga/9407013))

Generalization of this from abelian to non-abelian [[conformal blocks]] to "generalized theta functions" appears in

* {#BeauvilleLaszlo93} [[Arnaud Beauville]], [[Yves Laszlo]], _Conformal blocks and generalized theta functions_, Comm. Math. Phys. __164__ (1994), 385 - 419, [euclid](http://projecteuclid.org/euclid.cmp/1104270837), [alg-geom/9309003](http://arxiv.org/abs/alg-geom/9309003), [MR1289330](http://www.ams.org/mathscinet-getitem?mr=1289330)

brief review is in

* {#Gawedzki99} [[Krzysztof Gawedzki]], section 5 of _Conformal field theory: a case study_ in Y. Nutku, C. Saclioglu, T. Turgut (eds.) _Frontier in Physics_ 102, Perseus Publishing (2000) ([hep-th/9904145](http://xxx.lanl.gov/abs/hep-th/9904145))

That the Riemann zeta functions are the local coordinate expressions of the covariantly constant sections of the [[Hitchin connection]] is due to

* {#Hitchin90} [[Nigel Hitchin]], remark 4.12 in _Flat connections and geometric quantization_, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. ([Euclid](http://projecteuclid.org/euclid.cmp/1104200841))

Relation to [[elliptic genera]] (see also at _[[Jacobi form]]_)

* {#KL95} [[Kefeng Liu]], section 2.4 of _On modular invariance and rigidity theorems_, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 ([EUCLID](http://projecteuclid.org/euclid.jdg/1214456221), [pdf](http://www.math.ucla.edu/~liu/Research/loja.pdf))

Theta functions for higher dimensional varieties and their relation to [[automorphic forms]] is due to

* [[André Weil]], _Sur certaines groups d'operateur unitaires_, Acta. Math.  111 (1964), 143-211

see [Gelbhart 84, page 35 (211)](Langlands+program#Gelbhart84) for review.

Further developments here include

* {#Kudla77} [[Stephen Kudla]], _Relations between automorphic forms produced by theta-functions_, in _Modular Functions of One Variable VI_, Lecture Notes in Math. 627, Springer, 1977, 277–285.

* {#Kudla78} [[Stephen Kudla]], _Theta functions and Hilbert modular forms_,Nagoya Math. J. 69 (1978) 97-106

* {#Stopple95} [[Jeffrey Stopple]], _Theta and $L$-function splittings_, Acta Arithmetica LXXII.2 (1995) ([pdf](http://matwbn.icm.edu.pl/ksiazki/aa/aa72/aa7221.pdf))

* Yum-Tong Siu, _Theta functions in higher dimensions_ ([[SiuHigherTheta.pdf:file]])

[[!redirects theta functions]]
[[!redirects theta-function]]
[[!redirects theta-functions]]

[[!redirects Theta function]]
[[!redirects Theta functions]]
[[!redirects Theta-function]]
[[!redirects Theta-functions]]

[[!redirects theta-line bundle]]
[[!redirects theta-line bundles]]
[[!redirects theta line bundle]]
[[!redirects theta line bundles]]
[[!redirects theta line-bundle]]
[[!redirects theta line-bundles]]


theta functions - contents

**[[theta function]]**, **[[modular form]]**, **[[automorphic form]]**, **[[automorphic representation]]**

* [[Jacobi theta function]]

* [[Riemann theta function]]

* [[Dedekind eta function]]

**[[Mellin transform]]**

**[[zeta function]]**, **[[L-function]]**

* [[Riemann zeta function]]

* [[Dirichlet L-function]]

* [[Artin L-function]]

* [[automorphic L-function]]

* [[Ruelle zeta function]], [[Selberg zeta function]]

**[[eta function]]**

**in physics**

* [[conformal blocks]]

* [[zeta function regularization]]

* [[vacuum energy]]


theta vacuum

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Vacua
+-- {: .hide}
[[!include vacua -- contents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[Yang-Mills theory]] on 4d [[Minkowski spacetime]] the _theta vacuum_ is a [[vacuum state]] which is a [[superposition]] of each [[Yang-Mills instanton]]-sector weighted by a [[complex phase]] proportional to the [[theta angle]] times the [[instanton number]].

## Related concepts

* [[instanton sea]]

[[!include states and observables -- content]]

## References

Survey is in

* [[Richard Healey]], _Gauge Theory and the Theta Vacuum_, 2007 ([web](http://philsci-archive.pitt.edu/4403/), [pdf](http://philsci-archive.pitt.edu/4403/1/Madrid_paper_%28for_publication%29.pdf))

Detailed argument for the theta vacuum from [[chiral symmetry breaking]] is offered in

* [[Curtis Callan]], R.F. Dashen, [[David Gross]], _The Structure of the Gauge Theory Vacuum_, Phys.Lett. 63B (1976) 334-340 ([spire](http://inspirehep.net/record/3673?ln=en))

* [[Giovanni Morchio]], [[Franco Strocchi]], _Chiral symmetry breaking and theta vacuum structure in QCD_, Annals Phys.324:2236-2254, 2009 ([arXiv:0907.2522](https://arxiv.org/abs/0907.2522))

Discussion in a context of [[algebraic quantum field theory|algebraic]] [[non-perturbative quantum field theory]]:

* [[Franco Strocchi]], §8 of: _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

[[!redirects theta vacua]]

[[!redirects theta-vacuum]]
[[!redirects theta-vacua]]


Theta-space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Internal $(\infty,1)$-Categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _$\Theta_n$-space_ is one model for the notion of _[[(∞,n)-category]]_. A $\Theta_n$-space may be thought of as a [[globe category|globular]] $n$-category _up to coherent homotopy_, a globular $n$-category _[[internalization|internal]]_ to the [[(∞,1)-category]] [[∞Grpd]].

Concretely, a _$\Theta_n$-space is a [[simplicial presheaf]] on the [[Theta category|Theta_n category]], hence a "[[cellular object|cellular space]]" that satisfies

1. the globular [[Segal condition]] as a [[weak homotopy equivalence]];

1. and a completeness condition analogous to that of [[complete Segal spaces]].

In fact for $n = 1$ $\Theta_n = \Delta$ is the simplex category and a $\Theta_1$-space is the same as a [[complete Segal space]].

Noticing that a presheaf of _sets_ on $\Theta_n$ which satisfies the cellular [[Segal condition]] is equivalently a [[strict n-category]], $Theta_n$-spaces may be thought of as [[n-categories]] [[internal category in an (infinity,1)-category|internal to]] the [[(∞,1)-category]] [[∞Grpd]], defined in the [[cell category|cellular]] way.

## Overview

There is a [[cartesian closed category|cartesian closed]] [[category with weak equivalences]] $\Theta_n Sp_k^{fib}$ of **$(n+k,n)$-$\Theta$-spaces** for all

* $0 \leq n \leq \infty$;

* $-2 \leq k \leq \infty$

as the [[category of fibrant objects]] in a [[model category]] $\Theta_n Sp_k$,
being a [[Bousfield localization of model categories|left Bousfield localization]] of the injective [[model structure on simplicial presheaves]] on the $n$th [[Theta category]].

