topological quantum field theory

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### Context
#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
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#### Topological physics
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[[!include topological physics -- contents]]
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#Contents#
* table of contents
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## Idea

A **topological quantum field theory** is a [[quantum field theory]] which
-- as a [[FQFT|functorial quantum field theory]] -- is a functor on a flavor of the  [[(∞,n)-category of cobordisms]] $Bord_n^S$, where the [[n-morphism]]s are [[cobordism]]s without any non-topological further structure $S$ -- for instance no [[Riemannian metric]] structure -- but possibly some "topological structure", such as  [[Spin structure]] or similar.

For more on the general idea and its development, see [[FQFT]] and [[extended topological quantum field theory]].

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###### Remark

Often topological _quantum_ field theories are just called _topological field theories_ and accordingly the abbreviation TQFT is reduced to TFT. Strictly speaking this is a misnomer, which is however convenient and very common. It should be noted, however, that TQFTs may have classical counterparts which would better deserve to be called TFTs. But they are not usually.

=--

## Non-topological QFTs

In contrast to topological QFTs, non-topological quantum field theories in the [[FQFT]] description are $n$-functors on $n$-categories $Bord^S_n$ whose morphisms are manifolds with extra $S$-structure, for instance

* $S =$ conformal structure $\to$ [[conformal field theory]]

* $S =$ [[Riemannian manifold|Riemannian structure]] $\to$ "euclidean QFT"

* $S =$ [[pseudo-Riemannian metric|pseudo-Riemannian structure]] $\to$ "relativistic QFT"

## Examples
 {#Examples}

* [[2d TQFT]]

  * [[TCFT]]

  * [[2d Chern-Simons theory]]

* [[3d TQFT]]

  * [[Dijkgraaf-Witten theory]]

  * [[Chern-Simons theory]]

  * [[Reshetikhin-Turaev model]]

  * [[Turaev-Viro model]]

* [[4d TQFT]]

  * [[topological Yang-Mills theory]]

  * [[topologically twisted D=4 super Yang-Mills theory ]]

  * [[4d Chern-Simons theory]]

  * [[Yetter model]]

## Homotopy QFTs

These somehow lie between the previous two types. There is some simple extra structure in the form of a 'characteristic map' from the manifolds and bordisms to a 'background space' $X$.
In many of the simplest examples, this is taken to be the [[classifying space]] of a group, but this is not the only case that can be considered.  The topic is explored more fully in [[HQFT]].

## Related concepts

* [[cohomological field theory]], [[TCFT]]

* [[FQFT]], [[extended topological quantum field theory]]

* [[topological phase of matter]]

* [[topological quantum computation]]

## References
 {#References}

See also the references at [[2d TQFT]], [[3d TQFT]] and [[4d TQFT]].

Discussion of [[action functionals]] for [[topological field theories]] via [[equivariant ordinary differential cohomology]]:

* Joe Davighi, Ben Gripaios, [[Oscar Randal-Williams]], _Differential cohomology and topological actions in physics_ ([arXiv:2011.05768](https://arxiv.org/abs/2011.05768))

### Origin in physics
 {#OriginInPhysics}

The concept originates in the guise of [[cohomological quantum field theory]] motivated from TQFTs appearing in [[string theory]] in

* {#Witten88} [[Edward Witten]], _Topological quantum field theory_, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386. ([euclid:1104161738](http://projecteuclid.org/euclid.cmp/1104161738))

* {#Witten91} [[Edward Witten]], _Introduction to cohomological field theory_, International Journal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 ([[WittenCQFT.pdf:file]])

* {#CordesMooreRamgoolam94} Stefan Cordes, [[Gregory Moore]], Sanjaye Ramgoolam, _Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories_, Nucl. Phys. Proc. Suppl.41:184-244,1995 ([arXiv:hep-th/9411210](http://arxiv.org/abs/hep-th/9411210))

and in the discussion of [[Chern-Simons theory]] ("Schwarz-type TQFT") in

* [[Edward Witten]], _Quantum Field Theory and the Jones Polynomial_ Commun. Math. Phys. 121 (3) (1989) 351–399. MR0990772 ([project EUCLID](http://projecteuclid.org/euclid.cmp/1104178138))

Review:

* Danny Birmingham, [[Matthias Blau]], Mark Rakowski, [[George Thompson]], *Topological field theory*, Physics Reports **209** 4–5 (1991) 129-340 \[doi:10.1016/0370-1573(91)90117-5\]

See also:

* [[Markus Banagl]], *Positive topological quantum field theories*, Quantum Topology **6** 4 (2015) 609-706 [[arxiv/1303.4276](http://arxiv.org/abs/1303.4276), [doi:10.4171/qt/71](https://doi.org/10.4171/qt/71)]

### Global (1-functorial) TQFT

The [[FQFT]]-[[axioms]] for global (i.e. [[functorial field theory|1-functorial]]) TQFTs are due to:

* {#Atiyah89} [[Michael Atiyah]], *Topological quantum field theories*, Publications Mathématiques de l’IHÉS **86** (1989) 175-186 [[numdam:PMIHES_1988__68__175_0](http://www.numdam.org/item?id=PMIHES_1988__68__175_0)]

Exposition of the conceptual ingredients:

* [[John Baez]], _Quantum Quandaries: a Category-Theoretic Perspective_ ([arXiv:quant-ph/0404040](http://arxiv.org/abs/quant-ph/0404040))

More technical lecture notes:

* [[Daniel Freed]], *Lectures on topological quantum field theory*, in: *Integrable Systems, Quantum Groups, and Quantum Field Theories*, NATO ASI Series **409** (1992) [[doi:10.1007/978-94-011-1980-1_5](https://doi.org/10.1007/978-94-011-1980-1_5), [pdf](http://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf), [[Freed-TQFTLectures.pdf:file]]]

* {#Quinn95} [[Frank Quinn]], *Lectures on axiomatic topological quantum field theory*, in [[Dan Freed]], [[Karen Uhlenbeck]] (eds.) *Geometry and Quantum Field Theory* **1** (1995) [[doi:10.1090/pcms/001](https://doi.org/10.1090/pcms/001)]

* {#Walker06} [[Kevin Walker]], _TQFTs_, 2006 ([pdf](http://canyon23.net/math/tc.pdf))

* [[Mikhail Khovanov]] (notes by [[You Qi]]), §2 in: *Introduction to categorification*, lecture notes, Columbia University (2010, 2020) [[web](https://www.math.columbia.edu/~khovanov/cat2020/), [web](https://you-qi2121.github.io/mypage/categorificationnotes.html), full:[pdf](https://www.dropbox.com/scl/fi/wdesax1c8il6tgwbi20t4/KhovanovYouQi-Categorification.pdf?rlkey=l5cm3khnzu604ijdnl06o89od&dl=0)]

  > (with an eye towards [[link homology]])

An introduction specifically to [[2d TQFTs]] is in

* [[Joachim Kock]], _Frobenius algebras and 2D topological quantum field theories_, No. 59 of LMSST, Cambridge University Press, 2003., (full information [here](http://mat.uab.es/~kock/TQFT.html)).

See also the references at _[[HQFT]]_.

Relation to [[cutting and pasting of manifolds|cut-and-paste-ivariants]]:

* [[Carmen Rovi]], Matthew Schoenbauer, *Relating Cut and Paste Invariants and TQFTs*, The Quarterly Journal of Mathematics **73** 2 (2022) 579–607 [[arXiv:1803.02939](https://arxiv.org/abs/1803.02939), [doi:10.1093/qmath/haab044](https://doi.org/10.1093/qmath/haab044)]

See also:

* {#Torzewska22} [[Fiona Torzewska]], *Topological quantum field theories and homotopy cobordisms* [[arXiv:2208.14504](https://arxiv.org/abs/2208.14504)]

* [[Fiona Torzewska]], *Topological Quantum Field Theories and Homotopy Cobordisms*, [talk at](CQTS##TorzewskaDec2023) [[CQTS]] (Dec 2023) [slides:[[Torzewska-TQFTandHomCob.pdf:file]], video:[YT](https://youtu.be/7rtJ61EPL-M)]

### Local ($n$-functorial) TQFT

The local [[FQFT]] formulation (i.e. [[n-functor|n-functorial]]) together with the [[cobordism hypothesis]] was suggested in

*  [[John Baez]], [[James Dolan]], _Higher dimensional algebra and Topological Quantum Field Theory_ J.Math.Phys. 36 (1995) 6073-6105 ([arXiv:q-alg/9503002](http://arxiv.org/abs/q-alg/9503002))

and formalized and proven in

* [[Jacob Lurie]], [[On the Classification of Topological Field Theories]]; _TQFT and the Cobordism Hypothesis_ ([video](http://www.ma.utexas.edu/video/dafr/lurie/), [notes](http://www.ma.utexas.edu/users/plowrey/dev/rtg/notes/perspectives_TQFT_notes.html))

This also shows how [[TCFT]] fits in, which formalizes the original proposal of 2d [[cohomological quantum field theory]].

Lecture notes:

* {#Teleman14} [[Constantin Teleman]], *Five lectures on topological field theory*, in *Geometry and Quantization of Moduli Spaces*, CRM Advanced Courses in Mathematics, Birkhäuser (2016) [[doi:10.1007/978-3-319-33578-0_3](https://doi.org/10.1007/978-3-319-33578-0_3), [pdf](http://math.berkeley.edu/~teleman/math/barclect.pdf), [[Teleman-LecturesOnTFT.pdf:file]]]

A discussion amplifying the aspects of [[higher category theory]] is in

* [[Anton Kapustin]], _Topological field theory, higher categories, and their applications_, survey for ICM 2010, ([arxiv/1004.2307](http://arxiv.org/abs/1004.2307))

See also

* Mark Feshbach, [[Alexander Voronov]], _A higher category of cobordisms and topological quantum field theory_ [[arxiv/1108.3349](http://arxiv.org/abs/1108.3349)]

Indication of local [[quantization]] in the context of [[infinity-Dijkgraaf-Witten theory]] is in

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

[[!redirects topological quantum field theories]]
[[!redirects TQFT]]
[[!redirects TQFTs]]
[[!redirects TFT]]
[[!redirects TFTs]]

[[!redirects topological field theory]]
[[!redirects topological field theories]]


> under construction

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###Context###
#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
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#### Differential cohomology
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[[!include differential cohomology - contents]]
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This page collects some links and other material related to the article

* [[Sergei Gukov]], [[Anton Kapustin]],

  _Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories_

  ([arXiv:1307.4793](http://arxiv.org/abs/1307.4793))

The article considers the generalization of a phenomenon in [[spontaneous symmetry breaking]] in abelian ([[higher gauge theory|higher]]) [[gauge theory]], that was considered earlier in ([Banks-Seiberg 08](#BanksSeiberg08)), to _[[nonabelian cohomology|nonabelian]]_ [[higher gauge theory]], hence involving not just [[differential 1-form]] [[gauge potentials]] but 2-form gauge potentials ("nonabelian [[B-fields]]"), hence to [[principal 2-connections]].

