Segal-Carlsson completion theorem

[[!redirects Segal-Carlsson theorem]]
[[!redirects Segal theorem]]

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## Idea

The statement known as _Segal's conjecture_ (due to [[Graeme Segal]] in the 1970s, then proven by [Carlsson 84](#Carlsson84)) characterizes the [[stable cohomotopy]] [[cohomology group|groups]] $\pi^\bullet_{st}(B G)$ of the [[classifying space]] $B G$ of a [[finite group]] $G$ as the [[formal completion]] $\widehat \pi^\bullet_S(B G)$ at the [[augmentation ideal]] (i.e. when regarded as a [[ring of functions]]: its restriction to the [[infinitesimal neighbourhood]] of the basepoint) of the [[ring]] $\pi^\bullet_{st,G}(\ast)$ of $G$-[[equivariant stable cohomotopy]] [[cohomology groups|groups]] of the point, the latter also being [[isomorphism|isomorphic]] to the [[Burnside ring]] $A(G)$ of $G$:

$$
  A(G)
   \simeq
  \pi^\bullet_{st,G}(\ast)
    \overset{ \text{completion} \atop \text{projection} }{\longrightarrow}
  \widehat \pi^\bullet_{st,G}(\ast)
    \simeq
  \pi^\bullet_{st}(B G)
  \,.
$$

This statement is the direct analogue of the [[Atiyah-Segal completion theorem]], which makes the analogous statement for the [[generalized cohomology]] not being ([[equivariant stable cohomotopy|equivariant]]) [[stable cohomotopy]] but ([[equivariant K-theory|equivariant]]) [[complex K-theory]] (with the role of the [[Burnside ring]] then being the [[representation ring]] of $G$).

[[!include Segal completion -- table]]

## Related concepts

* [[equivariant stable cohomotopy]]

* [[equivariant homotopy theory]]

* [[equivariant cohomology]]

## Applications

A [[proof]] of the [[Sullivan conjecture]] follows with the Segal-Carlsson completion theorem

## References

The statement was proven in

* {#Carlsson84} [[Gunnar Carlsson]], _Equivariant Stable Homotopy and Segal's Burnside Ring Conjecture_,  Annals of Mathematics Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 189-224 ([jstor:2006940](https://www.jstor.org/stable/2006940), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/carlsson.pdf))

see also

* Erkki Laitinen, _On the Burnside ring and stable cohomotopy of a finite group_,  Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 ([jstor:24491306](https://www.jstor.org/stable/24491306), [[Laitinen79.pdf:file]])

* {#Lueck05} [[Wolfgang Lück]], _The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups_ ([arXiv:math/0504051](https://arxiv.org/abs/math/0504051))

Review includes

* {#May96} [[Peter May]], section XX 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. Comeza˜na, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/alaska1.pdf))

Application to proof of the [[Sullivan conjecture]] is due to

* {#Carlsson91} [[Gunnar Carlsson]], _Equivariant stable homotopy and Sullivan's conjecture_. Invent. Math. 103: 497–525, 1991 ([dml:143867](https://eudml.org/doc/143867), [pdf](http://www.digizeitschriften.de/download/PPN356556735_0103/PPN356556735_0103___log38.pdf))

See also

* Wikipedia, _[Segal conjecture](https://en.wikipedia.org/wiki/Segal_conjecture)_

[[!redirects Segal's conjecture]]

[[!redirects Segal conjecture]]


Segal-topos

A **Segal topos** is a model for an [[(infinity,1)-topos]] in terms of [[Segal categories]] or [[complete Segal spaces]].


segment object

In [section 4](http://arxiv.org/PS_cache/math/pdf/0502/0502155v2.pdf#page=11)
of

* Clemens Berger, [[Ieke Moerdijk]], _The Boardman-Vogt resolution of operads in monoidal model categories_ ([arXiv](http://arxiv.org/abs/math.AT/0502155))

the following definition is given:

Let $H$ be a [[monoidal model category]] and write $pt$ for the tensor unit in $H$ (not necessarily the terminal object).

A **[[segment object|segment (object)]]** $I$ in a [[monoidal model category]] $H$ is

* a factorization

  $$
    pt \amalg pt \stackrel{[0 , 1]}{\to}
    I \stackrel{\epsilon}{\to} pt
  $$

  of the [[codiagonal morphism]]

  $$
    pt \amalg pt \stackrel{[Id , Id]}{\to} pt
  $$

  from the [[coproduct]] of $pt$ with itself that
  sends each component identically to $pt$.

* together with an associative morphsim

  $$
    \vee : I \otimes I \to I
  $$

  which has 0 as its _neutral_ and 1 as
  its _absorbing_ element, and for which
  $\epsilon$ is a counit.

If $H$ is equipped with the structure of a [[model category]] then a segment object is an **interval** in $H$ if

$$
  [0, 1]\colon pt \amalg pt \to I
$$

is a cofibration and $\epsilon : I \to pt$ a weak equivalence.

The axioms of a segment are expressed by the commutativity of the following five diagrams (all isomorphisms being induced by the symmetric monoidal structure):

$$\array{
(H\otimes H)\otimes H&\to^\sim&H\otimes(H\otimes H)\\\downarrow^{\vee\otimes H}&&\downarrow_{H\otimes\vee}\\H\otimes H&\overset{\vee}{\leftarrow} H\overset{\vee}{\longleftarrow}&H\otimes H
}$$

$$\array{I\otimes H&\rightarrow^{0\otimes H}& H\otimes H&\leftarrow^{H\otimes 0}&H\otimes I\\&\searrow_\sim&\downarrow_\vee&\swarrow_\sim&\\&&H&&
}$$

$$\array{&&I\otimes H&\rightarrow^{1\otimes H}&H\otimes H&\leftarrow^{H\otimes 1}&H\otimes I&&\\&\swarrow^{I\otimes\epsilon}&\downarrow&&\downarrow_\vee&&\downarrow&\searrow^{\epsilon\otimes I}&\\I\otimes I&\rightarrow^\sim&I&\rightarrow^1&H&\leftarrow^1&I&\leftarrow^\sim&I\otimes I}$$

$$\array{H\otimes H&\rightarrow^{\epsilon\otimes\epsilon}&I\otimes I&\quad&I&\rightarrow^0&H\\\downarrow^\vee&&\downarrow_\sim&\quad&\downarrow_1&\searrow^{id}&\downarrow_\epsilon\\H&\rightarrow^\epsilon&I&\quad&H&\rightarrow^\epsilon&I}$$


Seiberg duality

> under construction

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## Idea

Seiberg duality (named after ([Seiberg](#Seiberg))) is a version of [[electric-magnetic duality]] in [[supersymmetry|supersymmetric]] [[gauge theory]].

For supersymmetric [[QCD]] it identifies in the infrared (long distance limit, and only there) the [[quarks]] and [[gluons]] in a theory with $N_f$ quark flavors and $SU(N_c)$ [[gauge group]] for

$$
  N_f \gt N_c +1
$$

with [[soliton]]s in a theory of $N_f$ quark flavors and gauge group $SU(\tilde N_c)$, where

$$
  \tilde N_c = N_f - N_c
  \,.
$$

## Properties

### Realization in string theory

Seiberg duality follows from phenomena in [[string theory]], where gauge theories arise as the [[worldvolume]] theories of [[D-branes]] ([[geometric engineering of quantum field theory]]). Seiberg duality is obtained for gauge theories of [[D-branes]] that stretch between two [[NS5-brane]]s. The duality operation corresponds to exchanging the two NS5-branes.

* [[Amihay Hanany]], [[Edward Witten]], _Type-IIB superstrings, BPS monopoles and three-dimensional gauge dynamics_ , Nucl. Phys. B 492 (1997) 152 ([hep-th/9611230]()).

* S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici and A. Schwimmer, _Brane dynamics and $N = 1$ supersymmetric gauge theory, Nucl. Phys. B 505 (1997) 202 ([hep-th/9704104]())

See also _[[string theory results applied elsewhere]]_.

### Formalization by derived quiver categories

Seiberg duality is formalized by [[equivalence of categories|equivalences]] of [[derived categories]] of [[quiver]] representations.

* David Berenstein, [[Michael Douglas]], _Seiberg Duality for Quiver Gauge Theories_ ([arXiv:hep-th/0207027](http://arxiv.org/abs/hep-th/0207027))

* Subir Mukhopadhyay, Koushik Ray, _Seiberg duality as derived equivalence for some quiver gauge theories_ ([arXiv:hep-th/0309191](http://arxiv.org/abs/hep-th/0309191))

## Examples

### Toric duality

_Toric Duality_ is Seiberg duality for $N=1$ theories with [[toric variety|toric]] [[moduli spaces]].

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 ([arXiv:hep-th/0003085](https://arxiv.org/abs/hep-th/0003085))

* [[Chris Beasley]], [[M. Ronen Plesser]], _Toric duality is Seiberg duality_, J. High Energy Phys. 12 (2001) 001 ([hep-th/0109053](https://arxiv.org/abs/hep-th/0109053))

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _Phase structure of D-brane gauge theories and toric duality_ , J. High Energy Phys. 08 (2001) 040 ([hep-th/0104259](https://arxiv.org/abs/hep-th/0104259))

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], [[Angel Uranga]], _Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys. 12 (2001) 035 ([hep-th/0109063](https://arxiv.org/abs/hep-th/0109063))

* [[Bo Feng]], S. Franco, [[Amihay Hanany]], [[Yang-Hui He]], _Unhiggsing the del Pezzo_, J. High Energy Phys. 08 (2003) 058 ([hep-th/0209228](https://arxiv.org/abs/hep-th/0209228))

* S. Franco and [[Amihay Hanany]], _Toric duality, Seiberg duality and Picard-Lefschetz transformations_ , Fortschr. Phys. 51 (2003) 738 ([hep-th/0212299](https://arxiv.org/abs/hep-th/0212299))

### From D-branes on del Pezzo singularities

* [[Christopher Herzog]], _Seiberg Duality is an Exceptional Mutation_ ([arXiv:hep-th/0405118](http://arxiv.org/abs/hep-th/0405118))

* Subir Mukhopadhyay, Koushik Ray, _Seiberg duality as derived equivalence for some quiver gauge theories_ Journal of High Energy Physics Volume 2004 JHEP02(2004)

### For exceptional gauge groups
 {#ExamplesForExceptionalGaugeGroups}

Seiberg duality for [[gauge group]]s which are [[exceptional Lie group]]s:

* [[Jacques Distler]], [[Andreas Karch]], _$N=1$ Dualities for Exceptional Gauge Groups and Quantum Global Symmetries_ ([arXiv:hep-th/9611088](http://arxiv.org/abs/hep-th/9611088))

But see

* Peter Cho, _Moduli in Exceptional SUSY Gauge Theories_ ([arXiv:hep-th/9712116](http://arxiv.org/abs/hep-th/9712116))

### Chiral and non-chiral duals

* P. Pouliot, _Chiral Duals of Non-Chiral SUSY Gauge Theories_ ([arXiv:hep-th/9507018](http://arxiv.org/abs/hep-th/9507018))

### Duality cascade

Due to

* [[Igor Klebanov]], [[Matthew Strassler]], _Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi$SB-Resolution of Naked Singularities_ ([arXiv](http://arxiv.org/abs/hep-th/0007191))

A review is in

* [[Matthew Strassler]], _The Duality Cascade_ ([arXiv:hep-th/0505153](http://arxiv.org/abs/hep-th/0505153))

Discussion in connection with non-conformal variants of [[AdS/CFT]] is in

* Eduardo Conde, [[Jérôme Gaillard]], [[Carlos Núñez]], Maurizio Piai, Alfonso V. Ramallo, _Towards the string dual of tumbling and cascading gauge theories_ ([arXiv:1112.3346](http://arxiv.org/abs/1112.3346))

### With few supercharges

* a 3d [[Yang-Mills theory|Yang-Mills]] [[Chern-Simons theory]] with two [[supercharges]] ($N = 1$ SUSY in 3d)

  (Adi Armoni, Amit Giveon, Dan Israel, Vasilis Niarchos, 2009)

* a non-supersymmetric theory in $4d$;

  (Adi Armoni, Dan Israel, Gregory Moraitis, Vasilis Niarchos, 2008).

