Supergroup

Short description: Algebraic structure used in theoretical physics

The concept of supergroup is a generalization of that of group. In other words, every supergroup carries a natural group structure, but there may be more than one way to structure a given group as a supergroup. A supergroup is like a Lie group in that there is a well defined notion of smooth function defined on them. However the functions may have even and odd parts. Moreover, a supergroup has a super Lie algebra which plays a role similar to that of a Lie algebra for Lie groups in that they determine most of the representation theory and which is the starting point for classification.

## Contents

* 1 Details
* 2 Examples
* 3 Notes
* 4 References

## Details

More formally, a Lie supergroup is a supermanifold G together with a multiplication morphism μ:G×G->G, an inversion morphism i:G->G and a unit morphism e:1->G which makes G a group object in the category of supermanifolds. This means that, formulated as commutative diagrams, the usual associativity and inversion axioms of a group continue to hold. Since every manifold is a supermanifold, a Lie supergroup generalises the notion of a Lie group.

There are many possible supergroups. The ones of most interest in theoretical physics are the ones which extend the Poincaré group or the conformal group. Of particular interest are the orthosymplectic groups Osp(M|N)[1] and the superunitary groups SU(M|N).

An equivalent algebraic approach starts from the observation that a supermanifold is determined by its ring of supercommutative smooth functions, and that a morphism of supermanifolds corresponds one to one with an algebra homomorphism between their functions in the opposite direction, i.e. that the category of supermanifolds is opposite to the category of algebras of smooth graded commutative functions. Reversing all the arrows in the commutative diagrams that define a Lie supergroup then shows that functions over the supergroup have the structure of a Z2-graded Hopf algebra. Likewise the representations of this Hopf algebra turn out to be Z2-graded comodules. This Hopf algebra gives the global properties of the supergroup.

There is another related Hopf algebra which is the dual of the previous Hopf algebra. It can be identified with the Hopf algebra of graded differential operators at the origin. It only gives the local properties of the symmetries i.e., it only gives information about infinitesimal supersymmetry transformations. The representations of this Hopf algebra are modules. Like in the non-graded case, this Hopf algebra can be described purely algebraically as the universal enveloping algebra of the Lie superalgebra.

In a similar way one can define an affine algebraic supergroup as a group object in the category of superalgebraic affine varieties. An affine algebraic supergroup has a similar one to one relation to its Hopf algebra of superpolynomials. Using the language of schemes, which combines the geometric and algebraic point of view, algebraic supergroup schemes can be defined including super Abelian varieties.

## Examples

The Super-Poincaré group is the group of isometries of superspace (specifically, Minkowski superspace with 𝒩 supercharges, where often 𝒩 is taken to be 1). It is most often treated at the algebra level, and is generated by the Super-Poincaré algebra

The Super-conformal group is the group of conformal symmetries of superspace, generated by the super-conformal algebra.

## Notes

1. ↑ (M|N) is pronounced "M vertical bar N." Bosonic part of Osp(M|N) consists of the direct sum of Sp(N) and SO(M) Lie groups. See superspace for a general definition. (cf. Larus Thorlacius, Thordur Jonsson (eds.), M-Theory and Quantum Geometry, Springer, 2012, p. 263).

## References

* supergroup in nLab

* v
* t
* e

String theory

Background|

* Strings
* History of string theory
* First superstring revolution
* Second superstring revolution
* String theory landscape

Theory|

* Nambu–Goto action
* Polyakov action
* Bosonic string theory
* Superstring theory
* Type I string
* Type II string
* Type IIA string
* Type IIB string
* Heterotic string
* N=2 superstring
* F-theory
* String field theory
* Matrix string theory
* Non-critical string theory
* Non-linear sigma model
* Tachyon condensation
* RNS formalism
* GS formalism

String duality|

* T-duality
* S-duality
* U-duality
* Montonen–Olive duality

Particles and fields|

* Graviton
* Dilaton
* Tachyon
* Ramond–Ramond field
* Kalb–Ramond field
* Magnetic monopole
* Dual graviton
* Dual photon

Branes|

* D-brane
* NS5-brane
* M2-brane
* M5-brane
* S-brane
* Black brane
* Black holes
* Black string
* Brane cosmology
* Quiver diagram
* Hanany–Witten transition

Conformal field theory|

* Virasoro algebra
* Mirror symmetry
* Conformal anomaly
* Conformal algebra
* Superconformal algebra
* Vertex operator algebra
* Loop algebra
* Kac–Moody algebra
* Wess–Zumino–Witten model

Gauge theory|

* Anomalies
* Instantons
* Chern–Simons form
* Bogomol'nyi–Prasad–Sommerfield bound
* Exceptional Lie groups (G2, F4, E6, E7, E8)
* ADE classification
* Dirac string
* p-form electrodynamics

Geometry|

* Kaluza–Klein theory
* Compactification
* Why 10 dimensions?
* Kähler manifold
* Ricci-flat manifold
* Calabi–Yau manifold
* Hyperkähler manifold
* K3 surface
* G2 manifold
* Spin(7)-manifold
* Generalized complex manifold
* Orbifold
* Conifold
* Orientifold
* Moduli space
* Hořava–Witten domain wall
* K-theory
* Twisted K-theory

Supersymmetry|

* Supergravity
* Superspace
* Lie superalgebra
* Lie supergroup

Holography|

* Holographic principle
* AdS/CFT correspondence

M-theory|

* Matrix theory
* Introduction to M-theory

String theorists|

* Aganagić
* Arkani-Hamed
* Atiyah
* Banks
* Berenstein
* Bousso
* Cleaver
* Curtright
* Dijkgraaf
* Distler
* Douglas
* Duff
* Ferrara
* Fischler
* Friedan
* Gates
* Gliozzi
* Gopakumar
* Green
* Greene
* Gross
* Gubser
* Gukov
* Guth
* Hanson
* Harvey
* Hořava
* Gibbons
* Kachru
* Kaku
* Kallosh
* Kaluza
* Kapustin
* Klebanov
* Knizhnik
* Kontsevich
* Klein
* Linde
* Maldacena
* Mandelstam
* Marolf
* Martinec
* Minwalla
* Moore
* Motl
* Mukhi
* Myers
* Nanopoulos
* Naqvi
* Năstase
* Nekrasov
* Neveu
* Nielsen
* van Nieuwenhuizen
* Novikov
* Olive
* Ooguri
* Ovrut
* Polchinski
* Polyakov
* Rajaraman
* Ramond
* Randall
* Randjbar-Daemi
* Roček
* Rohm
* Scherk
* Schwarz
* Seiberg
* Sen
* Shenker
* Siegel
* Silverstein
* Sơn
* Staudacher
* Steinhardt
* Strominger
* Sundrum
* Susskind
* 't Hooft
* Townsend
* Trivedi
* Turok
* Vafa
* Veneziano
* Verlinde
* Verlinde
* Wess
* Witten
* Yau
* Yoneya
* Zamolodchikov
* Zamolodchikov
* Zaslow
* Zumino
* Zwiebach

0.00

(0 votes)

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Supergroup&oldid=637190"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template