Specifically, in ([Banks-Seiberg 08, (2.4)-(2.9)](#BanksSeiberg08)) is an argument saying that some properties of the low energy limit of abelian [[Yang-Mills theory|Yang-Mills like]] [[gauge theories]] in four dimensions with ([[Higgs mechanism|Higgs-like]]) [[spontaneous symmetry breaking]] to a discrete gauge [[cyclic group]] $\mathbb{Z}_p$ are described by a dual [[BF-theory]], hence a [[higher gauge theory]] represented by a [[principal 2-connection|2-connection]] instead of an ordinary [[connection on a bundle|connection]]. Here the duality is [[electric-magnetic duality]], but applied not to the 1-form [[gauge field]] but to the [[Higgs boson]]-like [[scalar field]], whose magentic dual in 4d is indeed a $(4 - (0+1) - 1 = 2)$-form  [[higher gauge field]]. See at _[[connection on a 2-bundle]]_ for more on this.

In the nonabelian case the proposed mechanism is no longer [[electric-magnetic duality]] of a [[Higgs boson]]-like [[field (physics)|field]], but instead to condensation of [[monopoles]].

The abelian phenomenon is reviewed and expanded on in its relation to higher dimensional [[QFT with defects|defect]] [[quantum operators]] in section III. Generalization to the nonabelian case is then in section IV. An outlook on the relation to structures such as in [[geometric Langlands duality]] is in section V.

Followups include

* [[Anton Kapustin]], [[Ryan Thorngren]], _Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement_ ([arXiV:1308.2926](http://arxiv.org/abs/1308.2926))

* [[Ryan Thorngren]], _Electric-Magnetic Duality of Topological Gauge Theories from Compactification_ ([arXiv:1309.1489](http://arxiv.org/abs/1309.1489))

***

> under construction

#Contents#
* table of contents
{:toc}

## II. Surface operators in abelian gauge theories

(...)

## III. TQFTs for abelian gapped phases
 {#TQFTsForAbelianGappedPhases}

> Here is an observation about how the idea in that section might be formalized. (Due to discussion among Domenico Fiorenza, Hisham Sati, and Urs Schreiber).

The [[moduli stack]] $\mathbf{B}\mathbb{Z}_n$ of "discrete [[gauge fields]]" for [[gauge group]] $\mathbb{Z}_n := \mathbb{Z}/n\mathbb{Z}$ with $n \in \mathbb{N} $ may be realized as the following [[homotopy fiber]] of smooth moduli stacks of [[circle n-bundle with connection|circle connections]]

$$
  \array{
    \mathbf{B}\mathbb{Z}_n &\to& \mathbf{B}U(1)_{conn}
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{(-)^n}}
    \\
    \ast &\stackrel{0}{\to}& \mathbf{B}U(1)_{conn}
  }
$$

(in [[Smooth∞Grpd]]). By construction, an element $X \to \mathbf{B}\mathbb{Z}_n$ of this [[homotopy fiber]]  is equivalently

* a 1-form gauge field $\tilde F_A \colon X \to \mathbf{B}U(1)_{conn}$ with local connection 1-form $A$;

* a 0-form gauge field $\phi \colon \tilde F_A \to 0$ trivializing $\tilde F_A$, with local component $\phi$ locally satisfying $d \phi = n A$.

This data is what motivates the discussion in the article.

The [[electric-magnetic duality]] may be formulate as a "differential higher [[Pontryagin duality]]"

$$
  \mathbf{B}^2 \hat \mathbb{Z}_n \simeq [\mathbf{B}\mathbb{Z}_n, \mathbf{B}^3 U(1)_{conn}]
  \,,
$$

where $\hat \mathbb{Z}_n \coloneqq [\mathbb{Z}_n, U(1)] \simeq \mathbb{Z}_n$ is the [[Pontryagin dual]] group.

For this [[moduli infinity-stack|higher moduli stack]] of discrete [[higher gauge theory|higher gauge fields]] we have a local description by smooth data as before: it fits into the [[homotopy fiber sequence]]

$$
  \array{
    \mathbf{B}^2\mathbb{Z}_n &\to& \mathbf{B}^2 U(1)_{conn}
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{(-)^n}}
    \\
    \ast &\stackrel{0}{\to}& \mathbf{B}^2 U(1)_{conn}
  }
  \,.
$$

In analogy to the above, this now describes the space of field configurations consisting of a [[circle 2-group]]-[[connection on a 2-bundle|principal 2-connection]] $B$ and a trivialization of its $n$th power by a 1-form connection $A$, i.e. $d A - n B = 0$.

This one may also understand as the vanishing condition on the 2-form curvature of a [[connection on a 2-bundle|2-connection]]. This is the observation that drives the second part of the article.

### A. Abelian Higgs phases

* [[Higgs mechanism]]

### B. Abelian confining phases

* [[confinement]]

### C. Oblique confining phases

(...)

### D. Mixed phases

(...)

## IV. TQFTs for nonabelian gapped phases

(...)

### A. Nonabelian Higgs phases

(...)

### B. Nonabelian confining phases

(...)

#### 1. Monopole condensation

A notce concerning the discussion of [[monopoles]] on the bottom right of p. 5.

Let $G$ be a [[connected topological space|connected]] [[Lie group]]. A [[subgroup]]

$$
  \Gamma_0 \hookrightarrow \pi_1(G)
$$

of its [[fundamental group]] defines a [[covering]] $t \colon H \to G$, exhibited by the following [[pasting diagram]] of [[homotopy pullbacks]] of [[smooth ∞-groupoids]]:

$$
  \array{
    \pi_1(G)/\Gamma_0 &\to& H &\to& \Pi(H) &\to& \mathbf{B}\pi_1(H) \simeq \mathbf{B}\Gamma_0
    \\
    \downarrow && \downarrow^{\mathrlap{t}} && \downarrow^{\mathrlap{\Pi(t)}} && \downarrow
    \\
    \ast &\stackrel{e}{\to}& G &\stackrel{}{\to}& \Pi(G) &\stackrel{\tau_1}{\to}& \mathbf{B} \pi_1(G)
  }
$$

Here $(-) \to \Pi(-)$ is the [[unit of an adjunction|unit]] of the [[shape modality]]. In the given low degree case the above diagram can be checked explicitly by standard means, but it may be worthwhile to note that the whole situation follows fully generally as by the [Galois theory in cohesive ](cohesive+%28infinity,1%29-topos+--+structures#GaloisTheory) ([Schreiber, section 3.8.6](#SchreiberCohesive)).

#### 2. Nonabelian gerbes and nonabelian B-fields

* [[connection on a 2-bundle]]

* [[clutching construction]]

#### 3. A TQFT for a nonabelian confining phase

* [[BF-theory]]

### C. A TQFT for a nonabelian oblique confining phase

(...)

## References

The argument relating aspects of the low energy limit of [[spontaneous symmetry breaking]] in ordinary [[gauge theory]] by [[electric-magnetic duality]] of the [[Higgs boson]]-like field to [[BF-theory|BF]]-type [[higher gauge theory]] is in section 2 of

* [[Tom Banks]], [[Nathan Seiberg]], _Symmetries and Strings in Field Theory and Gravity_, Phys. Rev. D83:084019,2011 ([arXiv:1011.5120](http://arxiv.org/abs/1011.5120))
 {#BanksSeiberg08}

following discussion of [[D-brane]]/[[NS5-brane]] phenomena in

* [[Juan Maldacena]], [[Gregory Moore]], [[Nathan Seiberg]], _D-brane Charges in Five-brane backgrounds_,  	JHEP 0110:005,2001 ([arXiv:hep-th/0108152](http://arxiv.org/abs/hep-th/0108152))

(...)

The formulation of [[principal 2-connections]] via their [[higher parallel transport]] used in section IV is from

* [[Urs Schreiber]],  [[Konrad Waldorf]], _Connections on non-abelian gerbes and their holonomy_, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. ([TAC](http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html), [arXiv:0808.1923](http://arxiv.org/abs/0808.1923), web)

expanding on

* [[John Baez]], [[Urs Schreiber]], _Higher gauge theory_, in A. Davydov et al. _Categories in Algebra, Geometry and Mathematical Physics_, Contemp. Math. 431, AMS, Providence, Rhode Island, 2007, pp. 7-30 ([arXiv:math/0511710](http://arxiv.org/abs/math/0511710), [web](http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos+--+references/#BaezSchreiber))

Some of the above comments on the formulation in [[smooth ∞-groupoids]] refer to

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_
 {#SchreiberCohesive}

category: reference


topological recursion

#Contents#
* taable of contents
{:toc}

## Idea

Topological recursion ([Chekhov-Eynard-Orantin 06](#ChekhovEynardOrantin06),[Eynard-Orantin 07](#EynardOrantin07)) is a universal [[recursion]] formula that controls asymptotic expansion of many [[integrable systems]] such as [[matrix models]] or the [[Hitchin system]] ([Baraglia-Huang 17](#BaragliaHuang17)).

[[Maxim Kontsevich]] and [[Yan Soibelman]] reformulated and slightly generalised it, seeing it as a [[quantization]] of certain quadratic Lagrangians in the [[cotangent bundle]] of some [[vector space]] (reviewed in [Andersen-Borot-Chekhov-Orantin 17, section 2](#AndersenBorotChekhovOrantin17)).