Discussed in

* Adi Armoni, _Two Examples of Seiberg Duality in Gauge Theories With Less Than Four Supercharges_ ([pdf](http://pyweb.swan.ac.uk/~pyarmoni/seiberg.pdf))

## Related entries

* [[quiver gauge theory]]

[[duality in physics]]

* [[S-duality]]

  * [[electric-magnetic duality]]

    * [[Montonen-Olive duality]]

  * **Seiberg duality**

  * [[geometric Langlands duality]]

## References

### Original articles

The original article is

* [[Nathan Seiberg]], _Electric-Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories_ ([arXiv:hep-th/9411149](http://arxiv.org/abs/hep-th/9411149))
 {#Seiberg}

The "cascade" of Seiberg dualities is due to

* [[Igor Klebanov]], [[Matthew Strassler]], _Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi$SB-Resolution of Naked Singularities_ ([arXiv:hep-th/0007191](http://arxiv.org/abs/hep-th/0007191))

### Lectures and reviews

Surveys and reviews include

* [[Matthew Strassler]], _The Duality Cascade_ (2005) ([arXiv:hep-th/0505153](http://arxiv.org/abs/hep-th/0505153))

* M. Chaichian, W.F. Chen, C. Montonen, _New Superconformal Field Theories in Four Dimensions and N=1 Duality_ ([arXiv:hep-th/0007240](http://arxiv.org/abs/hep-th/0007240))

* Flip Tanedo, _Notes on Seibergology_ ([pdf](http://www.lns.cornell.edu/~pt267/files/notes/Seibergology.pdf))

See also section 22 of

* [[Philip Argyres]], _Introduction to supersymmetry_ ([pdf](http://www.physics.uc.edu/~argyres/661/susy1996.pdf))

[[!redirects Seiberg dualities]]

[[!redirects toric duality]]


Seiberg-Witten curve -- references

### Seiberg-Witten curve
 {#ReferencesSeibergWittenCurve}

#### SW-curve in SW-theory

The notion of the *Seiberg-Witten curve* in [[Seiberg-Witten theory]] originates in:

* [[Nathan Seiberg]], [[Edward Witten]], pp. 37 in: _Monopoles, duality and chiral symmetry breaking in $N=2$
supersymmetric QCD, Nucl. Phys. B431 (1994) 484-550 [[hep-th/9408099](https://arxiv.org/abs/hep-th/9408099)]

Review:

* [[Taro Kimura]], §4.2.3.1 in: *Instanton Counting, Quantum Geometry and Algebra*, Mathematical Physics Studies, Springer (2021) [[doi:10.1007/978-3-030-76190-5_4](https://doi.org/10.1007/978-3-030-76190-5_4)]

and in relation to [[integrable systems]]:

* [[Andrei Marshakov]], *Seiberg-Witten Curves and Integrable Systems* [[arXiv:hep-th/9903252](https://arxiv.org/abs/hep-th/9903252)]

#### As M5-brane worldvolume
 {#ReferencesSWCurveAsM5Worldvolume}

Observation that under [[geometric engineering of QFT|geometric engineering]] of [[D=4 N=2 SYM]] on [D4/NS5 intersections](NS5-brane#D4EndingOnNS5) and further [[duality between M-theory and type IIA string theory|lift]] to a single [[M5-brane]], the SW-curve is identified with the M5-[[worldvolume]] transverse to the SYM-spacetime:

* {#KlemmLercheMayrVafaWarner96} [[Albrecht Klemm]], [[Wolfgang Lerche]], [[Peter Mayr]], [[Cumrun Vafa]], [[Nicholas Warner]], _Self-Dual Strings and $\mathcal{N}=2$ Supersymmetric Field Theory_, Nucl. Phys. B **477** (1996) 746-766 [[arXiv:hep-th/9604034](http://arxiv.org/abs/hep-th/9604034), doi:10.1016/0550-3213(96)00353-7]

* {#Witten97} [[Edward Witten]], *Solutions Of Four-Dimensional Field Theories Via M Theory*, Nucl. Phys. B **500** (1997) 3-42 [[arXiv:hep-th/9703166](http://arxiv.org/abs/hep-th/9703166), doi:10.1016/S0550-3213(97)00416-1]

Further early discussion:

* Karl Landsteiner, Esperanza Lopez, David A. Lowe, *$\mathcal{N}$=2 Supersymmetric Gauge Theories, Branes and Orientifolds*, Nucl. Phys. B **507** (1997) 197-226 [[arXiv:hep-th/9705199](https://arxiv.org/abs/hep-th/9705199), doi:10.1016/S0550-3213(97)00559-2]

* A. Brandhuber, [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], *M Theory And Seiberg-Witten Curves: Orthogonal and Symplectic Groups*, Nucl. Phys. B **504** (1997) 175-188 [[arXiv:hep-th/9705232](https://arxiv.org/abs/hep-th/9705232), doi:10.1016/S0550-3213(97)00531-2]

Review in:

* Csaba Csaki, Joshua Erlich, John Terning, pp. 4 of; *Seiberg-Witten Description of the Deconstructed 6D $(0,2)$ Theory*, Phys. Rev. D **67** 025019 (2003) [[arXiv:hep-th/0208095](https://arxiv.org/abs/hep-th/0208095), [doi:10.1103/PhysRevD.67.025019](https://doi.org/10.1103/PhysRevD.67.025019)]

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

* Ling Bao, Elli Pomoni, Masato Taki, Futoshi Yagi, p. 5 of: *M5-branes, toric diagrams and gauge theory duality*, J. High Energ. Phys. **2012** 105 (2012) [[arXiv:1112.5228](https://arxiv.org/abs/1112.5228), doi:10.1007/JHEP04(2012)105]

* [[Taro Kimura]], §4.5.2 in: *Seiberg–Witten Geometry*, Chapter 4 in: *Instanton Counting, Quantum Geometry and Algebra*, Mathematical Physics Studies, Springer (2021) [[doi:10.1007/978-3-030-76190-5_4](https://doi.org/10.1007/978-3-030-76190-5_4)]

#### Quantum SW-curve

On the "quantum SW curve", a kind of [[quantization|quantized]] SW curve:

* Alba Grassi, Jie Gu, [[Marcos Mariño]], *Non-perturbative approaches to the quantum Seiberg-Witten curve*,  J. High Energ. Phys. **2020** 106 (2020) [[arXiv:1908.07065](https://arxiv.org/abs/1908.07065), doi:10.1007/JHEP07(2020)106 ]

In relation to [[class S-theories]] and "[[M3-brane]]"-[[defect branes]] inside [[M5-branes]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], *Elliptic Quantum Curves of Class $\mathcal{S}_k$*,  J. High Energ. Phys. **2021** 28 (2021) [[arXiv:2008.05155](https://arxiv.org/abs/2008.05155), doi:10.1007/JHEP03(2021)028]

In relation to [[E-strings]] and [[D6-D8-brane bound states]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], [[Xin Wang]], *E-string Quantum Curve*, Nuclear Physics B **973** (2021) 115602 [[arXiv:2103.16996](https://arxiv.org/abs/2103.16996), [doi:10.1016/j.nuclphysb.2021.115602](https://doi.org/10.1016/j.nuclphysb.2021.115602)]

On why the SW-curve should be quantized this way, as seen from [[topological string theory]]:

* [[Mina Aganagic]], [[Robbert Dijkgraaf]], [[Albrecht Klemm]], [[Marcos Mariño]], [[Cumrun Vafa]], pp. 7 of: *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]], §2 in: *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]


Seiberg-Witten theory

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## Idea

_Seiberg-Witten theory_ studies the [[moduli space]] of [[vacua]] in [[N=2 D=4 super Yang-Mills theory]], in particular the [[electric-magnetic duality]] ([[Montonen-Olive duality]]) of the theory.

## Related concepts

* [[N=2 D=4 super Yang-Mills theory]]

* [[Coulomb branch]], [[Higgs branch]]

* [[Donaldson theory]]

* [[Chern-Simons theory]], [[topological Yang-Mills theory]]

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

* [[AGT correspondence]]

## References

### General

> For more and for general references see at _[[N=2 D=4 super Yang-Mills theory]]_.

The original articles:

* [[Nathan Seiberg]], [[Edward Witten]], _Monopole condensation, and confinement in $N=2$ supersymmetric Yang-Mills theory_, Nucl. Phys. B **426** (1994) 19-52;

  Erratum-ibid. B **430** (1994) 485-486,  ([arXiv:hep-th/9407087](https://arxiv.org
/abs/hep-th/9407087))

* [[Nathan Seiberg]], [[Edward Witten]], _Monopoles, duality and chiral symmetry breaking in $N=2$
supersymmetric QCD, Nucl. Phys. B431 (1994) 484-550 [[hep-th/9408099](https://arxiv.org/abs/hep-th/9408099)]

Review:

* [[Taro Kimura]], *Seiberg–Witten Geometry*, Chapter 4 in: *Instanton Counting, Quantum Geometry and Algebra*, Mathematical Physics Studies, Springer (2021) [[doi:10.1007/978-3-030-76190-5_4](https://doi.org/10.1007/978-3-030-76190-5_4)]

* Jürgen Einhorn, Thomas Friedrich, _Seiberg-Witten theory_ ([pdf](http://matwbn.icm.edu.pl/ksiazki/bcp/bcp39/bcp3925.pdf))

* [[Simon Donaldson]], _The Seiberg-Witten equations and 4-manifold topology_ ([pdf](http://www.ams.org/journals/bull/1996-33-01/S0273-0979-96-00625-8/S0273-0979-96-00625-8.pdf))

* [[Matilde Marcolli]], _Seiberg-Witten gauge theory_, [pdf](http://www.its.caltech.edu/~matilde/swcosi.pdf)

* piljin yi, _Seiberg-Witten theory -- with a view toward MQCD_ ([pdf](http://newton.kias.re.kr/~piljin/public_html/thesis/SW-MQCD.pdf))

* Wikipedia, _[Seiberg-Witten invariants](https://en.wikipedia.org/wiki/Seiberg-Witten_invariant)_

A useful discussion of the physical origins of the Seiberg-Witten equations for mathematicians:

* Siye Wu, _The Geometry and Physics of the Seiberg-Witten Equations_, Progress in mathematics **205** (2002)

In relation to [[integrable systems]]:

* [[Alexander Gorsky]], [[Igor Krichever]], [[Andrei Marshakov]], [[Andrei Mironov]], [[Andrey Morozov]], *Integrability and Seiberg-Witten Exact Solution*, Phys. Lett. B **355** (1995) 466-474 [[arXiv:hep-th/9505035](https://arxiv.org/abs/hep-th/9505035), doi:10.1016/0370-2693(95)00723-X]

Discussion of lifts of SW-invariants to [[M-theory]]:

* [[Neil Lambert]], [[Peter West]], _Monopole Dynamics from the M-Fivebrane_, Nucl.Phys. B556 (1999) 177-196 [[arXiv:hep-th/9811025](http://arxiv.org/abs/hep-th/9811025)]

A lift of Seiberg-Witten invariants to classes in [[circle group]]-[[equivariant stable cohomotopy]]:

* M. Furuta,  (2001), _Monopole Equation and the 11/8-Conjecture_ , Mathematical Research Letters 8: 279--291 [[doi](https://doi.org/10.4310/MRL.2001.v8.n3.a5)]

* Stefan Bauer, Mikio Furuta, _A stable cohomotopy refinement of Seiberg-Witten invariants: I_ [[arXiv:math/0204340](https://arxiv.org/abs/math/0204340)]

* Stefan Bauer, _A stable cohomotopy refinement of Seiberg-Witten invariants: II_ [[arXiv:math/0204267](https://arxiv.org/abs/math/0204267)]

[[!include Seiberg-Witten curve -- references]]

### Relation to Rozansky-Witten invariants

On relation between [[Rozansky-Witten invariants]] and [[Seiberg-Witten invariants]] of [[3-manifolds]]:

* [[Matthias Blau]], George Thompson, _On the Relationship between the Rozansky-Witten and the 3-Dimensional Seiberg-Witten Invariants_, Adv. Theor. Math. Phys. **5** (2002) 483-498 [[arXiv:hep-th/0006244](https://arxiv.org/abs/hep-th/0006244)]

[[!redirects Seiberg-Witten duality]]
[[!redirects Seiberg-Witten dualities]]

[[!redirects Seiberg-Witten invariant]]
[[!redirects Seiberg-Witten invariants]]

[[!redirects Seiberg-Witten curve]]
[[!redirects Seiberg-Witten curves]]

[[!redirects quantum Seiberg-Witten curve]]
[[!redirects quantum Seiberg-Witten curves]]


Seifert 3-manifold

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## Definition

A _Seifert 3-manifold_ is a [[3-manifold]] which is the total space of a [[circle bundle]] over a 2-dimensional [[orbifold]].