The [[weak equivalences]] in $\Theta_n Sp_k^{fib}$ are then (by the standard result discussed at [[Bousfield localization of model categories]]) just the objectwise weak equivalences in the standard [[model structure on simplicial sets]] $sSet_{Quillen}$.

## Definition

For $J$ a [[category]], write $\Theta J$ for the [[categorical wreath product]] over the [[simplex category]] $\Delta$ [Ber05](http://arxiv.org/abs/math/0512575).

Then with $\Theta_0 := {*}$ we have inductively

$$
  \Theta_n = \Theta \Theta_{n-1}
  \,.
$$

For $D = SPSh(C)^{inj}_{S}$ a [[model structure on simplicial presheaves]] on a category $C$ obtained by left [[Bousfield localization]] at a set of morphisms $S \subset Mor(SPSh(C)^{inj})$ from the global injective model structure, write

$$
  D-\Theta Sp := SPSh(\Theta C)^{inj}_{S_\Theta}
  \,,
$$

where $S_\Theta$ is the set of morphisms given by .... .

Set

$$
  \Theta_0 Sp_k := SSet_k
  \,,
$$

the left [[Bousfield localization]] of the standard [[model structure on simplicial sets]] such that fibrant objects are the [[Kan complex]]es that are [[homotopy n-type|homotopy k-type]]s. Then finally define inductively

$$
  \Theta_{n+1} Sp_k := (\Theta_n Sp_k)-\Theta Sp
  \,.
$$

Unwinding this definition we see that

$$
  \Theta_{n} Sp_k = SPSh(\Theta_n)^{inj}_{S_{n}}
  \,,
$$

for some set $S_n \subset Mor(SPSh(C))$ of morphisms.

## Properties

### Special values of $(n,k)$

+-- {: .query}
  I would have started $k$ at $-1$.  What does Rezk\'s notion do with $k = -2$?  ---Toby

  $-1$-groupoids are spaces which are either empty or contractible.  $-2$-groupoids are spaces which are contractible.  So $k=-2$ is the completely trivial case; it's included for completeness.  -- Charles

  I do know what a $(-2,0)$-category is, a triviality as you say.  But for $n \gt 0$, an $(n-2,n)$-category is the same as an $(n-2,n-1)$-category as far as I can see.  (Note: I say this *without* having worked through your version, but just thinking about what $(n,r)$-categories should be, as at [[(n,r)-category]].)  ---Toby

  I would say: $(n-2,n)$-category is a trivial concept, for every $n$, though $(n-2,n-1)$ isn't.  An $(n+1+k,n+1)$-category should amount to a category enriched over $(n+k,n)$-categories.  An $(-2,0)$-category is trivial (a point); an $(-1,1)$-category is a category enriched over the point, and so equivalent to the terminal category; an $(0,2)$-category is a category enriched over categories which are equivalent to the terminal category (and so equivalent to the terminal $2$-category, etc.) -- Charles R.

  <del>H\'m, that is a good argument.</del>

  (Sorry for not noticing before that you *are* Charles Rezk; for some reason I though of [[Charles Wells]].)  ---Toby

  [[David Roberts]]: I'm a little confused. The way I think about it, and I may have the indexing wrong, is that in an $(n,n+2)$-category $C$, for _all_ pairs of $n$-arrows $x,y$, there is a unique $n+1$-arrow between them. This implies that $x$ and $y$ are parallel, in particular, that $C$ has a single $(n-1)$-arrow.

  _Toby_:  Wait, I don\'t buy Charles\'s argument after all.  Yes, a $(-1,1)$-category is a category enriched over the point, but that doesn\'t make it necessarily the terminal category; it makes it a truth value.  If it has an object, then it\'s trivial, but it might be empty instead.  The difference between a $(-1,1)$-category and a $(-1,0)$-category is that every $0$-morphism in the latter must be invertible, which is no difference at all; that\'s why we have this repetition.  (And thereafter it propagates indefinitely.)

  Similarly, with David\'s argument, what if $C$ has no $n$-arrows at all?

  [[David Roberts]]: Yes - that should then be \'Assuming $C$ has an object, then it has a single $(n-1)$-arrow\'. Assuming I got the indexing right, I must stress. I think I grasp $(n,n+1)$-categories, but I'm not solid on these new beasties.

  Toby, I guess you are right.  I don't know what I was thinking.  -- Charles R.

  Thanks for joining, in, Charles. Toby is, by the way, our esteemed expert for -- if not the inventor of -- [[negative thinking]]. :-) - [[Urs Schreiber|USc]]

  _Toby_:  All right, so we allow $k = -2$, since $n$ might be $0$; but for an $(n-2,n)$-$\Theta$-space is the same as an $(n-1,n)$-$\Theta$-space for $n \gt 0$.  OK, I\'m happy with that; now to understand the definition!  (^_^)
  =--

### Cartesian monoidal and enriched structure
 {#CartesianMonoidalAndEnrichedStructure}

The [[model category]] $\Theta_n Sp_k$ is a [[cartesian monoidal category|cartesian]] [[monoidal model category]].

The idea is that $\Theta_n Sp_k$ is naturally an [[enriched model category]] over itself.

### $(n+1,r+1)$-$\Theta$-space of $(n,r)$-$\Theta$-spaces
 {#OfAll}

Here is the idea on how to implement the notion $(n+1,r+1)$-category of all $(n,r)$-categories in the context of Theta-spaces. At the time of this writing, this hasn't been spelled out in total.

As mentioned above regard $\Theta_k Sp_n$ as a category enriched over itself. Then define a presheaf $\mathbf{X}$ on $\Theta_{n+1}$ by setting

* $\mathbf{X}[0] = $ collection of objects of $\Theta_n Sp_k$

* $\mathbf{X}([m](\theta_1, \cdots, \theta_m))
   = \coprod_{a_0, \cdots, a_m}
     C(a_0,a_1)(\theta_1) \times \cdots \times
     C(a_{m-1},a_m)(\theta_m)
  $

This object satisfies the [[Segal conditions]] (its descent conditions) in all degrees except degree 0. A suitable localization operation ca-n fix this. The resulting object should be the "$(n+1,k+1)$-$\Theta$-space of $(n,k)$-$\Theta$-spaces".

### Homotopy hypothesis

The definition of weak $(n,r)$-categories modeled by $\Theta$-spaces does satisfy the [[homotopy hypothesis]]: there is an evident notion of groupoid objects in $\Theta_n Sp_k$ and the full subcategory on these models [[homotopy n-types]].

([Rez09, 11.25](#Rezk)).

### Relation to cellular sets

There is a [[model structure on cellular sets]] (see there), hence on set-valued presheaves on $\Theta_n$ (instead of simplicial presheaves) which  is [[Quillen equivalence|Quillen equivalent]] to the Rezk model structure on $\Theta_n$-spaces.

