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.