## Related concepts

* [[lens space]]

## References

See also:

* Wikipedia, _[Seifert fiber space](http://en.wikipedia.org/wiki/Seifert_fiber_space)_

On the [[3d-3d correspondence]] for Seifert manifolds:

* [[Du Pei]], _3d-3d correspondence for Seifert manifolds_, 2016 ([spire:1469350](https://inspirehep.net/literature/1469350), [pdf](https://inspirehep.net/files/d509ff9e32448da3a5674f286b93224a))

[[!redirects Seifert 3-manifolds]]

[[!redirects Seifert fiber space]]
[[!redirects Seifert fiber spaces]]


Seifert surface

* wikipedia [Seifert surface](http://en.wikipedia.org/wiki/Seifert_surface)

Related entries: [[knot]], [[Alexander polynomial]]

[[!redirects Seifert surfaces]]


Seiji Terashima

* [personal page](http://www2.yukawa.kyoto-u.ac.jp/~seiji.terashima/)

* [inspire page](https://inspirehep.net/authors/986336)

## Selected writings

Towards an enhancement of the [[BFSS matrix model]] by [[topological K-theory]] to account for the [[K-theory classification of D-brane charge]] ([[K-matrix model]]):

* {#AsakawaSugimotoTerashima01} [[Tsuguhiko Asakawa]], [[Shigeki Sugimoto]], [[Seiji Terashima]], *D-branes, Matrix Theory and K-homology*, JHEP 0203 (2002) 034 [[arXiv:hep-th/0108085](https://arxiv.org/abs/hep-th/0108085), [doi:10.1088/1126-6708/2002/03/034](https://doi.org/10.1088/1126-6708/2002/03/034)]

On [[Montonen-Olive duality]] in [[D=4 super Yang-Mills theory]] via [[ABJM-model]] as [[D3-brane]] model after [[double dimensional reduction]] followed by [[T-duality]]:

* [[Koji Hashimoto]], Ta-Sheng Tai, [[Seiji Terashima]], _Toward a Proof of Montonen-Olive Duality via Multiple M2-branes_, JHEP 0904:025 (2009) [[arxiv:0809.2137](https://arxiv.org/abs/0809.2137)]

On [[M2-M5-brane bound states]] in the [[BMN matrix model]]:

* {#AIST17b} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _Spherical transverse M5-branes from the plane wave matrix model_, JHEP 02 (2018) 076 $[$[arXiv:1711.07681](https://arxiv.org/abs/1711.07681), doi:10.1007/JHEP02(2018)076$]$

* {#AIST17a} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _On the transverse M5-branes in matrix theory_, Phys. Rev. D **96** 126003 (2017) $[$[arXiv:1701.07140](https://arxiv.org/abs/1701.07140), [doi:10.1103/PhysRevD.96.126003](https://doi.org/10.1103/PhysRevD.96.126003)$]$

category: people


Selberg integral

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## Idea

The __Selberg integral__ is a higher-dimensional generalization of the integral representation of Euler's beta function due Alte Selberg. Many further "Selberg-type" generalizations appear in the study of multidimensional generalizations of hypergeometric functions, arrangements of hyperplanes, [[Knizhnik-Zamolodchikov equation]], representation theory of quantum and affine Lie algebras and of vertex operator algebras, random matrix theory etc. There is also an elliptic generalization, see [[elliptic Selberg integral]].

$$
\int_0^1 \cdots \int_0^1 (u_1\cdots u_p)^{x-1} [(1-u_1)\cdots (1-u_p)]^{y-1} \left( \prod_{i\lt j} (u_j-u_i)\right)^{2 z} d u_1 \ldots d u_p =
\prod_{s = 1}^p \frac{\Gamma(1+s z) \Gamma(x+(s-1)z)\Gamma(y+(s-1)z)}{ \Gamma(1+z)\Gamma(x+y+(p+s-2)z)}
$$
where $p\gt 0$ is a positive integer, $Re x\gt 0$, $Re y\gt 0$ , $Re z \gt max\{-p^{-1},- Re x/(p-1), - Re y/(p-1)\}$. Notice that the discriminant $\prod_{i\lt j} (u_j-u_i)$ is the value of the standard [[Vandermonde determinant]].

## References

* wikipedia [Selberg integral](http://en.wikipedia.org/wiki/Selberg_integral)
* Alte Selberg, _Remarks on a multiple integral_, Norsk Matematisk Tidsskrift __26__: 71–78, [MR0018287](http://www.ams.org/mathscinet-getitem?mr=0018287)
* Atle Selberg – utdypning, [biography](http://snl.no/.nbl_biografi/Atle_Selberg/utdypning) in Norweigian
* Nils A. Baas, _Atle Selberg (1917-2007)_ [pdf](http://www.math.ntnu.no/~baas/Atle_Selberg_en.pdf)
* Peter J. Forrester, S. Ole Warnaar, _The importance of the Selberg integral_, Bull. Amer. Math. Soc. (N.S.) __45__ (2008), 489-534, [pdf](http://www.ams.org/journals/bull/2008-45-04/S0273-0979-08-01221-4/S0273-0979-08-01221-4.pdf), [arxiv/0710.3981](http://arxiv.org/abs/0710.3981)
* G. E. Andrews, R. Askey, R. Roy, _Special functions_, Enc. of Math. and its Appl. (1999)
* [[I. M. Gelfand]], M. M. [[Kapranov]], [[A. Zelevinsky]], _Discriminants, resultants and multidimensional determinants_, Birkhäuser 1994, 523 pp.
* J.-G. Luque, J.-Y. Thibon, _Hankel hyperdeterminants and Selberg integrals_, J. Phys. __A36__ (2003), 5267–5292,  MR1985318 (2004d:15011)
* K. Aomoto, _Jacobi polynomials associated with Selberg integrals_, SIAM
J. Math. Anal. 18 (1987) 545–549; _On the complex Selberg integral_, Q. J. Math. Oxford __38__ (1987) 385–399.
* R. S. Askey, _Some basic hypergeometric extensions of integrals of Selberg and Andrews_, SIAM J. Math. __11__ (1980) 938–951.


Selberg trace formula

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## Idea

The _Selberg trace formua_ ([Selberg 56](#Selberg56)) is an expression for certain sums of [[eigenvalues]] of the [[Laplace operator]] on a compact [[hyperbolic manifold|hyperbolic]] [[Riemann surface]] (recalled e.g. as [Bump, theorem 19](#Bump)).

It was introduced as a nonabelian generalization of the [[Poisson summation formula]] (e.g.[Voros-Balasz 86, p. 169](#VorosBalasz86)), thought of as a relation between the eigenvalues of the Laplacian and the lengths of closed [[geodesics]].

It motivates (e.g. [Bump, p.18](#Bump)) the definition of the Selberg [[zeta function of a Riemann surface]].

## Related concepts

* [[Riemann hypothesis]]

* [[number theory and physics]]

* [[Arthur-Selberg trace formula]]

* [[Duistermaat-Guillemin trace formula]]

* [[zeta function of a Riemann surface]]

* [[Selberg zeta function]]

## References

* {#Selberg56} [[Atle Selberg]], _Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series_, Journal of the Indian Mathematical Society 20 (1956) 47-87.

* {#Friedman04} [[Joshua Friedman]], _The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations_ ([arXiv:math/0410067](http://arxiv.org/abs/math/0410067))

* Wikipedia, _[Selberg trace formula](http://en.wikipedia.org/wiki/Selberg_trace_formula)_

* {#Bump} Bump, theorem 19 in _Spectral theory of $\Gamma \backslash SL(2,\mathbb{R})$_ ([[BumpSpectralTheory.pdf:file]])

* {#VorosBalasz86} A. Voros and N.L. Balasz, _Chaos on the pseudosphere_, Physics Reports 143 no. 3 (1986)

For [[Dirac operators]]:

* Jens Bolte, Hans-Michael Stiepan, _The Selberg trace formula for Dirac operators_ ([arXiv:math-ph/0607010](http://arxiv.org/abs/math-ph/0607010))


Selberg zeta function

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## Idea

Motivated by the resemblance of the [[Selberg trace formula]] to Weil's formula for the sum of zeros of the [[Riemann zeta function]], ([Selberg 56](#Selberg56)) defined for any compact [[hyperbolic manifold|hyperbolic]] [[Riemann surface]] a [[zeta function]]-like expression, the _[[Selberg zeta function of a Riemann surface]]_. (e.g. [Bump, below theorem 19](#Bump)). There is also a Selberg zeta function "of odd type" for odd-dimensional manifolds  ([Millson 78](#Millson78), [Bunke-Olbrich 94a, prop. 4.5](#BunkeOlbrich94a)).

## Definition

### For even-dimensional manifolds

(...)

### For odd-dimensional manifolds
 {#DefinitionForOdd}

Let

* $n \in \mathbb{N}$, $n \geq 1$;

* $G = SO(n,1)$ the [[Lorentz group]] or $G = Spin(n,1)$ its [[spin group]];

* $K = SO(n)$ the [[special orthogonal group]] or $K = Spin(n)$ the [[spin group]];

* $K \hookrightarrow G$ the canonical inclusion, exhibiting a [[maximal compact subgroup]];

* $G = K A N$ an [[Iwasawa decomposition]];

* $\Gamma \hookrightarrow G$ a [[discrete group|discrete]] [[subgroup]];

* $E$ a [[complex vector space|complex]] [[finite dimensional vector space]]

* $\chi \colon \Gamma \to U(E)$ a [[unitary representation]] of $\Gamma$.

Then the [[quotient]]

$$
  X \coloneqq \Gamma \backslash G / K
$$

is a [[hyperbolic manifold]] of odd [[dimension]] with [[fundamental group]] being $\pi_1(X) \simeq \Gamma$. Accordingly the representation $\chi$ is equivalently a [[flat vector bundle]] on $X$.

Write $Conj(\Gamma)$ for the set of [[conjugacy classes]] of $\Gamma$ and write

$$
  Prim(\Gamma) \hookrightarrow Conj(\Gamma)
$$

for the [[subset]] of elements $[g]$ for which $n_\Gamma(g) = 1$. Regarded as elements of the [[fundamental group]] as above, these elements correspond to paths which are [[prime geodesics]] in $X$.

**Definition**

The _Selberg zeta function_ $\zeta_\chi$ of this data is defined for $Re(s)\gt \rho \coloneqq (n-1)/2$ to be the [[infinite product]]

$$
  \zeta_\chi(s) =
  \underset{{[g] \in Prim(\Gamma)} \atop {[g] \neq 1}}{\prod}
  \;
  \underoverset{k = 0}{\infty}{\prod}
  \det\left(
    1 -
    e^{-(\rho + s)l(g)}
    S^k(Ad(g)^{-1}_{\mathbf{n}})
    \sigma(m)
    \;
    \chi(g)
  \right)
$$

(...) ([BunkeOlbrich 94a, def. 4.1](#BunkeOlbrich94a))

## Properties

### Analogy with Artin L-function
 {#AnalogyWithArtinLFunction}

That the Selberg zeta function is equivalently an [[Euler product]] of [[characteristic polynomials]] is due to ([Gangolli 77, (2.72)](#Gangolli77), [Fried 86, prop. 5](#Fried86)).

That it is in particular the Euler product of [[characteristic polynomials]] of the [[determinants]] of the [[monodromies]] of the [[flat connection]] corresponding to the given [[group representation]] (similar to the [[Ruelle zeta function]]) is ([Bunke-Olbrich 94, prop. 6.3](#BunkeOlbrich94)) for the even-dimensional case and ([Bunke-Olbrich 94a, def. 4.1](#BunkeOlbrich94a)) for the odd-dimensional case. (Or rather, the [[Ruelle zeta function]] ([Bunke-Olbrich 94a, def. 5.1](#BunkeOlbrich94a))).

This is [[analogy|analogous]] to the standard definition of an [[Artin L-function]] if one interprets a) a [[Frobenius map]] $Frob_p$ (as discussed there) as an element of the [[arithmetic fundamental group]] of an [[arithmetic curve]] and b) a [[Galois representation]] as a [[flat connection]].

So under this analogy the Selberg zeta function for hyperbolic 3-manifolds as well as the [[Artin L-function]] for a [[number field]] both are like an [[infinite product]] over primes ([[prime geodesics]] in one case, [[prime ideals]] in the other, see also at _[Spec(Z) -- As a 3-dimensional space containing knots](Spec%28Z%29#As3dSpaceContainingKnots)_) of determinants of monodromies of the given flat connection.

See at _[Artin L-function -- Analogy with Selberg zeta function](Artin%20L-function#AnalogyWithSelbergZeta)_ for more. This analogy has been highlighted in ([Brown 09](#Brown09), [Morishita 12, remark 12.7](#Morishita12)).