In fact the Theta-space model structure is the [simplicial completion](Cisinski+model+structure#SimplicialCompletion) of the [[Cisinski model structure]] on presheaves on $\Theta_n$ ([Ara](#Ara))

## Examples

For low values of $n,k$ this reproduces the following cases:

* for $n=0$ we have $\Theta_0 Sp_\infty = sSet_{Quillen}$ with its [[model structure on simplicial sets|standard model structure]] and hence $\Theta_0 Sp_\infty^{fib} = $ [[∞Grpd]].

* for $n=1$ objects in $\Theta_1 Sp_\infty^{fib} $ are [[complete Segal spaces]], hence [[(∞,1)-category|(∞,1)-categories]].

## Related concepts

* [[cellular set]]

* [[omega-category]]

* [[globular operad]], [[globular theory]],

## References

The notion of $\Theta$-spaces was introduced in

* [[Charles Rezk]], _A cartesian presentation of weak $n$-categories_ Geom. Topol. 14 (2010), no. 1, 521–571 ([arXiv:0901.3602](http://arxiv.org/abs/0901.3602))

  _Correction to "A cartesian presentation of weak $n$-categories"_ Geom. Topol. 14 (2010), no. 4, 2301–2304. MR 2740648 ([pdf](http://www.math.uiuc.edu/~rezk/cs-objects-correction.pdf))
 {#Rezk}

* [[Charles Rezk]], _Cartesian presentations of weak n-categories
An introduction to $\Theta_n$-spaces_ (2009) ([pdf](http://www.math.uiuc.edu/~rezk/northwestern-2009-n-cat-handout.pdf))

The definition of the categories $\Theta_n$ goes back to [[Andre Joyal]] who also intended to define [[n-category|n-categories]] using it. This has been achieved at about the same time by Simpson:

* [[Carlos Simpson]], _A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert-Van Kampen_ ([arXiv:alg-geom/9704006](https://arxiv.org/abs/alg-geom/9704006))

* [[Carlos Simpson]], _On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani's weak n-categories_ ([arXiv:math/9810058](https://arxiv.org/abs/math/9810058))

Discussion comparing $\Theta_{n+1}$-spaces to [[enriched (infinity,1)-categories]] in $\Theta_n$-spaces is in

* {#BergnerRezk} [[Julie Bergner]], [[Charles Rezk]], _Comparison of models for $(\infty,n)$-categories_ ([arXiv:1204.2013](http://arxiv.org/abs/1204.2013))

* {#BergnerRezk14} [[Julie Bergner]], [[Charles Rezk]], _Comparison of models for $(\infty,n)$-categories II_ ([arXiv:1406.4182](http://arxiv.org/abs/1406.4182))

The note on the $(n+1,k+1)$-$\Theta$-space of all $(n,k)$-$\Theta$-spaces comes from communication with [[Charles Rezk]] [here](http://mathoverflow.net/questions/5867/n1-r1-theta-space-of-n-r-theta-spaces).

Relation to simplicial completion of the [[Cisinski model structure]] [[model structure on cellular sets|on cellular sets]] is in

* [[Dimitri Ara]], _Higher quasi-categories vs higher Rezk spaces_ ([arXiv:1206.4354](http://arxiv.org/abs/1206.4354))
 {#Ara}

[[!redirects theta space]]
[[!redirects theta spaces]]
[[!redirects theta-space]]
[[!redirects theta-spaces]]
[[!redirects Theta space]]
[[!redirects Theta spaces]]
[[!redirects Theta-space]]
[[!redirects Theta-spaces]]
[[!redirects ϑ space]]
[[!redirects ϑ spaces]]
[[!redirects ϑ-space]]
[[!redirects ϑ-spaces]]
[[!redirects θ space]]
[[!redirects θ spaces]]
[[!redirects θ-space]]
[[!redirects θ-spaces]]

[[!redirects theta_n space]]
[[!redirects theta_n spaces]]
[[!redirects theta_n-space]]
[[!redirects theta_n-spaces]]
[[!redirects Theta_n space]]
[[!redirects Theta_n spaces]]
[[!redirects Theta_n-space]]
[[!redirects Theta_n-spaces]]
[[!redirects ϑ_n space]]
[[!redirects ϑ_n spaces]]
[[!redirects ϑ_n-space]]
[[!redirects ϑ_n-spaces]]
[[!redirects θ_n space]]
[[!redirects θ_n spaces]]
[[!redirects θ_n-space]]
[[!redirects θ_n-spaces]]

[[!redirects theta-n space]]
[[!redirects theta-n spaces]]
[[!redirects theta-n-space]]
[[!redirects theta-n-spaces]]
[[!redirects Theta-n space]]
[[!redirects Theta-n spaces]]
[[!redirects Theta-n-space]]
[[!redirects Theta-n-spaces]]
[[!redirects ϑ-n space]]
[[!redirects ϑ-n spaces]]
[[!redirects ϑ-n-space]]
[[!redirects ϑ-n-spaces]]
[[!redirects θ-n space]]
[[!redirects θ-n spaces]]
[[!redirects θ-n-space]]
[[!redirects θ-n-spaces]]


Thibault Damour

* [webpage](http://www.ihes.fr/~damour/)

Damour made the theoretical prediction ([Buonanno-Damour 98](#BuonannoDamour98), see [Damour 10](#Damour10)) of the [[gravitational wave]]-signal emitted by inspiralling [[relativistic binaries]] which has later been observed by [[LIGO]].

## Selected writings

On [[U-duality]]:

* [[Thibault Damour]], [[Marc Henneaux]], [[Hermann Nicolai]], _$E(10)$ and a 'small tension expansion' of M
theory_, Phys. Rev. Lett. 89, 221601 (2002) ([arXiv:hep-th/0207267](http://arxiv.org/abs/hep-th/0207267));

* {#Damour14} [[Thibault Damour]], _Quantum supersymmetric Bianchi IX  Cosmology and its hidden Kac-Moody Structure_, talk at [Gravitation, Solitons and Symmetries](http://www.lestudium-ias.com/#!registration-conference-21-may-gibbons/ci2z) ([[DamourSQC14.pdf:file]])

On prediction of [[gravitational wave]] profiles emitted by [[black hole]] [[relativistic binaries]] (as later detected by [[LIGO]]:

* {#Damour10} [[Thibault Damour]], _Gravitational Waves from Coalescing Binary Black Holes: Theoretical and Experimental Challenges_, talk in Munich, 2010 ([web](http://www.theorie.physik.uni-muenchen.de/MATH/seminars/asc_kolloquium/archiv_sose10/damour/video_damour/index.html), [youtube](https://www.youtube.com/watch?v=hVy3YBalpWE))

* {#BuonannoDamour98} [[Alessandra Buonanno]], [[Thibault Damour]], _Effective one-body approach to general relativistic two-body dynamics_, Phys.Rev. D59 (1999) 084006 ([arXiv:gr-qc/9811091](http://arxiv.org/abs/gr-qc/9811091))

