# Generalised geometry #2: Generalised vector space and bilinear form

Generalised linear algebra

In the first note we introduced one of the fundamental structures of generalised geometry, namely the generalised tangent bundle ${E \simeq TM \oplus T^{\star}M}$. In the extension of the standard tangent bundle ${TM}$ to ${TM \oplus T^{\star}M}$, we are simultaneously extending linear algebra to some notion of generalised linear algebra. This is a point nicely emphasised in Roberto Rubio’s lecture notes.

Think of it this way. In standard differential geometry we have the tangent bundle ${TM}$, whose sections are endowed with a Lie bracket satisfying the Leibniz rule

$\displaystyle [X, fY] = X(f)Y + f[X,Y]. \ \ \ \ \ (1)$

In generalised geometry, on the other hand, we are motivated to extend linear algebra to generalised linear algebra such that we replace the standard vector space ${V}$ of differential geometry with the generalised vector space

$\displaystyle \hat{V} \simeq V \oplus V^{\star}, \ \ \ \ \ (2)$

with ${V}$ a d-dimensional vector space and ${V^{\star}}$ its dual. We can denote the elements of this space ${X = x + \xi, Y = y + \varepsilon}$ with ${x,y \in V}$ and ${\xi, \varepsilon \in V^{\star}}$.

For intuition, realise that this extension can be motivated at its most basic when studying symplectic forms that represent a skew-symmetric analogue of an inner product: i.e., skew-symmetric isomorphisms ${V \rightarrow V^{\star}}$ or ${V^{\star} \rightarrow V}$. This also takes us into a study of linear presymplectic and Poisson structures, but such preliminary discussion is dropped in these notes.

Definition 1 (Canonical pairing and inner product signature) Given the generalised vector space ${\hat{V} = V \oplus V^{\star}}$, then for two generalised tangent vectors ${X, Y}$ there exists a canonical metric naturally equipped on this space such that

$\displaystyle \mathcal{I} = \langle X,Y \rangle = \langle x + \xi, y + \varepsilon \rangle = \frac{1}{2}(\iota_{x}\varepsilon + \iota_{y} \varepsilon), \ \ \ \ \ (3)$

which is a maximally indefinite inner product of signature ${(d,d)}$. Note that ${\iota_x}$ denotes contraction by ${x}$. The factor of ${1/2}$ is purely convention with no geometric meaning.

Next, define a subspace ${W \subset V \oplus V^{\star}}$. Then the orthogonal compliment for this pairing can be defined in the standard sense

$\displaystyle W^{\perp} = \{ u \in V + V^{\star} \mid \langle u, W \rangle = \}. \ \ \ \ \ (4)$

It is said that ${W}$ is isotropic if ${W \subset W^{\perp}}$. The idea of maximally isotropic subspaces raises interesting considerations, particularly globally defined structures with an integrability condition, which are often studied under the heading of linear Dirac structures. As a matter of introduction, if ${W}$ is isotropic but not constrained in an isotropic subspace than it is maximally isotropic.

The bilinear form (3) can be verified by taking a basis ${\{e_i : i = 1,...,d \}}$ for ${V}$ with dual basis ${\{e^i : i = 1,...,D \}}$ for ${V^{\star}}$. Then we can define ${\{ e_i \pm e^i ; i,...,d \}}$ as the total basis for ${V \oplus V^{\star}}$, wherein the inner product admits a diagonal form ${\pm 1}$.

Explicitly, consider the basis ${(e^i + e_i, e_i - e^i)}$ with the canonical pairing given by

$\displaystyle \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}, \ \ \ \ \ (5)$

in which the signature is ${(d,d)}$ with an orthogonal basis of ${d}$ vectors of positive length and ${d}$ vectors of negative length. It can then be shown that the dimension of the maximally isotropic subspace ${W}$ must be at most ${\dim V}$. In fact, this statement is true for any maximally isotropic subspace.

Proposition 2 For a vector space ${V}$, we have that ${\dim V}$ is the dimension of any maximally isotropic subspace of ${V + V^{\star}}$.

Proof: Let ${L}$ be a maximally isotropic subspace. Consider ${L^{\perp}}$ to be semidefinite such that ${Q(v) \leq 0}$ or ${Q(v) \geq 0}$ for all ${v \in L}$. Otherwise, consider a compliment ${C}$ so that ${L^{\perp} = L \oplus C}$, then if ${C}$ contains two vectors ${v,w}$ with ${Q(v) > 0}$ and ${Q(w) < 0}$, a suitable linear combination ${v \lambda w}$ would be null. As a result ${L \oplus \ span \ (v + \lambda w)}$ would be isotropic containing ${L}$. Additionally, as ${L^{\perp} \cap V = \{ 0 \}}$ we have

$\displaystyle \dim L^{\perp} = \dim (L^{\perp} \oplus V) - \dim V \leq 2d - d = d. \ \ \ \ \ (6)$

On the other hand, given that the pairing is non-degenerate, we write ${\dim L^{\perp} = 2d - \dim L}$ such that we now have

$\displaystyle 2d - \dim L \leq d, \ \ \ \ \ (7)$

which is to say ${\dim L \geq d}$ so ${\dim L = d}$. $\Box$

What we have found is that, as ${V + V^{\star}}$ is a vector space of dimension ${2d}$ with a symmetric pairing of signature ${(d,d)}$, there is a choice two maximally isotropic subspaces.

Remark 1 (Motivation) There is another way to motivate the inner product (3) by simply noting that ${TM}$ and ${T^{\star}M}$ are vector spaces, and, as they are related by a duality transformation (this will be made clear later) we obtain a natural bilinear symmetric form.

Brief remark: Clifford E modules

The bilinear form (3), or what is also referred to as the canonical metric or canonical pairing, should not be confused with a metric in the standard sense. To obtain an ordinary metric, we must introduce extra structure.

To make sense of this claim, consider the generalised bundle from note 1 [LINK]. Each fibre of ${E \simeq TM \oplus T^{\star}M}$ has an action on the corresponding fibre of the exterior bundle ${\wedge T^{\star}}$ given by

$\displaystyle (x + \xi) \cdot \omega = \xi \wedge \omega + \iota_x \omega. \ \ \ \ \ (8)$

It follows that

$\displaystyle (x + \xi) \cdot ((x + \xi) \cdot \omega) = \xi(x) \omega = (x + \xi, x + \xi)\omega, \ \ \ \ \ (9)$

which gives description of the exterior algebra in terms of the structure of a bundle of Clifford ${E}$ modules with respect to the natural bilinear on ${E}$. Typically, in the literature the corresponding Clifford algebras generated by ${(x + \xi)^2 = \xi(x)}$ is denoted Cliff(E).

To conclude: as we’ll see, the form ${( \ , \ )}$ on ${E}$ enables the reduction of the structure group of ${E}$ to ${O(d,d)}$. In fact, in the maths literature we see that further reduction to the structure ${SO(d,d)}$ is possible. This will be the topic of the next note, as we focus on symmetries and the structure group generators.