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6.7 The real semi-simple algebras 𝖘𝖔 (k,l)

The dimensional reduction from 10 to 3 dimensions of 𝒩 = 1 supergravity coupled to m Maxwell multiplets leads to a nonlinear sigma model 𝒢∕𝒦 (𝒢) with Lie(𝒢 )=𝔰𝔬 (8, 8 + m ) (see Section 7). To investigate the geometry of these cosets, we shall construct their Tits–Satake diagrams.

The 𝔰𝔬(n, ℂ ) Lie algebra can be represented by n × n antisymmetric complex matrices. The compact real form is 𝔰𝔬(k + l, ℝ ), naturally represented as the set of n × n antisymmetric real matrices. One way to describe the real subalgebras 𝔰𝔬 (k, l), aligned with the compact form 𝔰𝔬(k + l, ℝ), is to consider 𝔰𝔬(k,l) as the set of infinitesimal rotations expressed in Pauli coordinates, i.e., to represent the hyperbolic space on which they act as a Euclidean space whose first k coordinates, a x, are real while the last l coordinates b y are purely imaginary. Writing the matrices of 𝔰𝔬 (k, l) in block form as

( ) A iC X = − i Ct B , (6.174 )
where
A = − At ∈ ℝk ×k, B = − Bt ∈ ℝl×l, C ∈ ℝk ×l, (6.175 )
we may obtain a maximal Abelian subspace 𝔞 by allowing C to have nonzero elements only on its diagonal, i.e., to be of the form:
( a1 ⋅⋅⋅ 0) ( ) | . | a1 ⋅⋅⋅ ⋅⋅⋅ 0 | .. | | .. | C = |( 0 ⋅⋅⋅ al|) or C = ( . ) , (6.176 ) .. .. 0 ⋅⋅⋅ ak ⋅⋅⋅ 0 . ⋅⋅⋅ .
with k > l or l < k, respectively.

To proceed, let us denote by Hj the matrices whose entries are everywhere vanishing except for a 2 × 2 block,

( ) 0 1 , − 1 0

on the diagonal. These matrices have the following realisation in terms of the Kij (defined in Equation (6.83View Equation)):

2j− 1 2j Hj = K 2j − K 2j−1. (6.177 )
They constitute a set of 𝔰𝔬(k + l) commuting generators that provide a Cartan subalgebra; it will be the Cartan subalgebra fixed by the Cartan involution defined by the real forms that we shall now discuss.

6.7.1 Dimensions l = 2 q + 1 < k = 2 p

Motivated by the dimensional reduction of supergravity, we shall assume k = 2p, even. We first consider l = 2q + 1 < k. Then by reordering the coordinates as follows,

{x1,y1;⋅⋅⋅ ; xl,yl; xl+1,xl+2;⋅⋅⋅ ; x2p− 2,x2p−1; x2p}, (6.178 )
we obtain a Cartan subalgebra of 𝔰𝔬(2q + 1, 2p), with noncompact generators first, and aligned with the one introduced in Equation (6.177View Equation) by considering the basis {iH ,⋅⋅⋅ , iH , H ,⋅⋅⋅ , 1 l l+1 H } q+p30. These generators are all orthogonal to each other. Let us denote the elements of the dual basis by {fA |A = 1,⋅⋅⋅ ,p + q}, and split them into two subsets: {fa|a = 1,⋅⋅⋅ ,2q + 1} and {fα |α = 2q + 2,⋅⋅⋅ ,p + q}. The action of the Cartan involution on these generators is very simple,
θ[fa] = − fa, and θ[fα] = +f α. (6.179 )
The root system of 𝔰𝔬(2q + 1,2p) is B (p+q), represented by Δ = {±fA ± fB |A < B = 1,⋅⋅⋅ ,p + q} ∪ {±fA |A1, ⋅⋅⋅ ,p + q}. A simple root basis can be taken as:
{α1 = f1 − f2,⋅⋅⋅ , αp+q−1 = fp+q−1 − fp+q, αp+q = fp+q}.
It is then straigthforward to obtain the action of the Cartan involution on the simple roots:
θ[αA] = − αA for A = 1,⋅⋅⋅ ,2q, θ [α ] = − α − 2(α + ⋅⋅⋅ + α ), 2q+1 2q+1 2q+2 q+p θ[αA] = + αA for A = 2q + 2, ⋅⋅⋅ ,q + p.
The corresponding Tits–Satake diagrams are displayed in Figure 38View Image.
View Image

Figure 38: Tits–Satake diagrams for the 𝔰𝔬(2p,2q + 1) Lie algebra with q < p. If p = q + 1, all nodes are white.

From Equation (6.179View Equation) we also obtain without effort that the set of restricted roots consists of the 4q(2q + 1) roots {±fa ± fb}, each of multiplicity one, and the 4q + 2 roots {±fa }, each of multiplicity 2(p − q) − 1. These constitute a B2q+1 root system.

6.7.2 Dimensions l = 2 q + 1 > k = 2 p

Following the same procedure as for the previous case, we obtain a Cartan subalgebra consisting of 2p noncompact generators and q − p compact generators. The corresponding Tits–Satake diagrams are displayed in Figure 39View Image.

View Image

Figure 39: Tits–Satake diagrams for the 𝔰𝔬(2p,2q + 1) Lie algebra with q ≥ p. If q = p, all nodes are white.

The restricted root system is now of type B2p, with 4p(2p − 1) long roots of multiplicity one and 4p short roots of multiplicity 2 (q − p) + 1.

6.7.3 Dimensions l = 2 q, k = 2 p

Here the root system is of type Dp+q, represented by Δ = { ±fA ± fB|A < B = 1,⋅⋅⋅ , p + q}, where the orthonormal vectors fA again constitute a basis dual to the natural Cartan subalgebra of 𝔰𝔬(k + l). Now, k = 2p and l = 2q are both assumed even, and we may always suppose k ≥ l. The Cartan involution to be considered acts as previously on the fA:

θ[fa] = − fa, a = 1,⋅⋅⋅ , 2q (6.180 )
and
θ[fα] = +fα, α = 2q + 1, ⋅⋅⋅ , p + q for q < p. (6.181 )
The simple roots can be chosen as
{α1 = f1 − f2,⋅⋅⋅ , αp+q −1 = fp+q−1 − fp+q, αp+q = fp+q− 1 + fp+q},

on which the Cartan involution has the following action:

The corresponding Tits–Satake diagrams are obtained in the same way as before and are displayed in Figure 40View Image.

View Image

Figure 40: Tits–Satake diagrams for the 𝔰𝔬(2p,2q) Lie algebra with q < p − 1, q = p − 1 and q = p.

When q < p, the restricted root system is again of type B2q, with 4q(2q − 1) long roots of multiplicity one and 4q short roots of multiplicity 2(p − q). For p = q, the short roots disappear and the restricted root system is of D2p type, with all roots having multiplicity one.


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