brane

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> This article is about the general concept of branes. For more specific details see also also at _[[super p-brane]]_, _[[black brane]]_ and _[[D-brane]]_.

***

#Contents#
* table of contents
{:toc}

## Idea

The term _brane_ in formal high energy [[physics]], and in particular in [[string theory]], refers to entities that one thinks of as physical objects that generalize the notion _point particles_ to higher dimensional objects.

The term derives from the word _membrane_ that was originally used to describe 2-dimensional "particles". When the need was felt to speak also about 3-, 4- and higher dimensional such "particles" the usage "3-brane", "4-brane" etc. was introduced. Ordinary particles would be 0-branes in this counting, the strings in [[string theory]] would be 1-branes and membranes themselves 2-branes.

Generally, there are two different incarnations of branes

1. **fundamental $p$-branes** (as in "[[fundamental particle]]"): these are given by [[sigma-models]] on $(p+1)$-dimensional [[worldvolumes]] describing propagation of single $p$-dimensional objects on certain [[target spacetimes]]. If these sigma-models are required to exhibit manifest [[target spacetime]] [[supersymmetry]], then they are [[Green-Schwarz sigma models]], which are classified by a "[[brane scan]]" in [[super L-infinity algebra]] [[L-infinity algebra cohomology|cohomology]];

1. **[[black branes|black p-branes]]** (as in "[[black hole]]"): these are solutions to [[field theories]], typically [[supergravity]] theories, with [[singularities]] of [[dimension]] $(p+1)$. In analogy to how a [[charged black hole]] ($p = 0$) [[source|sources]] an [[electromagnetic field]] with [[field strength]] 2-form, so black $p$-branes source $(p+2)$-form [[higher gauge fields]] and hence appear in those [[supergravity]] theories where such exists.

The idea is that these two concepts match where a [[condensate]] of fundamental $p$-branes turns into a black $p$-brane.

Indeed, the classical [[no-hair theorem]] matches [[fundamental particles]] (i.e. 0-branes) characterized (via the [[Wigner classification]]) by just their [[mass]], electromagnetic [[charge]] and [[angular momentum]]  to [[black hole]] solutions of pure vacuum gravity. Accordingly, it is an old suggestion ([Einstein-Infeld-Hoffmann 39](#EinsteinInfeldHoffmann39)) that fundamental particles could be identified with singular solutions of vacuum gravity.

This matching generalizes to higher dimensional $p$-branes in higher dimensional [[supergravity]] and there is an exact correspondence between fundamental [[Green-Schwarz super p-branes]] and extremal [[BPS solutions|BPS]] black brane solutions.

In [[string theory]] there is a third incarnation of branes, known as

* **[[D-branes]]**: these are the admissible [[boundary conditions]] for the 2-dimensional [[sigma models]] describing [[open strings]].

One envisions that as one passes from [[perturbative string theory]] to the [[non-perturbative effect|non-perturbative]] version of the theory ("[[M-theory]]") these [[D-branes]] show [[back-reaction]] and turn into the [[UV-completion]] of the [[black branes]] seen in the [[supergravity]] [[effective field theory]]. This relation is key in the microscopic computation of [[black hole entropy]] for [[black holes in string theory]].

### Boundary conditions or D-branes
 {#Dbranes}

Some words on [[D-branes]]

#### In terms of the algebraic data of the QFT on the worldvolume

An abstractly defined $n$-dimensional [[quantum field theory]] is a consistent assignment of [[state]]-space and correlators to $n$-dimensional [[cobordisms]] with certain structure (topological structure, conformal structure, Riemannian structure, etc. see [[FQFT]]/[[AQFT]]). In an _open-closed QFT_ the cobordisms are allowed to have boundaries. See at _[[boundary field theory]]_ for more on this.

In this abstract formulation of QFT a **brane** is a type of data assigned by the QFT to boundaries of cobordisms.

##### In $2d$ rational CFT

A well understood class of examples is this one: among all 2-dimensional [[conformal field theory]], the case of _full rational 2d CFT_ has been understood completely, using [[FFRS-formalism]]. It is then a theorem that full 2-rational CFTs are classified by

1. a [[modular tensor category]] $\mathcal{C}$ (to be thought of as being the category of representations of the [[vertex operator algebra]] of the 2d CFT);

1. a special symmetric [[Frobenius algebra]] object $A$ [[internalization|internal]] to $\mathcal{C}$.

In this formulation a type of **brane** of the theory is precisely an $A$-[[module]] in $\mathcal{C}$ (an $A$-[[bimodule]] is a [[bi-brane]] or _defect line_ ):

the 2d cobordisms with boundary on which the theory defined by $A \in \mathcal{C}$ carry as extra structure on their connected boundary pieces a label given by an equivalence class of an $A$-module in $\mathcal{C}$. The assignment of the CFT to such a cobordism with boundary is obtained by

* triangulating the cobordism,

* labeling all internal edges by $A$

* labelling all boundary pieces by the $A$-module

* all vertices where three internal edges meet by the multiplication operation

* and all points where an internal edge hits a moundary by the corresponding [[action]] morphism

* and finally evaluating the resulting [[string diagram]] in $\mathcal{C}$.

So in this abstract algebraic formulation of QFT on the worldvolume, a brane is just the datum assigned by the QFT to the boundary of a cobordism. But abstractly defined QFTs may arise from [[quantization]] of [[sigma model]]s. This gives these boundary data a geometric interpretation in some space. This we discuss in the next section.

##### In $2d$ TFT

Another case where the branes of a QFT are under good mathematical control is [[TCFT]]: the [[(infinity,1)-category]]-version of a 2d [[TQFT]].

Particularly the [[A-model]] and the [[B-model]] are well understood.

* the branes of the B-model ("B-branes") form the the [[stable (infinity,1)-category]] of [[chain complex]]es of [[quasicoherent sheaves]] on the target space (often considered just in terms of its [[homotopy category of an (infinity,1)-category]], the [[derived category]] of quasicoherent sheaves);

*  the branes of the A-model form the [[Fukaya category]] of the target space.

(...)

#### In terms of geometric data of the $\sigma$-model background

An abstractly defined QFT (as a consistent assignment of state spaces and propagators to cobordisms as in [[FQFT]]) may be obtained by [[quantization]] from _geometric data_ :

Such a _[[sigma-model]] QFT_ is the [[quantization]] of an [[action functional]] on a space of maps $\Sigma \to X$ from a cobordism ("worldvolume") $\Sigma$ to some target space $X$ that may carry further geometric data such as a [[Riemannian metric]],  or other background [[gauge field]]s.

One may therefore try to match the geometric data on $X$ that encodes the $\sigma$-model with the algebraic data of the [[FQFT]] that results after quantization. This gives a geometric interpretation to many of the otherwise purely abstract algebraic properties of the worldvolume QFT.

It turns out that if one checks which geometric data corresponds to the $A$-modules in the above discussion, one finds that these tend to come from structures that look at least roughly like _submanifolds_ of the target space $X$. And typically these submanifolds themselves carry their own background [[gauge field]] data.

A well-understood case is the [[Wess-Zumino-Witten model]]: for this the target space $X$ is a simple [[Lie group]] $X = G$ and the background field is a [[circle n-bundle with connection|circle 2-bundle with connection]] (a [[bundle gerbe]]) on $G$, representing the background field that is known as the [[Kalb-Ramond field]].

In this case it turns out that branes for the sigma model on $X$ are given in the simplest case by conjugacy classes $D \subset G$ inside the group, and that these carry a [[twisted bundle|twisted vector bundle]] with the twist given by the Kalb-Ramond background bundle. These vector bundles are known in the [[string theory]] literature as _Chan-Paton vector bundles_ . The geometric intuition is that a QFT with certain boundary condition comes from a quantization of spaces of maps $\Sigma \to G$ that are restricted to take the boundary of $\Sigma$ to these submanifolds.

More generally, one finds that the geometric data that corresponds to the branes in the algebraically defined 2d QFT is given by cocycles in the twisted [[differential K-theory]] of $G$. These may be quite far from having a direct interpretation as submanifolds of $G$.

The case of rational 2d CFT considered so far is only the best understood of a long sequence of other examples. Here the collection of all [[D-branes]] -- identified with the collection of all internal modules over an internal frobenius algebra, forms an ordinary [[category]].

More generally, at least for 2-dimensional [[TQFT]]s analogous considerations yield not just categories but [[stable (∞,1)-categories]] of boundary condition objects. For instance for what is called the [[B-model]] 2-d [[TQFT]] the category of [[D-branes]] is the [[derived category]] of [[coherent sheaves]] on some Calabi-Yau space.

Starting with Kontsevich's [[homological algebra]] reformulation of [[homological mirror symmetry|mirror symmetry]] the study of (derived) D-brane categories has become a field in its own right in pure mathematics.

... lots of further things to say ...

### Fundamental or $\sigma$-model branes

In [[string theory]] one speaks apart from the [[D-branes]] also about **fundamental branes** . These are the objects $\Sigma$ in the $n$-dimensional [[sigma model]] themselves.

* For $n=0$ this describes the ordinary quantum mechanics of a point particles on $X$. And such point particles are the _fundamental particles_ for instance of the [[standard model of particle physics]].

* For $n=1$ this describes the quantum propagation of a [[string theory|string]], and accordingly one speaks of the _fundamental string_ or F1-brane (fundamental 1-brane).

* For $n=2$ this describes the quantum propagation of a membrane.

* There are good indications that there is a way to describe heterotic [[string theory]] not in terms of fundamental 1-branes but in terms of the [[sigma-model]] of a fundamental 5-brane -- the [[magnetic charge|magnetic dual]] of the 1-brane in 10-dimensions.

* etc.

[[!include brane scan]]

### Black branes

See _[[black brane]]_.

### Defect branes

See _[[defect brane]]_.

### Exotic branes

See _[[exotic brane]]_.

## Properties

### Worldvolume theories

[[!include table of branes]]

### The super-brane scan

If the worldvolume QFT of the fundamental branes (for instance the worldsheet 2d[[CFT]] of the string) is required to be a [[supersymmetric QFT]], specifically if the [[Green-Schwarz action functional]] is used only particular combinations of the dimension $dim \Sigma = p + 1$ of the worldvolume and $D = dim X$ of [[spacetime]] are possible.

The corresponding table has been called the **[[brane scan]]**

[[branescan.gif:pic]]

## Related concepts

* [[black brane]], [[solitonic brane]]

  [[Green-Schwarz sigma-model|fundamental brane]]

  [[probe brane]], [[flavor brane]]

* [[wrapped brane]]

* [[brane intersection]]

* [[D-brane]], [[fractional D-brane]]

* [[sigma-model]]

* [[double dimensional reduction]]

* [[particle]], [[superparticle]]

* [[string]], [[superstring]]

* [[membrane]], [[M2-brane]], [[M5-brane]], [[M-brane]]

* [[NS5-brane]]

* [[black brane]]

* [[electric eigenbrane]]. [[magnetic eigenbrane]]

* [[exotic brane]]

[[!include field theory with boundaries and defects - table]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

### Prehistory

* {#EinsteinInfeldHoffmann39} [[Albert Einstein]], [[Leopold Infeld]], B. Hoffmann, _The gravitational equations and the problem of motion_, Annals of Mathematics, Vol 39, No. 1, 1938

See also

* Wikipedia, _[Black hole electron](https://en.wikipedia.org/wiki/Black_hole_electron)_

The terminology "$p$-brane" originates in

* {#DuffInamiPopeSezginStelle88} [[Mike Duff]], T. Inami, [[Christopher Pope]], [[Ergin Sezgin]],  [[Kellogg Stelle]], _Semiclassical Quantization of the Supermembrane_, Nucl.Phys. B297 (1988) 515-538 ([spire:247064](http://inspirehep.net/record/247064))

For more history see

* [[Mike Duff]], _Thirty years of Erice on the brane_, Based on lectures at the International Schools of Subnuclear Physics 1987-2017 and the International Symposium "60 Years of Subnuclear Physics at Bologna", University of Bologna, November 2018 ([arXiv:1812.11658](https://arxiv.org/abs/1812.11658))

### General

* [[Pietro Fré ]], *The Branes: Three Viewpoints*, Chater 7  in: *Gravity, a Geometrical Course*, Springer (2013) [[spire:1242195](http://inspirehep.net/record/1242195), [doi:10.1007/978-94-007-5443-0_7](https://doi.org/10.1007/978-94-007-5443-0_7)]

* [[Greg Moore]], _What is... a brane?_, Notices of the AMS **52** 2 (2005) [[pdf](http://www.ams.org/notices/200502/what-is.pdf)]

* [[Joan Simón]], *Brane Effective Actions, Kappa-Symmetry and Applications* [[arXiv:1110.2422](http://arxiv.org/abs/1110.2422)]

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 6 of: _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press (2012)

### Boundary conditions / D-branes

(...)

See [[D-brane]].

For exhaustive details on D-branes in 2-dimensional rational [[CFT]] see the references given at

* [[FFRS-formalism]].

A classical text describing how the physics way to think of D-branes for the [[topological string]] leads to seeing that they are objects in [[derived categories]] (of [[coherent sheaves]] for the [[B-model]]) is reviewed in

* [[Paul Aspinwall]], _D-Branes on Calabi-Yau Manifolds_ ([arXiv](http://arxiv.org/abs/hep-th/0403166))

based on

* [[Michael Douglas]], _D-branes, Categories and $N=1$ Supersymmetry_, J.Math.Phys. 42 (2001) 2818-2843 ([arXiv:hep-th/0011017](https://arxiv.org/abs/hep-th/0011017))

* [[Paul Aspinwall]], Albion Lawrence, _Derived Categories and Zero-Brane Stability_ ([arXiv:hep-th/0104147](https://arxiv.org/abs/hep-th/0104147))

This can to a large extent be read as a dictionary from [[homological algebra]] terminology to that of D-brane physics.

