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I would be glad if someone can give me some (hopefully easy to understand!) references for learning about these groups Osp, USp and PSU and their representations.

I run into these mostly while studying about supersymmetry and I wonder if PSU groups are the same as what are called ``supergroups".

I frequently run into $Osp(3\vert4)$, $USp(2N)$, $PSU(2,2\vert 4)$, $SU(2,2)$ and $SU(2\vert 4)$.

(I don't understand this notation of having a "$\vert$" and I don't know the definition of any of these groups)

Some of the concepts about them that I face are,

  • The commuting subalgebra of the lie algebra of $SU(2,2,\vert m)$ is that of $SU(2,1\vert m-1)$. The generators of $SU(2,1\vert m-1)$ are related to those of $SU(2,2,\vert m)$ via some "obvious reduction". I don't know what that means.

  • The "bosonic subgroup" of $SU(2,1\vert m-1)$ is apparently $SU(2,1) \times U(m-1)$. I wonder what this means.

  • The supersymmetry generators are supposed to transform as "bifundamentals" under the bosonic subalgebra of $SU(2,2)$ and $SU(m)$. (I don't know what is a "bifundamental")

I am very confused about these concepts and would be grateful if someone one help traverse this subject.

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    $\begingroup$ shouldn't this be on physics? $\endgroup$
    – yoyo
    Mar 2 '11 at 16:47
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    $\begingroup$ Well. I came to these groups from physics literature but I guess at the end of the day it is a question of group/representation theory (probably of some exotic kind!) $\endgroup$
    – Anirbit
    Mar 3 '11 at 5:52
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Supergroups are a little tricky. It may be useful to learn about the corresponding Lie superalgebras instead. A possible place to start with learning the basics and notation is Kac's article, "A sketch of Lie superalgebra theory": http://projecteuclid.org/euclid.cmp/1103900590

In particular, you may be interested in the section on p.58 on real Lie superalgebras where you'll find the lowercase (Lie superalgebra) versions of what you want.

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    $\begingroup$ Thanks for the reference! I will take a look at your suggestion. Ever since posting this question I have also come across two other large-review-ish writings by Kac on such topics. But it seems you have a shorter reference! :P $\endgroup$
    – Anirbit
    Jun 1 '12 at 19:48
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This notation is about groups that can be viewed as subgroups of the group of matrices under matrix conjugation. Since some of names in your list are supergroups and others are just bosonic groups I will explain only the case of supergroups. I hope that the application to what I am going to say here for bosonic groups will become clear.

This names are telling us what are the constraints and projections that we are imposing in a general (super)matrix to obtain the desired subgroup.

We start with $M\in GL(p|q)$, i.e. General Linear supermatrix of the form $$ M=\begin{pmatrix}m&\psi\\\chi&n\end{pmatrix},\quad \left(\text{i.e.}\,M^{I}_{J}=\begin{pmatrix}m^{\alpha}_{\beta}&\psi^{\alpha}_{j}\\\chi^{i}_{\beta}&n^{i}_{j}\end{pmatrix}\right) $$ where $(m,n)$ are $GL(p)$ and $GL(q)$ bosonic matrices, while $(\chi,\psi)$ are $p\times q$ and $q\times p$ fermionic matrices. The group multiplication is by matrix multiplication while the group conjugation is given by $$ M'\rightarrow MM'M^{-1} $$ Note that if we do a pure bosonic transformation $M$ on $M'$ we have $$ M M' M^{-1}=\begin{pmatrix}mm'm^{-1}&m\psi' n^{-1}\\n \chi'm & nn'n\end{pmatrix} $$ so this means that both fermions are in the fundamental\anti-fundamental of $GL(p)$ and $GL(q)$. We can see this directly from the index structure $\psi^{\alpha}_{i}$ and $\chi^{i}_{\alpha}$, where index up means fundamental and index down means anti-fundamental.

The constraints that we are allowed to impose are:

  • Special constraint: We require that the superdeterminant vanishes, i.e. $$ Sdet M = 0 $$ this constraint get rid of the $U(1)$ generator $$ \begin{pmatrix}1_{p\times p}&0\\0&-1_{q\times q}\end{pmatrix} $$ that rotate $\psi$ and $\theta$ with opposite phases.

  • Unitary constraint: We require that $M$ preserves an hermitian metric $$ M \eta M^{\dagger}=\eta, \qquad \eta=\begin{pmatrix}\eta^{(r,s)}&0\\0&\eta^{(q,0)}\end{pmatrix} $$ where $\eta^{(r,s)}$ is a metric of signature $(r,s)$ ($r+s=p$) and $\eta^{(q,0)}$ is a metric with signature $(q,0)$. This constraint will restrict $m$ to be $U(r,s)$, $n$ to be $U(q)$ and also impose some reality conditions for the $\psi$ and $\chi$ that is compatible with $U(r,s)$ and $U(q)$ transformations.

  • Orthosymplectic constraint: We require that $M$ preserves a metric $$ M \eta M^{T}=\eta,\qquad \eta=\begin{pmatrix}\delta^{(r,s)}&0\\0&\Omega\end{pmatrix} $$ where $\delta$ is a symmetric metric (i.e. an orthogonal metric) of signature $(r,s)$ and $\Omega$ is an anti-symmetric metric (i.e. a symplectic metric). This will restrict $m$ to be $SO(r,s)$ and $n$ to be $Sp(q)$. It will also relate $\psi$ and $\chi$ such that we can express one in terms of the other.

  • Projection: Is the identification $M\cong \lambda M$. This get rid of the $U(1)$ generator $$ \begin{pmatrix}1_{p\times p}&0\\ 0& 1_{q\times q}\end{pmatrix} $$ that does not rotate $\psi$ and $\chi$ but can rotate objects that are in the fundamental of $gl(p)$ and/or $gl(q)$.

Note that you can combine different constraints, so for instance $PSU(2,2|4)$ will have, up to the P identification, $m$ being a $SU(2,2)$ while $n$ being a $SU(4)$. For more details see this lecture notes page 84 for bosonic groups and page 160 for supergroups.

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