* {#BuonannoDamour00} [[Alessandra Buonanno]], [[Thibault Damour]], _Binary black holes coalescence: transition from adiabatic inspiral to plunge_, IX Marcel Grossmann Meeting in Rome, July 2000 ([arXiv:gr-qc/0011052](http://arxiv.org/abs/gr-qc/0011052))

* {#DamourNagar} [[Thibault Damour]], [[Alessandro Nagar]], _An improved analytical description of inspiralling and coalescing black-hole binaries_, Phys.Rev.D79:081503,2009 ([arXiv:0902.0136](http://arxiv.org/abs/0902.0136))

* {#Damour14} [[Thibault Damour]], [[Alessandro Nagar]], _A new analytic representation of the ringdown waveform of coalescing spinning black hole binaries_, 	10.1103/PhysRevD.90.024054 ([arXiv:1406.0401](http://arxiv.org/abs/1406.0401))

* {#Damour16} [[Thibault Damour]], _Gravitational Waves and Binary Systems_, talk at IHES, Feb 2016 ([video recording](https://www.youtube.com/watch?v=qYong_D4VUI))

* {#Damour17} [[Thibault Damour]], _High-energy gravitational scattering and the general relativistic two-body problem_, Phys. Rev. D 97, 044038 (2018) ([arXiv:1710.10599](https://arxiv.org/abs/1710.10599))

* {#BiniDamour18} Donato Bini, [[Thibault Damour]], _Gravitational spin-orbit coupling in binary systems at the second post-Minkowskian approximation_, Phys. Rev. D 98, 044036 (2018) ([arXiv:1805.10809](https://arxiv.org/abs/1805.10809))

* {#Damour19} [[Thibault Damour]], _On the gravitational interaction of spinning bodies_, talk at _[Souriau2019](http://souriau2019.fr/?page_id=2)_  ([video recording](https://www.youtube.com/watch?v=8HO3DLKircs))

## Related $n$Lab entries

* [[relativistic binary]]

* [[gravitational wave]]

* [[U-duality]]

* [[exceptional generalized geometry]]

category: people


Thibault Decoppet

* [personal page](https://www.thibaultdecoppet.com/)

## Selected writings

On [[fiber functor|fiber]] [[2-functors]] for [[fusion 2-categories]]:

* {#DY23} [[Thibault Decoppet]], [[Matthew Yu]], *Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories* (2023) [[arXiv:2306.08117](https://arxiv.org/abs/2306.08117)]

## Related entries

* [[generalized global symmetry]]

category: people


thick subcategory

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

#Contents#
* table of contents
{:toc}

## Definition in (pre)triangulated categories

A non-[[empty category|empty]] [[full subcategory|full]] (pre-)[[triangulated subcategory]] is called _thick_ (or __épaisse__) if it is "closed under [[direct summands]]" (e.g. [Murayama, Def. 11](#Murayama), see [Stacks Project, Def. 13.6.1](#StacksProject05RA)).

## In abelian categories

A nonempty full subcategory $T$ of an [[abelian category]] $A$ is  __thick__ (in the strong sense) if for every exact sequence
$$
0 \longrightarrow M\longrightarrow M''\longrightarrow M'\longrightarrow 0
$$
in $A$, the object $M''$ is in $T$ iff $M$ and $M'$ are in $T$.

In words, it is an [[additive category|additive]] [[full subcategory]] closed under subobjects, quotients and extensions (e.g. [Dichev 2009](#Dichev09) and [[Pierre Gabriel]] in [[Des Categories Abeliennes]]).

In any abelian category all subobjects are [[kernels]] of [[epimorphisms]]; and all quotients are [[cokernels]] of [[monomorphisms]].

Another way to pack the same definition is that thick subcategory of an abelian category is a [[topologizing subcategory]] closed under extensions.

#### Remarks on terminology in abelian context

For some authors the thick subcategory is also called a _[[Serre subcategory]]_ (in a weak sense), the term which we reserve for (generally) a stronger notion.

For any subcategory of an [[abelian category]] $A$ one denotes by $\bar{T}$ the full subcategory of $A$ generated by all objects $N$ for which any (nonzero) [[subquotient]] of $N$ in $T$ has a (nonzero) [[subobject]] from $T$. This becomes an idempotent operation on the class of subcategories of $A$ where $T\subset \bar{T}$ iff $T$ is [[topologizing subcategory|topologizing]].
Moreover $\bar{T}$ is always thick in the stronger sense.
[[Serre subcategory|Serre subcategories]] in the strong sense are those (nonempty) subcategories which are stable under the operation $T\mapsto\bar{T}$. In full generality this is a bit
stronger than thick subcategory and the two senses agree if the ambient category is a full category of modules.

## Serre quotient category

Following the extensions of an early work of Serre by Grothendieck and Gabriel, for any thick subcategory $T$ in an [[abelian category]] $A$, one defines the (Serre) quotient category $A/T$ as the one having the same [[objects]] as $A$ and [[hom-sets]] given by

$$
  (A/T)(X,Y) := colim A(X',Y/Y')
$$

where the [[colimit]] runs through all [[subobjects]] $X'\subset X$, $Y'\subset Y$ such that $X/X' \in Ob T$, $Y'\in Ob T$. The quotient functor $Q \colon A\to A/T$ is obvious.

Notice that the set of morphisms is indeed [[small set|small]], so that the Serre quotient category exists as a [[locally small category]]. On the other hand, one can construct an equivalent [[localization]] by the Gabriel-Zisman  localizing at the class $\Sigma$ of all morphisms
whose kernel and cokernel are in $T$. Although
$\Sigma$ admits the [[calculus of fractions]],
this method does not guarantee the existence in general.

A basic example is the quotient of the category of abelian groups modulo the torsion groups. This category is equivalent to the category of $\mathbb{Q}$-vector spaces, by the functor which maps an abelian group $M$ to the scalar extension $M \otimes_{\mathbb{Z}} \mathbb{Q}$. (See the [Stacks Project, Tag 0B0J](http://stacks.math.columbia.edu/tag/0B0J) for a proof.)

### Localizing subcategories

A thick subcategory (here always in the strong sense) is said to be [[localizing subcategory|localizing]] if $T$ is thick and the Serre quotient functor $Q$ admits a right adjoint $A/T\to A$, often called the __section functor__. In other words $A/T$ is then a reflective subcategory of $A$. Every coreflective thick subcategory $T$ admits a section functor, and the converse holds if $A$ has injective envelopes. A thick subcategory $T\subset A$ is a coreflective iff $(T,F)$ is a [[torsion theory]] where

$$
F \coloneqq \{X\in Ob A\,|\,A(T,X) = 0\}
$$

## Thick subcategories and saturation

Recall that a class $\Sigma$ of morphisms with category of fractions $\mathcal{C}_\Sigma$ is called _saturated_ if $\Sigma$ coincides with the class of morphisms inverted by the canonical functor $\mathcal{C}\to \mathcal{C}_\Sigma$.