More recent similar material with the emphasis on the [[K-theory]] aspects is

* [[Richard Szabo]], _[[Szabo09.pdf:file]]_

### Fundamental branes

The "[[brane scan]]" table showing the consistent dimension pairs for the [[Green-Schwarz action functional]] was depicted in

* [[Michael Duff]], _Supermembranes: the first fifteen weeks_ CERN-TH.4797/87 ([scan] (https://iopscience.iop.org/article/10.1088/0264-9381/5/1/023/pdf))
{#Duff}

going back to

* A. Achúcarro, J. M. Evans, [[Paul Townsend]] and D. L. Wiltshire, _Super $p$-branes_ Physics Letters B Volume 198, Issue 4, 3 (1987)

Further developments are in

More along these lines is in

* [[Michael Duff]], S. Ferrara, _Four curious supergravities_ ([arXiv](http://arxiv.org/abs/1010.3173))

See also [[division algebras and supersymmetry]].

### Branes ending on branes

* [[Andrew Strominger]], _Open P-Branes_, Phys. Lett. B383:44-47,1996 ([arXiv:hep-th/9512059](http://arxiv.org/abs/hep-th/9512059))

[[!redirects branes]]

[[!redirects fundamental brane]]
[[!redirects fundamental branes]]

[[!redirects NS-fivebrane]]
[[!redirects NS-fivebranes]]

[[!redirects p-brane]]
[[!redirects p-branes]]


brane bound states -- table

**[[brane intersections]]/[[bound states]]/[[wrapped branes]]/[[polarized branes]]**

* [[D-branes]] and [[anti D-branes]] form bound states by [[tachyon condensation]], thought to imply the classification of [[D-brane charge]] by [[K-theory]]

* [[intersecting D-brane models|intersecting D-branes]]/[[fuzzy funnels]]:

  * [[Dp-D(p+2)-brane bound state]] and [[Yang-Mills monopoles]]

    * [[D0-D2 brane bound state]]

    * [[D1-D3 brane bound state]]

    * [[D2-D4 brane bound state]]

    * [[D3-D5 brane bound state]]

    * [[D4-D6 brane bound state]]

    * [[D6-D8-brane bound state]]

  * [[Dp-D(p+4)-brane bound state]] and [[Yang-Mills instantons]]

    * [[D0-D4 brane bound state]]

    * [[D1-D5 brane bound state]]

    * [[D2-D6 brane bound state]]

    * [[D3-D7 brane bound state]]

    * [[D4-D8 brane bound state]]

    * [[D5-D9 brane bound state]]

      [[small instanton]]

* Dp-D(p+6) brane bound state

  * [[D0-D6 brane bound state]]

  * [[D1-D7 brane bound state]]

  * [[D2-D8 brane bound state]]

* [NS5-D4-D2 bound states](NS5-brane#NS5D4D2BoundStates)

* [[Hanany-Witten effect]]

[[S-duality]]$\,$[[bound states]]:

  * [[(p,q)-string]]

  * [[(p,q)5-branes]]

  * [[string network]]

  * [[5-brane web]]

[[brane intersection|intersecting]]$\,$[[M-branes]]:

* [[M5-MO9 brane bound state]]

* [[M2-M5 brane bound state]]

  * [[M-string]], [[E-string]]

* [[membrane triple junction]]


brane bouquet

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#Contents#
* table of contents
{:toc}

## Idea

The generalization of the [[brane scan]] based on [[super L-infinity algebra]] [[algebra extension|extensions]]. As summarized in Figure 3 of ([Huerta, Sati, and Schreiber (2019)](#HSS19ade)):

\begin{imagefromfile}
        "file_name": "branebouquet.PNG",
        "width": 700
    \end{imagefromfile}

## References

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], *Super Lie n-algebras, higher WZW models, and super p-branes*, Intern. J. Geom. Meth. Mod. Phys. 12 (2015) 1550018 , ([arXiv:1308.5264](https://arxiv.org/abs/1308.5264))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], *Sphere valued supercocycles in M-theory and type IIA string theory*, J. Geom. Phys. 114 (2017) 91-108, ([arXiv:1606.03206](https://arxiv.org/abs/1606.03206)).

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], *T-Duality from super Lie n-algebra cocycles for super p branes*, ATMP Volume 22 (2018) Number 5, ([arXiv:1611.06536] (https://arxiv.org/abs/1611.06536))

* [[Hisham Sati]], [[Urs Schreiber]], *Higher T-duality in M-theory via local supersymmetry*, Physics Letters B, Volume 781, 10 June 2018, Pages 694-698, ([arXiv:1805.00233](https://arxiv.org/abs/1805.00233))

* [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]]. *Gauge enhancement via Parameterized stable homotopy theory*, Communications in Mathematical Physics 2019
([arXiv:1806.01115](https://arxiv.org/abs/1806.01115))

* [[John Huerta]] and [[Urs Schreiber]], *M-Theory from the superpoint*, Lett. in Math. Phys. 2018, ([arXiv:1702.01774](https://arxiv.org/abs/1702.01774))

* {#HSS19ade} [[John Huerta]], [[Hisham Sati]], and [[Urs Schreiber]]. *Real ADE-Equivariant (co)Homotopy and Super M-Branes*. Commun. Math. Phys. 371, 425–524 (2019). ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

* {#Sch19slides} [[Urs Schreiber]]. *Equivariant super homotopy theory*. Talk at Geometry in Modal Homotopy Type Theory Pittsburgh 2019. ([slides](https://ncatlab.org/schreiber/files/EquivariantSuperHomotopyTheory120314a.pdf))


brane intersection

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## Idea

[[intersection]] of [[branes]] in [[string theory]]

## Examples

[[!include brane bound states -- table]]

More:

[[M5-brane]] with [[KK-monopole]]:

$$
  (5\vert M5, KK)
  =
  \left\{
    \array{
      x_0 & - & - & - & - & x_5 & x_6 & x_7 & x_8 & x_9 & -
      \\
      x_0 & A_1 & A_2 & A_3 & z & x_5 & x_6 & x_7 & x_8 & x_9 & x_{10}
    }
  \right.
$$

([BREJS 97 (14)](#BREJS97))

For more see at _[Half NS5-branes](NS5-brane#NSHalfBranes)_

\linebreak

[[D4-branes]] ending on [[D6-branes]] from [[M5-branes]] in [[MK6-branes]]:

[Hashimoto 98](#Hashimotop98), see also [Smith 02 (245)](#Smith02)

## Related concepts

* [[wrapped brane]]

* [[polarized brane]]

* [[color branes and flavor branes]]

[[!include brane bound states -- table]]

## References

### General

* {#Smith02} [[Douglas J. Smith]], *Intersecting brane solutions in string and M-theory*, Class. Quant. Grav. **20** R233 (2003) [[arXiv:hep-th/0210157](https://arxiv.org/abs/hep-th/0210157), [doi:10.1088/0264-9381/20/9/203](https://doi.org/10.1088/0264-9381/20/9/203)]

* {#Hashimoto98} Akikazu Hashimoto, _Supergravity Solutions for Localized Intersections of Branes_, JHEP 9901 (1999) 018 ([arXiv:hep-th/9812159](https://arxiv.org/abs/hep-th/9812159))

For [[M-branes]]:

* [[Mees de Roo]], *Intersecting branes and Supersymmetry*, in: *Supersymmetry and Quantum Field Theory*, Proceedings of the D. Volkov Memorial Seminar Held in Kharkov, Ukraine, 5–7 January 1997, Springer (1998) [[arXiv:hep-th/9703124](https://arxiv.org/abs/hep-th/9703124), [doi:10.1007/BFb0105225](https://doi.org/10.1007/BFb0105225)]

* {#BREJS97} [[Eric Bergshoeff]], [[Mees De Roo]], [[Eduardo Eyras]], [[Bert Janssen]], [[Jan Pieter van der Schaar]], _Intersections involving waves and monopoles in eleven dimensions_, Class. Quantum Grav. **14** 2757 (1997) [[arXiv:hep-th/9704120](https://arxiv.org/abs/hep-th/9704120), [doi:0264-9381/14/10/005](http://iopscience.iop.org/0264-9381/14/10/005)]

* [[Eric Bergshoeff]], [[Joaquim Gomis]], [[Paul Townsend]], _M-brane intersections from worldvolume superalgebras_, Phys.Lett. B421 (1998) 109-118 ([arXiv:hep-th/9711043](http://arxiv.org/abs/hep-th/9711043))

* {#Townsend97} [[Paul Townsend]], section 4 of _M-theory from its superalgebra_ ([arXiv:hep-th/9712004](http://arxiv.org/abs/hep-th/9712004))

* Sergey A. Cherkis, _Supergravity Solution for M5-brane Intersection_ ([arXiv:hep-th/9906203](https://arxiv.org/abs/hep-th/9906203))

### Brane webs
 {#BraneWebs}

Intersections of branes of equal dimension may form regular patterns known as _brane webs_. This applies notably in [[type IIB string theory]] where [[(p,q)-strings]] may form "[[string webs]]" (references [below](#ReferencesStringWebs)) and [[(p,q)5-branes]] may form "[[5-brane webs]]" (references [further below](#References5BraneWebs))

#### General

* Oleg Lunin, _Brane webs and 1/4-BPS geometries_, JHEP09 (2008) ([arXiv:0802.0735](https://arxiv.org/abs/0802.0735))

#### String webs
 {#ReferencesStringWebs}

* {#Sen97} [[Ashoke Sen]], _String Network_, JHEP 9803:005, 1998 ([arXiv:hep-th/9711130](https://arxiv.org/abs/hep-th/9711130))

* Soo-Jong Rey, Jung-Tay Yee, _BPS Dynamics of Triple $(p,q)$ String Junction_, Nucl.Phys. B526 (1998) 229-240 ([arXiv:hep-th/9711202](https://arxiv.org/abs/hep-th/9711202))

* [[Keshav Dasgupta]], [[Sunil Mukhi]], _BPS Nature of 3-String Junctions_, Phys.Lett. B423 (1998) 261-264 ([arXiv:hep-th/9711094](https://arxiv.org/abs/hep-th/9711094))

Via [[tropical geometry]]:

* Koushik Ray, _String networks as tropical curves_, JHEP 0809:098, 2008 ([arXiv:0804.1870](https://arxiv.org/abs/0804.1870))

The lift to [[11d supergravity]]/[[M-theory]] via [[membrane triple junctions]] is discussed in

* {#KroghLee97} Morten Krogh, Sangmin Lee, _String Network from M-theory_, Nucl. Phys. B516:241-254, 1998 ([arXiv:hep-th/9712050](https://arxiv.org/abs/hep-th/9712050))

* {#Ramadevi99} P. Ramadevi, _Supergravity Solution for Three-String Junction in M-Theory_, JHEP 0006 (2000) 005 ([arXiv:hep-th/9906247](https://arxiv.org/abs/hep-th/9906247))

#### 5-brane webs
 {#References5BraneWebs}

* {#AharonyHanany97} [[Ofer Aharony]], [[Amihay Hanany]], section 3 of _Branes, Superpotentials and Superconformal Fixed Points_, Nucl.Phys.B504:239-271, 1997 ([arXiv:hep-th/9704170](https://arxiv.org/abs/hep-th/9704170))

* {#Kol97} [[Barak Kol]], _5d Field Theories and M Theory_, JHEP 9911 (1999) 026 ([arXiv:hep-th/9705031](https://arxiv.org/abs/hep-th/9705031))

* [[Ofer Aharony]], [[Amihay Hanany]], [[Barak Kol]], _Webs of $(p,q)$ 5-branes, Five Dimensional Field Theories and Grid Diagrams_, JHEP 9801:002,1998 ([arXiv:hep-th/9710116](http://arxiv.org/abs/hep-th/9710116))

* [[Francesco Benini]], [[Sergio Benvenuti]], [[Yuji Tachikawa]], _Webs of five-branes and $\mathcal{N}=2$ superconformal field theories_, JHEP 0909:052 (2009) [[arXiv:0906.0359](https://arxiv.org/abs/0906.0359)]

* Sung-Soo Kim, Masato Taki, Futoshi Yagi, _Tao Probing the End of the World_, Prog. Theor. Exp. Phys. 2015, 083B02 ([arXiv:1504.03672](https://arxiv.org/abs/1504.03672))

* Brice Bastian, Stefan Hohenegger, _Five-Brane Webs and Highest Weight Representations_ ([arXiv:1706.08750](https://arxiv.org/abs/1706.08750))

* Justin Kaidi, _$(p,q)$-strings probing five-brane webs_, J. High Energ. Phys. (2017) 2017: 87 ([arXiv:1708.03404](https://arxiv.org/abs/1708.03404))

* Michael Gutperle, Andrea Trivella, Christoph F. Uhlemann, _Type IIB 7-branes in warped $AdS_6$: partition functions, brane webs and probe limit_, JHEP04(2018)135 ([arXiv:1802.07274](https://arxiv.org/abs/1802.07274))

* Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, _Dualities and 5-brane webs for 5d rank 2 SCFTs_ ([arXiv:1806.10569](https://arxiv.org/abs/1806.10569))

[[!redirects brane intersections]]

[[!redirects intersecting brane]]
[[!redirects intersecting branes]]

[[!redirects coincident brane]]
[[!redirects coincident branes]]
[[!redirects stack of coincident branes]]
[[!redirects stacks of coincident branes]]

[[!redirects brane web]]
[[!redirects brane webs]]

[[!redirects 5-brane web]]
[[!redirects 5-brane webs]]

[[!redirects bound state of D-branes]]
[[!redirects bound states of D-branes]]


brane matrix models -- content

**[[matrix models]] for [[brane]] dynamics**:

| [[D-brane]] | [[matrix model]] |
|-------------|------------------|
| [[D0-brane]] | [[BFSS matrix model]], <br/> [[BMN matrix model]] |
| [[D(-1)-brane]] | [[IKKT matrix model]] |
| [[D4-brane]]  | [[nuclear matrix model]] |

| [[M-brane]] | [[matrix model]] |
|-------------|------------------|
| [[D2-brane]] | [[membrane matrix model]] |

See also:

* [[K-matrix model]]


brane scan

The **[[brane]] scan**.