There is some deep relation to the [[topological string]] [[B-model]] and [[Gromov-Witten invariants]] ([Bouchard-Klemm-Marino-Pasquetti 09](#BouchardKlemmMarinoPasquetti09)). It yields proofs of [[mirror symmetry]] in certain cases, valid at all [[genus of a surface|genera]] ([Eynard-Orantin 12](#EynardOrantin12), [Fang-Liu-Zong 13](#FangLiuZong13)).

A geometric refinement of topological recursion is known as _geometric recursion_ and developed in ([Andersen-Borot-Orantin 17](#AndersenBorotOrantin17)).

## Related concepts

* [[spectral curve]]

* [[matrix model]]

## References

### Original articles

The method was introduced in

* {#ChekhovEynardOrantin06} [[Leonid Chekhov]], [[Bertrand Eynard]], [[Nicolas Orantin]], _Free energy topological expansion for the 2-matrix model_, JHEP 0612:053,2006 ([arXiv:math-ph/0603003](https://arxiv.org/abs/math-ph/0603003))

* {#EynardOrantin07} [[Bertrand Eynard]], [[Nicolas Orantin]], _Invariants of algebraic curves and topological expansion_ ([arXiv:math-ph/0702045](https://arxiv.org/abs/math-ph/0702045))

### Review and exposition

Review and exposition includes

* [[Bertrand Eynard]], [[Nicolas Orantin]], _Topological recursion in random matrices and enumerative geometry_, J. Phys. A: Mathematical and Theoretical , 42(29), 2009 ([arXiv:0811.3531](https://arxiv.org/abs/0811.3531))

* [[Bertrand Eynard]], _A short overview of the "Topological recursion"_ ([arXiv:1412.3286](https://arxiv.org/abs/1412.3286))

* [[Gaëtan Borot]], _Lecture notes on topological recursion and geometry_, ([arXiv:1705.09986](https://arxiv.org/abs/1705.09986))

A textbook developing various topics related to topological recursion (in particular, discussing the links to enumeration of [[topological map|maps]]) is

* [[Bertrand Eynard]], _Counting Surfaces_, CRM Aisenstadt Chair lectures, Progress in Mathematical Physics 70, Birkhäuser, 2016

A series of lectures on topological recursion were delivered by Nicolas Orantin during the Institut Henri Poincaré's thematic trimester "[Combinatorics and interactions](http://combi17.math.cnrs.fr)", and are available online:

* [[Nicolas Orantin]], _Topological Recursion_, eight-part lecture series, 2017. [youtube](https://m.youtube.com/watch?v=qvBfeVLPZpE) (1/8)

The Kontsevich-Soibelman approach (and much more) is reviewed in

* {#AndersenBorotChekhovOrantin17} [[Jørgen Andersen]], [[Gaëtan Borot]], [[Leonid Chekhov]], [[Nicolas Orantin]], _The ABCD of topological recursion_ ([arXiv:1703.03307](https://arxiv.org/abs/1703.03307))

Discussion in the context of the [[Hitchin system]] includes

* {#BaragliaHuang17} [[David Baraglia]], Zhenxi Huang, _Special Kähler geometry of the Hitchin system and topological recursion_ ([arXiv:1707.04975](https://arxiv.org/abs/1707.04975))

reviewed in

* {#Baraglia18} [[David Baraglia]], _Topological recursion and Bergman tau functions for the Hitchin system_, talk at _[Tau Functions of Integrable Systems and Their Applications](http://www.birs.ca/events/2018/5-day-workshops/18w5025)_, Banff 2018 ([recording](http://www.birs.ca/events/2018/5-day-workshops/18w5025/videos/watch/201809060845-Baraglia.html))

See also

* {#Borot18} [[Gaëtan Borot]], _Quantum Airy structures and topological recursion_ talk at _[Geometry and Physics of Quantum Curves, Banff 2018](http://www.birs.ca/events/2018/5-day-workshops/18w5078)_ ([recording](http://www.birs.ca/events/2018/5-day-workshops/18w5078/videos/watch/201809100859-Borot.html))

### Relation to topological string and mirror symmetry

The relation to the [[topological string]] [[B-model]], [[Gromov-Witten invariants]] and [[mirror symmetry]] is due to

* {#BouchardKlemmMarinoPasquetti09} [[Vincent Bouchard]], [[Albrecht Klemm]], [[Marcos Marino]], [[Sara Pasquetti]], _Remodeling the B-model_, Commun.Math.Phys.287:117-178, 2009 ([arXiv:0709.1453](https://arxiv.org/abs/0709.1453))

* [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, _A matrix model for the topological string I: Deriving the matrix model_ ([arXiv:1003.1737](https://arxiv.org/abs/1003.1737))

* [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, _A matrix model for the topological string II: The spectral curve and mirror geometry_ ([arXiv:1007.2194](https://arxiv.org/abs/1007.2194))

* {#EynardOrantin12} [[Bertrand Eynard]], [[Nicolas Orantin]], _Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture_ ([arXiv:1205.1103](https://arxiv.org/abs/1205.1103))

* {#FangLiuZong13} Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, _All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds_ ([arXiv:1310.4818](https://arxiv.org/abs/1310.4818))

### Geometric refinement

* {#AndersenBorotOrantin17} [[Jørgen Andersen]], [[Gaëtan Borot]], [[Nicolas Orantin]], _Geometric recursion_, ([arXiv:1711.04729](https://arxiv.org/abs/1711.04729))

[[!redirects topological recursions]]


topological ring

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#Contents#
* table of contents
{:toc}

## Definition

A _topological ring_ is a [[ring]] [[internalization|internal to]] [[Top]], a [[ring object]] in [[Top]]:

a [[topological space]] $R$ equipped with the structure of a [[ring]] on its underlying [[set]], such that [[addition]] $+$ and [[multiplication]] $\cdot$ are [[continuous functions]].

Remarks:

* The structure of a topological ring $(R,+,\cdot)$ makes $R$ a [[uniform space]].

* It is automatic that negation in a topological unital ring is continuous, since it is the operation of multiplication with $-1$, and multiplication is continuous in each argument. Hence for $(R,+,\cdot)$ a topological ring, then $(R,+)$ is a [[topological group]].

* A _[[topological field]]_ is a topological ring $K$ whose underlying ring is in fact a [[field]] and such that reciprocation $(-)^{-1}: K \setminus \{0\} \to K \setminus \{0\}$ is continuous. This latter condition is the same as demanding that the subspace topology on $K \setminus \{0\}$ induced by the embedding $K \setminus \{0\} \hookrightarrow K$ coincide with the subspace topology induced by the embedding $K \setminus \{0\} \to K \times K: x \mapsto (x, x^{-1})$. More at [[topological field]].

+-- {: .num_remark}
###### Remark
In a topological ring, the closure of $\{0\}$ is an [[ideal]]. It follows that for a topological field $F$, either $0$ is a closed point (so that $F$ is $T_1$ and therefore [[Tychonoff space|completely regular Hausdorff]], by standard arguments in the theory of uniform spaces), or is a [[codiscrete space]].
=--

A [[topological algebra]] over a topological ring $R$ is a topological ring $S$ together with a topological ring map $R \to S$ that makes $S$ an $R$-algebra at the underlying set level (a topological [[associative algebra]]).

## Examples

* The [[real numbers]] form a topological field.

* Any [[pseudocompact ring]] such as the completed [[group ring]] of a [[profinite group]] is a topological ring.

*  For any prime $p$, the ring of [[p-adic integers]] is a topological ring.

* A [[Banach algebra]] is in particular a [[topological algebra]], hence a topological ring. Hence so is a [[C-star-algebra]].

## Related concepts

* [[topological monoid]]

* [[normed ring]]

* [[local field]]

* [[completion of a ring]]

* [[linear topological ring]]

* [[real homotopy theory]]

* [[topological vector space]]

## References

Lecture notes include

* [[Daniel Murfet]], _Topological rings_ ([pdf](http://therisingsea.org/notes/TopologicalRings.pdf))

See also

* [[eom]], _[Topological ring](https://www.encyclopediaofmath.org/index.php/Topological_ring)_

* [[Stacks Project]], _[Topological ring](http://stacks.math.columbia.edu/tag/07E8)_

* Wikipedia, _[Topological ring](https://en.wikipedia.org/wiki/Topological_ring)_

[[!redirects topological rings]]


topological site

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

# Contents
* table of contents
{:toc}

## Idea

A **topological site** is a [[site]] $(\mathcal{C}, J)$ that generalizes the site that underlies Giraud's [[gros topos]] of topological spaces.  The aim is to get in $Sh(\mathcal{C}, J)$ a well-behaved category of topological spaces.

## Definition

A site $(\mathcal{C}, J)$ is called _topological_ if $\mathcal{C}$ is a category of topological spaces and continuous maps, which is closed under open inclusions i.e. if $X\in\mathcal{C}$ and $U\in\mathcal{O}(X)$ then $U\hookrightarrow X$ is a morphism in $\mathcal{C}$ and $J$ is the [[open cover|open cover topology]] i.e. $J$ is generated by families $\{ U_i\hookrightarrow X\}$ where the $U_i$ are open and jointly cover $X$.

A topos $\mathcal{E}$ is called _topological_ when it is equivalent to a topos
$Sh(\mathcal{C}, J)$ with $(\mathcal{C}, J)$ a topological site.

### Remark
This definition follows Moerdijk-Reyes ([1984](#MR85)) but some variation is possible here e.g. one could additionally demand that $\mathcal{C}$ is closed under finite limits and contains the real numbers $\mathbb{R}$ (cf. Mac Lane-Moerdijk 1994). Instead of topological spaces one could use locales - this is  pursued by Fourman (1983,1984,2013). Compare also the approach taken in Montañez (2013).

## Examples

 * Let $\mathcal{C}$ have objects the locally compact subspaces of some $\mathbb{R}^n$, $n\in\mathbb{N}$ varying, and morphisms the $C^\infty$-maps. The topos $Sh(\mathcal{C}, J)$ is called the _euclidean topos_ in Moerdijk-Reyes ([1984](#MR85)) where its properties and in particular its relations with the [[Dubuc topos]] are explored.

