*3.1. Quadratic Polynomial*

Let us consider the matrix *X* = (*<sup>C</sup>*, *L*) = ⎛⎜⎜⎝*C*0 *L*0 *C*1 *L*1 *C*2 *L*2 *C*3 *L*3⎞⎟⎟⎠ ∈ *M*(4, 2, R) with the column

vectors *C*, *L* ∈ R4. Here, *<sup>M</sup>*(*<sup>m</sup>*, *n*, R) the set of *m* by *n* matrices with real coefficients. The entry of the map

$$\mathbb{S}: M(\mathbf{4}, \mathbf{2}, \mathbb{R}) \ni X \mapsto XfX^T \in \operatorname{Alt}(\mathbf{4}, \mathbb{R})\_{\operatorname{rk} \le 2}$$

is given by

$$\mathbb{S}\_{\mu\nu}(X) = (X \!\!/ X^T)\_{\mu\nu} = \mathbb{C}\_{\mu} L\_{\nu} - L\_{\mu} \mathbb{C}\_{\nu}.$$

which realizes the definition of *<sup>S</sup>μν*. The map S is *GL*(4, R) × *SL*(2, <sup>R</sup>)-equivariant, where we remark the accidental isomorphism of lower rank groups:

$$SL(2, \mathbb{R}) = \{ h \in M(2, \mathbb{R}) \mid \det h = 1 \} = Sp(2, \mathbb{R}) = \{ h \in M(2, \mathbb{R}) \mid hlh^T = l \}.$$

The action of *GL*(4, R) on Alt(4, R) is prehomogeneous [10]. The image Alt(4, <sup>R</sup>)rk≤2 is the complement of the open *GL*(4, R)-orbit Alt(4, <sup>R</sup>)rk=4, and its defining equation is given by the basic relative invariant, Pfaffian

$$\text{Pf}(S) = S\_{01}S\_{23} + S\_{02}S\_{31} + S\_{03}S\_{12}.$$

Then, the singular set Alt(4, <sup>R</sup>)rk≤2 = {*S* ∈ Alt(4, R) | Pf(*S*) = 0} is the zero locus of Pfaffian, and the open orbit Alt(4, <sup>R</sup>)rk=<sup>4</sup> has two connected components {*S* ∈ Alt(4, R) | ±Pf(*S*) > <sup>0</sup>}. The relation Pf(*S*) = 0 is considered as a Plücker relation of Grassmann manifold Grass(4, 2, <sup>R</sup>).

*3.2. Symmetry Breaking*

00

 0

We restrict the general linear group *GL*(4, R) to the subgrouop *O*(1, <sup>3</sup>). Let *g* = ⎛⎜⎜⎝10 0 0 0 −10 0 0 0 −1 0 −1⎞⎟⎟⎠ be the standard non-degenerate symmetric matrix with signature

(1, <sup>3</sup>). Define Lorentz group (indefinite orthogonal group of signature (1, 3)) by

$$O(1,3) = \{ l \in M(\mathbf{4}, \mathbb{R}) \mid l^T \emptyset l = \emptyset \}.$$

Gram matrix with respect to this metric is given by the map

$$\mathbb{G}: \mathcal{M}(\mathbf{4}, \mathbf{2}, \mathbb{R}) \ni X \mapsto X^T \mathbb{g}X \in \operatorname{Sym}(\mathbf{2}, \mathbb{R})$$

where Sym(*<sup>n</sup>*, R) is the set of real symmetric matrices of size *n*. The map G is *O*(1, 3) × *GL*(2, <sup>R</sup>)-equivariant:

$$\mathbb{G}(lXh) = h^T \mathbb{G}(X)h, \quad \forall l \in O(1,3), h \in GL(2,\mathbb{R}).$$

We define

$$\mathcal{Y}(\mathbb{R}) := \mathcal{M}(\mathfrak{A}, \mathfrak{Z}, \mathbb{R})\_{\mathrm{rk} = 2} \cap \mathbb{G}^{-1}(\mathrm{Sym}(\mathfrak{Z}, \mathbb{R})\_{\mathrm{rk} \leq 1})\_{\mathrm{rk}}$$

an *O*(1, 3) × *SL*(2, R)-invariant subset of *M*(4, 2, <sup>R</sup>). Moreover, let

$$\mathcal{S}^1 := \{ \mathbf{v} = \begin{pmatrix} v\_1 \\ v\_2 \end{pmatrix} \in \mathbb{R}^2 \mid v\_1^2 + v\_2^2 = 1 \}$$

and an analogue of Veronese map is defined by

$$\mathbb{V}\_2 \colon \mathbb{S}^1 \times \mathbb{R}^\times \ni (\mathbf{v}, -\rho) \mapsto -\rho \mathbf{v} \mathbf{v}^T \in \text{Sym}(\mathbf{2}, \mathbb{R})\_{\text{rk} \le 1}. \tag{7}$$