### Relation to the eta-function
 {#RelationToTheEtaFunction}

Under suitable conditions, the Selberg zeta function of odd type is an exponential of the [[eta function]] of a suitable [[Dirac operator]]

$$
  \zeta_S(0) = \exp\left(i \pi \eta_D(0)\right)
$$

([Millson 78](#Millson78), [Bunke-Olbrich 94a, prop. 4.5](#BunkeOlbrich94a), [Park 01, theorem 1.2](#Park01), [Guillarmou-Moroianu-Park 09](#GuillarmouMoroianuPark09)).

### Relation to analytic torsion
 {#RelationToAnalyticTorsion}

The [[Ruelle zeta function]] at 0 gives a power of [[analytic torsion]]

([Fried 86](#Fried86), [Bunke-Olbrich 94a, theorem 5.5.](#BunkeOlbrich94a))

* [analytic torsion -- relation to Selberg zeta function](analytic+torsion#RelationToSelbergZeta)

### Relation to prime geodesic asymptotics

The Selberg zeta function controls the [[asymptotics]] of [[prime geodesics]] via the [[prime geodesic theorem]] in direct [[analogy]] to how the [[Riemann zeta function]] controls the asymptotics of [[prime numbers]] via the [[prime number theorem]].

## Related concepts

* [[Ruelle zeta function]]

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

## References

The original article is

* {#Selberg56} [[Atle Selberg]], _Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series_, Journal of the Indian Mathematical Society 20 (1956) 47-87.

Review includes

* Wikipedia, _[Selberg zeta function](http://en.wikipedia.org/wiki/Selberg_zeta_function)_

* {#Watkins} [[Matthew Watkins]], citation collection on _[Selberg trace formula and zeta functions](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics4.htm)_

* {#Bump} Bump, below theorem 19 in _Spectral theory of $\Gamma \backslash SL(2,\mathbb{R})$_ ([[BumpSpectralTheory.pdf:file]])

* _Selberg and Ruelle zeta functions for compact hyperbolic manifolds_ ([pdf](http://www.math.uni-bonn.de/people/xenia/Oberseminar%202014/Oberseminar%202014.pdf))

Expression of the Selberg/Ruelle zeta function as an [[Euler product]] of [[characteristic polynomials]] is due to

* {#Gangolli77} Ramesh Gangolli, _Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one_, Illinois J. Math. Volume 21, Issue 1 (1977), 1-41. ([Euclid](http://projecteuclid.org/euclid.ijm/1256049498))

* {#Fried86} [[David Fried]], _The zeta functions of Ruelle and Selberg. I_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 19 no. 4 (1986), p. 491-517 ([Numdam](http://www.numdam.org/item?id=ASENS_1986_4_19_4_491_0))

The analogy with the [[Artin L-function]] is highlighted in

* {#Brown09} Darin Brown, _Lifting properties of prime geodesics_, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 ([euclid](http://projecteuclid.org/euclid.rmjm/1239113439))

* {#Morishita12} [[Masanori Morishita]], section 12.1 of _Knots and Primes: An Introduction to Arithmetic Topology_, 2012 ([web](https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false))

Discussion of the relation between, on the one hand, [[zeta function of an elliptic differential operator|zeta function]] of [[Laplace operators]]/[[eta function of a self-adjoint operator|eta funcstions]] of [[Dirac operators]] and, on the other hand, Selberg zeta functions includes

* {#DHokerPhong86} [[Eric D'Hoker]] [[Duong Phong]], _Communications in Mathematical Physics_, Volume 104, Number 4 (1986), 537-545 ([Euclid](http://projecteuclid.org/euclid.cmp/1104115166))

* {#Sarnak87} [[Peter Sarnak]], _Determinants of Laplacians_, Communications in Mathematical Physics, Volume 110, Number 1 (1987), 113-120. ([Euclid](http://projecteuclid.org/euclid.cmp/1104159171))

* [[Ulrich Bunke]], [[Martin Olbrich]], Andreas Juhl, _The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one, theta functions, trace formulas and the Selberg zeta function_, Annals of Global Analysis and Geometry February 1994, Volume 12, Issue 1, pp 357-405

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

and for odd-dimensional spaces also in

* {#Fried86} [[David Fried]], _Analytic torsion and closed geodesics on hyperbolic manifolds_, Invent. math. 84, 523-540 (1986) ([pdf](http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN356556735_0084&divID=LOG_0026&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN356556735_0084%7C&targetFileName=PPN356556735_0084_LOG_0026.pdf&))

* {#Millson78} [[John Millson]], _Closed geodesic and the $\eta$-invariant_, Ann. of Math., 108, (1978) 1-39 ([jstor](http://www.jstor.org/stable/1970928))

* {#BunkeOlbrich94a} [[Ulrich Bunke]], [[Martin Olbrich]], _Theta and zeta functions for odd-dimensional locally symmetric spaces of rank one_ ([arXiv:dg-ga/9407012](http://arxiv.org/abs/dg-ga/9407012))

* {#BunkeOlbrich94b} [[Ulrich Bunke]], [[Martin Olbrich]] _$\Gamma$-Cohomology and the Selbeg zeta function_ ([arXiv:dg-ga/9411004](http://arxiv.org/abs/dg-ga/9411004))

* [[Ulrich Bunke]], [[Martin Olbrich]], _Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group_ ([arXiv:dg-ga/9603003](http://arxiv.org/abs/dg-ga/9603003))

* {#BunkeOlbrich95} [[Ulrich Bunke]], [[Martin Olbrich]], _Selberg zeta and theta functions: a differential operator approach_, Akademie Verlag 1995

* {#Park01} Jinsung Park, _Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps_ ([arXiv:0111175](http://arxiv.org/abs/math/0111175))

* {#Friedman04} [[Joshua Friedman]], _The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations_ ([arXiv:math/0410067](http://arxiv.org/abs/math/0410067))

* {#Friedman06} [[Joshua Friedman]], _Regularized determinants of the Laplacian for cofinite Kleinian groups with finite-dimensional unitary representations_, Communications in Mathematical Physics ([arXiv:math/0605288](http://arxiv.org/abs/math/0605288))

* {#GuillarmouMoroianuPark09} Colin Guillarmou, Sergiu Moroianu, Jinsung Park, _Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds_ ([arXiv:0901.4082](http://arxiv.org/abs/0901.4082))

[[!redirects Selberg zeta function of a Riemann surface]]


Selcan Aksoy

[[!redirects a]]


selection monad

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## Idea

In a [[cartesian closed category]] , given an [[object]] $S$, the _selection monad_, also known as the _select monad_, is the [[endofunctor]] $J_S(X) \mapsto [[X, S], X]$ (where $[-,-]$ denotes the [[internal hom]]).

This is a [[strong monad]].

There is a monad [[homomorphism]] from the selection monad to the [[continuation monad]] for $S$, $K_S(X) = (X \to S) \to S$, which sends $\epsilon \in J_S(X)$ to $\bar{\epsilon} \in K_S(X)$, where $\bar{\epsilon}(p) = p(\epsilon(p))$.

If we understand the [[continuation monad]] as mapping an object to the generalized [[quantifiers]] over it, with $S$ a generalized [[truth value]], a selection function for a generalized quantifier is an element of its [[preimage]] under the monad morphism.

For instance, a selection functional for the [[supremum]] functional $sup: (X \to S) \to S$, when it exists, applied to a function, $p: X \to S$, gives a point in $X$ at which $p$ attains its maximum value.

Due to the resemblance of an [[algebra over a monad|algebra]], $J_S(A) \to A$, to [[Peirce's law]] in logic, $((p \Rightarrow q) \Rightarrow p) \Rightarrow p $, $J_S$ is also called the _Peirce monad_ in ([Escardó-Oliva 2012](#Escardó-Oliva2012)).

## Properties

* There is a [[distributive law]] $T J_S \Rightarrow J_S T$ for every strong monad $T$ ([Fiore 2019](#Fiore2019)).

## Related entries

* [[monad in computer science]]

## References

* §2.5 of Kieburtz, Richard B., Borislav Agapiev, and James Hook. _Three monads for continuations_. Oregon Graduate Institute of Science and Technology, Department of Computer Science and Engineering, 1992.

* [[Martín Escardó]] and Paulo Oliva, _Selection Functions, Bar Recursion, and Backward Induction_, Mathematical Structures in Computer Science, 20(2):127–168, 2010, ([pdf](https://www.cs.bham.ac.uk/~mhe/papers/selection-escardo-oliva.pdf))

* [[Martín Escardó]] and Paulo Oliva, _What Sequential Games, the Tychonoff Theorem and the Double-Negation Shift have in Common_, ([pdf](https://www.cs.bham.ac.uk/~mhe/papers/msfp2010/Escardo-Oliva-MSFP2010.pdf))

*  {#Escardó-Oliva2012} [[Martín Escardó]] and Paulo Oliva, _The Peirce translation_, Annals of Pure and Applied Logic, 163(6):681–692, 2012, ([pdf](https://www.cs.bham.ac.uk/~mhe/papers/peirce.pdf)).

* [[Jules Hedges]], _The selection monad as a CPS transformation_, ([arXiv:1503.06061](https://arxiv.org/abs/1503.06061))

* Martin Abadi, [[Gordon Plotkin]], _Smart Choices and the Selection Monad_, ([arXiv:2007.08926](https://arxiv.org/abs/2007.08926))

* [[Marcelo Fiore]], _Fast-growing clones_, ([Talk at CT 2019](http://conferences.inf.ed.ac.uk/ct2019/slides/fiore.pdf))

[[!redirects select monad]]


selection theorem

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## Idea

Selection theory asks if given a [[multi-valued function]] $F\colon X \to Y$ does there exist a continuous selector $f\colon X \to Y$, i.e. a single-valued [[continuous function]] such that $f(x) \in F(x)$ for all $x \in X$. A _selection theorem_ states that under certain assumptions on $X, Y, F$ there is indeed a selector. Normally, such assumptions include that $X$ is [[paracompact topological space|paracompact]], $Y$ is some subset of a [[topological vector space]], $F$ is a [[lower semicontinuous map]] (also called hemicontinuous), and $F(x)$ is [[convex set|convex]] for each $x \in X$.

More generally, one may ask if there is a [[multi-valued function]] $G\colon X \to Y$ such that $G(x) \subset F(y)$ for all $y\in X$ and $G$ nicer behaved than $F$, e.g. $G$ lower semicontinuous and $G(x)$ compact for every $x\in G$ or $G$ single-valued and measurable.

## Selection theorems

\begin{theorem}[Michael selection theorem]
  Let $X$ be [[paracompact topological space|paracompact]], $Y$ a [[Banach space]], $F$ a [[lower semicontinuous map]], and $F(x)$ [[inhabited set|nonempty]], [[convex set|convex]], and [[closed subset|closed]] for every $x\in X$. Then $F$ admits a selector.
\end{theorem}

## Related entries

* [[multi-valued function]]
* [[Michael's theorem]]

## References

Michael selection theorem appeared in

* [[Ernest Michael]], _Continuous selections. I_, Annals of Math. __63__ (2): 361–382 ([doi:10.2307/1969615](http://dx.doi.org/10.2307/1969615), MR0077107)

Overviews of selection theorems is found in

* Dušan Repovš, Pavel V. Semenov, _Continuous Selections of Multivalued Mappings_. Dordrecht: Kluwer Academic Publishers 1998. ISBN 0-7923-5277-7.
* Dušan Repovš, Pavel V. Semenov (2014). "Continuous Selections of Multivalued Mappings", In Hart, K. P.; van Mill, J.; Simon, P. (eds.). Recent Progress in General Topology. _III_. Berlin: Springer. pp. 711–749.  [arXiv:1401.2257](https://arxiv.org/abs/1401.2257).

See also

* Wikipedia, _[Selection theorem](https://en.wikipedia.org/wiki/Selection_theorem)_

[[!redirects selection theory]]
[[!redirects selector]]


self-adjoint

[[!redirects self adjoint]]

The term *self-adjoint* is used in different senses, not all related:

* A [[self-adjoint functor]] is a functor that is [[adjoint functor|adjoint]] to itself.

* A [[self-adjoint operator]] and [[self-adjoint extension]] on a [[Hilbert space]].

* A [[self-adjoint morphism]] in a [[dagger category]].

category: disambiguation

[[!redirects self adjoints]]
[[!redirects self-adjoint]]
[[!redirects self-adjoints]]


self-adjoint extension

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## Idea

Given a [[symmetric operator]] on a domain of some [[Hilbert space]], there may be several _extensions_ of it to a [[self-adjoint operator]]. These typically correspond to choices of _boundary conditions_ ([WeiJiang](#WeiJiang)).