Let $A$ be an abelian category, $T$ be a thick subcategory in the strong sense and let $\Sigma_T$ be the class of morphisms in $A$ such that their kernel and cokernel are in $T$. Then $\Sigma_T$ has a right and a left calculus of fractions and is saturated for the canonical functor $p:A\to A/T$. Furthermore, $p$ maps to zero objects precisely the objects in $T$.

Conversely, let $\Sigma$ be a saturated class of morphisms of $A$ with a calculus of fractions on the right and on the left. Then the full subcategory on the objects that are mapped to zero by the canonical $p:A\to A_\Sigma$ is thick. In other words, there is a bijection between thick subcategories in the strong sense and saturated classes of morphisms with a calculus of fractions on the right and on the left.

(For this material see [Schubert 1970](#Schubert70b), pp.105-107).

## Related concepts

* [[thick subcategory theorem]]

## References

### Abelian context

* [[Pierre Gabriel]], _Des catégories abéliennes_, Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448 ([numdam](http://www.numdam.org/item?id=BSMF_1962__90__323_0))

* [[eom]], *[Localization of categories](http://eom.springer.de/l/l060290.htm)*

* [[A. L. Rosenberg]], _Noncommutative algebraic geometry and representations of quantized algebras_, MIA __330__, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9

It is called "dense subcategory" (see page 165) in Popescu:

* N. Popescu, [[Abelian categories with applications to rings and modules]], London Mathematical Society Monographs 3, Academic Press 1973, xii + 470 pp

* {#Schubert70b} H. Schubert, _Kategorien II_ , Springer Heidelberg 1970.

Discussion for [[quiver representations]]:

* {#Dichev09} Nikolay Dimitrov Dichev, *Thick subcategories for quiver representations* 2009 ([pdf](https://d-nb.info/995877602/34))

### Triangulated context

* {#Murayama} Takumi Murayama, *The classification of thick subcategories and Balmer's reconstruction theorem* ([pdf](https://web.math.princeton.edu/~takumim/NeemanBalmer.pdf))

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, lecture 26, _Thick subcategories_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture26.pdf))

* {#StacksProject05RA} [[The Stacks Project]], *[Quotients of triangulated categories](https://stacks.math.columbia.edu/tag/05RA)*

[[!redirects thick subcategories]]
[[!redirects epaisse subcategory]]
[[!redirects epaisse subcategories]]
[[!redirects épaisse subcategory]]
[[!redirects épaisse subcategories]]
[[!redirects Serre quotient category]]

[[!redirects thick ideal]]
[[!redirects thick ideals]]


thick subcategory theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called the _thick subcategory theorem_ gives a a characterization of all [[thick subcategories]] of the [[stable homotopy category]] of [[p-localization|p-local]] [[finite spectra]].

This is a consequence of the [[nilpotence theorem]].

## Statement

For $n \in \mathbb{N}$, write $K(n)$ for the [[Morava K-theory]] [[spectrum]].

Say that a $p$-local finite spectrum $X$ has _type $n$_ if $K(n)_\ast(X) \neq 0$ but $K(k)_\ast(X) \simeq 0$ for all $k \lt n$. (e.g. [Lurie, def. 6](#Lurie)).

Write $Spectra^{fin}_p \hookrightarrow Spectra$ be the [[full subcategory]] of the [[(infinity,1)-category of spectra]] on the [[p-localization|p-local]] [[finite spectra]].

The **thick subcategory theorem** says that thick subcategories of $Spectra^{fin}_p$ are precisely the full subcategories on the spectra of type $\lt n$, for some $n$. (e.g. [Lurie, theorem 8](#Lurie)).

In other words, the thick subcategories are the [[kernels]] of $K(n)_\ast$ (for any given prime).

## Applications

The thick subcategory theorem serves to determine the [[prime spectrum of a symmetric monoidal stable (∞,1)-category]] of the [[stable homotopy theory]].

(For an exposition see [MazelGee 13, around slide 9](#MazelGee13)).

## References

The original article is

* [[Michael Hopkins]], [[Jeff Smith]], _Nilpotence and stable homotopy theory_ II. Ann. of Math. (2),148(1):1–49, 1998

Further discussion is in

* Alain Jeanneret, [[Peter Landweber]], [[Douglas Ravenel]], _A note on the thick subcategory theorem_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/jlr.pdf))

* {#Lurie} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, lecture 26, _Thick subcategories_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture26.pdf))

Some big-picture motivation is in

* {#MazelGee13} [[Aaron Mazel-Gee]], _You could've invented $tmf$_, April 2013 ([pdf slides](http://math.berkeley.edu/~aaron/writing/ustars-tmf-beamer.pdf))


Thierry Coquand

Thierry Coquand is a professor in computer science at the University of Gothenburg, Sweden.

* [webpage](http://www.cse.chalmers.se/~coquand/)

## Selected writings

Introducing the [[calculus of constructions]] that later forms the basis of the [[proof assistant]] *[[Coq]]*:

*  [[Thierry Coquand]], [[Gérard Huet]], _The Calculus of Constructions_, INRIA Rapport **530** (1986) [[inria:00076024](http://hal.inria.fr/inria-00076024/en/), [pdf](http://hal.inria.fr/docs/00/07/60/24/PDF/RR-0530.pdf)]

Introducing the notion of [[inductive types]] in [[type theory]] which later became the foundations of the [[calculus of inductive constructions]] used by [[Coq]]:

* [[Thierry Coquand]], [[Christine Paulin]], *Inductively defined types*, COLOG-88 Lecture Notes in Computer Science **417**, Springer (1990) 50-66 [[doi:10.1007/3-540-52335-9_47](https://doi.org/10.1007/3-540-52335-9_47)]

On [[homological algebra]] in [[constructive mathematics]] via [[type theory]]:

* {#CoquandSpiwack} [[Thierry Coquand]], [[Arnaud Spiwack]], *Towards constructive homological algebra in type theory*, in: *Towards Mechanized Mathematical Assistants. MKM Calculemus 2007*,  Lecture Notes in Computer Science **4573** Springer (2007) [[doi:10.1007/978-3-540-73086-6_4](https://doi.org/10.1007/978-3-540-73086-6_4), [pdf](https://hal.inria.fr/inria-00432525/document)]

Discussion of notions of [[finite sets]] in [[constructive mathematics]]:

* [[Arnaud Spiwack]], [[Thierry Coquand]], *Constructively Finite?*, in: *Contribuciones cientifícas en honor de Mirian Andrés Gómez*, Universidad de La Rioja (2010) 217-230 [ISBN:978-84-96487-50-5, [inria-00503917](https://inria.hal.science/inria-00503917)]

On [[identity types]] in [[dependent type theory]] ([[homotopy type theory]]):

* [[Thierry Coquand]], slides 26-28 of: *Equality and dependent type theory* (2011) [[pdf](https://www.cse.chalmers.se/~coquand/equality.pdf), [[Coquand-EqualityAndDependentTypeTheory.pdf:file]]]