The [[Green-Schwarz action functional|Green-Schwarz type]] super $p$-[[brane]] [[sigma-models]] (see at _[[table of branes]]_ for further links and see at _[[schreiber:The brane bouquet]]_ for the full classification):

| $\stackrel{d}{=}$ |  $p =$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |  8 |  9 |
|--|--------|---|---|---|---|---|---|---|----|----|
| 11 | |  |  [[M2-brane|M2]] |  |   | [[M5-brane|M5]] |   |   |   |   |
| 10 | [[D0-brane|D0]]  | [[F1-brane|F1]], [[D1-brane|D1]]  | [[D2-brane|D2]]  | [[D3-brane|D3]] | [[D4-brane|D4]]  | [[NS5-brane|NS5]], [[D5-brane|D5]] |  [[D6-brane|D6]]  | [[D7-brane|D7]]  | [[D8-brane|D8]]  | [[D9-brane|D9]]  |
| 9 | |   |   |  | [[super 4-brane in 9d|*]]  |  |   |   |   |   |
| 8 | |   |   | [[super 3-brane in 8d|*]] |   |  |   |   |   |   |
| 7 | |   | [[topological M2-brane|M2]]${}_{top}$  |  |   |  |   |   |   |   |
| 6 | |  [[little string|F1]]${}_{little}$, [[self-dual string|S1]]${}_{sd}$  |  | [[3-brane in 6d|S3]] |   |  |   |   |   |   |
| 5 | |  | [[super 2-brane in 5d|*]]   | |   |  |   |   |   |   |
| 4 | | [[super 1-brane in 4d|*]]  | [[super 2-brane in 4d|*]]   | |   |  |   |   |   |   |
| 3 | | [[super 1-brane in 3d|*]]  |  | |   |  |   |   |   |   |

(The first columns follow the [[exceptional spinors table]].)

The corresponding exceptional [[super L-∞ algebra]] [[∞-Lie algebra cohomology|cocycles]] (schematically, without prefactors):

| $\stackrel{d}{=}$ |  $p =$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |  8 |  9 |
|--|--------|---|---|---|---|---|---|---|----|----|
| 11 | |  |  $\Psi^2 E^2$ on [[super Poincare Lie algebra|sIso]](10,1) |  |   | $\Psi^2 E^5 + \Psi^2 E^2 C_3$ on [[m2brane]]  |   |   |   |   |
| 10 | | $\Psi^2 E^1$ on  [[super Poincare Lie algebra|sIso]](9,1) |  $B_2^2 + B_2 \Psi^2 + \Psi^2 E^2$ on [[type II supergravity Lie 2-algebra|StringIIA]] | $\cdots$ on [[type II supergravity Lie 2-algebra|StringIIB]]  | $B_2^3 + B_2^2 \Psi^2 + B_2 \Psi^2 E^2 + \Psi^2 E^4$  on [[type II supergravity Lie 2-algebra|StringIIA]] | $\Psi^2 E^5$ on [[super Poincare Lie algebra|sIso]](9,1) |  $B_2^4 + \cdots + \Psi^2 E^6$ on [[type II supergravity Lie 2-algebra|StringIIA]]  | $\cdots$ on [[type II supergravity Lie 2-algebra|StringIIB]]  | $B_2^5 + \cdots + \Psi^2 E^8$ in [[type II supergravity Lie 2-algebra|StringIIA]]  | $\cdots$ on [[type II supergravity Lie 2-algebra|StringIIB]]  |
| 9 | |   |   |  | $\Psi^2 E^4$ on  [[super Poincare Lie algebra|sIso]](8,1) |  |   |   |   |   |
| 8 | |   |   | $\Psi^2 E^3$ on  [[super Poincare Lie algebra|sIso]](7,1) |   |  |   |   |   |   |
| 7 | |   | $\Psi^2 E^2$ on  [[super Poincare Lie algebra|sIso]](6,1) |  |   |  |   |   |   |   |
| 6 | |  $\Psi^2 E^1$ on  [[super Poincare Lie algebra|sIso]](5,1)  |  | $\Psi^2 E^3$ on  [[super Poincare Lie algebra|sIso]](5,1) |   |  |   |   |   |   |
| 5 | |  | $\Psi^2 E^2$ on  [[super Poincare Lie algebra|sIso]](4,1)   | |   |  |   |   |   |   |
| 4 | | $\Psi^2 E^1$ on  [[super Poincare Lie algebra|sIso]](3,1) | $\Psi^2 E^2$ on  [[super Poincare Lie algebra|sIso]](3,1)   | |   |  |   |   |   |   |
| 3 | | $\Psi^2 E^1$ on  [[super Poincare Lie algebra|sIso]](2,1)  |  | |   |  |   |   |   |   |

### The Brane molecule

Furthermore, there exists a more general classification of possible supermembranes in spacetime with $S$ spatial dimensions and $T$ time dimensions, appearing in ([Blencowe-Duff 88](#BlencoweDuff88)). In this sense, the brane scan is but the $T=1$ branch of the brane molecule. The objects appearing here are expected to be related to other generalizations of string theory. See [[D=12 supergravity]] and [[bosonic M-theory]].

[[branemoleculepc.png:pic]]

Compare:

* {#BlencoweDuff88} [[Miles Blencowe]], [[Mike Duff]], _Supermembranes and the Signature of Space-time_, Nucl. Phys. B310 (1988) 387-404 ([spire:262142](inspirehep.net/record/262142), 10.1016/0550-3213(88)90155-1, [pdf](http://inspirehep.net/record/262142/files/cer-000099708.pdf))

* *[[brane bouquet]]*


branes > history

< [[branes]]
[[!redirects branes -- history]]


Branislav Jurco

**Branislav Jurčo** is a Czech [[mathematical physics|mathematical physicist]] ([[group]] theory, [[quantum groups]], noncommutative gauge theories, [[integrable systems]], [[nonabelian cocycle]]s in [[string theory]], [[Batalin-Vilkovisky formalism]]) and category theorist ([[gerbes]], classifying 2-spaces and topoi, [[higher gauge theory|higher gauge theories]], [[twisted K-theory]], [[operad]]s), born in former Czechoslovak Republic.

* [institute page](https://www.mff.cuni.cz/en/faculty/organizational-structure/people?hdl=5368)

* [GoogleScholar page](https://scholar.google.de/citations?user=05IMCCwAAAAJ&hl=de)

* [InSpire page](https://inspirehep.net/authors/2280915)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=193829)

## Selected writings

On [[quantum optics]]:

* Jan Peřina, Zdeněk Hradil, Branislav Jurčo, [[Quantum optics and fundamentals of physics]], Kluwer, Dodrecht-Boston 1994

and [[integrable systems]]:

* [[Branislav Jurčo]], *On quantum integrable models related to nonlinear quantum optics. An algebraic Bethe ansatz approach*,  J. Math. Phys. 30, 1739 (1989) ([doi:10.1063/1.528262](http://dx.doi.org/10.1063/1.528262))

On the [[Yang-Baxter equation]] and [[integrable systems]]:

* Branislav Jurčo, _Classical Yang-Baxter equations and quantum integrable systems ([[Gaudin models]])_, pp. 1616-6361 in Springer Lect. Notes in Physics __370__ (1990) [doi](http://dx.doi.org/10.1007/3-540-53503-9)

A model for the [[supergravity C-field]] in terms of [[nonabelian bundle 2-gerbe|nonabelian]] [[bundle 2-gerbes]]:

* [[Paolo Aschieri]], [[Branislav Jurčo]], *Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory*, JHEP 0410:068 (2004) [[arXiv:hep-th/0409200](https://arxiv.org/abs/hep-th/0409200), [doi:10.1088/1126-6708/2004/10/068](https://doi.org/10.1088/1126-6708/2004/10/068)]

On [[fiber bundles]] ([[principal bundles]], [[vector bundles]]), [[topological K-theory]] applications to [[D-branes]]:

* [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin  Schottenloher]], _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Springer Lecture Notes in Physics __726__, 2008, ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf), [doi:10.1007/978-3-540-74956-1](https://link.springer.com/book/10.1007/978-3-540-74956-1))

On [[classifying toposes]]

* [[Igor Baković]], Branislav Jurčo, _The classifying topos of a topological bicategory_, Homology, homotopy and applications 12(1), 279-300 (2010), [published files](http://www.intlpress.com/HHA/v12/n1/a14) [arxiv/0902.1750](https://arxiv.org/abs/0902.1750) [cafe](http://golem.ph.utexas.edu/category/2009/02/bakovi_and_juro_on_classifying.html)

On [[nonabelian bundle 2-gerbes]]:

* [[Branislav Jurčo]], *Nonabelian bundle 2-gerbes*,  International Journal of Geometric Methods in Modern Physics **08** 01 (2011) 49-78 [[arXiv:0911.1552](http://arxiv.org/abs/0911.1552), [doi:10.1142/S0219887811004963](https://doi.org/10.1142/S0219887811004963)]

On [[string field theory]]:

* Branislav Jurčo, Korbinian Muenster, _Type II superstring field theory: geometric approach and operadic description_, JHEP 1304:126 (2013) ([arXiv/1303.2323](http://arxiv.org/abs/1303.2323), doi)

On [[string field theory]] in terms of [[L-infinity algebras|$L_\infty$-algebras]]:

* {#DoubekJurcoMuenster13} [[Martin Doubek]], [[Branislav Jurčo]], Korbinian Muenster, _Modular operads and the quantum open-closed homotopy algebra_, J. High Energ. Phys. 2015, 1–55 (2015) ([arXiv:1308.3223](https://arxiv.org/abs/1308.3223), doi:10.1007/JHEP12(2015)158)

On [[Riemannian geometry]] on [[Courant algebroids]] and relation to [[supergravity]] [[equations of motion]]:

* [[Branislav Jurčo]], [[Jan Vysoky]], _Courant Algebroid Connections and String Effective Actions_, Proceedings of Tohoku Forum for Creativity, Special volume: Noncommutative Geometry and Physics IV ([arXiv:1612.01540](https://arxiv.org/abs/1612.01540))

On [[schreiber:Higher Structures|higher structures]] in [[M-theory]]:

* [[Branislav Jurčo]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]], _Higher Structures in M-Theory_,  Introduction to _[Higher Structures in M-Theory, LMS-EPSRC Durham Symposium, August 2018](http://www.maths.dur.ac.uk/lms/109/index.html)_, Fortsch. d. Phys. 2019 ([arXiv:1903.02807](https://arxiv.org/abs/1903.02807))

On [[higher structures]] in [[string theory]] and [[M-theory]]:

* [[Branislav Jurčo]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]], _Higher Structures in M-Theory_, Introduction to _[[Higher Structures in M-Theory 2018]]_, Fortsch. d. Phys. **67** (2019) 8-9 [[arXiv:1903.02807](https://arxiv.org/abs/1903.02807), [doi:10.1002/prop.201910001](https://doi.org/10.1002/prop.201910001)]

On [[quantum L-infinity algebras]] for the [[BV-BRST formalism]]:

* [[Martin Doubek]], [[Branislav Jurčo]], [[Ján Pulmann]], *Quantum L∞ Algebras and the Homological Perturbation Lemma*  Comm. Math. Phys. **367** (2019) 215–240 [[arXiv:1712.02696](https://arxiv.org/abs/1712.02696), [doi:10.1007/s00220-019-03375-x](https://doi.org/10.1007/s00220-019-03375-x)]

* [[Branislav Jurčo]], [[Tommaso Macrelli]], [[Christian Sämann]], [[Martin Wolf]], *Loop Amplitudes and Quantum Homotopy Algebras*, Journal of High Energy Physics **2020** 3 (2020) [doi:10.1007/JHEP07(2020)003, [arXiv:1912.06695](https://arxiv.org/abs/1912.06695)]

* [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], *Perturbative Quantum Field Theory and Homotopy Algebras*, PoS (CORFU2019) 199 [[arXiv:2002.11168](https://arxiv.org/abs/2002.11168), [doi:10.22323/1.376.0199](https://doi.org/10.22323/1.376.0199)]

On the [[classical double copy]] ([[KLT relations]], relating [[gravity]] to [[Yang-Mills theory]]) via [[L-infinity algebras|$L_\infty$-algebras]]:

* [[Leron Borsten]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], _Double copy from homotopy algebras_, Fortschritte der Physik, **69** 8--9 (2021) 2100075 [[arXiv:2102.11390](https://arxiv.org/abs/2102.11390), [doi:10.1002/prop.202100075](https://doi.org/10.1002/prop.202100075)]

On [[higher gauge theory]]:

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

category: people

[[!redirects Branislav Jurčo]]
[[!redirects B. Jurčo]]


Branko Juran

## Selected writings

On [[MO]]:

* [[Branko Juran]], _Thom spaces and the Oriented Cobordism Ring_, 2020 ([pdf](http://www.math.uni-bonn.de/people/daniel/2020/CharClasses/Vortrag4.pdf), [[JuranMO.pdf:file]])

On [[orbifold cohomology]] seen in [[global equivariant homotopy theory]]:

* {#Juran20} [[Branko Juran]], _Orbifolds, Orbispaces and Global Homotopy Theory_ ([arXiv:2006.12374](https://arxiv.org/abs/2006.12374))

category: people


Brauer group

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

#Contents#
* table of contents
{:toc}

## Idea

For $R$ a [[ring]], the _Brauer group_ $Br(R)$ is the [[group]] of [[Morita equivalence]] classes of [[Azumaya algebras]] over $R$.