## Applications

### In quantum mechanics

In [[quantum mechanics]] the [[observable]]s are supposed to be [[self-adjoint operator]]s, in particular the [[Hamiltonian]]. The physical input often directly provides only a [[symmetric operator]], which encodes _local_ information about the [[dynamics]] of the system . The choice of its self-adjoint extension corresponds to choices of _boundary conditions_ on the [[state]]s of the system, hence _global_ information about the _[[kinematics]]_ .

See the [References on applications in quantum mechanics](#ReferencesApplicationsQM) below.

## References

See also [[Friedrichs extension]].

### General

Part VI, "Self-adjoint extension theory of symmetric operators" in

* [[Konrad Schmüdgen]], _Unbounded self-adjoint operators on Hilbert space_, Springer GTM 265, 2012

* {#WeiJiang} Guangshenh Wei, Yaolin Jiang, _A characterization of positive self-adjoint extensions and its application to ordinary differential operators_ Proceedings of the American Mathematical Society,  Volume  133, Number 10 ([jstor](http://www.jstor.org/pss/4097913)) ([pdf](http://www.ams.org/journals/proc/2005-133-10/S0002-9939-05-07837-8/S0002-9939-05-07837-8.pdf))

### For applications in quantum theory
 {#ReferencesApplicationsQM}

An exposition and motivation by means of the simple case of a quantum particle in an infinitely deep well potential is given in

* {#BonneauFarautValent01} Guy Bonneau, Jacques Faraut, Galliano Valent, _Self-adjoint extensions of operators and the teaching of quantum mechanics_, [arXiv:quant-ph/0103153](http://arxiv.org/abs/quant-ph/0103153)

A discussion in the context of [[local nets of observables]] in [[algebraic quantum field theory]]:

*  [[Hans-Jürgen Borchers]], [[Jakob Yngvason]], *Local nets and self-adjoint extensions of quantum field operators*, Letters in Mathematical Physics, **21** (1991) 151–155 [[doi:10.1007/BF00401649](https://doi.org/10.1007/BF00401649)]

Applications to [[quantum field theory]] of [[anyons]] is discussed in

* G. Amelino-Camelia, D. Bak, _Schrödinger self-adjoint sxtension and quantum field theory_, [arXiv:hep-th/9406213](http://arxiv.org/abs/hep-th/9406213)

category: analysis, physics
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self-adjoint functor

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## Idea

A *self-adjoint functor* is an [[endofunctor]] $F : C \to C$ such that $F \dashv F$, so that $F$ is both [[left adjoint]] and [[right adjoint]] to itself.

## Examples

- The [[duality involution]] $({-})^{op} : Cat \to Cat$ is self-adjoint. More generally, this is true of the underlying category of any [[2-category]] with a duality involution.
- If a category $C$ has [[biproducts]], then the composite $\oplus \circ \Delta_n$ of the (discrete $n$-ary) [[diagonal functor]] $\Delta_n$ with the ($n$-ary) biproduct functor $\oplus$ is self-adjoint.

## Functors self-adjoint on the left

There is a similar phenomenon involving a change of variance. A functor $F : C^{op} \to C$ is called **self-adjoint on the left** if $F \dashv F^{op}$. In this case, we have a natural isomorphism $C(F A, B) \cong C(F B, A)$. (Conversely, we may talk about functors **self-adjoint on the right** if $C(A, F B) \cong C(B, F A)$.)

- The contravariant powerset functor $\mathcal{P}: Set \to Set^{op}$ is left-adjoint to $\mathcal{P}^{op} : Set^{op} \to Set$, i.e. self-adjoint on the right.
- More generally, in a symmetric monoidal closed category $(C, \otimes, I, \multimap)$, for a fixed object $A$, the functor $(-) \multimap A$ is self-adjoint on the right.

## Related concepts
 {#RelatedConcepts}

* [[adjoint triple]]

* [[ambidextrous adjunction]]

[[!redirects self-adjoint functors]]

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self-adjoint morphism

# Contents
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## Definition

A [[morphism]] $f$ in a [[dagger-category]] is **self-adjoint** if

$$
  f^\dagger = f
  \,.
$$

For appropriate choices of dagger categories this restricts to various notions of self-adjointness:

* a **self-adjoint [[matrix]]** is a self-adjoint morphism in the $\dagger$-category whose objects are $\mathbb{C}^n$s and whose morphisms are [[linear map]]s and the dagger-operation is transposition followed by [[complex conjugation]];

* a *bounded* self-adjoint [[linear operator]] on a [[Hilbert space]] is defined everywhere and equals its own adjoint. The [[self-adjoint operator]] in general, unbounded case, is however not precisely equal to its adjoint as their domains may differ. As [[unbounded operator]]s do not form an algebra this notion is nevertheless not a special case of the above. See [[self-adjoint operator]].

## Related concepts

* **self-adjoint morphisms**

* [[unitary morphism]]

* [[compact self-adjoint operator]]

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self-adjoint operator

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## Definition

Let $A: H\to H$  be an [[unbounded operator]] on a [[Hilbert space]] $H$. An unbounded operator $A^*$ is its __[[adjoint operator|adjoint]]__ if

* $(A x|y) = (x|A^*y)$ for all $x\in dom(A)$ and $y\in dom(A^*)$; and
* every $B$ satisfying the above property for $A^*$ is a restriction of $A$.

An adjoint does not need to exist in general.

An unbounded operator is **symmetric** if $dom(A)\subset dom(A^*)$ and $A x = A^*x$ for all $x\in dom(A)$ (one also writes $A\subset A^*$).

The domain of $A^*$ is the set of all vectors $y\in H$ such that the linear functional
$x\mapsto (Ax|y)$ is bounded on $dom(A)$.

The graph $\Gamma_A\subset H\oplus H$ satisfies $\Gamma_{A^*} = \tau(\Gamma_A)^\perp$ where $\perp$ denotes the [[orthogonal complement]] and $\tau$ denotes the transposition of the direct summands changing the sign of one of the factors, i.e. $x\oplus y\mapsto -y\oplus x$. An unbounded operator $A$ is __closed__ if $\Gamma_A$ is closed subspace of $H\oplus H$. An operator $B$ is a closure of an operator $A$ if $\Gamma_B$ is a closure of operator $\Gamma_A$. It is said that $B$ is an __extension__ of $A$ and one writes $B\supset A$ if $\Gamma_B\supset \Gamma_A$. The closure of an unbounded operator does not need to exist.

For any unbounded operator $A$ with a dense $dom(A)\subset H$, if the adjoint operator $A^*$ exists, then $A^*$ is closed, and if $(A^*)^*$ exists then it coincides with a closure of $A$.

An [[unbounded operator]] $A : H\to H$ on a Hilbert space $H$ is __self-adjoint__ if

* it has a densely defined domain $dom(A)\subset H$
* $A = A^*$, i.e. $dom(A^*)= dom(A)$ and $A x = A^* x$ for all $x\in dom(A)$

An (unbounded) operator is __essentially self-adjoint__ if it is symmetric and its [[spectrum]] (as a subspace of the [[complex plane]]) is contained in the [[real line]]. Alternatively, it is symmetric if its closure is self-adjoint.

A __Hermitean__ (or __hermitian__) operator is a [[bounded operator|bounded]] symmetric operator (which is necessarily self-adjoint), although some authors use the term for any self-adjoint operator.

For a bounded operator $A: H\to K$ between Hilbert spaces, define the [[Hermitean conjugate]] operator $A^*: K\to H$ by $(Ax|y)_H = (x|A^*y)_K$, for all $x\in K$, $y\in H$. Distinguish it from the concept of the [[transposed operator]] $A^T: K^*\to H^*$ between the [[dual vector space|dual spaces]].

In an arbitrary $*$-[[star-algebra|algebra]], a __self-adjoint__ or __hermitian__ element is any element $A$ such that $A^* = A$.

## Related concepts

* [[self-adjoint extension]]

* [[unitary operator]]

* [[Hermitian form]], [[Hermitian matrix]]

* [[quantum observable]], [[daseinisation]]

* [[positive operator]]

## References

Original articles:

* {#vonNeumann30} [[John von Neumann]], §I in: *Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren*, Math. Ann. **102** (1930) 49–131 [[doi:10.1007/BF01782338](https://doi.org/10.1007/BF01782338)]

See also:

* A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988

* A. B. Antonevič, Ja. B. Radyno, Funkcional'nij analiz i integral'nye uravnenija, Minsk 1984

* S. Kurepa, _Funkcionalna analiza, elementi teorije operatora_, Školska knjiga, Zagreb 1981.

* Reed, M.; Simon, B.: _Methods of modern mathematical physics_. Volume 1, Functional Analysis

* Walter Rudin, _Functional analysis_

* Konrad Schmuedgen, _Unbounded self-adjoint operators on Hilbert space_, Springer GTM 265

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self-distributive operation

# Self-distributive operations
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## Definition

Let $\cdot\colon M \times M\to M$ be a binary operation, i.e. $(M,\cdot)$ is a [[magma]]. We say that the operation $\cdot$ is

* __left self-distributive__ if for all $x,y,z\in M$, $x\cdot(y\cdot z) = (x\cdot y)\cdot (x\cdot z)$;
* __right self-distributive__ if for all $x,y,z\in M$, $(y\cdot z)\cdot x = (y\cdot x)\cdot (z\cdot x)$.

See also [[shelf]].

## Examples

* The binary operation in any [[semilattice]] is self-distributive on both sides, following from associativity, commutativity, and idempotence.

* The operations in a [[rack]] (and hence also in a [[quandle]]) are self-distributive on the side on which they act.  In particular, this includes the operation of [[conjugation]] in a [[group]].

* A [[Laver table]] is the multiplication table of a self-distributive operation.

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self-dual higher gauge theory

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## Idea

The standard [[action functional]] for the [[higher U(1)-gauge field]] given by a [[circle n-bundle with connection]] $(P, \nabla)$ over a [[pseudo-Riemannian manifold|(pseudo)]] [[Riemannian manifold]] $(X,g)$ is

$$
  \nabla \mapsto \int_X F_\nabla \wedge \star_g F_{\nabla}
  \,,
$$

where $F_\nabla$ is the [[curvature]] $(n+1)$-form. If the [[dimension]]

$$
  dim X = 4 k + 2
$$

then the [[Hodge star]] operator squares to $+1$ (Lorentzian signature) or $-1$ (Euclidean signature) on $\Omega^{k+1}(X)$. Therefore it makes sense in these dimensions to impose the _self-duality_ or _chirality_ constraint

$$
  \pm F_\nabla = \star F_\nabla
  \,.
$$

With this duality constraint imposed, one speaks of **self-dual higher gauge fields** or **chiral higher gauge fields** or **higher gauge fields with self-dual curvature**. (These are a higher degree/dimensional generalization of what in [[Yang-Mills theory]] are called [[Yang-Mills instanton]] field configurations.)

Since imposing the self-duality constraint on the fields that enter the above [[action functional]] makes that functional vanish identically, self-dual higher gauge theory is notorious for being subtle in that either it does not have a [[Lagrangian field theory]] description, or else a somewhat intricate indirect one (e.g. [Witten 96, p. 7](#Witten96), [Belov-Moore 06a](#BelovMooreI)). But instead one may regard the self-duality condition rather as part of the quantum theory ([Witten 96, p.8](#Witten96), [Witten 99, section 3](#Witten99) [DMW 00b, page 3](#MW00)), namely as a choice of [[polarization]] of the [[phase space]] of an unconstrained theory in one dimension higher. By such as "[[holographic principle]]" the [[partition function]] of the self-dual theory on an $X$ of dimension $4 k +2$ is given by the [[state]] (wave function) of an abelian  [[higher dimensional Chern-Simons theory]] in dimension $4 k + 3$.

The way this works is understood in much mathematical detail for $k = 0$, see at _[[AdS3-CFT2 and CS-WZW correspondence]]_. Motivated by this it has been proposed and studied in a fair bit of mathematical detail for $k = 1$ ([Witten 96](#Witten96), [[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 02]]), see at _[[M5-brane]]_ and _[[6d (2,0)-superconformal QFT]]_ and [AdS7/CFT6](AdS-CFT#AdS7CFT6). In both these cases the [[higher gauge fields]] are [[cocycle]]s in [[ordinary differential cohomology]]. In ([DMW 00](#MW00), [Belov-Moore 06b](#BelovMooreII)) it is suggested that similarly taking the self-dual fields to be cocycles in ([[differential K-theory|differential]]) [[complex K-theory]] produces the [[RR-fields]] of [[type II superstring theory]] in dimension 10.

## Definition (outline)
 {#Definition}

The [[holographic principle|holographic]] definition of self-dual abelian higher gauge field theory in dimension $4k+2$ given in ([Witten 96](#Witten96), [Witten 99](#Witten99)) is in outline the following.