On the [[structure identity principle]]:

* [[Thierry Coquand]], [[Nils Anders Danielsson]], *Isomorphism is equality*, Indagationes Mathematicae **24** 4 (2013) 1105-1120 [[doi:10.1016/j.indag.2013.09.002](https://doi.org/10.1016/j.indag.2013.09.002)]

On [[weakly constant functions]] and [[propositional truncation]] in [[homotopy type theory]]:

* [[Nicolai Kraus]], [[Martín Escardó]], [[Thierry Coquand]], [[Thorsten Altenkirch]], *Notions of Anonymous Existence in Martin-Löf Type Theory*,  Logical Methods in Computer Science **13** 1 [doi:10.23638/LMCS-13(1:15)2017, [arXiv:1610.03346](https://arxiv.org/abs/1610.03346)]

On [[synthetic algebraic geometry]]:

* [[Felix Cherubini]], [[Thierry Coquand]], [[Matthias Hutzler]], *A Foundation for Synthetic Algebraic Geometry* (2023) [[arXiv:2307.00073](https://arxiv.org/abs/2307.00073)]

## Related entries

* [[calculus of constructions]]

* [[Coq]]

* [[cubical type theory]]

category: people


thin > history

< [[thin]]
[[!redirects thin -- history]]


thin category

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

# Thin categories
* table of contents
{: toc}

## Definition

A __thin category__ or __posetal category__ is a [[category]] in which, given a [[pair]] of [[objects]] $x$ and $y$ and any two [[morphisms]] $f, g \,\colon\, x \to y$, the morphisms $f$ and $g$ are equal:

$$
      x \underoverset{\quad g \quad}{f}{\rightrightarrows} y
      \;\;\;\implies\;\;\;
      f = g
$$

Another synonym is *[[(0,1)-category]]*.

## Properties
 {#Properties}

### Relation to order theory
 {#RelationToOrderTheory}

Up to [[isomorphism of categories|isomorphism]], a thin category [[relation between type theory and category theory|is]] a [[preordered set]] ("[[proset]]").  Up to [[equivalence of categories|equivalence]], a thin category [[relation between type theory and category theory|is]] the same thing as a [[partially ordered set]] ("[[poset]]").

For more on this see at *[[relation between preorders and (0,1)-categories]]*.

## Examples

A [[poset]] is a thin category. In particular so are [[lattice|(semi)lattices]], [[Heyting algebra|Heyting algebras]] and [[frame|frames]].

## Related concepts

* [[symmetric proset]]

* [[specialization order]]

* [[separation axioms in terms of lifting properties]]

[[!redirects thin category]]
[[!redirects thin categories]]


thin element

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

#Contents#
* table of contents
{:toc}

## Idea

"Thin elements" occur in various contexts.  The basic idea is that  many [[higher category theory|higher categorical structures]] have cells (of [[geometric shapes for higher structures|various shapes]]).  However, in some contexts, some of these cells should be regarded as "more canonical," "more [[equivalence]]-like," or "more identity-like" than others.  They are ''really'' there because of lower dimensional features. The cells with this property are frequently called **thin** and the extra information of which cells are thin is called a **thin structure** or a **stratification**.

## Thin cells in higher groupoids

In the simplicial [[singular complex]] $S X$ of a [[topological space]] $X$, a singular simplex $f: \Delta^n \to X$ is called **thin** if it factors through a [[retraction]] $r: \Delta^n \to \Lambda^{n-1}_i$ to some horn of $\Delta^n$.  Then the well known [[Kan complex|Kan]] condition on $S X$ can be strengthend to say that every horn in $S X$ has a _thin_ filler.  There are also, usually, other non-thin fillers which depend on $X$.

It is difficult to decide the laws that these these thin elements should satisfy.  Keith Dakin's axioms on thin elements in a simplicial  set $K$ with subsets $T_n \subseteq K_n, n \geq 1$, of elements called _thin_ are

1. Degenerate elements are thin;

1. Any horn has a unique thin filler;

1. If all faces but one of a thin element are thin, then so is the remaining face.

Ashley's theorem, conjectured by Dakin, is that the category of such  $T$-[[T-complex|complexes]] is equivalent to that of [[crossed complexes]].

As for relations with complexes having cells of different shapes, Brown and Higgins proved that cubical $T$-complexes are equivalent to crossed complexes, and to strict cubical $omega$-groupoids with connections.  In that theory, the notion of thin element is crucial because it can be proved that any (multiple) composition of thin elements is thin, and because of the relation of thin elements to commutative shells.  David Jones in his thesis developed the notion of polyhedral $T$-complex, and gave sufficient axioms for these to be equivalent to crossed complexes.

## Thin structures in double categories

Among other things, a **thin structure** on a [[double category]] can be seen as giving the information of which squares should be thought of as _commuting_ rather than simply _containing a 2-cell_.  The most general definition seems to be that a thin structure on a double category $D$ is a morphism of double categories $\Theta: \square C \to D$ where $C$ is any category and $\square C$ is the double category of commuting squares in $C$.

Note that $D$ contains the categories $D_v, D_h$ of vertical and horizontal edges respectively, and it is often convenient to take $C$ to be say $D_v$ and for $\Theta$ to be the identity on $D_v$. Then $\Theta$ defines a function $\Gamma: D_v \to D$ to the set of squares of $D$ which is called a _connection_, and also a functor $\gamma: D_v \to D_h$ which is called the _holonomy_ of the connection $\Gamma$. The laws which $\Gamma$ satisfies allow $\Theta$ to be recovered from $\Gamma$ in some cases.

Thus some thin structures  contain a way to represent any vertical arrow as a horizontal arrow (or vice versa, depending on one's conventions).  Thin structures can equivalently be described by "foldings" and "connection pairs."  It turns out that when they exist, they are unique up to a (suitably weak) notion of equivalence; see also [[framed bicategory]].

## Simplicial nerves for higher categories

Following on the work mentioned above for thin elements in Kan complexes, one can consider thin elements in [[nerves]] of higher categories.  In "The algebra of oriented simplices" Street constructed the simplicial nerve for a [[strict omega-category]], which comes equipped with a collection of thin simplices that are identities (rather than merely cells).  Thus this nerve is not just a simplicial set, but a [[stratified simplicial set]] (a simplicial set equipped with a collection of "thin simplices" containing all degeneracies).

Street and Roberts conjectured that the nerves of strict $\omega$-categories could be characterized as stratified simplicial sets such that
1. every [[complicial horn]] has a unique thin filler, and
1. for such a thin filler, if all faces but the one missing face in the complicial horn are thin, then so is the last one.

Such stratified simplicial sets are called [[complicial sets]].  This conjecture was later proved by Verity.