## Properties

### Relation to categories of modules
 {#RelationToCatsOfModules}

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

For $R$ a commutative [[ring]], let $Alg_R$ or $2Vect_R$ (see at [[2-vector space]]/[[2-module]]) be the [[2-category]] whose

* [[objects]] are $R$-[[associative algebra|algebras]];

* [[morphisms]] are [[bimodules]] over $R$-algebras;

* [[2-morphisms]] are bimodule homomorphisms.

=--

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

This may be understood as the 2-category of (generalized) [[2-vector bundles]] over $Spec R$, the [[Isbell duality|formally dual]] [[space]] whose [[function algebra]] is $R$. This is a [[braided monoidal category|braided monoidal 2-category]].

=--

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

Let

$$
  \mathbf{Br}(R) \coloneqq Core(Alg_R)
$$

be its [[Picard 3-group]], hence the maximal [[infinity-group|3-group]] inside (which is hence a [[braided 3-group]]), the [[core]] on the invertible objects, hence the [[2-groupoid]] whose

* [[objects]] are algebras which are invertible up to [[Morita equivalence]] under tensor product;

* [[morphisms]] are [[Morita equivalences]];

* [[2-morphisms]] are invertible bimodule homomorphisms.

=--

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

This may be understood as the 2-groupoid of (generalized) [[line 2-bundles]] over $Spec R$ (for instance [[holomorphic line 2-bundles]] in the case of [[higher complex analytic geometry]]), inside that of all [[2-vector bundles]].

=--

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

The [[homotopy groups]] of $\mathbf{Br}(R)$ are the following:

* $\pi_0(\mathbf{Br}(R))$ is the Brauer group of $R$;

* $\pi_1(\mathbf{Br}(R))$ is the [[Picard group]] of $R$;

* $\pi_2(\mathbf{Br}(R))$ is the [[group of units]] of $R$.

=--

See for instance ([Street](#Street)).

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

Analogous statements hold for (non-commutative) [[superalgebras]], hence for $\mathbb{Z}_2$-[[graded algebras]]. See at _[superalgebra -- Picard 3-group, Brauer group](super+algebra#Picard2Groupoid)_.

=--

### Relation to étale cohomology
 {#RelationToEtaleCohomology}

The Brauer group of a [[ring]] $R$ is a [[torsion]] subgroup of the second [[etale cohomology]] group of $Spec R$ with values in the [[multiplicative group]] $\mathbb{G}_m$

$$
  Br(X) \hookrightarrow H^2_{et}(X, \mathbb{G}_m)
  \,.
$$

This was first stated in ([Grothendieck 68](#Grothendieck68)) (see also [Grothendieck 64, prop. 1.4](#Grothendieck64) and see at _[algebraic line n-bundle -- Properties](algebraic+line+n-bundle#Properties)_). Review discussion is in ([Milne, chapter IV](#Milne)). A detailed discussion in the context of [[nonabelian cohomology]] is in ([Giraud](#Giraud)).

A theorem stating conditions under which the Brauer group is precisely the [[torsion]] subgroup of $H^2_{et}(X, \mathbb{G}_m)$ is due to ([Gabber](#Gabber)), see also the review in ([de Jong](#deJong)). For more details and more literature on this see ([Bertuccioni](#Bertuccioni)).

This fits into the following pattern

* $H^0_{et}(R, \mathbb{G}_m) = R^\times$ ([[group of units]])

* $H^1_{et}(R, \mathbb{G}_m) = Pic(R)$ ([[Picard group]]: iso classes of invertible $R$-modules)

* $H^2_{et}(R, \mathbb{G}_m)_{tor} = Br(R)$ ([[Brauer group]]: [[Morita equivalence]] classes of [[Azumaya algebras]] over $R$) (the torsion equivalence classes of the [[Brauer stack]])

It is therefore natural to regard all of $H^2_{et}(R, \mathbb{G}_m)$ as the "actual" Brauer group. This has been called the "[[bigger Brauer group]]" ([Taylor 82](#Taylor82), [Caenepeel-Grandjean 98](#CaenepeelGrandjean98), [Heinloth-Schöer 08](#HeinlothSchoeer08)). The bigger Brauer group has actually traditionally been implicit already in the term "[[formal Brauer group]]", which is really the [[formal geometry]]-version of the bigger Brauer group.

### Relation to derived étale cohomology
 {#RelationToDerivedEtaleCohomology}

More generally, this works for $R$ a (connective) [[E-infinity ring]]
(the following is due to [Antieau-Gepner 12](#AntieauGepner12), see [Haugseng 14](#Haugseng14) for more).

Let $GL_1(R)$ be its [[infinity-group of units]].
If $R$ is [[connective spectrum|connective]], then the first [[Postnikov tower|Postnikov stage]] of the [[Picard group|Picard]] [[infinity-groupoid]]

$$
  Pic(R) \coloneqq Mod(R)^\times
$$

is

$$
  \array{
   \mathbf{B}_{et} GL_1(-) &\to& Pic(-)
   \\
   && \downarrow
   \\
   && \mathbb{Z}
  }
  \,,
$$

where the top morphism is the inclusion of locally free $R$-modules.

So $H^1_{et}(R, GL_1)$ is not equal to $\pi_0 Pic(R)$, but it is off only by $H^0_{et}(R, \mathbb{Z}) = \prod_{components of R} \mathbb{Z}$.

Let $Mod_R$ be the [[(infinity,1)-category]] of $R$-[[module spectra|modules]].

There is a notion of $Mod_R$-[[enriched (infinity,1)-category]], of "$R$-linear $(\infty,1)$-categories".

$Cat_R \coloneqq Mod_R$-modules in [[presentable (infinity,1)-categories]].

Forming module $(\infty,1)$-categories is then an [[(infinity,1)-functor]]

$$
  Alg_R \stackrel{Mod}{\to} Cat_R
$$

Write $Cat'_R \hookrightarrow Cat_R$ for the image of $Mod$.  Then define the [[Brauer group|Brauer]] [[infinity-group]] to be

$$
  Br(R) \coloneqq (Cat'_R)^\times
$$

One shows ([Antieau-Gepner 12](#AntieauGepner12)) that this is exactly the Azumaya $R$-algebras modulo Morita equivalence.

**Theorem** (B. Antieau, D. Gepner)

1. For $R$ a connective $E_\infty$ ring, any Azumaya $R$-algebra $A$ is étale locally trivial: there is an [[etale topology|etale cover]] $R \to S$ such that $A \wedge_R S \stackrel{Morita \simeq}{\to} S$.

   (Think of this as saying that an Azumaya $R$-algebra is étale-locally a Matrix algebra, hence Morita-trivial: a "bundle of compact operators" presenting a (torsion) $GL_1(R)$-2-bundle).

1. $Br : CAlg_R^{\geq 0} \to Gpd_\infty$ is a sheaf for the [[etale cohomology]].

**Corollary**

1. $Br$ is [[connected object in an (infinity,1)-topos|connected]]. Hence $Br \simeq \mathbf{B}_{et} \Omega Br $.

1. $\Omega Br \simeq Pic$, hence $Br \simeq \mathbf{B}_{et} Pic$

[[Postnikov tower]] for $GL_1(R)$:

$$
  for\; n \gt 0: \pi_n GL_1(S) \simeq \pi_n
$$

hence for $R \to S$ étale

$$
  \pi_n S \simeq \pi_n R \otimes_{\pi_0 R} \pi_0 S
$$

This is a [[quasi-coherent sheaf]] on $\pi_0 R$ of the form $\tilde N$ (quasicoherent sheaf associated with a module), for $N$ an $\pi_0 R$-module. By vanishing theorem of higher cohomology for quasicoherent sheaves

$$
  H_{et}^1(\pi_0 R, \tilde N) = 0; for p \gt 0
$$

For every [[(infinity,1)-sheaf]] $G$ of [[infinity-groups]], there is a [[spectral sequence]]

$$
  H_{et}^p(\pi_0 R; \tilde \pi_q G) \Rightarrow \pi_{q-p} G(R)
$$

(the second argument on the left denotes the $qth$ Postnikov stage). From this one gets the following.

* $\tilde \pi_0 Br \simeq *$

* $\tilde \pi_1 Br \simeq \mathbb{Z}$;

* $\tilde \pi_2 Br \simeq \tilde \pi_1 Pic \simeq \pi_0 GL_1 \simeq \mathbb{G}_m$

* $\tilde \pi_n Br$ is quasicoherent for $n \gt 2$.

there is an [[exact sequence]]

$$
  0 \to
  H_{et}^2(\pi_0 R, \mathbb{G}_m)
  \to \pi_0 Br(R)
  \to H_{et}^1(\pi_0 R, \mathbb{Z})
  \to 0
$$

(notice the inclusion $Br(\pi_0 R) \hookrightarrow H_{et}^2(\pi_0 R, \mathbb{G}_m)$)

this is [[split exact sequence|split exact]] and so computes $\pi_0 Br(R)$ for connective $R$.

Now some more on the case that $R$ is not connective.

Suppose there exists $R \stackrel{\phi}{\to} S$ which is a faithful
[[Galois extension]] for $G$ a [[finite group]].

**Examples**

1. (real into complex [[K-theory spectrum]]) $KO \to KU$ (this is $\mathbb{Z}_2$)

1. [[tmf]] $\to tmf(3)$

Give $R \to S$, have a [[fiber sequence]]

$$
  Gl_1(R/S) \stackrel{fib}{\to} GL_1(R) \to GL_1(S)
  \to
  Pic(R/S) \stackrel{fib}{\to} Pic(R) \to Pic(S)
  \to
  Br(R/S) \stackrel{fib}{\to} Br(R) \to Br(S) \to \cdots
$$

**Theorem** (descent theorems) (Tyler Lawson, David Gepner) Given $G$-Galois extension $R \stackrel{\simeq}{\to} S^{hG}$ ([[homotopy fixed points]])

1. $Mod_R \stackrel{\simeq}{\to} Mod_S^{hG}$

1. $Alg_R \stackrel{\simeq}{\to} Alg_S^{hG}$

it follows that there is a homotopy fixed points spectral sequence

$$
  H^p(G, \pi_\bullet \Sigma^n GL_1(S))
  \Rightarrow
  \pi_{-n} GL_1(S)
$$

**Conjecture** The spectral sequence gives an Azumaya $KO$-algebra $Q$ which is a nontrivial element in $Br(KO)$ but becomes trivial in $Br(KU)$.

## Related concepts

* [[Brauer stack]]

* [[formal Brauer group]]

* [[group of units]]/[[multiplicative group]], [[Picard group]]

* [[Azumaya algebra]]

* [[Brauer ∞-group]]

[[!include moduli of higher lines -- table]]

## References

Brauer groups are named after [[Richard Brauer]].

Original discussion includes

* {#Grothendieck64} [[Alexander Grothendieck]], _Le groupe de Brauer: II. Théories cohomologiques_. Séminaire Bourbaki, 9 (1964-1966), Exp. No. 297, 21 p. ([Numdam](http://www.numdam.org/item?id=SB_1964-1966__9__287_0))

* {#Grothendieck68} [[Alexandre Grothendieck]], _Le groupe de Brauer_, _Dix exposés sur la cohomologie des schémas_, Masson and North-Holland, Paris and Amsterdam, (1968), pp. 46–66.

An introduction is in

* Pete Clark, _On the Brauer group_  (2003) ([pdf](http://math.uga.edu/~pete/trivial2003.pdf))

See also

* [[John Duskin]], _The Azumaya complex of a commutative ring_, in: Categorical algebra and its applications (Louvain-La-Neuve, 1987), 107–117, Lecture Notes in Math. __1348__, Springer 1988.

* {#Street} [[Ross Street]], _Descent_, Oberwolfach preprint (sec. 5, _Brauer groups_) [pdf](http://www.math.mq.edu.au/~street/Descent.pdf); _Some combinatorial aspects of descent theory_, Applied categorical structures __12__ (2004) 537-576, [math.CT/0303175](http://arxiv.org/abs/math/0303175) (sec. 12, _Brauer groups_)

The relation to [[cohomology]]/[[etale cohomology]] is discussed in

* [[James Milne]], _Étale cohomology_, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, New Jersey (1980)
 {#Milne}

* [[Jean Giraud]], _Cohomologie non abelienne_, Die Grundlehren der mathematischen Wissenschaften, vol. 179, Springer-Verlag, Berlin, 1971.
 {#Giraud}

* [[Ofer Gabber]], _Some theorems on Azumaya algebras_, Ph. D. Thesis, Harvard University, 1978, Groupe de Brauer, Lecture Notes in Mathematics, vol. 844, Springer-Verlag, Berlin, 1981, pp. 129–209.

* [[Aise Johan de Jong]], _A result of Gabber_ ([pdf](http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf))
 {#deJong}

* Inta Bertuccioni, _Brauer groups and cohomology_, Archiv der Mathematik, vol. 84 Number 5 (2005)
 {#Bertuccioni}

Brauer groups of [[superalgebras]] are discussed in

* [[C. T. C. Wall]], _Graded Brauer groups_, J. Reine Angew. Math. 213 (1963/1964), 187-199.