Imagine the self-dual $2k$-form [[field (physics)|field]] on $\Sigma$ to be coupled to [[sources]] given by $(2k+1)$-form gauge fields, hence by [[cocycles]] in [[ordinary differential cohomology]] $[\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}]$ of degree $(2k+2)$. One may try to write down a [[Lagrangian]] for this coupling (and that is what is discussed in ([Belov-Moore 06I](#BelovMooreI))) and the resulting [[partition function]] is then locally a function of the [[source fields]], hence a function on the [[intermediate Jacobian]] $[\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}]$ and globally (since the [[action functional]] will not be strictly [[gauge invariance|gauge invariant]]) a holomorphic [[section]] of a certain [[holomorphic line bundle]] on that space.

However, no choice of [[Lagrangian]] will enforce a strictly self-dual gauge field (the anti-self dual component will remain in the space of fields, hence the partition function will depend on it). Therefore the idea is to turn this around, reject the idea that self-dual higher gauge theory is directly itself a [[Lagrangian quantum field theory]], and instead _define_ the partition function of the self-dual higher gauge theory (and thereby, indirectly, the self-dual higher gauge theory itself!) as being a certain section of a certain line bundle on the [[intermediate Jacobian]].

The question then is: which line bundle? Some plausibility arguments show that it must be a [[Theta characteristic]] bundle, hence a [[square root]] of the [[canonical bundle]] on the [[intermediate Jacobian]].  These have 1-dimensional spaces of holomorphic sections ([[theta functions]]) and hence any choice of that already uniquely fixes the would-be [[partition function]], too, up to a choice of global factor. (For that reason later authors often regard the partition function as a line bundle, somewhat abusing the terminology.)

The claim then is that the correct choice of [[Theta characteristic]] to use is a [[quadratic refinement]] of the [[intersection pairing]] (the [[Beilinson-Deligne cup-product]])

$$
  [\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}]
  \times
  [\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}]
  \stackrel{\cup}{\longrightarrow}
  [\Sigma, \mathbf{B}^{4k+3}\mathbb{G}^\times_{conn}]
  \stackrel{\int}{\longrightarrow}
  \mathbf{B}\mathbb{G}^\times_{conn}
  \,.
$$

Such [[quadratic refinement]] turns out to be given for $k = 0$ by a [[Spin structure]] (leading to "[[Spin Chern-Simons theory]]") and for $k =1$ by a [[Wu structure]] (and the latter case is the actual example of interest in ([Witten 96](#Witten96))). This is what the central theorem of ([[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 02]]) establishes rigorously.

One notices now that while the self-dual $(4k+2)$-dimensional gauge field theory thus itself is not a [[Lagrangian quantum field theory]], it arises [[holographic principle|holographically]] form a field theory in dimension $4k+3$ that is, namely the [[intersection pairing]] above is the [[Lagrangian]] for [[higher dimensional Chern-Simons theory]] and the [[holomorphic line bundle]] which it gives rise to on the [[intermediate Jacobian]], is the [[prequantum line bundle]] of Chern-Simons theory, whose holomorphic sections are its [[quantum states]] (see at _[[quantization of Chern-Simons theory]]_). From this perspective the [[square root]] [[quadratic refinement]] is the [[metaplectic correction]] in the [[geometric quantization]] of Chern-Simobs theory.

This kind of relation

| self-dual $(4k+2)$d gauge theory |   | $(4k+3)$d Chern-Simons theory |
|---|---|---|
| [[sources]] | $\leftrightarrow$ |  [[field (physics)|fields]] |
| [[partition function]]/[[correlator]] | $\leftrightarrow$ | [[wavefunction]]/[[quantum state]] |
| [[conformal blocks]] | $\leftrightarrow$ | [[space of quantum states]] |

is the hallmark of the [[holographic principle]].

## Properties

### Holographic relation to higher Chern-Simons theory

#### Idea and examples

Generally, [[higher dimensional Chern-Simons theory]]
in dimension $4k+3$ (for $k \in \mathbb{N}$) is [[holographic principle|holographically]] related to self-dual higher gauge theory in dimension $4k+2$ (at least in the abelian case).

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

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

* $(k=2)$: [[11-dimensional Chern-Simons theory]] (of [[field (physics)|fields]] which are [[cocycles]] in ([[twisted differential K-theory|twisted differential]]) [[complex K-theory]]) is related to a parts of a [[type II string theory]] on its boundary (or that of the space-filling [[D9-brane]], if one wishes) ([Belov-Moore 06b](#BelovMooreII)).

#### Conformal structure from polarization
 {#ConformalStructureFromPolarization}

We indicate why [[higher dimensional Chern-Simons theory]] is -- if holographically related to anything -- holographically related to [[self-dual higher gauge theory]].

The [[phase space]] of [[higher dimensional Chern-Simons theory]] in [[dimension]] $4k+3$ on $\Sigma \times \mathbb{R}$ can be identified with the space of [[curvature|flat]] $2k+1$-forms on $\Sigma$. The [[presymplectic form]] on this space is given by the pairing

$$
  (\delta B_1, \delta B_2) \mapsto
  \int_\Sigma \delta B_1 \wedge \delta B_2
$$

obtained as the [[integration of differential forms]] over $\Sigma$ of the [[wedge product]] of the two forms.

The [[geometric quantization]] of the theory requires that we choose a [[polarization]] of the [[complexification]] of this space (split the space of forms into "[[canonical coordinates]]" and their "[[canonical momenta]]").

One way to achieve this is to choose a [[conformal structure]] on $\Sigma$. The corresponding [[Hodge star operator]]

$$
  \star : \Omega^{2k+1}(\Sigma) \to \Omega^{2k+1}(\Sigma)
$$

provides the polarization by splitting into self-dual and anti-self-dual forms:

notice that (by [this Formula](Hodge+star+operator#eq:HodgeSquareOnRiemannian) formulas at [[Hodge star operator]]) we have on mid-dimensional forms

$$
  \star \star B = (-1)^{(2k+1)(4k+3)} B
  = - B
  \,.
$$

Therefore it provides a [[complex structure]] on $\Omega^{2k+1}(\Sigma) \otimes \mathbb{C}$.

We see that the symplectic structure on the space of forms can equivalently be rewritten as

$$
  \begin{aligned}
    \int_X B_1 \wedge B_2
    & = - \int_X B_1 \wedge \star \star B_2
  \end{aligned}
  \,.
$$

Here on the right now the [[Hodge star operator|Hodge inner product]] of $B_1$ with $\star B_2$ appears, which is invariant under applying the Hodge star to both arguments.

We then decompose $\Omega^{2k+1}(\Sigma)$ into the $\pm i$-[[eigenspace]]s of $\star$: say $B \in \Omega^{2k+1}(\Sigma)$ is  _imaginary self-dual_ if

$$
  \star B = i B
$$

and _imaginary anti-self-dual_ if

$$
  \star B = - i B
  \,.
$$

Then for imaginary self-dual $B_1$ and $B_2$ we find that the symplectic pairing is

$$
  \begin{aligned}
    (B_1, B_2)
    &=
    -i \int_X B_1 \wedge \star B_2
    \\
    & =
    -i \int_X (\star B_1) \wedge \star (\star B_2)
    \\
    & =
    +i \int_X B_1 \wedge \star B_2
  \end{aligned}
  \,.
$$

Therefore indeed the symplectic pairing vanishes on the self-dual and on the anti-selfdual forms. Evidently these provide a decomposition into [[Lagrangian subspaces]].

(See also at _[[Serre duality]]_.)

Therefore a [[state]] of higher Chern-Simons theory on $\Sigma$ may locally be thought of as a function of the self-dual forms on $\Sigma$. Under holography this is (therefore) identified with the [[correlator]] of a [[self-dual higher gauge theory]] on $\Sigma$.

### Partition function
 {#PartitionFunction}

By the above discussion (...) the [[partition function]] of self-dual higher gauge theory is given by (a multiple of) the unique [[holomorphic section]] of a [[square root]] of the [[line bundle]] classified by the [[secondary intersection pairing]]. ([Witten 96](#Witten96), [Hopkins-Singer](#HopkinsSinger)).

### Cohomological description

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

## Examples

### Chiral boson in 2d (on the string)
 {#ChiralBosonin2d}

For $k = 0$ the self-dual theory abelian is that of a [[scalar field]] $\phi$ on a real-2-dimensional surface $\Sigma$ such that $\mathbf{d}\phi$ is self-dual. For any [[complex structure]] on $\Sigma$, making it a [[complex torus]], this makes $\phi$ a "chiral" boson, in the sense of a chiral half of the $U(1)$-[[WZW model]].

Original discussion is due to [Srivastava 1989](#Srivastava89), [GGR 1992](#GGR92). A quick review as a warm-up for the higher dimensional case is in  [Witten 96, section 2](#Witten96). More discussion is in [GBMNV](#GBMNV) [Miao & Müller-Kirsten 2000](#MiaoMüller-Kirsten00), see also the references at _[[AdS3-CFT2 and CS-WZW correspondence]]_.

### Chiral 2-form in 6d (on the M5-brane)

The [[worldvolume]] theory of the [[M5-brane]], the
[[6d (2,0)-superconformal QFT]], contains a self-dual 2-form field. Its [AdS7-CFT6 holographic](AdS-CFT#AdS7CFT6) description by 7-dimensional Chern-Simons theory is due to ([Witten 96](#Witten96)).

[[!include relation between 5d Maxwell theory and self-dual 3-forms in 6d -- section]]

### RR-Fields in 10d (on the "9-brane")
 {#RRFieldsin10d}

The [[RR-field]] in [[type II string theory]] (by the idea of [[K-theory classification of D-brane charge]]) are meant to be self-dual as [[cocycles]] in ([[differential K-theory|differential]]) [[topological K-theory]] (cf. at *[[pre-metric electromagnetism]]* the *[section on RR-fields](pre-metric+electromagnetism#RRFieldInGravitationalBackground)*).

The proper discussion of these fields hence involves generalizing the above story from [[ordinary cohomology]] and [[principally polarized variety|principally polarized]] [[intermediate Jacobians]] to an analog in [[Whitehead-generalized cohomology]] such as [[topological K-theory]] and an appropriate generalized form of intermediate Jacobians (cf. *[[Hodge-filtered differential cohomology]]*).

(Of course one may, as a warmup, "approximate" K-theory classes on a 10-manifold by integral cohomology lifts of the degree-5 component of their [[Chern character]] and hence discuss that as abelian self-dual higher gauge theory in dimension 10. This is discussed in ([Witten 99, section 4](#Witten99))).

The relevant analog of the [[intermediate Jacobian]] for K-theory on a 10-manifold is naturally taken to be

$$
  K(X)\otimes_{\mathbb{Z}} \mathbb{R}/ K(X)
$$

(where the action is via the [[Chern character]]/realification $K \to K \otimes \mathbb{R}$).

This is proposed and discussed in ([Witten 99, section 4.3](#Witten99), [Moore-Witten 99, section 3](#MooreWitten99), [DMW 00, section 7.1](#MW00), [Belov-Moore 06b, section 5](#BelovMooreII), [MPS 11](#MSP11)):

where for ordinary cohomology the ([[quadratic refinement]] of the) [[intersection product]] (= [[Deligne-Beilinson cup product]] followed by [[fiber integration in ordinary differential cohomology]]) provided the [[symplectic structure]]/[[polarized variety|polarization]] of the intermediate Jacobian, here it is tensoring of [[Dirac operators]] followed by [[fiber integration in differential K-theory]], hence the [[index of a Dirac operator|index]] map in K-theory.

See at _[intermediate Jacobian -- For complex K-theory](intermediate+Jacobian#ForComplexKTheory)_.