## Properties

+-- {: .num_prop #T1-mod}
###### Proposition

Let $A$ be a $T_U$-propositional theory, and let $A_0$ be the corresponding locale in $Set$ (so $Sh(A_0)$ classifies $A$-models in Grothendieck toposes). Let $Sh(\mathcal{C}, J)$ be a topological topos. Then the object of $A$-models in $Sh(\mathcal{C}, J)$ is given by the sheaf $Cts(-, A_0) : \mathcal{C}^{op}\to Set$ which assigns to a space $X$ in $\mathcal{C}$ the set of all continuous maps from $X$ to $A_0$.
=--

This occurs as cor.1.5 in Moerdijk-Reyes ([1984](#MR85)). Here $T_U$ is the pendant for locales of the $T_1$-separation property:  a locale $X$ is $T_U$ if for all locale maps $f,g:Y\to X$ the relation $f\leq g$ implies $f=g$ (cf. the [[Elephant]] II, p.501).

+-- {: .num_prop #Fanthm}
###### Proposition
Let $Sh(\mathcal{C}, J)$ be a topological topos. Then the [[Fan theorem]] holds in $Sh(\mathcal{C}, J)$. If furthermore all $X\in\mathcal{C}$ are [[locally compact space|locally compact]] then [[Bar induction]] holds as well.
=--

This is contained in props.1.7-8 in Moerdijk-Reyes ([1984](#MR85)).

## Related entries

* [[Johnstone's topological topos]]

* [[open cover]]

* [[continuous truth]]

* [[gros topos]]

## References

* {#SGA4} [[Michael Artin|M.Artin]], [[Alexander Grothendieck|A.Grothendieck]], [[J. L. Verdier]] (eds.), _Théorie des Topos et Cohomologie Etale des Schémas - SGA 4_ , LNM **269** Springer Heidelberg 1972.

* [[Michael Fourman]], _T$_1$ Spaces over Topological Sites_ , JPAA **27** (1983) pp.223-224. [link](https://core.ac.uk/download/pdf/82251219.pdf)

* [[Michael Fourman]], _Continuous Truth I: non-constructive objects_ , pp.161-180 in Lolli, Longo, Marcja (eds.), _Proc. Logic Colloquium, Florence 1982_ , Elsevier Amsterdam 1984. ([draft](http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman84-continuous-truth-I.pdf))

* [[Michael Fourman]], _Continuous Truth II: reflections_ , pp,153-167 in LNCS **8071** Springer Heidelberg 2013. ([draft](http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman2013-continuous-truth-II.pdf))

* Gerrit Van Der Hoeven, [[Ieke Moerdijk]], _Sheaf models for choice sequences_ , APAL **27** (1984) pp.63-107.

* {#Johnstone79}[[Peter Johnstone]], _On a topological topos_ , Proc. London Math. Soc. (3) **38** (1979) pp.237–271.

* {#MM94} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (pp.113, 325ff, 416)

* {#MR84}[[Ieke Moerdijk]], [[Gonzalo E. Reyes]], _Smooth Spaces versus Continuous Spaces in Models of Synthetic Differential Geometry_ , JPAA **32** (1984) pp.143-176. [link](https://www.sciencedirect.com/science/article/pii/0022404984900495)

* R. Montañez, _Topoi generated by topological spaces_ , Talk CT15 Aveiro 2015. ([pdf-slides](http://sweet.ua.pt/dirk/ct2015/slides/Montanez.pdf))

* Example A2.1.11 in the [[Elephant]]


topological space

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

# Topological spaces
* table of contents
{: toc}

## Idea
 {#Idea}

The notion of _topological space_ aims to axiomatize the idea of a _[[space]]_ as a collection of [[points]] that hang together ("[[cohesive topos|cohere]]") in a _[[continuous function|continuous]]_ way. Roughly speaking, a *topology* on a [[set]] "of points" prescribes which [[subsets]] are to be considered "[[neighborhoods]]" of the points they contain. Various conditions or [[axioms]] must be satisfied in order for such neighborhood systems to form a topology, but one of the most important is that for any two neighborhoods of a point, their [[intersection]] must also be a neighborhood of that point.

Many notions of [[spaces]] used in [[mathematics]] have [[underlying topological spaces]], such as: [[manifolds]], [[schemes]], [[probability spaces]], etc.

The concept of a topology, gradually refined over the latter half of the 19th century and the first two decades of the 20th, was developed to capture what it means abstractly for a mapping between sets of points to be "[[continuous function|continuous]]". Intuitively, the idea of *bending*, *twisting* or *crumpling* a continuous body applies to continuous mappings, because they preserve neighborhood relations (in a suitable sense), but *tearing*, for instance, does not.

For example, the surface of a [[torus]] or doughnut is topologically equivalent to the surface of a mug: the surface of the mug can be deformed _continuously_ into the surface of a torus. Abstractly speaking: the continuous [[cohesion]] among the collections of points of the two surfaces is the same. Similarly, a [[circle]] and a [[square]] are considered equivalent from the standpoint of their topologies.

Some one-dimensional shapes with different topologies: the Mercedes-Benz symbol, a line, a circle, a complete graph with 5 nodes, the skeleton of a cube, and an asterisk (or, if you'll permit the one-dimensional approximation, a starfish). On the other hand, a circle has the same topology as a line segment with a wormhole at its finish which teleports you to its start; or more prosaically: The [[circle]] is [[homeomorphism|homeomorphic]] to the [[closed interval]] with *endpoints identified*.

There is a generalization of the notion of topological spaces to that of _[[locales]]_, which consists of dropping the assumption that all [[neighbourhoods]] are explicitly or even necessarily supported by points. For this reason, the theory of locales is sometimes called "pointless topology". In this form, the definition turns out to be quite fundamental and can be naturally motivated from just pure [[logic]] -- as the formal dual of _[[frames]]_ --  as well as, and [[duality|dually]], from [[category theory]] in its variant as [[topos theory]] -- by the notion of _[[(0,1)-toposes]]_.

Topological spaces are the objects studied in _[[topology]]_. But types of topological spaces exist in such great and wild profusion that in practice it is often more convenient to replace strict topological equivalence by a notion of [[weak equivalence]], namely of [[weak homotopy equivalence]]. From this point of view, topological spaces  support also _[[homotopy theory]]_.

Topological spaces equipped with extra [[property|properties]] and [[structure]] form the fundament of much of [[geometry]]. For instance a topological space locally isomorphic to a [[Cartesian space]] is a _[[manifold]]_. A topological space equipped with a notion of [[smooth functions]] into it is a [[diffeological space]]. The intersection of these two notions is that of a _[[smooth manifold]]_ on which [[differential geometry]] is based. And so on.

## Definitions
 {#Definitions}

We present first the

* [standard definition](#StandardDefinition)

and then a list of different

* [equivalent definitions](#AlternateDefinitions).

Finally we mention genuine

* [variants of the notion](#Variants).

### Standard definition
 {#StandardDefinition}

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

A **topological space** is a [[set]] $X$ equipped with a set of [[subsets]]
$U \subset X$, called the **[[open sets]]**, which are closed under

1. finite [[intersections]]
1. arbitrary [[unions]].

=--

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

The word 'topology' sometimes means the [[topology|study of topological spaces]] but here it means the collection of open sets in a topological space. In particular, if someone says 'Let $T$ be a topology on $X$', then they mean 'Let $X$ be equipped with the structure of a topological space, and let $T$ be the collection of open sets in this space'.

=--

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

Since $X$ itself is the intersection of zero subsets, it is open, and since the [[empty set]] $\emptyset$ is the union of zero subsets, it is also open. Moreover, every open subset $U$ of $X$ contains the empty set and is contained in $X$

$$
  \emptyset \subset U \subset X
  \,,
$$

so that the topology of $X$ is determined by a [[poset of open subsets]] $Op(X)$ with [[bottom]] element $\bot = \emptyset$ and [[top]] element $\top = X$.

Since by definition the elements in this [[poset]] are closed under finite [[meets]] (intersection) and arbitrary [[joins]] (unions), this poset of open subsets defining a topology is a _[[frame]]_, the _[[frame of opens]]_ of $X$.

=--

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

A [[homomorphism]] between topological spaces $f : X \to Y$ is a **[[continuous function]]**:

a [[function]] $f:X\to Y$ of the underlying [[sets]] such that the [[preimage]] of every open set of $Y$ is an open set of $X$.

=--

Topological spaces with continuous maps between them form a [[category]], usually denoted _[[Top]]_.

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

The definition of [[continuous function]] $f : X \to Y$ is such that it induces a homomorphism of the corresponding [[frames of opens]] the other way around

$$
  Op(X) \leftarrow Op(Y) : f^{-1}
  \,.
$$

And this is not just a morphism of [[posets]] but even of [[frames]].
For more on this see at _[[locale]]_.

=--

### Alternate equivalent definitions
 {#AlternateDefinitions}

There are many equivalent ways to define a topological space.  A non-exhaustive list follows:

* A set $X$ with a [[frame]] of open sets (the standard definition, given above), called a *topology* on $X$.

* A set $X$ with a co-frame of [[closed sets]] (the complements of the open sets), satisfying dual axioms: closure under finite unions and arbitrary intersections. This is sometimes called a co-topology on $X$.

* A pair $(X, int)$, where $int\colon P(X) \to P(X)$ is a [[lex functor|left exact]] [[comonad]] on the [[power set]] of $X$ (the "[[interior]] operator"). In more nuts-and-bolts terms, this means for all subsets $A, B$ of $X$ we have
$$A \subseteq B \Rightarrow int(A) \subseteq int(B), \;\;\; int(A) \subseteq A, \;\;\; int(A) \subseteq int(int(A)), \;\;\; int(A \cap B) = int(A) \cap int(B), \;\;\; int(X) = X.$$
The open sets are exactly the fixed points of $int$. The first three of these conditions say $int$ is a [[closure operator|coclosure operator]].

* A pair $(X, cl)$ where $cl$ is a [[rex functor|right exact]] [[Moore closure]] operator satisfying axioms dual to those of $int$.  The closed sets are the fixed points of $cl$. Such an operator is sometimes called a *Kuratowski closure operator* (compare Kuratowski's closure-complement problem at [[closed subspace]]).