* [[Pierre Deligne]], _Notes on spinors_ in _[[Quantum Fields and Strings]]_

* [[Peter Donovan]], [[Max Karoubi]], _Graded Brauer groups and K-theory with local coefficients_, Publications Math. IHES 38 (1970), 5-25 ([pdf](http://www.math.jussieu.fr/~karoubi/Donavan.K.pdf))

Refinement to [[stable homotopy theory]] and [[Brauer ∞-groups]] is discussed in

* {#Szymik11} [[Markus Szymik]], _Brauer spaces for commutative rings and structured ring spectra_ ([arXiv:1110.2956](http://arxiv.org/abs/1110.2956))

* {#BakerRichterSzymik12} [[Andrew Baker]], [[Birgit Richter]], [[Markus Szymik]], _Brauer groups for commutative $\mathbb{S}$-algebras_, J. Pure Appl. Algebra 216 (2012) 2361–2376 ([arXiv:1005.5370](http://arxiv.org/abs/1005.5370))

Unification of all this in a theory of [[(infinity,n)-modules]] is in

* {#Haugseng14} [[Rune Haugseng]], _The higher Morita category of $E_n$-algebras_ ([arXiv:1412.8459](http://arxiv.org/abs/1412.8459))

The "bigger Brauer group" is discussed in

* {#Taylor82} J. Taylor, _A bigger Brauer group_ Pacific J. Math. 103 (1982), 163-203 ([projecteuclid](https://projecteuclid.org/euclid.pjm/1102724219))

* {#CaenepeelGrandjean98} S. Caenepeel, F. Grandjean, _A note on Taylor's Brauer group_. Pacific J. Math. 186 (1998), 13-27

* {#HeinlothSchoeer08} [[Jochen Heinloth]], Stefan Schröer, _The bigger Brauer group and twisted sheaves_ ([arXiv:0803.3563](http://arxiv.org/abs/0803.3563))

See also

* [[Jochen Heinloth]], [[Marc Levine]], Stefan Scröer, _Forschungsseminar: Brauer groups and Artin stack_, 07 ([pdf](https://www.uni-due.de/~mat903/sem/brauer.pdf))

The observation that passing to [[derived algebraic geometry]] makes also the non-torsion elements in $H^2_{et}(-,\mathbb{G}_m)$ be represented by (derived) [[Azumaya algebras]] is due to

* {#Toen10} [[Bertrand Toën]], _Derived Azumaya algebras and generators for twisted derived categories_ ([arXiv:1002.2599](http://arxiv.org/abs/1002.2599))

Related MO discussion includes

* [Brauer groups and K-theory](http://mathoverflow.net/questions/87345/brauer-groups-and-k-theory)

Systematic discussion of Brauer groups in [[derived algebraic geometry]] is in

* {#AntieauGepner12} [[Benjamin Antieau]], [[David Gepner]], _Brauer groups and étale cohomology in derived algebraic geometry_, Geom. Topol. 18 (2014) 1149-1244 ([arXiv:1210.0290](http://arxiv.org/abs/1210.0290))

For the Brauer-Picard 2-group of a tensor category, see

* [[Alexei Davydov]], [[Dmitri Nikshych]], _Braided Picard groups and graded extensions of braided tensor categories_, ([arXiv:2006.08022](https://arxiv.org/abs/2006.08022))

[[!redirects Brauer groups]]

[[!redirects bigger Brauer group]]
[[!redirects bigger Brauer groups]]


Brauer induction theorem

+-- {: .rightHandSide}
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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Brauer induction theorem_ ([Brauer 46](#Brauer46)) states that, for [[ground field]] the [[complex numbers]], [[finite dimensional vector space|finite-dimensional]] [[linear representations]] $V$ of a [[finite group]] $G$ all arise as [[virtual representation|virtual combinations]] of [[induced representations]] $ind_{H}^G \big(W\big)$ of just 1-dimensional representations $W$, $dim_{\mathbb{C}}(W) =1$:

\[
  \label{GenericExampleOfBrauerInduction}
  [V]
  \;=\;
  \underset{
    \mathclap{
    {
      H_i \hookrightarrow G
    }
    \atop
    {
      {W_i \in Rep(H_i)\,,}
      \atop
      { dim(W_i) = 1 }
    }
    }
  }{\sum}
  \,
  n_i
  \,
  \Big[
    ind_{H_i}^G \big(W_i\big)
  \Big]
  \,,
  \phantom{AAA}
  a_i \in \mathbb{Z}
  \,.
\]

In other words, this says that the [[representation ring]] $R_{\mathbb{C}}(G)$ is generated from [[isomorphism classes]] $\left[ind_{H}^big(W\big)\right]$ of  [[induced representations]] $ind_{H}^G \big(W\big)$ of 1-dimensional representations $W$ of [[subgroups]] $H \subset G$.

This may be thought of as (implying) a [[splitting principle]] for [[linear representations]] ([Symonds 91](#Symonds91)), for more on this see at _[[characteristic classes of linear representations]]_ the section _[splitting principle](characteristic+class+of+a+linear+representation#SplittingPrinciple)_.

Brauer induction generalizes the immediate statement that [[finite dimensional vector space|finite-dimensional]] [[permutation representations]] are all [[direct sums]] of [[induced representations]] of the _[[trivial representation|trivial]]_ 1-dimensional representation; see at _[[induced representation of the trivial representation]]_.

The analogous statement holds true also for [[ground ring]] the [[quaternions]], while for [[ground field]] the [[real numbers]] one has to induce not just from 1-dimensional but also from 2-dimensional representations.

Of course, the expansions (eq:GenericExampleOfBrauerInduction) are not unique. But one may find [[functor|functorial]] choices that satisfy good extra properties, see below _[Snaith's explicit Brauer induction](#SnaithExpansion)_ and _[Symond's explicit Brauer induction](#SymondExplicitBrauerInduction)_.

<br/>

## Properties

### Snaith's explicit Brauer induction
 {#SnaithExpansion}

(...)

### Symonds' explicit Brauer induction
 {#SymondExplicitBrauerInduction}

We state an explicit and [[natural transformation|natural]] choice of Brauer induction due to [Symonds 91](#Symonds91) (Prop. \ref{SymondsExplicitBrauerInd}) below. Its main property is good compatibility with the [[total Chern classes of linear representations]] via a certain multiplicative transfer map on [[integral cohomology]] (the latter recalled as Lemma \ref{TransferEvens} below).

+-- {: .num_lemma #TransferEvens}
###### Lemma
**(Evens' multiplicative transfer)**

For $G$ a [[finite group]] and $H \subset G$ a [[subgroup]], there is a [[linear map]]

$$
  \mathcal{N}_H^G
  \;\colon\;
  \underset{k \in \mathbb{N}}{\prod} H^{2k}\big( B H, \mathbb{Z}\big)
  \longrightarrow
  \underset{k \in \mathbb{N}}{\prod} H^{2k}\big( B G, \mathbb{Z}\big)
$$

from the [[cohomology ring]] of the [[classifying space]] of $H$ to that of $G$ which is multiplicative in that its respects the product structure, hence the [[cup product]], on both sides

$$
  \mathcal{N}_H^G\big( \alpha \smile \beta\big)
  \;=\;
  \mathcal{N}_H^G\big( \alpha\big)  \smile \mathcal{N}_H^G\big( \beta\big)
  \,.
$$

=--

This is due to [Evens 63](#Evens63). There, the maps themselves are introduced on the bottom of p. 7, while their multiplicativity is stated as Prop. 4 on p. 10.

+-- {: .num_prop #SymondsExplicitBrauerInd}
###### Proposition
**(Symonds' explicit Brauer induction)**

For $G \in $ [[FinGrp]]
there is a [[linear map]] ([[homomorphism]] of [[abelian groups]])

$$
  R_{\mathbb{C}}\big( G \big)
  \overset
  {L}
  {\longrightarrow}
  \underset{
    [H \subset G]
  }{\prod}
  \mathbb{Z}
  \left[
    1dRep_{\mathbb{C}}\big( H \big)_{/\sim}
  \right]
$$

from the underlying abelian group of the [[representation ring]] to the [[product group]] of the [[free abelian groups]] that are spanned by the [[isomorphism classes]] of 1-dimensional representations over all [[conjugacy classes]] of [[subgroup]] $H \subset G$,

such that

1. $L$ is a [[natural transformation]] of [[functors]] $FinGrp^{op} \to Ab$,

   hence $L\big( f^\ast V\big) = f^\ast( L(V) )$;

1. $L$ is a [[section]] of the [[natural transformation]]

    $$
      \underset{
       [H \subset G]
      }{\prod}
      R^{1d}_{\mathbb{Z}}\big( H\big)
       \overset
       {\sum ind}
       {\longrightarrow}
      R_{\mathbb{C}}\big( G \big)
   $$

   which applies [[induced representation|induction]] and then sums everything up, in that the [[composition]] $\big( \sum ind \big) \circ L$ is the [[identity morphism|identity]]:

   $$
     \big( \sum ind \big) \circ L(V)
     \coloneqq
     \underset{
       [H \subset G]
     }{
       \sum
     }
     ind_H^G\left[ L(V)_H  \right]
     \;=\;
     V
   $$

1. $L$ is compatible with the [[total Chern classes of linear representations]]

   $$
     R_{\mathbb{C}}\big( G \big)
     \overset{c}{\longrightarrow}
     \underset{ k \in \mathbb{N} }{\prod}
     H^{2k}\big( B G, \mathbb{Z} \big)
   $$

   via their multiplicative transfer $\mathcal{N}_H^G$ (Lemma \ref{TransferEvens}) in that

   $$
     c
     \big(
       V
     \big)
     \;=\;
     \underset{ [H \subset G] }{\smile}
     \mathcal{N}_H^G
     \Big(
       c
       \big(
         L(V)_H
       \big)
     \Big)
     \,,
   $$

   hence in that the following [[commuting diagram|diagram commutes]]:

   $$
     \array{
       R_{\mathbb{C}}\big( G\big)
         &\overset{L}{\longrightarrow}&
       \underset{
         [H \subset G]
       }{\prod}
       \mathbb{Z}
       \left[
         1dRep_{\mathbb{C}}\big( H \big)_{/\sim}
       \right]
       \\
       {}^{\mathllap{c}}\Big\downarrow
       &&
       \Big\downarrow
       {}^{
         \underset{
           [H \subset G]
         }{\prod}
         \left(
           c
           \circ
           \mathcal{N}_H^G
         \right)
       }
       \\
       \underset{
         k \in \mathbb{N}
       }{\prod}
       H^{2k}\big( B G, \mathbb{Z} \big)
       &
         \underset{
           \underset{ [H \subset G] }{\smile}
         }{\longleftarrow}
       &
       \underset{
         [H \subset G]
       }{\prod}
       \underset{
        k \in \mathbb{Z}
       }
       {\Prod}
       H^{2k}\big( B G, \mathbb{Z}\big)
     }
   $$

1. a 1-dimensional representation $W \in 1dRep\big(G\big)_{/\sim} \subset R_{\mathbb{C}}\big(G\big)$ is sent to the [[tuple]] $L(W) = (W,0,0, \cdots)$ whose component over $G \subset G$ is $V$ itself, and all whose other components vanish;

1. in contrast, if $V \in Rep_{\mathbb{C}}\big( G \big)_{/\sim}$ has no 1-dimensional [[direct sum|direct summand]], then the $G$-compnents of $L(V)$ is zero;

=--

([Symonds 91, Prop. 2.1, Theorem 2.2, and Theorem 2.4](#Symonds91))

(...)

## References

Due to

* {#Brauer46} [[Richard Brauer]], ... (1946)

See also

* Wikipedia, _[Brauer's theorem on induced characters](https://en.wikipedia.org/wiki/Brauer%27s_theorem_on_induced_characters)_

In the context of [[characteristic classes of linear representations]]:

* {#Symonds91} [[Peter Symonds]], _A splitting principle for group representations_, Comment. Math. Helv. (1991) 66: 169 ([dml:140229](https://eudml.org/doc/140229), [doi:10.1007/BF02566643](https://doi.org/10.1007/BF02566643))

based on

* {#Evens63} [[Leonard Evens]], _A Generalization of the Transfer Map in the Cohomology of Groups_, Transactions of the American Mathematical Society Vol. 108, No. 1 (Jul., 1963), pp. 54-65 ([doi:10.1090/S0002-9947-1963-0153725-1](https://doi.org/10.1090/S0002-9947-1963-0153725-1), [jstor:1993825](https://www.jstor.org/stable/1993825))

and

* {#Kroll87} Ove Kroll, _An Algebraic Characterisation of Chern Classes of Finite Group Representations_, Bulletin of the LMS, Volume19, Issue3 May 1987 Pages 245-248 ([doi:10.1112/blms/19.3.245](https://doi.org/10.1112/blms/19.3.245))

[[!redirects Brauer's theorem]]

[[!redirects Brauer's theorem on induced characters]]

[[!redirects Brauer induction]]
[[!redirects Brauer inductions]]

[[!redirects explicit Brauer induction]]
[[!redirects explicit Brauer inductions]]


Brauer stack

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

#Contents#
* table of contents
{:toc}

## Idea

It is traditional to speak, for a suitable [[scheme]] $X$, of its _[[Picard group]]_ and of its _[[Brauer group]]_. Moreover, it is a classical fact that under suitable conditions the former admits itself a canonical geometric structure that makes it the _[[Picard scheme]]_ of $X$. Still well known, if maybe less commonly highlighted, is that this is just the [[0-truncated|0-truncation]] of the [[Picard stack]] of $X$, which is simply the [[mapping stack]] $[X, \mathbf{B}\mathbb{G}_m]$ into the [[delooping]] of the [[multiplicative group]]. In this form this applies immediately also to more general context such as [[E-∞ geometry]] ("[[spectral geometry]]") and gives a concept of _[[Picard ∞-stack]]_ ("derived Picard stack").
Given this and the [relation of the Brauer group to étale cohomology](Brauer+group#RelationToEtaleCohomology) it is clear that the [[Brauer group]] similarly arises as the [[torsion subgroup]] of the 0-truncation of the [[∞-stack]] which _ought_ to be called the **Brauer stack**, given as the [[mapping stack]]

$$
  \mathbf{Br}(X) \coloneqq [X,\mathbf{B}^2 \mathbb{G}_m]
$$

into the second [[delooping]] of the [[multiplicative group]] (modulating [[line 2-bundles]], for instance [[holomorphic line 2-bundles]] in [[higher complex analytic geometry]]), where $X$ is a [[scheme]] or may itself be a [[derived scheme]], [[algebraic stack]], etc. Indeed, just as the [[Picard stack]] turns under [[Lie integration]] (evaluation on [[infinitesimally thickened points]]) and 0-truncation into what is commonly called the _[[formal Picard group]]_, so this Brauer $\infty$-stack similarly gives what is commonly called the _[[formal Brauer group]]_.