## Related concepts

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

* [[non-Lagrangian field theory]]

* [[Perry-Schwarz action]]

* [[Kervaire invariant]]

* [[bosonization]]

* [[higher dimensional Chern-Simons theory]]

* [[holographic principle]], [[AdS-CFT]]

* [[dual graviton]]

## References

### General

Original articles on the general issue include

* {#HenneauxTeitelboim88} [[Marc Henneaux]], [[Claudio Teitelboim]], _Dynamics of chiral (self-dual) $p$-forms_, Physics Letters B Volume 206, Issue 4, 2 June 1988, Pages 650-654 (doi:10.1016/0370-2693(88)90712-5)

* {#PastiSorokinTonin06} [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On Lorentz Invariant Actions for Chiral P-Forms_, Phys.Rev. D55 (1997) 6292-6298 ([arXiv:hep-th/9611100](https://arxiv.org/abs/hep-th/9611100))

* {#Witten99} [[Edward Witten]], _Duality Relations Among Topological Effects In String Theory_, JHEP 0005:031,2000 ([arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086))

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

Surveys include

* [[Greg Moore]], _A Minicourse on Generalized Abelian Gauge Theory, Self-Dual Theories, and Differential Cohomology_, Lectures at Simons Center for Geometry and Physics (2011) ([pdf](http://www.physics.rutgers.edu/~gmoore/SCGP-Minicourse.pdf))

* {#Szabo12} [[Richard Szabo]], _Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology_ ([arXiv:1209.2530](http://arxiv.org/abs/1209.2530))

An approach based on [[supergeometry]]:

* [[C. A. Cremonini]], [[Pietro Grassi]], _Self-Dual Forms in Supergeometry I: The Chiral Boson_ ([arXiv:2012.10243](https://arxiv.org/abs/2012.10243))

### For the M5-sigma model

The [[sigma-model]] description of the (single) M5-brane of [[Green-Schwarz action functional]]-type was found in covariant form in

* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. 78 (1997) 4332-4334 ([arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149))

* {#PST97} [[Paolo Pasti]], [[Dmitri Sorokin]] and M. Tonin, _Covariant Action for a D=11 Five-Brane with the Chiral Field_, Phys. Lett. B398 (1997) 41 ([arXiv:hep-th/9701037](https://arxiv.org/abs/hep-th/9701037))

following

* [[Paul Townsend]], section 3.3 of _D-branes from M-branes_, Phys. Lett. B373 (1996) 68-75 ([arXiv:hep-th/9512062](https://arxiv.org/abs/hep-th/9512062))

  (which did not yet have the [[Hopf-Wess-Zumino term]])

and following the non-covariant form of the [[self-dual higher gauge field|self-duality mechanism]] ([[Perry-Schwarz action]]) of

* {#PerrySchwarz96} [[Malcolm Perry]], [[John Schwarz]], _Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory_, Nucl. Phys. B489 (1997) 47-64 ([arXiv:hep-th/9611065](http://arxiv.org/abs/hep-th/9611065))

* {#Schwarz97} [[John Schwarz]], _Coupling a Self-Dual Tensor to Gravity in Six Dimensions_, Phys.Lett.B395:191-195,1997

* [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], _World-Volume Action of the M Theory Five-Brane_ ([arXiv:hep-th/9701166](http://arxiv.org/abs/hep-th/9701166))

Discussion in the [[superembedding approach]] is in

* {#Sorokin99} [[Dmitri Sorokin]], Section 5.2 of _Superbranes and Superembeddings_, Phys.Rept.329:1-101, 2000 ([arXiv:hep-th/9906142](http://arxiv.org/abs/hep-th/9906142))

Discussion of the equivalence of these superficially different action functionals is in

* [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On the equivalence of different formulations of the M Theory five--brane_, Phys. Lett. B408 (1997) 135-141 ([arXiv:hep-th/9703127](http://arxiv.org/abs/hep-th/9703127))

Discussion in the [[D'Auria-Fré formulation of supergravity]]:

* {#ACDGMNRT22} [[Laura Andrianopoli]], [[C. A. Cremonini]], [[Riccardo D'Auria]], [[Pietro A. Grassi]], R. Matrecano, R. Noris, [[Lucrezia Ravera]], [[Mario Trigiante]], *M5-brane in the superspace approach*, Phys. Rev. D **106** 2 (2022) 026010 [[arXiv:2206.06388](https://arxiv.org/abs/2206.06388), [doi:10.1103/PhysRevD.106.026010](https://doi.org/10.1103/PhysRevD.106.026010)]

### For ordinary $U(1)$-higher gauge fields / ordinary differential cohomology

Self-duality for higher abelian gauge fields (in [[ordinary differential cohomology]]):

Original reference on self-dual/chiral fields include

* [[Xavier Bekaert]], [[Marc Henneaux]], _Comments on Chiral $p$-Forms_,  Int. J. Theor. Phys. 38:1161-1172, 1999 ([arXiv:hep-th/9806062](http://arxiv.org/abs/hep-th/9806062))

* [[Mans Henningson]], [[Bengt Nilsson]], Per Salomonson, _Holomorphic factorization of correlation functions in (4k+2)-dimensional (2k)-form gauge theory_ ([arXiv:hep-th/9908107](http://arxiv.org/abs/hep-th/9908107))

* [[Mans Henningson]], _The quantum Hilbert space of a chiral two-form in $d = 5 + 1$ dimensions_ ([arxiv:hep-th/0111150](http://arxiv.org/abs/hep-th/0111150))

The chiral boson in 2d:

* {#Srivastava89} Prem P. Srivastava, *Quantization of self-dual field revisited*, Phys. Rev. Lett. **63** (1989) 2791 [[doi:10.1103/PhysRevLett.63.2791](https://doi.org/10.1103/PhysRevLett.63.2791)]

* {#GGR92} H. O. Girotti, M. Gomes, and V. O. Rivelles, *Chiral bosons through linear constraints*, Phys. Rev. D **45** (1992) R3329(R) [[doi:10.1103/PhysRevD.45.R3329](https://doi.org/10.1103/PhysRevD.45.R3329)]

* {#GBMNV} [[Luis Alvarez-Gaumé]], [[Jean-Benoit Bost]], [[Greg Moore]], Philip Nelson, [[Cumrun Vafa]], _Bosonization On Higher Genus Riemann Surfaces_, Commun. Math. Phys. 112 (1987) 503 [[doi:10.1007/BF01218489](https://doi.org/10.1007/BF01218489), [pdf](https://core.ac.uk/download/pdf/76393468.pdf)]

* {#MiaoMüller-Kirsten00} Yan-Gang Miao and H. J. W. Müller-Kirsten, *Self-duality of various chiral boson actions*, Phys. Rev. D **62** (2000) 045014 [[doi:10.1103/PhysRevD.62.045014](https://doi.org/10.1103/PhysRevD.62.045014)]

A quick exposition of the basic idea is in

* [[Jacques Distler]], _Actions for self-dual gauge fields_ ([blog](http://golem.ph.utexas.edu/~distler/blog/archives/000809.html))

A precise formulation of the phenomenon in terms of [[ordinary differential cohomology]] and [[differential K-theory]]:

* [[Daniel Freed]], [[Greg Moore]], [[Graeme Segal]], *The Uncertainty of Fluxes*, Commun. Math. Phys. **271**  (2007) 247-274 [[arXiv:hep-th/0605198](http://arxiv.org/abs/hep-th/0605198), [doi:10.1007/s00220-006-0181-3](https://doi.org/10.1007/s00220-006-0181-3)]

* [[Daniel Freed]], [[Greg Moore]], [[Graeme Segal]], *Heisenberg Groups and Noncommutative Fluxes*,  Annals Phys. **322** (2007) 236-285 [[arXiv:hep-th/0605200](http://arxiv.org/abs/hep-th/0605200), [doi:10.1016/j.aop.2006.07.014](https://doi.org/10.1016/j.aop.2006.07.014)]

* {#BBSS15} [[Christian Becker]], [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Abelian duality on globally hyperbolic spacetimes_, Commun. Math. Phys. **349** (2017) 361-392 [[arXiv:1511.00316](https://arxiv.org/abs/1511.00316), [doi:10.1007/s00220-016-2669-9](https://doi.org/10.1007/s00220-016-2669-9)]

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

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

* [[Edward Witten]], _Duality Relations Among Topological Effects In String Theory_, JHEP 0005:031,2000 ([arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086))

Conceptual aspects of this are also discussed in section 6.2 of

* S. L. Lyakhovich, [[A. A. Sharapov]], _Quantizing non-Lagrangian gauge theories: an augmentation method_, JHEP 0701 047 (2007) [[arXiv:hep-th/0612086](http://arxiv.org/abs/hep-th/0612086)]

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

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

Discussion of [[S-duality]] in 6d self-dual higher gauge theory via [[non-commutative geometry|non-commutative]]-deformation:

* {#MathaiSati14} [[Varghese Mathai]], [[Hisham Sati]], _Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality_, J. Geom. Phys. 92:240-251, 2015 ([arXiv:1404.2257](https://arxiv.org/abs/1404.2257))

Discussion of the [[conformal blocks]] and [[geometric quantization]] of self-dual higher gauge theories is in

* [[Kiyonori Gomi]], _An analogue of the space of conformal blocks in $(4k+2)$-dimensions_ ([pdf](http://www.math.kyoto-u.ac.jp/~kgomi/papers/acb.pdf))

For the case of nonabelian self-dual 1-form gauge fields see the references at _[[Yang-Mills instanton]]_.

Another proposal for a Lagrangian for self-dual higher gauge fields, at the cost of an unusual kinds of decoupled [[auxiliary fields]]:

* {#Sen20} [[Ashoke Sen]], *Self-dual forms: Action, Hamiltonian and Compactification*,  Journal of Physics A: Mathematical and Theoretical, **53** 8 (2020) [[arXiv:1903.12196](https://arxiv.org/abs/1903.12196), [doi:10.1088/1751-8121/ab5423](https://doi.org/10.1088/1751-8121/ab5423)]

Enhancement to [[superspace]] (in [[D'Auroa-Fre formulation of supergravity]]):

* [ACDGMNRT22](#ACDGMNRT22)

Further discussion:

* {#Hull23} [[Chris Hull]], *Covariant Action for Self-Dual $p$-Form Gauge Fields in General Spacetimes* [[arXiv:2307.04748](https://arxiv.org/abs/2307.04748)]

### For RR-fields / differential K-theory

[[!include self-duality for pregeometric RR-fields -- references]]

### For the 11d supergravity C-field

[[!include self-duality for pregeometric C-field -- references]]

[[!redirects self-dual higher gauge theories]]

[[!redirects higher self-dual gauge theory]]
[[!redirects higher self-dual gauge theories]]

[[!redirects self-dual higher gauge field]]
[[!redirects self-dual higher gauge fields]]

[[!redirects higher self-dual gauge field]]
[[!redirects higher self-dual gauge field]]

[[!redirects self-dual field theory]]

[[!redirects chiral boson]]
[[!redirects chiral bosons]]


self-dual object

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#### Monoidal categories
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#Contents#
* table of contents
{:toc}

## Idea

In a [[monoidal category]] a _self-duality_ on a [[dualizable object]] $X$ is a choice of [[equivalence]] $X \simeq X^\ast$ with its [[dual object]].

## Properties

### Relation to $\dagger$-compact structure
 {#RelationToDaggerCompactStructure}

If each object $X$ of a [[compact closed category]] is equipped with a self-duality structure $h_X : X \simeq X^\ast$, then sending morphisms to their [[dual morphisms]] but with these identifications pre- and postcomposed

$$
  (-)^\dagger \;\colon\;
  (X \stackrel{f}{\longrightarrow} Y)
  \mapsto
  (Y \stackrel{h_Y}{\to} Y^\ast \stackrel{f^\ast}{\longrightarrow} X^\ast \stackrel{h_X^{-1}}{\to} X)
$$

constitutes a [[dagger-compact category]] structure.

See for instance ([Selinger, remark 4.5](#Selinger)).

Applied for instance to the category of finite-dimensional [[inner product spaces]] this dagger-operation sends [[matrices]] to their [[transposed matrix]].

## Graphical language
 {#GraphicalLanguage}

In terms of [[string diagrams]] (following Joyal and Street's conventions for [[braided monoidal categories]]), [Selinger](#Selinger) argues that the isomorphism $h_X : X \simeq X^\ast$ should be depicted as a half-twist.  In particular, for a [[tortile category]] equipped with a self-duality structure, the [[coherence]] condition

$$
(X^\ast \stackrel{h_{X^\ast}}{\to} X^{\ast\ast} \stackrel{h_X^\ast}{\to} X^\ast) \; =\;
(X^\ast \stackrel{\theta_{X^\ast}}{\longrightarrow} X^\ast)
$$

decomposes a full [[twist]] into a pair of half-twists.

This is a special case of half-twists as described by [Egger 2011](#Egger11).