The fiber product of two maps

$$\begin{aligned} \mathbb{G}: \mathcal{Y}(\mathbb{R}) &\longrightarrow \text{Sym}(\mathbf{2}, \mathbb{R})\_{\text{rk}\leq 1\prime} & \qquad \mathbb{C} &\mapsto \mathbb{G}(\mathbb{C}),\\ \mathbb{V}\_{2}: \mathbb{S}^{1} \times \mathbb{R}^{\times} &\to \text{Sym}(\mathbf{2}, \mathbb{R})\_{\text{rk}\leq 1\prime} & \qquad (\mathbf{v}, -\rho) \mapsto -\rho \mathbf{v} \mathbf{v}^{T} \end{aligned}$$

is defined by

$$\begin{split} Z(\mathbb{R}) &:= \mathcal{Y}(\mathbb{R}) \times\_{\text{Sym}(2,\mathbb{R})\_{\text{rk}\geq 1}} (\mathbb{S}^1 \times \mathbb{R}^\times) \\ &= \{ (X, \mathbf{v}, -\rho) \in M(\mathbf{4}, \mathbf{2}, \mathbb{R})\_{\text{rk}=2} \times \mathbb{S}^1 \times \mathbb{R}^\times \mid \mathbb{G}(X) = -\rho \mathbf{v} \mathbf{v}^T \}\_{\text{r}} \end{split}$$

then we obtain a real counterpart of (3):

$$\begin{array}{c} \begin{array}{ccc} Z(\mathbb{R}) & \xrightarrow{\Psi\_{2}} & \begin{array}{c} \Psi\_{2} \\ \xrightarrow{\cdot} \\ \mathbb{G} \end{array} \\ \mathbb{G}^{1} \times \mathbb{R}^{\times} & \xrightarrow{\cdot} & \begin{array}{c} \text{Sym}(\text{2},\mathbb{R})\_{\text{rk}\leq 1} \end{array} \end{array} \end{array} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \text{Y}(\mathbb{R}) \end{array} \end{array} \end{array}$$

*3.3. Tensor T* ˆ

> The map

$$\mathbb{T}: \operatorname{Alt}(4,\mathbb{R}) \ni \mathcal{S} \mapsto -\operatorname{Sg}\mathcal{S} \in \operatorname{Sym}(4,\mathbb{R})$$

has been defined to be compatible with *T* ˆ *ν μ* = <sup>−</sup>*SμσSνσ*. This map is *O*(1, <sup>3</sup>)-equivariant

$$\mathbb{T}(ISI^T) = l\mathbb{T}(\mathbb{S})I^T, \quad \forall l \in O(1,3)$$

We replace *φ* by Φ, and V by V4 given as follows:

$$\begin{aligned} \Phi: & Z(\mathbb{R}) \ni (X, \mathbf{v}, -\rho) \mapsto (X \!/ \mathbf{v}, -\rho) \in M(\mathbf{4}, \mathbf{1}, \mathbb{R})\_{\mathbf{rk} = 1} \times \mathbb{R}^{\times}, \\ \mathbb{V}\_{\mathbf{4}}: & \mathbb{R}^{\mathbf{4}} \times \mathbb{R}^{\times} \ni (\mathbf{w}, -\rho) \mapsto \rho \mathbf{w} \mathbf{w}^{T} \in \text{Sym}(\mathbf{4}, \mathbb{R})\_{\mathbf{rk} \le 1}. \end{aligned}$$

**Theorem 2.** (T ◦ S)(*X*)=(V4 ◦ <sup>Φ</sup>)(*<sup>X</sup>*, **v**, −*ρ*) *for all* (*<sup>X</sup>*, **v**, −*ρ*) ∈ *<sup>Z</sup>*(R)*.*

$$\begin{array}{l} \mathsf{Proof.} \ (\mathsf{T}\circ\mathsf{S})(X) = (X\!\!/\!X^{T})\!\!g(\!\!X\!\!/\!X^{T})^{T} = X\!\!f\!\!\mathsf{G}(X)\!\!f^{T}X^{T} = -X\!\!f\!\!\rho\mathbf{v}\mathbf{v}^{T}\!\!f^{T}X^{T} \\\ \mathsf{I} = \mathsf{V}\_{4}((X\!\!\!\mathsf{V},-\rho)) = (\mathsf{V}\_{4}\circ\mathsf{O})((X,\mathsf{v},-\rho)). \ \ \ \ \ \ \ \end{array}$$

This theorem is illustrated as

$$\begin{array}{ccccc} Z(\mathbb{R}) & \longrightarrow & M(\mathsf{4}, 2, \mathbb{R})\_{\mathrm{rk} = 2} \times \mathbb{S}^{1} \times \mathbb{R}^{\times} & \longrightarrow & M(\mathsf{4}, 1, \mathbb{R})\_{\mathrm{rk} = 1} \times \mathbb{R}^{\times} \\ \mathrm{\ddot{v}}\_{2} & & \Big| & \Big| & \Big| \mathbb{V}\_{4} \\ \mathrm{\ddot{Y}}(\mathbb{R}) & \longrightarrow & M(\mathsf{4}, 2, \mathbb{R})\_{\mathrm{rk} = 2} \xrightarrow{\mathbb{S}} \mathrm{Alt}(\mathsf{4}, \mathbb{R})\_{\mathrm{rk} = 2} & \xrightarrow{\mathbb{T}} & \mathrm{Sym}(\mathsf{4}, \mathbb{R}) \end{array}$$