However, while therefore the terminology "Brauer stack" is the evident continuation of a traditional pattern (which in the other direction continues with the [[group of units]] and the mapping scheme $[X,\mathbb{G}_m]$), it seems that this terminology has never been introduced in the literature (at time of this writing). (?)

## References

[[!redirects Brauer stacks]]
[[!redirects Brauer stacks]]

[[!redirects Brauer ∞-stack]]
[[!redirects Brauer ∞-stacks]]


Brauer ∞-group

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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}

## Idea

The refinement of the concept of [[Brauer group]] from [[algebra]] to [[higher algebra]] and [[stable homotopy theory]].

## Definition

([Szymik 11, def. 3.8, prop. 5.5](#Szymik11))

## Properties

### Relation to Picard $\infty$-group and $\infty$-group of units

Given an [[E-∞ ring]] $E$,
the [[looping]] of the Brauer $\infty$-group is the [[Picard ∞-group]] ([Szymik 11, theorem 5.7](#Szymik11)).

$$
  \Omega Br(E) \simeq Pic(E).
$$

The [[looping]] of that is the [[∞-group of units]] ([Sagave 11, theorem 1.2](#Sagave11)).

$$
  \Omega^2 Br(E) \simeq \Omega Pic(E) \simeq GL_1(E)
  \,.
$$

## References

* {#Szymik11} [[Markus Szymik]], _Brauer spaces for commutative rings and structured ring spectra_ ([arXiv:1110.2956](http://arxiv.org/abs/1110.2956))

* {#BakerRichterSzymik12} [[Andrew Baker]], [[Birgit Richter]], [[Markus Szymik]], _Brauer groups for commutative $\mathbb{S}$-algebras_, J. Pure Appl. Algebra 216 (2012) 2361–2376 ([arXiv:1005.5370](http://arxiv.org/abs/1005.5370))

* {#Sagave11} [[Steffen Sagave]], _Spectra of units for periodic ring spectra_ ([arXiv:1111.6731](http://arxiv.org/abs/1111.6731))

[[!redirects Brauer ∞-groups]]

[[!redirects Brauer infinity-group]]
[[!redirects Brauer infinity-groups]]


Bravais lattice

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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A *Bravais lattice* (terminology originating in the study of [[crystals]] in [[solid state physics]]) is the *equivalence class* of a [[crystal]] [[lattice (discrete subgroup)|lattice]] with *maximal* [[point symmetry]].

Mathematically, this the [[equivalence class]] of the corresponding [[crystallographic group]] $\Lambda \rtimes G_{pt} \;\subset\; \mathbb{R}^d \rtimes \mathrm{O}(d)$ (for $G_{pt} \coloneqq Stab_{O(d)}(\Lambda)$ the full [[stabilizer group]]) under [[conjugation]] by ambient [[linear transformations]] $\phi \,\in\,$ [[GL(n)|$GL(d)$]]:

$$
  \Lambda \rtimes G_{pt}
  \;\;\sim\;\;
  \phi(\Lambda) \rtimes \big( \phi \circ G_{pt} \circ \phi^{-1} \big)
  \,.
$$

(eg. [Schwarzenberger 1972, p. 325](#Schwarzenberger72))

Variant definitions exist, for instance it is of interest to retain handedness and divide out only by [[SL(n)|$SL(d)$]].

All crystal lattices with non-maximal point symmetry may be understood as unions of affine transformations of Bravais lattices.

## Related concepts

* [[crystal]], [[lattice (discrete subgroup)]]

* [[crystallographic group]]

* [[Brillouin torus]]

## References

The concept goes back to:

* {#Bravais1850}  [[Auguste Bravais]], *Mémoire sur les Systèmes Formés par les Points Distribués Régulièrement sur un Plan ou dans L'espace*, J. Ecole Polytech. 19 (1850) 1 $[$[ark:12148/bpt6k96124j](https://gallica.bnf.fr/ark:/12148/bpt6k96124j)$]$

A good review is in:

* [[Helmut G. F. Winkler]], *Hundert ]ahre Bravais Gitter*, Die Naturwissenschaften **37** 17 (1950) $[$[doi:10.1007/BF00738360](https://doi.org/10.1007/BF00738360), [pdf](https://link.springer.com/content/pdf/10.1007/BF00738360.pdf)$]$

Further discussion:

* {#Schwarzenberger72} [[Rolph Ludwig Edward Schwarzenberger]], *Classification of crystal lattices*, Mathematical Proceedings of the Cambridge Philosophical Society, **72** 3 (1972) 325-349 $[$[doi:10.1017/S0305004100047162](https://doi.org/10.1017/S0305004100047162)$]$

* M. Pitteri, G. Zanzotto, *On the Definition and Classification of Bravais Lattices*, Acta Cryst. A **52** (1996) 830-838 $[$[doi:10.1107/S0108767396005971](https://doi.org/10.1107/S0108767396005971)$]$

Lecture notes and textbook accounts:

* {#Engel86} [[Peter Engel]], Section 7 of: *Geometric Crystallography -- An Axiomatic Introduction to Crystallography*, D. Reidel Publishing (1986) $[$[doi:10.1007/978-94-009-4760-3](https://doi.org/10.1007/978-94-009-4760-3)$]$

* [[Peter Engel]], [[Louis Michel]], [[Marjorie Senechal]], Chapter 1 of: *Lattice Geometry*, IHES/P/04/45 (2004) $[$[pdf](https://cds.cern.ch/record/859509/files/cer-002542451.pdf), [cds:859509](https://cds.cern.ch/record/859509)$]$

* [[Sheng San Li]], Section 1.2 in: *Semiconductor Physical Electronics*, Springer (2006)   $[$[doi:10.1007/0-387-37766-2_4](https://doi.org/10.1007/0-387-37766-2_4)$]$

* {#Souvignier08} Bernd Souvignier, around Def. 54 in: *Group theory applied to crystallography*, Summer School on Mathematical and Theoretical Crystallography (2008) $[$[pdf](https://www.math.ru.nl/~souvi/krist_09/cryst.pdf)$]$

* {#Muller13} Ulrich Müller, Def. 6.15 in: *Symmetry Relationships between Crystal Structures*, Oxford University Press (2013) $[$[pdf](https://www.xray.cz/kryst/grupy-struktury.pdf)$]$

See also:

* Wikipedia, *[Bravais lattice](https://en.wikipedia.org/wiki/Bravais_lattice)*

[[!redirects Bravais lattices]]

[[!redirects Bravais class]]
[[!redirects Bravais classes]]

[[!redirects hexagonal lattice]]
[[!redirects hexagonal lattices]]

[[!redirects honeycomb lattice]]
[[!redirects honeycomb lattices]]

[[!redirects crystal lattice]]
[[!redirects crystal lattices]]


brave new algebra

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

#Contents#
* table of contents
{:toc}

## Idea

The term _brave new algebra_ has been coined to refer to the [[higher algebra]] of [[E-infinity rings]] ("brave new rings"), amplifying the step from more traditional [[algebra]].

Specifically, the term is typically used for the particular presentation of [[E-infinity rings]] by way of [[highly structured ring spectra]] such as ([[commutative monoids|commutative]]) [[monoids]] with respect to a [[symmetric monoidal smash product of spectra]] (such as available on [[orthogonal spectra]] ([[orthogonal ring spectra]]), [[symmetric spectra]] ([[symmetric ring spectra]]) and [[S-modules]]).

According to [Greenlees 06, p. 1](#Greenlees06):

> The phrase 'brave new rings' was coined by [[Friedhelm Waldhausen|F.Waldhausen]], presumably to capture both an optimism about the possibilities of generalizing rings to ring spectra, and a proper awareness of the risk that the new step in abstraction would take the subject dangerously far from its justification in examples.

In consequence people also speak for instance of _brave new Hopf algebroids_ when referring to the [[Hopf algebroids]] of (dual) $E$-[[Steenrod algebras]] for some [[ring spectrum]] $E$ as appearing in the [[Adams spectral sequence]]. See at _[Steenrod algebra -- Hopf algebroid structure](#Steenrod+algebra)_.

## Related entries

* [[higher algebra]], [[higher linear algebra]]

* [[E-infinity geometry]]

## References

Introductions and expositions include

* [[Peter May]], _Brave new worlds in stable homotopy theory_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/WorldsNWU.pdf))

* {#Greenlees06} [[John Greenlees]], _First steps in brave new commutative algebra_ ([arXiv:math/0609453](http://arxiv.org/abs/math/0609453))

Monographs on the technical core of "brave new algebra" in terms of [[symmetric monoidal categories of spectra]] include

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], [[Michael Mandell]] _Modern foundations for stable homotopy theory_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newfirst.pdf))

* {#EKMM} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May|P. May]], _Rings, modules and algebras in stable homotopy theory_, AMS Mathematical Surveys and Monographs Volume 47 (1997) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

For more references see at _[[ring spectrum]]_ and at _[[higher algebra]]_.

[[!redirects brave new algebras]]

[[!redirects brave new ring]]
[[!redirects brave new rings]]

[[!redirects brave new Hopf algebra]]
[[!redirects brave new Hopf algebras]]

[[!redirects brave new Hopf algebroid]]
[[!redirects brave new Hopf algebroids]]


Braxton Collier

* [MO page](http://mathoverflow.net/users/29293/braxton-collier)

## Related pages

* [[Courant Lie 2-algebroid]]

category: people


Brayton Gray

* [personal page](http://homepages.math.uic.edu/~brayton/)

* [institute page](https://mscs.uic.edu/profiles/brayton/)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6538)

## Selected writings

On [[topology|topological]] [[homotopy theory]]:

* [[Brayton Gray]], *Homotopy Theory: An Introduction to Algebraic Topology*, Academic Press (1975) $[$[ISBN:978-0-12-296050-5](https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/64/suppl/C), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/gray.pdf)$]$

category: people


Bredon cohomology

+-- {: .rightHandSide}
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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _Bredon cohomology_ after ([Bredon 67a](#Bredon67a), [Bredon 67a](#Bredon67a)) is the flavor of [[ordinary cohomology|ordinary]] $G$-[[equivariant cohomology]] which uses the "fine" [[equivariant homotopy theory]] of [[topological G-spaces]] that by [[Elmendorf's theorem]] is equivalent to the [[homotopy theory]] of [[(∞,1)-presheaves]] over $G$-[[orbit category]], instead of  the "coarse" [[Borel model structure|Borel homotopy theory]]. See at _[Equivariant cohomology -- Idea](equivariant+cohomology#Idea)_ for more motivation.

For more technical details see there _[equivariant cohomology -- Bredon equivariant cohomology](equivariant+cohomology#Bredon)_.

## Definition
 {#Definition}

Let $G$ be a [[compact Lie group]], write $Orb_G$ for its [[orbit category]] and write $PSh_\infty(Orb_G)$ for the [[(∞,1)-category of (∞,1)-presheaves]] over $Orb_G$. By [[Elmendorf's theorem]] this is [[equivalence of (∞,1)-categories|equivalent]] to the [[homotopy theory]] of [[topological G-spaces]] with weak equivalences the $H$-[[fixed point]]-wise [[weak homotopy equivalences]] for all closed subgroups $H$ ("the [[equivariant homotopy theory]]"):

$$
  \mathbf{H}^{Orb_G^{op}} \coloneqq  L_{fpwe} G Top \simeq PSh_\infty(Orb_G)
  \,.
$$

A [[spectrum object]] $E \in Stab(\mathbf{H}^{Orb_G^{op}})$ in the [[(∞,1)-topos]] $\mathbf{H}^{Orb_G^{op}}$ is what is called a [[spectrum with G-action]] or, for better or worse, a "[[naive G-spectrum]]".

For $X$ a [[G-space]], then its [[cohomology]] in $\mathbf{H}^{Orb_G^{op}}$ with [[coefficients]] in such $A$ might be called _generalized Bredon cohomology_ (in the "generalized" sense of [[generalized (Eilenberg-Steenrod) cohomology]]).

Specifically for $n \in \mathbb{N}$ and $A \in Ab(Sh(Orb_G))$ an [[abelian sheaf]] then there is an [[Eilenberg-MacLane object]]

$$
  K(n,A) \in \mathbf{H}^{Orb_G^{op}}
$$

whose [[categorical homotopy groups in an (infinity,1)-topos|categorical homotopy groups]] are concentrated in degree $n$ on $A$.

Then _ordinary Bredon cohomology_ (in the "ordinary" sense of [[ordinary cohomology]]) in degree $n$ with [[coefficients]] in $A$ is cohomology in $\mathbf{H}^{Orb_G^{op}}$ with coefficients in $K(n,A)$:

$$
  H_G^n(X,A) \simeq \pi_0 \mathbf{H}^{Orb_G^{op}}(X,A)
$$

(see the general discussion at _[[cohomology]]_).

If here $X$ is presented by a [[G-CW complex]] and hence is cofibrant in the [[model category]] structure that presents the [[equivariant homotopy theory]] (see at _[[Elmendorf's theorem]]_ for details), then the [[derived hom space]] on the right above is equivalently given by the ordinary $G$-[[fixed points]] of the ordinary [[mapping space]] of the [[topological space]] underlying the [[G-spaces]].

$$
  H_G^n(X,A) \simeq \pi_0 [X,A]_G
  \,.
$$

In this form ordinary Bredon cohomology is expressed in ([Bredon 67a, p. 3](#Bredon67a), [Bredon 67b, Theorem (2.11), (6.1)](#Bredon67b)), review in in ([Greenlees-May, p. 10](#GreenleesMay)).