* A set $X$ together with, for each $x \in X$, a [[filter]] $N_x$ on $X$, i.e., a collection of inhabited subsets of $X$ closed under finite intersections and also upward-closed ($U \in N_x$ and $U \subseteq V$ together imply $V \subseteq N_x$). If $U \in N_x$, we call $U$ a *neighborhood* of $x$. The remaining conditions on these neighborhood systems are that $x \in U$ for every $U \in N_x$, and that for every $U \in N_x$, there exists $V \in N_x$ such that $V \subseteq U$ and $V$ is a neighborhood of each point it contains. In this formulation, a subset $U \subseteq X$ is *open* if it is a neighborhood of every point it contains.

The next two definitions of topological space are at a higher level of abstraction, but the underlying idea that connects them with the neighborhood system formulation is that we say a filter $F$ on $X$ *converges* to a point $x \in X$ if $N_x \subseteq F$. The point then is to characterize properties of convergence abstractly.

* A [[relational beta-module]]; that is, a [[lax algebra]] of the [[monad]] $\beta$ of [[ultrafilters]] on the [[(1,2)-category]] [[Rel]] of sets and [[binary relations]].  More explicitly, this means a set $X$ together with a relation called "[[convergence]]" between ultrafilters and points satisfying certain axioms.  This exhibits it as a special sort of [[generalized multicategory]], and also as a special sort of [[pseudotopological space]]. However, the equivalence of this definition to the traditional definition of a topological space requires the [[ultrafilter principle]] to be true.

* A set with a [[convergence relation]] between [[nets]] or [[filters]] (not just ultrafilters) and points, or even between transfinite [[sequences]] and points, satisfying appropriate axioms.

The following are not definitions, but they provide alternative ways to present a topological space.

* A [[topological base|(topological) base]] on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$ whose union is all of $X$, and such that whenever $B, C \in \mathcal{B}$ and $x \in B \cap C$, there exists $D \in \mathcal{B}$ such that $D \subseteq B \cap C$ and $x \in D$.

If $\mathcal{B}$ is a base on $X$, then it is easily shown that the collection of all unions of subcollections of $\mathcal{B}$ is a topology on $X$.

* A set $X$ with any collection of subsets whatsoever, to be thought of as a [[subbase]] for a topology.

From the fact that the intersection of any collection of topologies is also a topology, there is a smallest topology that contains a given subbase $\mathcal{S}$. It consists of all possible unions of all possible finite intersections of members of $\mathcal{S}$. This is called the *topology generated by* the subbase.

### Variations
 {#Variants}

Historically, the notion of topological space (see the historical references given [there](topology#ReferencesHistoricalOrigins)) involving [[neighbourhoods]] was first developed by [[Felix Hausdorff]] in 1914 in his seminal text on [[set theory]] and [[topology]], *Fundamentals of Set Theory* (*[[Grundzüge der Mengenlehre]]*). Hausdorff's definition originally contained the $T_2$-[[separation axiom]] (now known as the definition of *[[Hausdorff spaces]]*). This axiom was in effect removed by [[Kazimierz Kuratowski]] in 1922, who defined general topological spaces in terms of [[closure operators]] that preserve finite [[unions]]. The usual [[open set]] formulation was widely popularized by Bourbaki in their 1940 treatise (without identifying a single author behind this notion).

However, in more modern treatments that emphasize [[category theory|category theoretic]] methods, particularly to address needs of [[homotopy theory]], it becomes important to consider not just the category [[Top]] of all topological spaces, but [[convenient categories of topological spaces]] that are better behaved, especially with regard to function spaces and [[cartesian closed category|cartesian closure]]. Thus many texts work with [[nice topological spaces]] (such as [[sequential topological spaces]]) and/or a [[nice category of spaces|nice-]] or [[convenient category of topological spaces]]  (such as [[compactly generated spaces]]), or indeed to directly use a model of $\infty$-[[infinity-groupoid|groupoids]] (such as [[simplicial sets]]).

On the other hand, when doing [[topos theory]] or working in [[constructive mathematics]], it is often more appropriate to use [[locales]] than topological spaces.

Some applications to [[analysis]] require more general [[convergence spaces]] or other generalisations.

In [[dependent type theory]], one could also have a [[topological space]] be a general [[type]] instead of an [[h-set]]. For most kinds of topological spaces in dependent type theory, the [[T0-space|$T_0$]]-[[separation axiom]] forces the type to be an h-set.

### In dependent type theory
 {#DependentTypeTheory}

In [[dependent type theory]], given a type $X$, the type of all [[subtypes]] of $X$, the [[powerset]] of $X$, is defined as the [[function type]]
$$\mathcal{P}(X) \coloneqq X \to \Omega$$
where $\Omega$ is the [[type of all propositions]] with the type reflector type family $P:\Omega \vdash \mathrm{El}_\Omega(P) \; \mathrm{type}$. In the [[inference rules]] for the [[type of all propositions]], one has an operation $(-)_\Omega$ which takes a proposition $P$ and turns it into an element of the [[type of all propositions]] $P_\Omega:\Omega$.

The **local membership relation** $x \in_A B$ between elements $x:A$ and material subtypes $B:\mathcal{P}(A)$ is defined as
$$x \in_A B \coloneqq \mathrm{El}_\Omega(B(x))$$

Arbitrary unions and intersections of subtypes could be defined in [[dependent type theory]]:

* Given a type $A$ and a type $I$, there is an function $\bigcap:(I \to \mathcal{P}(A)) \to \mathcal{P}(A)$ called the $I$-indexed [[intersection]], such that for all families of subtypes $B:I \to \mathcal{P}(A)$, $\bigcap_{i:I} B(i)$ is defined as
$$\left(\bigcap_{i:I} B(i)\right)(x) \coloneqq (\forall i:I.x \in_A B(i))_\Omega$$
for all $x:A$, where
$$\forall x:A.B(x) \coloneqq \left[\prod_{x:A} B(x)\right]$$
is the [[universal quantification]] of a [[type family]] and $[T]$ is the propositional truncation of $T$.

* Given a type $A$ and a type $I$, there is an function $\bigcup:(I \to \mathcal{P}(A)) \to \mathcal{P}(A)$ called the $I$-indexed [[union]], such that for all families of subtypes $B:I \to \mathcal{P}(A)$, $\bigcup_{i:I} B(i)$ is defined as
$$\left(\bigcup_{i:I} B(i)\right)(x) \coloneqq (\exists i:I.x \in_A B(i))_\Omega$$
for all $x:A$, where
$$\exists x:A.B(x) \coloneqq \left[\sum_{x:A} B(x)\right]$$
is the [[existential quantification]] of a [[type family]] and $[T]$ is the propositional truncation of $T$.

In [[dependent type theory]], however, one cannot quantify over arbitrary types, since one could only quantify over elements of a type. Instead, one has to use a [[Tarski universe]] $(U, \mathrm{El}_U)$, where the elements of $U$ represent $U$-[[small types]], and then quantify over $U$. In the case of topological spaces, instead of the [[open sets]] being closed under arbitrary unions, the open sets are only closed under all $U$-small unions $\bigcup_{i:\mathrm{El}_U(I)} B(i)$ for $I:U$.

\begin{definition}
Given a [[Tarski universe]] $(U, \mathrm{El}_U)$, a **[[topological space]]** is a type $X$ with a **$U$-small topology**, a type of subtypes $O(X)$ with canonical [[embedding]] $i_O:O(X) \hookrightarrow \mathcal{P}(X)$, called the open sets of $X$, which are closed under [[finite type|finite]] [[intersections]] and $U$-[[small type|small]] unions.
\end{definition}

Given a topological space $(X, O(X))$, we define the membership [[relation]] between elements $x:X$ and open sets $V:O(X)$:

$$x:X, V:O(X) \vdash x \in V \; \mathrm{type}$$

by

$$x \in V \coloneqq \mathrm{El}_\Omega((i_O(V))(x))$$

By definition of the type of all propositions and its type reflector, $x \in V$ is always a [[h-proposition]] for all $x:X$ and $V:O(X)$.

## Examples

### Special cases

* [[finite topological space]]

* [[discrete topological space]]

* [[codiscrete topological space]]

* [[cofinite topology]]

* [[order topology]] [[specialization topology]], [[Scott topology]]

* [[compact topological space]]

* [[paracompact topological space]], [[sigma-compact topological space]], [[Lindelöf topological space]]

* [[spectral topological space]]

* [[CW-complex]]

* [[manifold]]

...

### Specific examples

* [[empty space]]

* [[point space]]

* [[Sierpinski space]]

* [[simplex]]

* [[circle]]

* [[Cantor space]]

* [[long line]], [[line with two origins]]

* [[K-topology]]

* [[projective space]], [[real projective space]], [[complex projective space]]

* [[classifying space]]

...

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

The [[Cartesian space]] $\mathbb{R}^n$ with its standard notion of open subsets [[topological base|generated from]]: [[unions]] of [[open balls]] $D^n \subset \mathbb{R}^n$.

=--

## Related concepts

* [[separation axioms]]

* [[finer topology]], [[coarser topology]]

* [[first countable topological space]], [[second countable topological space]], [[separable topological space]], [[Hausdorff topological space]], [[topological manifold]]

* [[sigma-topological space|$\sigma$-topological space]]

* [[topological property]]

* [[locale]], [[topos]]

* [[ionad]]

* [[proximity space]], [[uniform space]], [[syntopogenous space]]

* [[quotient topology]], [[quotient topological space]]

* [[subspace topology]], [[topological subspace]]

* [[product topological space]]

* [[effective topological space]], [[equilogical space]]

* [[topological G-space]]

* [[filter space]]

* [[connected topological spaces]], [[simply connected topological space]]

* [[nilpotent topological space]]

* [[fractal]]

* [[sheaf on a topological space]]

[[!include topology -- references]]

[[!redirects topological space]]
[[!redirects topological spaces]]
[[!redirects topological structure]]
[[!redirects topological structures]]

[[!redirects small topology]]
[[!redirects small topologies]]

[[!redirects small topological structure]]
[[!redirects small topological structures]]


topological stack

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

#Contents#
* table of contents
{:toc}

## Idea

A **topological stack** is a [[geometric stack]] on the [[site]] [[Top]]: a [[topological groupoid]] regarded as presenting an object in the [[(2,1)-sheaf]] [[(2,1)-topos]] $Sh_{(2,1)}(Top)$.