A [[weak complicial set]] is a stratified simplicial set satisfying the above conditions except that thin fillers need not be unique. Street conjectured that these represent the nerves of _weak_ $\omega$-categories, where now the thin simplices are supposed to be the _equivalences_ rather than the identities.  But since every complicial set is weak complicial, one should also require that the collection of thin simplices is "maximal" in some sense; Street gave one proposed characterization.  Weak complicial sets have been studied extensively by Verity.

## References

Relevant references for thinness in the context of  [[simplicial T-complexes]] include:

* [[Ronnie Brown]], _An Introduction to simplicial T-complexes_,  [Esquisses Math. 32 (1983) Part 1](http://ehres.pagesperso-orange.fr/Cahiers/brownem32.pdf)

* M.K. Dakin, _Kan complexes and multiple groupoid structures_ Ph.D Thesis, University of Wales, Bangor, 1977. [Esquisses Math. (1983) 32 Part 2](http://ehres.pagesperso-orange.fr/Cahiers/dakinEM.pdf)

* N. Ashley, _Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan_, Ph.D Thesis University of Wales, Bangor, 1978; Dissertationes Math., 165, (1989), 11 – 58.
[Esquisses Math. (1983)  32 Part 3](http://ehres.pagesperso-orange.fr/Cahiers/AshleyEM32.pdf)

Verity's work can be found in

*   D. R. Verity, 2005, Complicial Sets , available from : [arXiv:math.CT/0410412](http://arxiv.org/abs/math.CT/0410412).

*   D. Verity, 2006, Weak complicial sets I: basic homotopy theory , available from : [arXiv:math/0604414](http://arxiv.org/abs/math/0604414).

*    D. R. Verity, 2006, Weak complicial sets. III. Enriched and internal quasi-category theory , (in preparation).

*    D. R. Verity, 2007, Weak complicial sets. II. Nerves of complicial Gray-categories , in Categories in algebra, geometry and mathematical physics , volume 431 of Contemp. Math., 441--467, Amer. Math. Soc., Providence, RI.

[[!redirects thin elements]]
[[!redirects thin]]


thin homotopy

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

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

## Idea ##

A _thin homotopy_ between paths $f,g: I \to X$ in a [[topological space]] $X$ (with $I = [0,1]$ the standard [[interval]]) is a [[homotopy (as a transformation)|homotopy]] $I\times I \to X$ which, roughly speaking, has zero area.

## Definition

### Thin homotopies in smooth spaces ##

A (smooth) [[homotopy (as a transformation)|homotopy]] $F: I \times I \to X$ between smooth paths in a [[smooth space]] $X$  is called **thin** if the rank of its differential $d F(s,t): T_{s,t} I \times I \to T_{F(s),F(t)} X$ is less than 2 for all $s,t \in I \times I$.

(More to go here...)

### Thin homotopies in topological spaces ##

The following is taken from

* [[K. A. Hardie]], [[K. H. Kamps]], [[R.W. Kieboom]], _A homotopy 2-groupoid of a Hausdorff space_, Appl. Cat. Str.**8** (2000).

We define a _finite tree_ to be a one-dimensional finite polyhedron.

A homotopy $F:I\times I \to X$ between paths $F(-,0)$ and $F(-,1)$ in the [[topological space]] $X$ is called **thin** if $F$ factors through a finite tree,
$$
  I\times I \stackrel{F_0}{\to} T \stackrel{F_1}{\to} X
$$
such that the paths $F_0(-,0):I\to T$, $F_0(-,1):I\to T$ are piecewise-linear.

When $X$ is [[Hausdorff space|Hausdorff]], points, paths and thin homotopies in $X$ form a [[bigroupoid]].

## Applications

### Path n-groupoids

The definition of [[path groupoid]]s and [[path n-groupoid]]s as strict or [[semi-strict infinity-category|semi-strict]] [[n-category|n-groupoids]] typically involves taking morphisms to be thin homotopy classes of paths. See there for more details.

### Parallel transport

The [[parallel transport]] of a [[connection on a bundle]] is an assignment of fiber-homomorphisms to paths in a manifold that is invariant under thin homotopy.

[[!redirects thin homotopies]]


things to be categorified

This page contains ideas of concepts and constructions which might be profitably [[vertical categorification|categorified]].

###Galois theory for algebraic extensions###

Jeff Morton and John Baez apparently

>... figured out how to categorify the algebraic integers in any algebraic extension of the rationals, getting an "algebraic extension" of the category of finite sets. We figured out the beginnings of a theory that associates a "Galois 2-group" to any such algebraic extension.

###Coalgebra###

The [[Yoneda embedding]] of a category $\mathcal{C}$ to $F(\mathcal{C}) = [\mathcal{C}^{op}, Set]$ is a 2-coalgebra for the 2-endofunctor $F$.

Like with the powerset functor there surely can't be a terminal 2-coalgebra for $F$. What about $G(\mathcal{C}) = 1 + A \times \mathcal{C}$, where $A$ is a fixed category? The terminal 2-coalgebra would have as objects finite or infinite lists of objects of $A$ with lists of arrows of $A$ as morphisms.

###Structure Types###

A sketch of [2-structure types](http://golem.ph.utexas.edu/category/2008/04/2structure_types.html).

###Riemann-Hilbert problem###

For any [[representation]] of the [[fundamental group]] of a punctured [[Riemann surface]], we can find a linear differential equation with holomorphic coefficients, such that the monodromies of the solutions realize this representation: [discussion](http://golem.ph.utexas.edu/category/2007/08/justificatory_narratives.html#c011603).


third fractional Pontryagin class

## Idea

The next [[Pontryagin class]] after

* [[first fractional Pontryagin class]]

* [[second fractional Pontryagin class]]

## Related concepts

[[!include higher spin structure - table]]


third stable homotopy group of spheres

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

#Contents#
* table of contents
{:toc}

## Idea

The third [[stable homotopy group of spheres]] (the third [[stable stem]]) is the [[cyclic group]] of [[order of a group|order]] [[24]]:

\[
  \label{QuaternionicHopfFibrationGeneratingpi3}
  \array{
    \pi_3^s &\simeq& \mathbb{Z}/24
    \\
    [h_{\mathbb{H}}] &\leftrightarrow& [1]
  }
\]

where the generator $[1] \in \mathbb{Z}/24$ is represented by the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$.

\begin{imagefromfile}
        "file_name": "K3CobordismBetween24ThreeSpheres.jpg",
        "web": "nlab",
        "width": 260,
        "unit": "px",
        "float": "right",
        "margin": {
            "top": -40,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

## Properties

### As the third framed bordism group

Under the [[Pontrjagin-Thom isomorphism]], identifying the [[stable homotopy groups of spheres]] with the [[bordism ring]] $\Omega^{fr}_\bullet$ of [[stable framing|stably framed]] manifolds (see at _[[MFr]]_), the generator (eq:QuaternionicHopfFibrationGeneratingpi3) is represented by the [[3-sphere]] (with its left-invariant framing induced from the identification with the [[Lie group]] [[SU(2)]] $\simeq$ [[Sp(1)]] )

$$
  \array{
    \pi_3^s & \simeq & \Omega_3^{fr}
    \\
    [h_{\mathbb{H}}] & \leftrightarrow & [S^3_{fr=1}]
    \,.
  }
$$

Moreover, the relation $24 \cdot [S^3_{Lie}] \,\simeq\, 0$ is represented by the [[bordism]] which is the [[complement]] of 24 [[open balls]] inside [[generalized the|the]] [[K3]]-manifold (e.g. [Wang-Xu 10, Sec. 2.6](#WangXu10), [Bauer 10](#Bauer10), [SP 17](#SP17)).