The definition of Bredon cohomology which is more popular ([Bredon 67a, p. 2](#Bredon67a), [Bredon 67b, I.6](#Bredon67b)) is a [[chain complex]]-model for this: regarding $X$ again as a presheaf on the [[orbit category]], define a presheaf of chain complexes

$$
  C_\bullet(X) \;\colon\; Orb_G^{op}\longrightarrow Ch_\bullet
$$

by

$$
  C_n(X)(G/H) \coloneqq H_n((X^n)^H, (X^{n-1})^H, \mathbb{Z})
  \,,
$$

where on the right we have the [[relative homology]] of the [[CW complex]] decomposition underlying the [[G-CW complex]] $X$ in degrees as indicated. The [[differential]] on these chain complexes is defined in the obvious way (...).

Then one has an expression for _ordinary Bredon cohomology_  similar to that of [[singular cohomology]] as follows:

$$
  H_G^n(X,A) \simeq H_n(Hom_{Orb_G}(C_\bullet(X), A))
  \,.
$$

(due to [Bredon 67](#Bredon67), see e.g. ([Greenlees-May, p. 9](#GreenleesMay))).

More generally there is $RO(G)$-graded [[equivariant cohomology]] with [[coefficients]] in [[genuine G-spectra]]. This is _also_ sometimes still referred to as "Bredon cohomology". For more on this see at _[equivariant cohomology -- Bredon cohonology](equivariant%20cohomology#Bredon)_.

## Examples

### With coefficients in representation ring

[[!include incarnations of rational equivariant topological K-theory -- table ]]

## Related concepts

* [[equivariant cohomology]], [[equivariant homotopy theory]]

* [[Elmendorf's theorem]]

* [[twisted Bredon cohomology]]

* [[global equivariant homotopy theory]]

* [[global equivariant stable homotopy theory]]

* [[orbifold cohomology]]

[[!include equivariant cohomology -- table]]

## References

The original text:

* {#Bredon67b} [[Glen Bredon]], _[[Equivariant cohomology theories]]_, Springer Lecture Notes in Mathematics Vol. 34. 1967 ([doi:10.1007/BFb0082690](https://link.springer.com/book/10.1007/BFb0082690))

announced in

* {#Bredon67a} [[Glen Bredon]], _Equivariant cohomology theories_, Bull. Amer. Math. Soc. Volume 73, Number 2 (1967), 266-268. ([educlid:1183528794](https://projecteuclid.org/euclid.bams/1183528794))

Also:

* {#Illman72} [[Sören Illman]], Chapter III in: *Equivariant Algebraic Topology*, Princeton University 1972 ([[Illman_EquivariantAlgebraicTopology1972.pdf:file]])

* [[Sören Illman]], _Equivariant singular homology and cohomology_, Bull. Amer. Math. Soc. Volume 79, Number 1 (1973), 188-192 ([euclid:bams/1183534324](https://projecteuclid.org/euclid.bams/1183534324))

Review:

* {#Blumberg17} [[Andrew Blumberg]], section 1.4 of _Equivariant homotopy theory_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

* {#tomDieck87} [[Tammo tom Dieck]], Section II.9 of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

* {#GreenleesMay} [[John Greenlees]], [[Peter May]], pages 9-10 of _[[Equivariant stable homotopy theory]]_, chapter 8, pages 277-323 in: [[Ioan James]] (ed.), _[[Handbook of Algebraic Topology]]_, North-Holland, Amsterdam, 1995. ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7), [pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

* {#May96} [[Peter May]], section I.4 of _Equivariant homotopy and cohomology theory_ CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf), [[MayEtAlEquivariant96.pdf:file]])

* Paolo Masulli, section 2 of _Equivariant homotopy: $KR$-theory_, Master thesis (2011) ([pdf](http://www.math.ku.dk/~jg/students/masulli.msthesis.2011.pdf))

The [[Eilenberg-MacLane objects]] over the [[orbit category]] are discussed in detail in

* {#Lewis92} [[L. Gaunce Lewis, Jr.]], _Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen suspension theorems_, Topology Appl., 48 (1992), no. 1, pp. 25–61 (doi:10.1016/0166-8641(92)90120-O)

Equivalence of Bredon cohomology of [[topological G-spaces]] $X$ to [[abelian sheaf cohomology]] of the [[topological quotient space]] $X/G$ with [[coefficients]] a "[[locally constant sheaf]] except for dependence on [[isotropy groups]]":

* {#Honkasalo88} [[Hannu Honkasalo]], _Equivariant Alexander-Spanier cohomology_,  Mathematica Scandinavia,  63, 179-195, 1988 ([doi:10.7146/math.scand.a-12232](https://doi.org/10.7146/math.scand.a-12232))

  (for [[finite groups]])

* {#Honkasalu90} [[Hannu Honkasalo]], _Equivariant Alexander-Spanier cohomology for actions of compact Lie groups_, Mathematica Scandinavica Vol. 67, No. 1 (1990), pp. 23-34 ([jstor:24492569](https://www.jstor.org/stable/24492569))

  (for [[compact Lie groups]])

* {#Honkasalo96} [[Hannu Honkasalo]], _Sheaves on fixed point sets and equivariant cohomology_, Math. Scand. __78__ (1996), 37--55 ([jstor:](https://www.jstor.org/stable/24492815))

  (reformulation in [[topos theory]])

See also at _[[orbifold cohomology]]_.

Equivalent formulation using the [[Grothendieck construction]] for a certain [[Cat]]-valued [[presheaf]] on the [[orbit category]]"

* [[Ieke Moerdijk|I. Moerdijk]], J-A. Svensson, _The equivariant Serre spectral sequence_, Proc. Amer. Math. Soc. __118__ (1993), no. 1, 263--278 [doi:10.2307/2160037](http://dx.doi.org/10.2307/2160037)

Further remarks on this and on the [[twisted cohomology]]-version is in

* Goutam Mukherjee, N. Pandey, _Equivariant cohomology with local coefficients_ ([pdf](http://www.ams.org/proc/2002-130-01/S0002-9939-01-06377-8/S0002-9939-01-06377-8.pdf))

* Amiya Mukherjee, Goutam Mukherjee, _Bredon-Illman cohomology with local coefficients_, The Quarterly Journal of Mathematics, Volume 47, Issue 2, June 1996, Pages 199–219 ([doi:10.1093/qmath/47.2.199](https://doi.org/10.1093/qmath/47.2.199))

* {#Honkasalo97} [[Hannu Honkasalo]], _A sheaf-theoretic approach to the equivariant Serre spectral sequence_, J. Math. Sci. Univ. Tokyo 4 (1997), 53--65 ([pdf](http://journal.ms.u-tokyo.ac.jp/pdf/jms040103.pdf))

* Samik Basua, Debasis Sen, _Representing Bredon cohomology with local coefficients_, Journal of Pure and Applied Algebra Volume 219, Issue 9, September 2015, Pages 3992-4015 ([doi:10.1016/j.jpaa.2015.02.001](https://doi.org/10.1016/j.jpaa.2015.02.001))

On cases where [[prime field|$\mathbb{Z}/p$]]-[[equivariant cohomology|equivariant]] [[Bredon cohomology]] groups  are [[free modules]] over the Bredon cohomology of the point:

* [[Kevin K. Ferland]], [[L. Gaunce Lewis, Jr.]], *The $RO(G)$-Graded Equivariant Ordinary Homology of $G$-Cell Complexes with Even-Dimensional Cells for $G=\mathbb{Z}/p$*, Memoirs of the AMS **167** (2004) [[ams:memo-167-794](https://bookstore.ams.org/memo-167-794), 978-ISBN:1-4704-0392-8]

Specifically on [[Kronholm's freeness theorem]] for $\mathbb{Z}/2$-equivariant Bredon cohomology:

* [[William C. Kronholm]], *A Freeness Theorem for $RO(\mathbb{Z}/2)$-graded Cohomology*, Topology and its Applications **157** 5 (2010) 902-915 [[arXiv:0908.3825](https://arxiv.org/abs/0908.3825), [doi:10.1016/j.topol.2009.12.006](https://doi.org/10.1016/j.topol.2009.12.006)]

* [[Eric Hogle]], [[Clover May]], *The freeness theorem for equivariant cohomology of $Rep(C_2)$-complexes*, Topology and its Applications **285** 1 (2020) 107413 [[arXiv:2005.07300](https://arxiv.org/abs/2005.07300), [doi:10.1016/j.topol.2020.107413](https://doi.org/10.1016/j.topol.2020.107413)]

* {#DuggerHazelMay24} [[Daniel Dugger]], [[Christy Hazel]], [[Clover May]], *Equivariant $\mathbb{Z}/\ell$-modules for the cyclic group $C_2$*, Journal of Pure and Applied Algebra **228** 3 (2024) 107473 [[arXiv:2203.05287](https://arxiv.org/abs/2203.05287), [doi:10.1016/j.jpaa.2023.107473](https://doi.org/10.1016/j.jpaa.2023.107473)]

[[!redirects Bredon-Illman cohomology]]

[[!redirects Bredon equivariant cohomology]]

[[!redirects Kronholm's freeness theorem]]


Breit-Wigner distribution

Due to decay, an unstable quantum-mechanical system cannot be in the eigenstate of the Hamiltonian. Effectively, this means that the system is described by a wave-packet which ranges over a distribution of energies (or frequencies, as in the field of spectroscopy). In the simplest model of an exponential decay, the distribution of energies is given by
so called Breit-Wigner distribution (and to some extent this is true for
most short-living metastable system, e.g. resonances in high energy physics).

Given some quantities $E_0$ and $\Gamma$ of the dimension of energy the __Breit-Wigner energy distribution__

$$
|\rho(E)|^2 = \frac{\Gamma/2\pi}{\frac{1}{2}\Gamma^2+(E-E_0)^2}
$$

which should be in principle damped all the way to $0$ from some energy $E_{min}$ as the spectrum of any Hamiltonian must be bounded below. The plot of the Breit-Wigner curve as seen in spectroscopy is often called Breit-Wigner curve.

Various effects in nature lead to modifed, asymmetric and other spectral curves to which the Breit-Wigner is merely the crudest approximation.

## Literature

#### Classical Breit-Wigner

* Anthony Sudbery, _Quantum mechanics and the particles of nature_, formula 3.237 and around.
* G. Breit, E. Wigner, Phys. Rev. 49, 519 (1936).

#### Corrected curves

Assymetric line shapes in autoionization and Rydberg atoms are studied in the famous

* Ugo Fano, _Effects of configuration interaction on intensities and phase shifts_, Phys. Rev. 124, 1866 (1961) [doi](https://doi.org/10.1103/PhysRev.124.1866)
* wikipedia: [Fano resonance](https://en.wikipedia.org/wiki/Fano_resonance)
* A. Bianconi, _Ugo Fano and shape resonances in X-ray and Inner Shell Processes_ AIP Conference Proceedings (2002): (19th Int. Conference Roma June 24–28, 2002) [cond-mat/0211452](http://arXiv.org/abs/cond-mat/0211452)
* VV Sokolov, VG Zelevinsky, _Dynamics and statistics of unstable quantum states_, Nuclear Physics A504:3 (1989) 562-588 doi; _Simple mode on a highly excited background: Collective strength and damping in the continuum_, Phys. Rev. C 56, 311 (1997)

category: physics

[[!redirects Breit-Wigner curve]]
[[!redirects Breit-Wigner energy distribution]]


Brendan Fong

* [webpage](http://www.brendanfong.com/)

## Selected writings

On [[hypergraph categories]]:

* {#FongDC} [[Brendan Fong]], *Decorated Cospans*, Theory and Applications of Categories, Vol. 30, 2015, No. 33, pp 1096-1120 ([arxiv:1502.00872](https://arxiv.org/abs/1502.00872))

* {#FongThesis} [[Brendan Fong]], *The Algebra of Open and Interconnected Systems*, Ph.D. Thesis ([arxiv:1609.05382](https://arxiv.org/abs/1609.05382))

* {#FongSpivak} [[Brendan Fong]] and [[David Spivak]], *Hypergraph Categories* ([arxiv:1806.08304](https://arxiv.org/abs/1806.08304))

On [[category theory]]:

* {#FongSpivak18} [[Brendan Fong]], [[David Spivak]], _An invitation to applied category theory_, 2018 ([web](http://math.mit.edu/~dspivak/teaching/sp18/), [pdf](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf))

category: people


Brendan McKay

* [webpage](https://users.cecs.anu.edu.au/~bdm/)

## Selected writings

On [[cochain cohomology]] of [[graph complexes]]:

* [[Dror Bar-Natan]], [[Brendan McKay]], _Graph cohomology -- An overview and some computations_ ([[BarNatanMcKayGraphCohomology.pdf:file]])

category: people


Brent Yorgey

* [webpage](http://www.cis.upenn.edu/~byorgey/)

## related $n$Lab pages

* [[species]]

category: people


Brett McInnes

* [webpage](http://www.math.nus.edu.sg/~matmcinn/)

## Selected writings

* {#McInnes99a} [[Brett McInnes]], _The Semispin Groups in String Theory_, J. Math. Phys. 40:4699-4712, 1999 ([arXiv:hep-th/9906059](https://arxiv.org/abs/hep-th/9906059))

* {#McInnes99b} [[Brett McInnes]], _Gauge Spinors and String Duality_,  Nucl. Phys. B577:439-460, 2000 ([arXiv:hep-th/9910100](https://arxiv.org/abs/hep-th/9910100))

  (on [[semi-spin groups]] and [[central product]] [[Spin(n1).Spin(n2)]]-groups and their role as [[subgroups]] of [[gauge groups]] in [[heterotic string theory]]/[[Green-Schwarz anomaly cancellation]])

## Related $n$Lab entries

* [[semi-spin group]]

  * [[SemiSpin(16)]]

  * [[SemiSpin(32)]]

* [[Spin(n1).Spin(n2)]]

* [[heterotic string theory]]/[[Green-Schwarz anomaly cancellation]]

category: people


Brian C. Hall

* [institute page](https://math.nd.edu/people/faculty/brian-c-hall/)