## Related concepts

* [[inner product]]

## References

* {#Selinger} [[Peter Selinger]], _Autonomous categories in which $A \simeq A^\ast$_, talk at QPL 2012 ([[SelingerSelfDual.pdf:file]])

* [[Matthew B. Young]], §3.1 in *Self-Dual Hall modules*, PhD thesis, Stony Brook (2013) [[pdf](https://www.math.stonybrook.edu/alumni/2013-Matthew-Young.pdf), [[Young-HallModules.pdf:file]]]

  > (in a context relating to [[orientifold]]-[[BPS state|BPS]]-algebras)

* Chenjing Bu, Def. 3.2 in: *Enumerative invariants in self-dual categories. I. Motivic invariants* [[arXiv:2302.00038](https://arxiv.org/abs/2302.00038)]

On half-twists as [above](#GraphicalLanguage)

* {#Egger11} [[Jeff Egger]], *On involutive monoidal categories*, Theory and Applications of Categories **25** 14 (2011) 368-393 [[tac:25-14](http://www.tac.mta.ca/tac/volumes/25/14/25-14abs.html)]

{#InTQFT} Under the [[cobordism hypothesis]] (which is a theorem certainly for the relevant case $n = 1$), self-dual objects in [[symmetric monoidal (infinity,1)-categories|symmetric monoidal $\infty$-categories]] correspond equivalently to *un*-oriented 1-dimensional [[TQFTs]]:

* [[Jacob Lurie]]: Ex. 2.4.28 in: *[[On the Classification of Topological Field Theories]]*, Current Developments in Mathematics **2008** (2009) 129-280 [[arXiv:0905.0465](http://arxiv.org/abs/0905.0465), [doi:10.4310/CDM.2008.v2008.n1.a3](https://dx.doi.org/10.4310/CDM.2008.v2008.n1.a3), [euclid:cdm/1254748657](https://projecteuclid.org/euclid.cdm/1254748657)]

* [[Mikhail Khovanov]] (notes by [[You Qi]]), Thm. 4 on [p. 7 ](https://www.math.columbia.edu/~khovanov/cat2020/notes/2_TQFT.pdf#page=7) in §2 of: *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)]

* {#Teleman16} [[Constantin Teleman]], Rem. 1.7 in: *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]]]

same non-topological [[functorial field theory]]

* {#GradyPavlov21} [[Daniel Grady]], [[Dmitri Pavlov]], Thm 6.1.5 in: *The geometric cobordism hypothesis* [[arXiv:2111.01095](https://arxiv.org/abs/2111.01095)]

[[!redirects self-dual objects]]

[[!redirects self-dual]]

[[!redirects self-duality]]
[[!redirects self-dualities]]


self-dual string

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#Contents#
* table of contents
{:toc}

## Idea

The [[6d (2,0)-superconformal QFT]] admits a [[soliton]]/[[black brane]] solution which is a [[string]] charged under the [[self-dual higher gauge field|self-dual]] [[B-field]].

Externally this is the [[boundary field theory|boundary]] of the [[M2-brane]] [[brane intersection|ending]] on the [[M5-brane]]. As such this is "the string in [[M-theory]]" and hence is also called the _[[M-string]]_.

## Properties

### Elliptic genus and M5-brane charge

The [[elliptic genus]] of the self-dual string at the boundary of the [[M2-brane]] on the [[M5-brane]] has been suggested to be directly analogous to the [[Todd genus]] of the boundary of the [[open string|open]] [[superstring]] ending on [[D-branes]] and hence has been suggested to be the _[[M5-brane charge]]_ in ([Sati 10](#Sati10)). Discussion of the computation of this elliptic genus includes ([Hohenegger-Iqbal 13](#HoheneggerIqbal13)).

## Related concepts

* [[3-brane in 6d]]

[[!include table of branes]]

[[!include genera and partition functions - table]]

## References

As a [[black brane]] the self-dual string in 6d was maybe first seen in

* [[Michael Duff]], [[Jian Xin Lu]], *Black and super $p$-branes in diverse dimensions*, Nucl. Phys. B **416** (1994) 301-334 [[arXiv:hep-th/9306052](http://arxiv.org/abs/hep-th/9306052)]

The appearance of self-dual string in six dimensions with an [[ADE classification]] was first observed in

* {#Witten95} [[Edward Witten]], _Some Comments On String Dynamics_ ([arXiv:hep-th/9507121](http://arxiv.org/abs/hep-th/9507121))

As a [[Green-Schwarz sigma-model]] the [[brane scan]]-cocycle for the superstring in 6d is given in

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], (III.7.15) of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific, 1991

  (in [[D'Auria-Fré formulation of supergravity]])

Further discussion includes

* [[John Schwarz]], _Self-Dual Superstring in Six Dimensions_ ([arXiv:hep-th/9604171](https://arxiv.org/abs/hep-th/9604171))

* [[Paul S. Howe]], [[Neil Lambert]], [[Pete West]], _The Self-Dual String Soliton_, Nucl. Phys. B **515** (1998) 203-216 [[arXiv:hep-th/9709014](http://arxiv.org/abs/hep-th/9709014), doi:10.1016/S0550-3213(97)00750-5]

* {#Sati10} [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]]_, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 [arXiv:1001.5020](http://arxiv.org/abs/1001.5020),

In view of [[M-strings]] and the [[M-string elliptic genus]]:

* {#HoheneggerIqbal13} [[Stefan Hohenegger]], [[Amer Iqbal]], _M-strings, Elliptic Genera and N=4 String Amplitudes_ ([arXiv:1310.1325](http://arxiv.org/abs/1310.1325))

[[!redirects self-dual strings]]

[[!redirects self-dual string in 6d]]


self-dual Yang-Mills theory

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#Contents#
* table of contents
{:toc}

## Idea

A variant of [[Yang-Mills theory]] in which the [[field strength]]/[[curvature]] 2-form of the [[Yang-Mills field]] is constrained to be [[Hodge star operator|Hodge]] self-dual.

## Definition

### Via an action functional
 {#ViaActionFunctional}

Let $(X, g)$ be a ([[pseudo-Riemannian manifold|pseudo]]) [[Riemannian manifold]] of [[dimension]] 4. Write $\star \colon \Omega^2(X) \to \Omega^2(X)$ for the corresponding [[Hodge star operator]]. Its square is $\star^2 = +1$ for Euclidean [[signature of a quadratic form|signature]] and $\star^2 = -1$ for Lorentzian signature. Decompose (possibly after [[complexification]])

$$
  \Omega^2(X) \simeq \Omega^2(X)_+ \oplus \Omega^2(X)_-
$$

into the [[direct sum]] of [[eigenspaces]] of $\star$, the _self-dual_ and the _anti-self-dual forms_.

Let $G$ be a [[Lie group]]. Write $\mathfrak{g}$ for the corresponding [[Lie algebra]]. Let $\langle -,-\rangle$ be a binary [[invariant polynomial]] on the Lie algebra.

Accordingly we have

$$
  \Omega^2(X, \mathfrak{g}) \simeq \Omega^2(X, \mathfrak{g})_+ \oplus \Omega^2(X, \mathfrak{g})_-
  \,.
$$

The configuration space of self-dual Yang-Mills theory on $(X,g)$ is that of pairs $(\nabla, \mathcal{G})$ with

* $\nabla \in \mathbf{H}^1_{conn}(X,G)$ is a $G$-[[principal connection]] over $X$;

* $\mathcal{G} \in \Omega^2(X, \mathfrak{g})_-$ is an anti-self-dual 2-form.

The [[action functional]] of the theory is

$$
  (\nabla, \mathcal{G})
  \mapsto
  \int_X
  \langle
    (F_\nabla)_- \wedge \mathcal{G}
  \rangle
  dvol_g
  \,.
$$

### Via BV-complexes
 {#ViaBVComplexes}

Let $X$ be an oriented [[smooth manifold]] of [[dimension]] 4 equipped with a [[conformal structure]] with [[Hodge star]] operator $\star_g$. Let $G$ be a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$.

Let $P \to X$ be a $G$-[[principal bundle]] and write $\mathfrak{g}_P \coloneqq P \times_G \mathfrak{g}$ for the [[associated bundle]] via the [[adjoint action]] of the group on its Lie algebra. Fix a $G$-[[principal connection]] $\nabla_0$ on $P$ with self dual curvature $F_{\nabla_0} = 0 \in \Omega_-^2(X, \mathfrak{g}_P)$.

Consider then the [[chain complex]]

$$
  \array{
    &
    \Omega^0(X, \mathfrak{g}_P)
    &\stackrel{d_{\nabla_0}}{\to}&
    \Omega^1(X, \mathfrak{g}_P)
    &\stackrel{P_- \circ d_{\nabla_0}}{\to}&
    \Omega^2_-(X, \mathfrak{g}_P)
    \\
    \\
    deg =  & 1 & & 0 && -1
  }
  \,,
$$

where

* $d_\nabla$ is the [[de Rham differential]] coupled to the connection, hence the [[covariant derivative]] of $\nabla$;

* $P_-$ is the [[projection]] onto anti-self dual 2-forms.

This is a [[derived L-infinity algebroid]]  model for perturbations of self-dual connections about $\nabla_0$:

a [[field (physics)|field configuration]] is an element in degree 0, hence a differential 1-form $A \in \Omega^1(X, \mathfrak{g}_P)$, which is in the kernel of the differential, hence of self-dual curvature. A [[gauge transformation]] of this is an element $\lambda \Omega^1(X, \mathfrak{g}_P)$ transforming

$$
  A \mapsto A + d_{\nabla_0} \lambda
  \,.
$$

Consider then the [[action functional]] on this complex of fields which is simply zero. Then the corresponding local [[BV-complex]] (with local [[antibracket]] taking values in the [[densities]] on $X$) is

$$
  \array{
    &
    \Omega^0(X, \mathfrak{g}_P)
      &\stackrel{d_{\nabla_0}}{\to}&
    \Omega^1(X, \mathfrak{g}_P)
      &\stackrel{P_- \circ d_{\nabla_0}}{\to}&
    \Omega^2_-(X, \mathfrak{g}_P)
    \\
    & && \oplus && \oplus
    \\
    & && \Omega^2_-(X, \mathfrak{g}_P)
     &\to&
     \Omega^3(X, \mathfrak{g}_P)
    &\to&
      \Omega^4(X, \mathfrak{g}_P)
    \\
    \\
    deg =  & 1 & & 0 && -1 && -2
  }
  \,,
$$

This formulaton of self-dual Yang-Mills theory is considered in
([Costello-Gwilliam, section 4.12.3](#CostelloGwilliam)). There the grading is such that the Lie algebra of gauge transformations $\Omega^1(X,\mathfrak{g}_P)$ is in degree 0, whereas what is displayed above is the "delooped deived $L_\infty$-algebra".

## Properties

### Of the action functional

If one changes the action functional of self-dual Yang-Mills theory by adding a term

$$
 \cdots +  \epsilon \int_X \langle \mathcal{G} \wedge \mathcal{G}\rangle
$$

for some non-vanishing $\epsilon \in \mathbb{R}$, then it becomes equivalent to that of ordinary [[Yang-Mills theory]] in the form

$$
  \nabla \mapsto \frac{1}{\epsilon}
  \int_X
  \left(
    \langle F_\nabla \wedge \star F_\nabla \rangle
    -
    \langle F_\nabla \wedge F_\nabla \rangle
  \right)
  dvol_g
  \,.
$$

### Via the Penrose-Ward twistor transform

Solutions to the [[equations of motion]] of self-dual Yang-Mills theory are naturally produced by seding cohomology classes on [[twistor space]] through the [[Penrose-Ward twistor transform]]. See there for more details.

## Related concepts

* [[Bogomolny equation]], [[Nahm equation]]

* [[self-dual higher gauge theory]]

## References

### Via action functional

The action functional [above](#ViaActionFunctional) is due to

* Gordon Chalmers, Warren Siegel, _T-Dual Formulation of Yang-Mills Theory_, (1997) ([arXiv:hep-th/9712191](http://arxiv.org/abs/hep-th/9712191))

briefly reviewed at the beginning of

* M.V. Movshev, _A note on self-dual Yang-Mills theory_, [arXiv:0812.0224](http://arxiv.org/abs/0812.0224)

* [[Nigel Hitchin|N.J. Hitchin]], _The self-duality equations on a Riemann surface_, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126, [MR887284](http://www.ams.org/mathscinet-getitem?mr=887284) [doi](http://dx.doi.org/10.1112/plms/s3-55.1.59)

For self-dual [[super Yang-Mills theory]] a discusion is in

* E. Sokatchev, _An action for $N=4$ supersymmetric self-dual Yang-Mills theory_ ([arXiv:hep-th/9509099](http://arxiv.org/abs/hep-th/9509099))

See also

* H. J. de Vega, _Nonlinear multiplane wave solutions of self-dual Yang-Mills theory_ ([EUCLID](http://projecteuclid.org/euclid.cmp/1104161520))

### Via BV complex

The description via [[BV-complexes]] is amplified in the context of [[factorization algebras of observables]] in section 4.12.3 of

* [[Kevin Costello]], [[Owen Gwilliam]], _Factorization algebras in perturbative quantum field theory_ ([wiki](http://math.northwestern.edu/~costello/factorization_public.html), early/partial draft [pdf](http://math.northwestern.edu/~gwilliam/factorization.pdf))
 {#CostelloGwilliam}

[[!redirects self-dual Yang-Mills equation]]