### Via the fourth $(SU,fr)$-bordism group
 {#ViaTheFourthSuFrBordismGroup}

Equivalently, the elements of $\pi_3^s \,\simeq\, \Omega^{fr}_3$ are detected by half the [[Todd classes]] of [[coboundary|cobounding]] manifolds with [[special unitary group]]-[[tangential structure]] on their [[stable tangent bundle]] (elements of the [[MSUFr]]-[[bordism ring]]):

We have the following [[short exact sequence]] of the [[MSU]]-, [[MSUFr]]- and [[MFr]]-[[bordism rings]] ([Conner-Floyd 66, p. 104](#ConnerFloyd66))

\[
  \label{HalfToddClassesOnShortExactSequenceOfSUFrBordismRings}
  \array{
  0
  \to
  &
  \Omega^{SU}_{8\bullet+4}
  &
  \overset{i}{\longrightarrow}
  &
  \Omega^{SU,fr}_{8\bullet+4}
  &
  \overset{\partial}{
    \longrightarrow
  }
  &
  \Omega^{fr}_{8\bullet + 3}
  &
  \simeq
  &
  \pi^s_{8\bullet+3}
  \\
  &
  \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}}
  &&
  \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}}
  &&
  \big\downarrow{}^{}
  &&
  \big\downarrow{}^{e_{\mathbb{R}}}
  \\
  0
  \to
  &
  \mathbb{Z}
  &\overset{\;\;\;\;\;}{\hookrightarrow}&
  \mathbb{Q}
  &\overset{\;\;\;\;}{\longrightarrow}&
  \mathbb{Q}/\mathbb{Z}
  &=&
  \mathbb{Q}/\mathbb{Z}
  }
\]

which produces from half the [[Todd class]] of cobounding $(SU,fr)$-manifolds the [[KO]]-theoretic [[Adams e-invariant]] $e_{\mathbb{R}}$ ([Adams 66, p. 39](e-invariant#Adams66)) of the boundary manifold in $\Omega^{fr}_{8k + 3} \simeq \pi^s_{8k+3}$. For $k = 0$ this detects the third stable homotopy group of spheres, by the following:

+-- {: .num_prop #AdamseRInvariantDetectsThirdStableHomotopyGroupOfSpheres}
###### Proposition
**([Adams 66, Example 7.17  and p. 46](e-invariant#Adams66))**

In degree 3, the [[KO]]-theoretic [[e-invariant]] $e_{\mathbb{R}}$ takes the value $\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z}$ on the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$  and hence reflects the full [[third stable homotopy group of spheres]]:

$$
  \array{
    \pi^s_3
      &
      \underoverset{
        \simeq
      }{
        e_{\mathbb{R}}
      }{
        \;\;\longrightarrow\;\;
      }
      &
    \mathbb{Z}/24
    &
    \subset
    &
    \mathbb{Q}/\mathbb{Z}
    \\
    [h_{\mathbb{H}}]
    &&\mapsto&&
    \left[\tfrac{1}{24}\right]
  }
$$

while $e_{\mathbb{C}}$ sees only "half" of it (by [Adams 66, Prop. 7.14](e-invariant#Adams66)).

=--

## Related concepts

* [[quaternionic Hopf fibration]]

* [[24 branes transverse to K3]]

* [[first stable homotopy group of spheres]]

* [[second stable homotopy group of spheres]]

## References

The original computation:

* [[Vladimir Abramovich Rokhlin]], _On a mapping of the $(n+3)$-dimensional sphere into the $n$-dimensional sphere_,  (Russian) Doklady Akad. Nauk SSSR (N.S.) 80, (1951). 541–544

with a mistake (in the unstable range) corrected in

* [[Vladimir Abramovich Rokhlin]], _New results in the theory of four-dimensional manifolds_, (Russian) Doklady Akad. Nauk SSSR (N.S.) 84, (1952). 221–224.

French translations are in:

* Lucien Guillou, Alexis Marin (eds.), _A la Recherche de la Topologie Perdue: I. Du côté de chez Rohlin. II. Le côté de Casson_, Progress in Mathematics 62, Birkhäuser Boston 1985 (ISBN:0817633294, 9780817633295)

Review:

* {#WangXu10} [[Guozhen Wang]], [[Zhouli Xu]], Section 2.6 of: _A survey of computations of homotopy groups of Spheres and Cobordisms_, 2010 ([[WangXuHomotopyGroupsOfSpheres2010.pdf:file]])

See also:

* Mehmet Kirdar, *On the First, the Second and the Third Stems of the Stable Homotopy Groups of Spheres* [[arXiv:2107.06103](https://arxiv.org/abs/2107.06103)]

More on the computation via the [[MFr|framed cobordism ring]] and the [[K3-manifold]] giving the [[cobordism]] that witnesses the [[order of a group|order]] of 24:

* {#Bauer10} [[Tilman Bauer]], answer to: _third stable homotopy group of spheres via geometry?_, 2010 ([MO:a/44885](https://mathoverflow.net/a/44885/381))

* {#SP17} [[Chris Schommer-Pries]], answer to: _Nilpotence of the stable Hopf map via framed cobordism_, 2017 ([MO:a/218053](https://mathoverflow.net/a/218053/381))

Via [[immersions]] of 3-spheres into Euclidean 4-space

* A. Szűcs, _Two Theorems of Rokhlin_, Journal of Mathematical Sciences 113, 888–892 (2003) ([doi:10.1023/A:1021208007146](https://doi.org/10.1023/A:1021208007146))

* Tobias Ekholm, Masamichi Takase, _Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions_, 	Bulletin of the London Mathematical Society 43 (2011) 251--266 ([arXiv:0903.0238](https://arxiv.org/abs/0903.0238))

Discussion of [[geometric string structure|geometric]] string bordism in degreee 3 as a means to speak (via the [[Pontryagin-Thom theorem]]) about the [[third stable homotopy group of spheres]]:

* [[Domenico Fiorenza]], [[Eugenio Landi]], *Integrals detecting degree 3 string cobordism classes* [[arXiv:2209.12933](https://arxiv.org/abs/2209.12933)]

* [[Domenico Fiorenza]], *String bordism invariants in dimension 3 from $U(1)$-valued TQFTs*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#FiorenzaMar2023)]

[[!redirects 3rd stable homotopy group of spheres]]

[[!redirects third stable stem]]
[[!redirects 3rd stable stem]]