## Definition

Let [[Top]] be the [[category]] of [[compactly generated spaces]] and [[continuous function]]. When equipped with a [[Grothendieck topology]] given by usual [[open cover]]s this becomes a [[subcanonical coverage|subcanonical]] [[large site]].

Consider the [[(2,1)-topos]] [[(2,1)-sheaves]]=[[stack]]s

$$
  TopStack := Sh_{(2,1)}(Top)
$$

of [[groupoid]]s on $\mathrm{Top}$; by [[Yoneda embedding|Yoneda]] $\mathrm{Top}$ is a full [[sub-(∞,1)-category|sub-(2,1)-category]]

$$
  Top \hookrightarrow TopStack
  \,.
$$

By analogy with the case of [[algebraic stack]]s one says that a morphism of 1-stacks $f:X\to Y$ in $\mathrm{TopStack}$ is a **[[representable morphism of stacks]]** if for any morphism of 1-stacks $T\to Y$ from a (stack associated to a) topological space $T$ to $Y$ the pullback $T\times_Y X$ is (2-isomorphic to the stack associated to) a [[topological space]].

Let $P$ be a property of a map of topological spaces. $P$ is said to be __invariant under change of base__ if for all $f: Y \to X$ with property $P$, if $g:Z \to X$ is any continuous map, the induced map $Z \times_X Y \to Z$ also has property $P$. $P$ is said to be __invariant under restriction__ if this holds whenever $g$ is an embedding. A property $P$ which is invariant under restriction is said to be __local on the target__ if any $f: Y \to X$ for which there exists an open cover $\left(U_\alpha \to X\right)$ such that the induced map $\coprod_{\alpha}  {U_\alpha  }  \times_{X} Y \to \coprod_\alpha  {U_\alpha  } $ has property $P$, must also have property $P$.

Examples of such properties are being an [[open map]], [[covering]] map, [[closed map]], [[local homeomorphism]] etc.

A [[representable morphism of stacks|representable map]] $f:\X \to \Y$ of stacks is said to have property $P$ if for any map $T \to \Y$ from a topological space, the induced map $T \times_{\Y} \X \to T$ has property $P$

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

A 1-[[stack]] $X$ of [[groupoid]]s over $\mathrm{Top}$ having a representable epimorphism from a [[topological space]] $X_0 \to X$ is a __topological stack__. Such an representable epimorphism is called an [[atlas]] (or [[chart]]).

=--

This is what is called  __pretopological stack__  in [Noohi](#Noohi) . The terminology __topological stack__  is reserved for those stacks whose [[atlas]] can be chosen to belong to a class of "local fibrations"; there are axioms which the class of local fibrations have to satisfy; there are several natural choices of this class which modify the variant of topological stacks considered.

Any map from a topological space $S$ to a topological stack $X$ is representable (i.e. diagonal $X\to X\times X$ is always representable). For a topological stack $Y$, if $P$ is invariant under restriction and local on the target, a representable morphism $f : X \to Y$ of 1-stacks has this property if there exists an atlas $T\to Y$ such that the induced map $X\times_Y T\to T$ has property $P$.

If $X_0 \to X$ is an atlas for a topological stack, then $X_0 \times_{X} X_0 \rightrightarrows X_0$ is a [[topological groupoid]], $\mathbf{X}$. The [[stackification]] of the presheaf of groupoids $T \mapsto Hom((T^{id},\mathbf{X}))$ is (2-iso to) $X$ (where $T^{id}$ is $T$ considered as a [[topological groupoid]] with only identity arrows).

Conversely, given a topological groupoid $G$, we can consider the stackification of $Hom(blank,G):= \left[ G\right]$. By direct inspection, one sees that $\left[ G\right](T)$ is the groupoid of principal G-bundles over $T$, $Bun_G(T)$. The canonical map $(G_0)^id \to G$ yields a map $a:G_0 \to \left[ G\right]$. If $p:T \to \left[ G\right]$ is any map from a space, then $T \times_{\left[ G\right]} G_0$ is the total space of the principal $G$-bundle over $T$ which $p$ corresponds to via Yoneda. If under the correspondence between [[principal bundle]]s and generalized homomorphims $p$ corresponds to a map $T^{id} \to G$, then $p$ factors through the map $a:G_0 \to \left[ G\right]$. If $p$ instead corresponds to a map $T_U \to G$ where $U \to T$ is a cover, then $p$ factors through $a$ locally, hence, $a$ is an epimorphism. Therefore an alternative definition of a topological stack is:

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

A 1-[[stack]] $X$ of [[groupoid]]s over $\mathrm{Top}$ is a __topological stack__ if it is [[equivalence in an (infinity,1)-category|equivalent]] to the stack  $G Bund$ of groupoid-[[principal bundle]] for some [[topological groupoid]] $G$.

=--

By the [[Yoneda lemma]], $Hom(T,Bun_G) \cong Bun_G(T)$ for all $T$. Moreover, if $H$ is another topological groupoid, $Hom(Bun_H,Bun_G) \cong Bun_G(H)$, where $Bun_G(H)$ is the groupoid of principal $G$-bundles over $H$. In fact, one can use this to show that the 2-category of topological stacks is equivalent to the [[bicategory]] of topological groupoids and principal bundles. One may also show that topological stacks are equivalent to the [[bicategory of fractions]] of topological groupoids with respect to formally inverting Morita-equivalences.

## Related concept

* [[geometric stack]]

  * [[algebraic stack]]

  * **topological stack**, [[topological groupoid]], [[topological infinity-groupoid]]

  * [[differentiable stack]], [[Lie groupoid]]

* [[geometric ∞-stack]]

* [[cohesive (∞,1)-topos|cohesive ∞-groupoid]]

  * [[discrete ∞-groupoid]]

  * [[Euclidean-topological ∞-groupoid]]

  * [[smooth ∞-groupoid]]

  * [[synthetic differential ∞-groupoid]]

## References

Basics

* {#Noohi}[[Behrang Noohi]], *Foundations of Topological Stacks I* ([arXiv:math.AG/0503247](http://arxiv.org/abs/math.AG/0503247))

* {#Carchedi11} [[David Carchedi]], _Categorical Properties of Topological and Diffentiable Stacks_, PhD thesis, Universiteit Utrecht, 2011 ([dspace:1874/208971](https://dspace.library.uu.nl/handle/1874/208971), [pdf](http://math.gmu.edu/~dcarched/Thesis_David_Carchedi.pdf). [[CarchediDifferentiableStacks.pdf:file]])

* [[David Carchedi]], *Compactly Generated Stacks: A Cartesian Closed Theory of Topological Stacks*, Advances in Mathematics **229** 6 (2012) 3339-3397 [[doi:10.1016/j.aim.2012.02.006](http://dx.doi.org/10.1016/j.aim.2012.02.006), [arXiv:0907.3925](http://arxiv.org/abs/0907.3925)]

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

On [[geometric realization of topological stacks]]:

* {#Noohi}[[Behrang Noohi]], *Homotopy types of topological stacks*, Advances in Mathematics Volume 230, Issues 4–6, July–August 2012, Pages 2014-2047 ([arXiv:0808.3799](https://arxiv.org/abs/0808.3799))

* [[Johannes Ebert]], *The homotopy type of a topological stack* ([arXiv:0901.3295](https://arxiv.org/abs/0901.3295))

On [[mapping stacks]] of [[topological stacks]]:

* {#Noohi08} [[Behrang Noohi]], _Mapping stacks of topological stacks_ ([arXiv:0809.2373](http://arxiv.org/abs/0809.2373))

and in the special case of [[free loop stacks]]:

* [[Kai Behrend]], [[Gregory Ginot]], [[Behrang Noohi]], [[Ping Xu]], _String topology for stacks_ [[arXiv:0712.3857](https://arxiv.org/abs/0712.3857)]

[[!redirects topological stack]]
[[!redirects topological stacks]]


topological string

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#Contents#
* table of contents
{:toc}

## Idea

In the broad sense of the word, a _topological string_ is a 2-dimensional [[TQFT]]. In its refined form this goes by the name _[[TCFT]]_. The "C" standing for _[[conformal field theory]]_ points to what historically was the main inspiration and still is the default meaning of _topological strings_: the _[[A-model]]_ and _[[B-model]]_ 2d TQFTs, which are each obtained by a "topological twisting" of [[2d SCFTs]].

Accordingly, much of "physical" [[string theory]] has its analogs in _topological string theory_. Notably the toplogical analogs of the [[D-branes]] of the physical string --  the [[A-branes]] and [[B-branes]] -- have been studied in great (mathematical) detail, giving rise to [[homological mirror symmetry]] and, eventually, the notion of [[TCFT]] itself.

Also the perspective of [[string theory]] as the dimensional reduction of a conjectured [[UV-completion]] of  [[11-dimensional supergravity]] -- "M-theory" -- has its analog for topological strings, going, accordingly, by the term _[[topological M-theory]]_.