## Selected writings

On [[Hall coherent states]]:

* [[Brian Charles Hall]], _The Segal-Bargmann coherent state transform for Lie groups_. J. Funct. Anal. __122__:103–151, 1994 ([doi:10.1006/jfan.1994.1064](http://dx.doi.org/10.1006/jfan.1994.1064))

* [[Brian Charles Hall]], _Holomorphic methods in analysis and mathematical physics_, First Summer School in Analysis and Mathematical Physics (Cuernavaca Morelos, 1998). Contemp. Math., __260__:1–59, 2000;

* [[Brian Charles Hall]], _Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type_, Comm. Math. Phys., __226__:233–268, 2002 ([doi:10.1007/s002200200607](http://dx.doi.org/10.1007/s002200200607))

* B. Driver, [[Brian Hall]], _Yang-Mills theory and the Segal-Bargmann transform_, Commun. Math. Phys. 201, 249–290 (1999) ([MR2000c:58064](http://www.ams.org/mathscinet-getitem?mr=1682238))

On [[Lie theory]] ([[Lie groups]], [[Lie algebras]], [[linear representations]]):

* {#Hall15} [[Brian C. Hall]], *Lie Groups, Lie Algebras, and Representations*, Springer 2015 ([doi:10.1007/978-3-319-13467-3](https://doi.org/10.1007/978-3-319-13467-3))

category: people

[[!redirects Brian Hall]]

[[!redirects Brian Charles Hall]]


Brian Conrad

* [webpage](http://math.stanford.edu/~conrad/)

## Selected writings

On the [[modularity theorem]]:

* {#BreuilConradDiamondTaylor2001} [[Christophe Breuil]], [[Brian Conrad]], [[Fred Diamond]], [[Richard Taylor]], *On the modularity of elliptic curves over $\mathbf{Q}$: Wild 3-adic exercises*, Journal of the American Mathematical Society **14** 4 (2001) 843–939 $[$[doi:10.1090/S0894-0347-01-00370-8](https://doi.org/10.1090/S0894-0347-01-00370-8)$]$

category: people


Brian Day

Brian John Day (1945-2012) was an Australian [[category theory|category theorist]].

* [Publications](http://maths.mq.edu.au/~street/Day.pub.html)

* [Obituary by Ross Street](https://www.austms.org.au/Publ/Gazette/2013/May13/Day.pdf)

## Selected writings

On [[enriched functor categories]]:

* [[Brian Day]], [[Max Kelly]], *Enriched functor categories*, in: *Reports of the Midwest Category Seminar III*, Lecture Notes in Mathematics **106** (1969) 178-191 [[doi:10.1007/BFb0059146](https://doi.org/10.1007/BFb0059146), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0059146.pdf)]

On [[limits]] and [[colimits]] and [[exponential objects]] in the [[category]] [[Top]] of [[topological spaces]]:

* {#DayKelly} [[Brian Day]], [[G. Max Kelly]], *On topological quotients preserved by pullback or products*, Mathematical Proceedings of the Cambridge Philosophical Society **67** 3 (1970) 553 - 558  ([doi:10.1017/S0305004100045850](https://doi.org/10.1017/S0305004100045850))

Proving [[Day's reflection theorem]]:

* {#Day72} [[Brian Day]], *A reflection theorem for closed categories*, Journal of Pure and Applied Algebra **2** 1 (1972) 1-11. [doi:10.1016/0022-4049(72)90021-7]

On [[monoidal 2-categories]], [[braided monoidal 2-categories]], [[sylleptic monoidal 2-categories]] and [[symmetric monoidal 2-categories]] (and [[pseudomonoids]]):

* [[Brian Day]], [[Ross Street]], *Monoidal Bicategories and Hopf Algebroids*, Advances in Mathematics Volume 129, Issue 1, 15 July 1997, Pages 99-157 ([doi:10.1006/aima.1997.1649](https://doi.org/10.1006/aima.1997.1649))

On lax [[braided monoidal categories]] with possibly non-invertible [[braiding]]:

* [[Brian Day]], [[Elango Panchadcharam]], [[Ross Street]], _Lax braidings and the lax centre_, Contemporary Mathematics **441** 1 (2007) [[pdf](http://science.mq.edu.au/~street/laxcentre.pdf), [[DayPanchadcharamStreet-LaxBraidings.pdf:file]]]

## Related entries

* [[Day convolution]]

[[!redirects B. Day]]
[[!redirects Brian Day]]
[[!redirects B. J. Day]]

category:people


Brian G. Wybourne

* [Wikipedia entry](https://en.wikipedia.org/wiki/Brian_Wybourne)

## Selected writings

On the "[[Gruppenpest]]" (applications of [[group theory]] to [[quantum physics]]):

* [[Brian G. Wybourne]], *The "The Gruppen Pest" yesterday, today, and tomorrow*, International Journal of Quantum Chemistry Symp. **7** (1973) 35-43 [[doi:10.1002/qua.560070706](https://doi.org/10.1002/qua.560070706)]

category: people

[[!redirects Brian Wybourne]]


Brian Greene

* [Wikipedia entry](#http://en.wikipedia.org/wiki/Brian_Greene)

## selected writings

* [[Brian Greene]], Yakov Kanter, _Small Volumes in Compactified String Theory_, Nucl.Phys.B497:127-145,1997 ([arXiv:hep-th/9612181](https://arxiv.org/abs/hep-th/9612181))

* [[Brian Greene]], [[Calin Lazaroiu]], _Collapsing D-Branes in Calabi-Yau Moduli Space: I_, Nucl.Phys. B604 (2001) 181-255 ([arXiv:hep-th/0001025](https://arxiv.org/abs/hep-th/0001025))

## related $n$Lab entries

* [[flop transition]]

category: people


Brian Lawrence

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=225223)

## Selected writings

On the [[Mordell conjecture]]:

* {#LawrenceVenkatesh18} [[Brian Lawrence]], [[Akshay Venkatesh]]: _Diophantine problems and $p$-adic period mappings_,  Invent. math. **221** (2020) 893–999  [[arxiv:1807.02721](https://arxiv.org/abs/1807.02721), [doi:10.1007/s00222-020-00966-7](https://doi.org/10.1007/s00222-020-00966-7)]

* {#Balakrishnan19} [[Jennifer Balakrishnan]],  A. J. Best, F. Bianchi, [[Brian Lawrence]], J. S. Müller, N. Triantafillou, J. Vonk: *Two recent $p$-adic approaches to the (effective) Mordell conjecture*, in: *Arithmetic L-Functions and Differential Geometric Methods* Progress in Mathematics **338** Birkhäuser (2021) [[arxiv:1910.12755](https://arxiv.org/abs/1910.12755), [doi:10.1007/978-3-030-65203-6_2]( https://doi.org/10.1007/978-3-030-65203-6_2)]

category: people


Brian Munson

* [website](http://palmer.wellesley.edu/~munson/)

## Selected writings

On [[Goodwillie calculus]]:

* [[Brian Munson]], _Introduction to the manifold calculus of Goodwillie-Weiss_,  Morfismos, Vol . 14, No. 1, 2010, pp. 1-60 ([arXiv:1005.1698](https://arxiv.org/abs/1005.1698))

On [[cubical objects]] in [[homotopy theory]], such as in [[n-excisive functors]] and [[Goodwillie calculus]]:

* {#MunsonVolic15} [[Brian Munson]], [[Ismar Volic]], _Cubical homotopy theory_, Cambridge University Press, 2015  ([pdf](http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf), [doi:10.1017/CBO9781139343329](https://doi.org/10.1017/CBO9781139343329))

## Related $n$-Lab entries

* [[Blakers-Massey theorem]]

* [[Goodwillie calculus]]

category: people


Brian Osserman

* [webpage](https://www.math.ucdavis.edu/~osserman/)

## Selected writings

On [[sesquilinear forms]]:

* [[Brian Osserman]], _A primer on sesquilinear forms_ ([[OssermanSesquilinearForms.pdf:file]])

category: people


Brian Parshall

__Brian Parshall__ is an American algebraist. His "research interest center on representation theory of finite groups, Lie algebras, and related mathematical objects".

* faculty [page](http://artsandsciences.virginia.edu/mathematics/people/faculty/bjp8w.html), personal [website](http://www.math.virginia.edu/Faculty/Parshall)

* Bangming Deng, Jie Du, [[Brian Parshall]], Jianpan Wang, _Finite dimensional algebras and quantum groups_, Mathematical Surveys and Monographs __150__, Amer. Math. Soc. 2008. xxvi+759 pp. [MR2009i:17023](http://www.ams.org/mathscinet-getitem?mr=2457938)

* B. Parshall, J. Wang, _Quantum linear groups_, Memoirs AMS __439__

category:people

[[!redirects B. Parshall]]


Brian Sanderson

* [webpage](http://homepages.warwick.ac.uk/~maaac/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=42445)

## Selected writings

On [[semi-simplicial sets]] and their [[geometric realization]]:

*  {#RourkeSanderson71} [[Colin Rourke]],  [[Brian Sanderson]], _Δ-Sets I: Homotopy Theory_. The Quarterly Journal of Mathematics 22: 321–338 (1971) ([doi:10.1093/qmath/22.3.321](https://doi.org/10.1093/qmath/22.3.321), [pdf](http://www.maths.ed.ac.uk/~aar/papers/deltars.pdf))

Discussion of [[equivariant homotopy theory|equivariant]] [[configuration spaces of points]] in [[Euclidean G-spaces]] and the [[Cohomotopy charge map]]-equivalence:

* {#RourkeSanderson00} [[Colin Rourke]], [[Brian Sanderson]], _Equivariant Configuration Spaces_, J. London Math. Soc. 62 (2000) 544-552 ([arXiv:math/9712216](https://arxiv.org/abs/math/9712216))

category: people


Brian Serot

* [obituary](https://web.physics.indiana.edu/faculty/serot.shtml)

## Selected writings

Inclusion of the [[pion]] and the [[rho-meson]] onto the [[Walecka model]] of [[quantum hadrodynamics]] (the _QHD-II model_):

* [[Brian Serot]], _A relativistic nuclear field theory with $\pi$ and $\rho$ mesons_, Physics Letters B Volume 86, Issue 2, 24 (1979), Pages 146-150 (doi:10.1016/0370-2693(79)90804-9)

* T Matsui, [[Brian Serot]], _The pion propagator in relativistic quantum field theories of the nuclear many-body problem_, Annals of Physics Volume 144, Issue 1, November 1982, Pages 107-167 (doi:10.1016/0003-4916(82)90106-3)

More on the [[quantum hadrodynamics]]:

* [[Brian Serot]], [[John Dirk Walecka]], _The Relativistic Nuclear Many Body Problem_, Adv. Nucl. Phys. 16 (1986) 1-327 ([spire:207866](https://inspirehep.net/literature/207866))

* [[Brian Serot]], _Quantum hadrodynamics_, Reports on Progress in Physics, Volume 55, Number 11 (1992) ([doi:10.1088/0034-4885/55/11/001](https://doi.org/10.1088/0034-4885/55/11/001))

* [[Brian Serot]], [[John Dirk Walecka]], _Recent Progress in Quantum Hadrodynamics_,  Int. J. Mod. Phys. E6:515-631, 1997 ([arXiv:nucl-th/9701058](http://arxiv.org/abs/nucl-th/9701058))

category: people


Brian Swingle

* [webpage](https://quics.umd.edu/people/brian-swingle)

## Selected writings

On [[string-net models]] via [[tensor networks]]:

* [[Zheng-Cheng Gu]], [[Michael Levin]], [[Brian Swingle]], [[Xiao-Gang Wen]], *Tensor-product representations for string-net condensed states*, Phys. Rev. B **79** (2009) 085118 $[$[doi:10.1103/PhysRevB.79.085118](https://doi.org/10.1103/PhysRevB.79.085118)$]$

Identifying [[AdS/CFT in condensed matter physics]] and [[holographic entanglement entropy]] in [[renormalization]] of [[tensor network states]] in [[solid state physics]]:

* {#Swingle09} [[Brian Swingle]], _Entanglement Renormalization and Holography_, Phys. Rev. D 86, 065007 (2012) ([arXiv:0905.1317](https://arxiv.org/abs/0905.1317))

* {#Swingle12} [[Brian Swingle]], _Constructing holographic spacetimes using entanglement renormalization_ ([arXiv:1209.3304](https://arxiv.org/abs/1209.3304), [spire:1185813](http://inspirehep.net/record/1185813))

On [classically controlled quantum computation](quantum+computation#ClassicalControlQuantumData) on [[NISQ]]-machines in terms of [[holographic entanglement entropy|holographic]] [[tensor network]] states:

* {#KimSwingle17} [[Isaac H. Kim]], [[Brian Swingle]], *Robust entanglement renormalization on a noisy quantum computer* ([arXiv:1711.07500](https://arxiv.org/abs/1711.07500))

On [[AdS/CFT in condensed matter physics]]:

* Mike Blake, Yingfei Gu, [[Sean A. Hartnoll]], Hong Liu, [[Andrew Lucas]], Krishna Rajagopal, [[Brian Swingle]], [[Beni Yoshida]], *Snowmass White Paper: New ideas for many-body quantum systems from string theory and black holes* [[arXiv:2203.04718](https://arxiv.org/abs/2203.04718)]

category: people


Brian Willett

* [InSpire page](https://inspirehep.net/authors/1064788)

## Selected writings

On [[generalized global symmetries]]:

* {#GKSW14} [[Davide Gaiotto]], [[Anton Kapustin]], [[Nathan Seiberg]], [[Brian Willett]], *Generalized Global Symmetries*  J. High Energ. Phys. **2015** 172 (2015) [[arXiv:1412.5148](https://arxiv.org/abs/1412.5148), doi:10.1007/JHEP02(2015)172]

category: people