*3.4. Grassmann and Flag Manifold*

The flag manifold is realized as an incidence variety of the product of two Grassmann manifold:

$$\begin{aligned} \text{Flag}(4; 1, 2, \mathbb{R}) &= \{ (V\_1, V\_2) \mid \dim V\_1 = 1, \dim V\_2 = 2, V\_1 \subset V\_2 \subset \mathbb{R}^4 \} \\ \text{Flag}(4; 1, 2, \mathbb{R}) &= \{ (V\_1, V\_2) \in \text{Gauss}(4, 1, \mathbb{R}) \times \text{Grass}(4, 2, \mathbb{R}) \mid V\_1 \subset V\_2 \}. \end{aligned}$$

For (*<sup>X</sup>*, **v**) ∈ *M*(4, 2, <sup>R</sup>)rk=<sup>2</sup> × *S*1, two column vectors of *X* spans a two-dimensional subspace *V*2, and a column vector *X J***v** generate a one-dimensional subspace *V*1 in *V*2. The map

$$\begin{array}{ccccc}\text{Grass}(4,2,\mathbb{R}) & \longleftarrow & \text{Flag}(4;1,1,2,\mathbb{R}) & \longrightarrow & \text{Grass}(4,1,\mathbb{R})\\\text{V}\_{2} & \longleftarrow & (\text{V}\_{1},\text{V}\_{2}) & \longleftrightarrow & \text{V}\_{1} \\\text{X} & \longleftarrow & (\text{X},\text{v}) & \longmapsto & \text{XJv} \end{array}$$

is the double fibration.

 =

•

### *3.5. The Interpretation of the Off-Shell Condition*

The vectors *C* and *L* in Clebsch parametrization should satisfy the following off-shell conditions [2]:

$$\mathbb{C}\_{\nu}\mathbb{C}^{\vee} = 0, \quad L\_{\nu}\mathbb{C}^{\mu} = 0, \quad L\_{\nu}L^{\mu} = -\rho. \tag{8}$$

We put*R* ¯ := 0 0 0 −*ρ*. Then, the condition (8) is written as

$$\mathbb{G}(X) = \mathbb{R}$$

In particular, in the case **v** = 01 ∈ *S*1, we compute the maps V2, Φ and V4: V2((**<sup>v</sup>**,−*ρ*))−*ρ***vv***<sup>T</sup>R*¯,

 =

• (**<sup>w</sup>**, −*ρ*) = <sup>Φ</sup>((*<sup>X</sup>*, **v**, −*ρ*)) = (*X J***<sup>v</sup>**, <sup>−</sup>*ρ*)=(*<sup>C</sup>*, <sup>−</sup>*ρ*), this implies **w** = *C*, *T*ˆ.

$$\mathbb{V}\_4((\mathbf{w}, -\rho)) = \rho \mathbf{w} \mathbf{w}^T = \rho \mathbb{C} \mathbb{C}^T = \mathbb{T}$$

This coincides with the result in [2]. An unnatural *J* in the definition of Φ is for the sake of compatibility with the existing formula.

We now remark *R* ∈ Sym(2, <sup>R</sup>)rk≤1.

$$\begin{array}{ccc} & Z(\mathbb{R}) & & \\ \Vdash & & \mid \\ \lnot \mathbb{G}^{-1}(\bar{\mathbb{R}}) & \longrightarrow & \Vdash \\ & \mid & \mid \mathbb{G} \\ \check{\mathbb{R}} & \in & \text{Sym}(\hat{\mathbf{2}}, \mathbb{R})\_{\text{rk}=1} \end{array}$$

The group *GL*(2, R) acts on Sym(2, <sup>R</sup>)rk=<sup>1</sup> and the stabilizer at *R*¯ is a Borel subgroup

$$B = \left\{ \begin{pmatrix} a & 0 \\ c & d \end{pmatrix} \right\} \subset GL(\mathbf{2}, \mathbb{R}).$$

Then, G : *Y*(R) −→ Sym(2, <sup>R</sup>)rk=<sup>1</sup> is a *GL*(2, <sup>R</sup>)-equivariant bundle. We regard the off-shell condition specifies a fiber of this bundle. A symmetry is hidden in the horizontal direction of this bundle, the group action of *GL*(2, <sup>R</sup>). Of course, form the Clebsch parametrization point of view, the role of *C* and *L* is not the same; the off-shell condition specifies the special isotropic direction for *C*: the choice of this direction is controlled by the homogeneous space *GL*(2, <sup>R</sup>)/*B*.