## Classification

[[!include 2d TQFT -- table]]

## Related concepts

* [[TCFT]]

  * [[A-model]]

  * [[B-model]]

* [[CP^N sigma-model]]

* [[quantum cohomology ring]]

* [[topological M-theory]], [[topological membrane]]

## References
 {#References}

### General

Review:

* {#NeitzkeVafa04} [[Andrew Neitzke]], [[Cumrun Vafa]], _Topological strings and their physical applications_, talk at [Simons Workshop in Mathematics and Physics 2004](http://insti.physics.sunysb.edu/conf/simonsworkII/) ([hep-th/0410178](http://xxx.lanl.gov/abs/hep-th/0410178))

* {#AntoniadisHohenegger07} I. Antoniadis, S. Hohenegger, _Topological Amplitudes and Physical Couplings in String Theory_, Nucl.Phys.Proc.Suppl.171:176-195,2007 ([arXiv:hep-th/0701290](https://arxiv.org/abs/hep-th/0701290))

* Marcel Vonk, _A mini-course on topological strings_ ([arXiv:hep-th/0504147](http://arxiv.org/abs/hep-th/0504147))

* [[Andrew Neitzke]], _Nonperturbative topological strings_, 2005 ([pdf](http://people.maths.ox.ac.uk/lmason/Tws/Neitzke.pdf))

On the relation to [[integrable systems]]:

* [[Mina Aganagic]], [[Robbert Dijkgraaf]], [[Albrecht Klemm]], [[Marcos Marino]], [[Cumrun Vafa]], *Topological Strings and Integrable Hierarchies*, Commun. Math. Phys. **261** (2006) 451-516 [[arXiv:hep-th/0312085](https://arxiv.org/abs/hep-th/0312085), [doi:10.1007/s00220-005-1448-9](https://doi.org/10.1007/s00220-005-1448-9)]

* [[Mina Aganagic]], [[Miranda Cheng]], [[Robbert Dijkgraaf]], Daniel Krefl, [[Cumrun Vafa]], *Quantum Geometry of Refined Topological Strings*, J. High Energ. Phys. **2012** 19 (2012) [[arXiv:1105.0630](https://arxiv.org/abs/1105.0630), doi:10.1007/JHEP11(2012)019]

On the relation to [[topological M-theory]]/the [[topological membrane]]:

* [[Jan de Boer]], [[Paul de Medeiros]], Sheer El-Showk, Annamaria Sinkovics, _Open $G_2$ Strings_ ([arXiv:hep-th/0611080](http://arxiv.org/abs/hep-th/0611080))

On [[non-perturbative effects]] and [[Stokes phenomena]] in [[topological string]] [[string scattering amplitude|amplitudes]]:

* Murad Alim, Arpan Saha, [[Jörg Teschner]], Iván Tulli, *Mathematical structures of non-perturbative topological string theory: from GW to DT invariants* ([arXiv:2109.06878](https://arxiv.org/abs/2109.06878))

See also:

* Wikipedia [topological string theory](http://en.wikipedia.org/wiki/Topological_string_theory)

* Lotte Hollands, _Topological Strings and Quantum Curves_ ([arXiv:0911.3413](https://arxiv.org/abs/0911.3413))

* Mina Aganagić, [[Cumrun Vafa]], _Large $N$ duality, mirror symmetry, and a Q-deformed A-polynomial for knots_, [arxiv/1204.4709](http://arxiv.org/abs/1204.4709)

* Min-xin Huang, _Recent Developments in Topological String Theory_ ([arXiv:1812.03636](https://arxiv.org/abs/1812.03636))

### Relation to black hole microstate counting

Disucssion of [[black holes in string theory]] via the topological string' [[Gopakumar-Vafa invariants]]:

* [[Rajesh Gopakumar]], [[Cumrun Vafa]], _M-Theory and Topological Strings--I_ ([arXiv:hep-th/9809187](https://arxiv.org/abs/hep-th/9809187))

* [[Rajesh Gopakumar]], [[Cumrun Vafa]], _M-Theory and Topological Strings--II_ ([arXiv:hep-th/9812127](https://arxiv.org/abs/hep-th/9812127))

### Relation to physical string amplitudes

The following includes discussion of [[superstring]] [[string scattering amplitudes]] in terms of topological string scattering amplitudes (for review see [NeitzkeVafa04, section 6](#NeitzkeVafa04) and [Antoniadis-Hohenegger 07](#AntoniadisHohenegger07):

* {#BCOV93} [[Michael Bershadsky]], [[Sergio Cecotti]], [[Hirosi Ooguri]], [[Cumrun Vafa]], *Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes*, Commun. Math. Phys. **165** (1994) 311-428 [[arXiv:hep-th/9309140](http://arxiv.org/abs/hep-th/9309140), [doi:10.1007/BF02099774](https://doi.org/10.1007/BF02099774)]

* I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor, _Topological Amplitudes in String Theory_, Nucl.Phys. B413 (1994) 162-184 ([arXiv:hep-th/9307158](http://xxx.lanl.gov/abs/hep-th/9307158))

* K.S. Narain, N. Piazzalunga, A. Tanzini, _Real topological string amplitudes_, JHEP (2017) 2017:80 ([arXiv:1612.07544](https://arxiv.org/abs/1612.07544))

### Computation via topological recursion

Computation via [[topological recursion]] in [[matrix models]] and all-[[genus of  a surface|genus]] proofs of [[mirror symmetry]] is due to

* {#BouchardKlemmMarinoPasquetti09} [[Vincent Bouchard]], [[Albrecht Klemm]], [[Marcos Marino]], [[Sara Pasquetti]], _Remodeling the B-model_, Commun.Math.Phys.287:117-178, 2009 ([arXiv:0709.1453](https://arxiv.org/abs/0709.1453))

* [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, _A matrix model for the topological string I: Deriving the matrix model_ ([arXiv:1003.1737](https://arxiv.org/abs/1003.1737))

* [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, _A matrix model for the topological string II: The spectral curve and mirror geometry_ ([arXiv:1007.2194](https://arxiv.org/abs/1007.2194))

* {#EynardOrantin12} [[Bertrand Eynard]], [[Nicolas Orantin]], _Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture_ ([arXiv:1205.1103](https://arxiv.org/abs/1205.1103))

* {#FangLiuZong13} Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, _All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds_ ([arXiv:1310.4818](https://arxiv.org/abs/1310.4818))

[[!redirects topological strings]]

[[!redirects topological string theory]]
[[!redirects topological string theories]]


topological submersion

A **topological submersion** is a map in [[Top]] generalising the sort of map that is called a [[submersion]] in [[Diff]].

There are two definitions of a topological submersion $p\colon Y \to X$:

* Each point in $Y$ has a neighbourhood $U$ such that $p\big|_U\colon U \simeq p(U) \times Z \to p(U)$ is projection  on the first factor. Sometimes $Z$ is required to be a [[cartesian space]] $\mathbb{R}^n$, but this is a bit restrictive.

* Each point $p$ of $Y$ has a local section $\sigma\colon V \to Y$ with $x\in V$ and $p = \sigma(x)$.

The second definition includes the first as a special case.

Surjective topological submersions form a singleton [[Grothendieck pretopology]] on [[Top]].

[[!redirects topological submersion]]
[[!redirects topological submersions]]


topological T-duality

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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Recall that _geometric_ [[T-duality]] is an operation acting on tuples roughly consisting of

* a [[smooth manifold]] $X$ with the structure of a torus-[[principal bundle]] $T^n \to X \to X/T^n$ -- modelling [[spacetime]]

* equipped with a [[circle n-bundle with connection|circle 2-bundle with connection]] -- modelling the [[Kalb-Ramond field]]

* and in [[twisted K-theory]] refined to elements in [[differential K-theory|differential twisted K-theory]] -- modelling the [[RR-field]]

* and notably equipped with a (pseudo)[[Riemannian metric]] -- modelling the field of [[gravity]].

The idea of **topological T-duality** (due to [Bouwknegt-Evslin-Mathai 04](#BouwknegtEvslinMathai04), [Bouwknegt-Hannabus-Mathai 04](#BouwknegtHannabusMathai04)) is to disregard the [[Riemannian metric]] and the connection and study the remaining "topological" structure.

While the idea of [[T-duality]] originates in [[string theory]], topological T-duality has become a field of study in pure mathematics in its own right.

In the language of [[bi-brane]]s a topological T-duality transformation is a bi-brane of a special kind between the two gerbes involved. The induced [[integral transform]] (on sheaves of sections of, or K-classes of) (twisted) vector bundles is essentially the [[Fourier-Mukai transformation]]. More on the bi-brane interpretation of (topological and non-topological) T-duality is in ([Sarkissian-Schweigert 08](#SarkissianSchweigert08)).

## Definition

Two tuples $(X_i \to B, G_i)_{i = 1,2}$ consisting of a $T^n$-bundle $X_i$ over a manifold $B$ and a line [[bundle gerbe]] $G_i \to X_i$ over $X$ are **topological T-duals** if there exists an isomorphism $u$ of the two [[bundle gerbes]] pulled back to the [[fiber product]] [[correspondence]] space $X_1 \times_B X_2$:

$$
  \array{
    && pr_1^* G_1 && \stackrel{u}{\leftarrow}
    && pr_2^* G_2
    \\
    & \swarrow && \searrow && \swarrow && \searrow
    \\
    G_1 &&&& X_1 \times_B X_2 &&&& G_2
    \\
    & \searrow && \swarrow && \searrow && \swarrow
    \\
    && X_1 &&&& X_2
    \\
    &&& \searrow && \swarrow
    \\
    &&&& B
  }
$$

of a certain prescribed [[integral transform]]-form ([Bunke-Rumpf-Schick 08, p. 9](#BunkeRumpfSchick08))).

## Related concepts

* [[T-duality]]

  * **topological T-duality**

  * [[differential T-duality]]

  * [[T-duality 2-group]]

* [[spherical T-duality]]

## References

### General

The concept of topological T-duality was introduced on the level of [[differential form]]-data in

* {#BouwknegtEvslinMathai04} [[Peter Bouwknegt]], [[Jarah Evslin]], [[Varghese Mathai]], _T-Duality: Topology Change from H-flux_, Commun. Math. Phys. 249:383-415, 2004 [[hep-th/0306062](http://arxiv.org/abs/hep-th/0306062), [doi:10.1007/s00220-004-1115-6](https://doi.org/10.1007/s00220-004-1115-6)]

* {#BouwknegtHannabusMathai04} [[Peter Bouwknegt]], [[Keith Hannabus]], [[Varghese Mathai]], _T-duality for principal torus bundles_, JHEP 0403 (2004) 018 ([hep-th/0312284](http://arxiv.org/abs/hep-th/0312284))

In these papers the $U(1)$-[[gerbe]] ([[circle 2-bundle with connection]]) does not appear, but an integral [[differential 3-form]], representing its [[Dixmier-Douady class]] does. Note that if the [[Eilenberg-MacLane spectrum|integral]] [[cohomology group]] $H^3(X,\mathbb{Z})$ of $X$ has [[torsion]] in dimension three, not all gerbes will arise in this way.

The formalization with the above topological/[[homotopy theory|homotopy theoretic]] data originates in

* {#BunkeSchick04} [[Ulrich Bunke]], [[Thomas Schick]], _On the topology of T-duality_, Rev. Math. Phys. **17** (2005) 77-112 [[arXiv:math/0405132](https://arxiv.org/abs/math/0405132), [doi:10.1142/S0129055X05002315](https://doi.org/10.1142/S0129055X05002315)]

* {#BunkeRumpfSchick08} {#BunkeRumpfSchick08} [[Ulrich Bunke|U. Bunke]], P. Rumpf, [[Thomas Schick]], _The topology of $T$-duality for $T^n$-bundles_,  Rev. Math. Phys. **18** 1103 (2006) [[arXiv:math.GT/0501487](http://arxiv.org/abs/math.GT/0501487), [doi:10.1142/S0129055X06002875](https://doi.org/10.1142/S0129055X06002875)]

* [[Ulrich Bunke]], [[Markus Spitzweck]], [[Thomas Schick]], _Periodic twisted cohomology and T-duality_, Astérisque No. 337 (2011), vi+134 pp. ISBN: 978-2-85629-307-2

A refined version of this using [[smooth stacks]] is due to

* [[Ulrich Bunke]], [[Thomas Nikolaus]], _T-Duality via Gerby Geometry and Reductions_ ([arXiv:1305.6050](http://arxiv.org/abs/1305.6050))

* {#Nikolaus14} [[Thomas Nikolaus]], _[[T-Duality in K-theory and Elliptic Cohomology]]_, talk at _String Geometry Network Meeting_, Feb 2014, ESI Vienna ([website](http://www.ingvet.kau.se/juerfuch/conf/esi14/esi14_34.html))

There is also [[C*-algebra|C*-algebraic]] version of toplogical T-duality, .e. in [[noncommutative topology]], which sees also topological T-duals in [[non-commutative geometry]]:

* [[Varghese Mathai]], [[Jonathan Rosenberg]], _T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology_ ([arXiv:hep-th/0401168](http://arxiv.org/abs/hep-th/0401168))

The equivalence of the [[C*-algebra|C*-algebraic]] to the Bunke-Schick version, when the latter exists, is discussed in

* Ansgar Schneider, _Die lokale Struktur von T-Dualitnätstripeln_ ([arXiv:0712.0260](https://arxiv.org/abs/0712.0260))

Introduction and review:

* [[Jonathan Rosenberg]], *Topology, $C^*$-algebras, and string duality*, Regional Conference Series in Mathematics **111**, Amer. Math. Soc. (2009)  [[doi:10.1090/cbms/111](https://doi.org/10.1090/cbms/111), [ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:05632075&type=pdf&format=complete)]

* §1 in [Waldorf 2022](#Waldorf22)

Another discussion that instead of [[noncommutative geometry]] uses [[topological groupoids]] is in

* Calder Daenzer, _A groupoid approach to noncommutative T-duality_ ([arXiv:0704.2592](http://arxiv.org/abs/0704.2592))

Comments in relation to [[T-folds]]:

* [[Peter Bouwknegt]], [[Ashwin S. Pande]], *Topological T-duality and T-folds*, Advances in Theoretical and Mathematical Physics **13** 5 (2009) [[arXiv:0810.4374](https://arxiv.org/abs/0810.4374), [doi:10.4310/ATMP.2009.v13.n5.a6](https://dx.doi.org/10.4310/ATMP.2009.v13.n5.a6)]

The [[bi-brane]] perspective on T-duality is amplified in

* {#SarkissianSchweigert08} Gor Sarkissian, [[Christoph Schweigert]], _Some remarks on defects and duality_ ([arXiv:0810.3159](http://arxiv.org
/abs/0810.3159))

Discussion for non-free torus actions (physically: [[KK-monopoles]]) is in

* {#Pande06} [[Ashwin S. Pande]], _Topological T-duality and Kaluza-Klein Monopoles_, Adv. Theor. Math. Phys. **12** (2007) 185-215 [[arXiv:math-ph/0612034](https://arxiv.org/abs/math-ph/0612034), [doi:10.4310/ATMP.2008.v12.n1.a3](https://dx.doi.org/10.4310/ATMP.2008.v12.n1.a3)]

Discussion in [[rational homotopy theory]]/[[dg-geometry]] is in

* [[Ernesto Lupercio]], Camilo Rengifo, [[Bernardo Uribe]], _T-duality and exceptional generalized geometry through symmetries of dg-manifolds_ ([arXiv:1208.6048](https://arxiv.org/abs/1208.6048))

and a derivation of the rules of topological T-duality from analysis of the [[super p-brane]] super-cocycles in super rational homotopy theory (with a [[doubled supergeometry]]) is given in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]_, [ATMP Volume 22 (2018) Number 5](http://www.intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0022/0005/) ([arXiv:1611.06536](https://arxiv.org/abs/1611.06536), [doi:10.4310/ATMP.2018.v22.n5.a3](https://dx.doi.org/10.4310/ATMP.2018.v22.n5.a3))

reviewed in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:T-duality in rational homotopy theory via strong homomotopy Lie algebras]]_, [Geometry, Topology and Mathematical Physics Journal, Volume 1 (2018)](http://geo-top.org/GTMP/index.php/volume-1-2018/)  ([arXiv:1712.00758](https://arxiv.org/abs/1712.00758))

* {#Fiorenza18} [[Domenico Fiorenza]], _T-duality in rational homotopy theory_, talk at _[38th Srni Winter School on Geometry and Physics](http://conference.math.muni.cz/srni/files/archiv/2018/)_, 2018 ([[FiorenzaSrni2018.pdf:file]])

and further expanded on on

* [[Hisham Sati]], [[Urs Schreiber]], p. 7 of: *[[schreiber:Cyclification of Orbifolds]]* [[arXiv:2212.13836](https://arxiv.org/abs/2212.13836)]

See also

* [[Domenico Fiorenza]], Mattia Coloma, Eugenio Landi, _A very short note on the (rational) graded Hori map_ ([arXiv:2003.13066](https://arxiv.org/abs/2003.13066))

Comprehensive discussion in [[higher differential geometry]]:

* {#Alfonsi19} [[Luigi Alfonsi]], _Global Double Field Theory is Higher Kaluza-Klein Theory_, Fortsch. d. Phys. 2020 ([arXiv:1912.07089](https://arxiv.org/abs/1912.07089), [doi:10.1002/prop.202000010](https://doi.org/10.1002/prop.202000010))

  (relating [[Kaluza-Klein compactification]] on [[principal ∞-bundles]] to [[double field theory]], [[T-folds]], [[non-abelian T-duality]], [[type II geometry]], [[exceptional geometry]], ...)

* [[Luigi Alfonsi]], _The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective_ ([arXiv:2007.04969](https://arxiv.org/abs/2007.04969))

### Equivariant refinement

Discussion in the [[equivariant cohomology|equivariant]] generality, ie. as an equivalence of [[twisted equivariant K-theory]] groups:

* [[Tom Dove]], [[Thomas Schick]], *Equivariant Topological T-Duality* [[arXiv:2310.06064](https://arxiv.org/abs/2310.06064)]

### Geometric refinement
 {#ReferencesGeometricRefinement}

On possible geometric refinement of topological T-duality via some form of [[differential cohomology]]:

via [[differential K-theory]] classes:

* {#KahleValentino} [[Alexander Kahle]], [[Alessandro Valentino]], _[[T-Duality and Differential K-Theory]]_, Communications in Contemporary Mathematics, **16** 02 (2014) [[arXiv:0912.2516](http://arxiv.org/abs/0912.2516), [doi:10.1142/S0219199713500144](https://doi.org/10.1142/S0219199713500144)]

using [[adjusted Weil algebra|adjusted]] [[principal 2-connections]]:

* [[Hyungrok Kim]], [[Christian Saemann]], *Non-Geometric T-Duality as Higher Groupoid Bundles with Connections* [[arXiv:2204.01783](https://arxiv.org/abs/2204.01783)]

* {#Waldorf22} [[Konrad Waldorf]], *Geometric T-duality: Buscher rules in general topology* [[arXiv:2207.11799](https://arxiv.org/abs/2207.11799)]

* [[Hyungrok Kim]], [[Christian Saemann]], *T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections* [[arXiv:2303.16162](https://arxiv.org/abs/2303.16162)]


topological twist

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#### Algebraic Quantum Field Theory
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#### Functorial quantum field theory
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[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A procedure for producing ([[Lagrangians]] for) [[topological quantum field theories]] from non-topological  but [[supersymmetry|supersymmetric]] [[QFT|QFTs]].

## Examples

* Topological twists of the [[type II string theory|type II]] [[superstring]] are the [[A-model]], [[B-model]] and [[half-twisted model]] versions of the [[topological string]].

* For the topological twist of [[D=4 super Yang-Mills theory]] see at _[[topologically twisted D=4 super Yang-Mills theory]]_.

* A topological twist of [[D=3 N=4 super Yang-Mills theory]] is [[Rozansky-Witten theory]].

* [[topologically twisted supergravity]]

## Related concepts

* [[cohomological field theory]], [[TCFT]]

## References

The influential idea goes back to

* {#Witten91} [[Edward Witten]], _Introduction to cohomological field theory_, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 ([[WittenCQFT.pdf:file]])

Mathematical formalization and classification is discussed in

* {#Costello11} [[Kevin Costello]], *Notes on supersymmetric and holomorphic field theories in dimension 2 and 4*, talk at *[Geometric and Algebraic Structures in Mathematics](https://www.math.stonybrook.edu/dennisfest/)*, Stony Brook (2011) [[arXiv:1111.4234](https://arxiv.org/abs/1111.4234)]

* {#ElliottSafronov18} [[Chris Elliott]], [[Pavel Safronov]], _Topological twists of supersymmetric observables_ ([arXiv:1805.10806](https://arxiv.org/abs/1805.10806))

* {#EagerSaberiWalcher18} Richard Eager, Ingmar Saberi, [[Johannes Walcher]], _Nilpotence varieties_ ([arXiv:1807.03766](https://arxiv.org/abs/1807.03766))

For more see the [references](topologically+twisted+D%3D4+super+Yang-Mills+theory#References) at _[[topologically twisted D=4 super Yang-Mills theory]]_.

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