Generalised geometry #3: Symmetries

When doing generalised linear algebra, we want to study transformations that preserve the canonical pairing from the last note (of signature ${O(d,d)}$):

$\displaystyle O(V \oplus V^{\star}) = \{A \in GL(V \oplus V^{\star}): \langle A_v, A_w \rangle = \langle v, w \rangle \ \text{for all} \ v,w \in V \oplus V^{\star}. \} \ \ \ \ \ (1)$

The group of linear transformations that preserve the inner product is the orthogonal group. What we’ll see is that the linear endomorphisms of $E \simeq TM \oplus T^{\star}M$ preserving this bilinear form is the orthogonal group ${O(E) = O(d,d)}$. The corresponding Lie algebra, which represents the algebra of infinitesimal transformations, is denoted by ${\mathfrak{o}(V \oplus V^{\star})}$ and, as we’ll see, ${\mathfrak{s}\mathfrak{o}(V \oplus V^{\star})}$. This is the Lie algebra of endomorphisms of ${E}$ which are skew adjoint with respect to the bilinear form.

Elements ${R \in \mathfrak{o}(V \oplus V^{\star})}$ satisfy ${\langle Rv, w \rangle + \langle v, Rw \rangle = 0}$. Using the splitting of the direct sum ${V \oplus V^{\star}}$ we may write ${R}$ in matrix form

$\displaystyle R = \begin{pmatrix} A & B \\ \beta & D \\ \end{pmatrix}, \ \ \ \ \ (2)$

with ${A : V \rightarrow V}$, ${B : V^{\star} \rightarrow V}$, ${\beta : V \rightarrow V^{\star}}$, and ${D: V^{\star} \rightarrow V^{\star}}$. Note that these are all linear maps. So we find that the condition on R takes the form

$\displaystyle \langle (AX + \beta \xi) + (BX + D\xi), Y + \varepsilon \rangle + \langle X + \xi, (AY + \beta \varepsilon + (BY + D\varepsilon) \rangle = 0. \ \ \ \ \ (3)$

For ${X = Y = 0}$ we have ${\varepsilon (\beta \xi) + \xi (\beta \varepsilon) = \varepsilon (\beta \xi) + \varepsilon (\beta^{\star} \xi) = 0}$, which must hold for all ${\varepsilon, \xi}$. This implies ${\beta^{\star} = -\beta}$, with ${\beta^{\star}}$ the adjoint of ${\beta}$ (i.e., it is the pullback). Similarly, consider ${\xi = \varepsilon = 0}$; then ${B^{\star} = -B}$. For the case ${Y=0}$, ${\xi = 0}$, it follows ${D = - A^{\star}}$. It is from these conditions that satisfy (3) that we often read in the literature the Lie algebra consisting of matrices of the form

$\displaystyle R = \begin{pmatrix} A & B \\ \beta & -A^T \\ \end{pmatrix}, \ \ \ \ \ (4)$

where, to be clear, ${A \in End(V)}$ is an arbitrary matrix generating ${GL(d) \subset O(d,d)}$. Now, for anyone who has studied some double field theory, all of this will start looking familiar. Moreover, note that B is a two-form such that ${B \in \wedge^2 (\mathbb{R}^d)^{\star}}$ generates what we call B-transformations. In the context of generalised geometry, ${\beta \in \wedge^2 \mathbb{R}}$ are ${\beta-transformations}$, with ${\beta}$ an element of ${\wedge^2 \mathbb{R}^d}$. In the maths literature, ${B}$ and ${\beta}$ are noted as being skew such that

$\displaystyle B(X,Y) = (B(X))(Y), \ \beta (\xi, \varepsilon) = \varepsilon (\beta(\xi)), \ \ \ \ \ (5)$

where the skewness implies alternativity.

Any endomorphisms ${L}$ of E that is skew adjoint with respect to the bilinear form can be viewed as a skew-symmetric bilinear form via ${X,Y \rightarrow (LX,Y)}$. Under this identification we therefore have

$\displaystyle \mathfrak{so}(V \oplus V^{\star}) \simeq \wedge^2 (V \oplus V^{\star}) \simeq \wedge^2 V \oplus End(V) \oplus \wedge^2 V^{\star}. \ \ \ \ \ (6)$

Structure group generators

Let’s now look at the three primary orthogonal transformations belonging to ${O(d,d;\mathbb{Z})}$. (What follows is a generalisation of the study of O(d,d) transformations on the level of the string spectrum. See, for instance, [Hitc10] and [Rub18]). The structure group ${O(d,d)}$ reduces to ${SO(d,d)}$ generated by the following elements:

1. ${\blacksquare}$ Diffeomorphisms: It is important that from (4), when ${B, \beta = 0}$ we get for any ${A \in End(V \oplus V^{\star})}$ the element

$\displaystyle h_A = \begin{pmatrix} A^{-1} & 0 \\ 0 & -A^T \end{pmatrix}, \ \ \ \ \ (7)$

where ${\det h_A = +1}$. Although the connection may not be immediately obvious, we will see in a much later discussion in what way this transformation relates to diffeomorphisms on the target space. For now, observe that its action on ${X}$ takes the form

$\displaystyle A(x + \xi) = Ax - A^T \xi, \ \ \ \ \ (8)$

which yields a skew-symmetric action of ${A}$ on ${X \in V \oplus V^{\star}}$. Taking the exponential ${e^A \in SO(V \oplus V^{\star})}$ of (7) we get

$\displaystyle e^A (x + \xi) = e^A x + e^{-A^T} \xi = e^A x + ((e^A)^{T})^{-1} \xi. \ \ \ \ \ (9)$

In the most general case for any ${P \in GL(V \oplus V^{\star})}$ we have an ${SO(V \oplus V^{\star})}$ action

$\displaystyle P(x + \xi) = Px + (P^T)^{-1}. \ \ \ \ \ (10)$

2. ${\blacksquare}$ B-transformations: From (4) consider transformations parameterised by an anti-symmetric ${d\times d}$ matrix with integer entries. The two-form ${B \in \wedge^2 V^{\star}}$ corresponding to the orthogonal transformation

$\displaystyle h_B = \begin{pmatrix} 1 & 0 \\ B & 1 \\ \end{pmatrix}. \ \ \ \ \ (11)$

acts as an endomorphisms of the generalised vector space ${V \oplus V^{\star}}$ such that

$\displaystyle B(x + \xi) = Bx = \iota_x B \ \ \ \ \ (12)$

and so its exponential ${e^B \in SO(V \oplus V^{\star})}$ then acts

$\displaystyle e^B (x + \xi) = (1 + B)(x + \xi) = x + \xi + \iota_x B, \ \ \ \ \ (13)$

with the two-form ${B}$ having components ${b_{ij}}$. In the maths literature this B-transformation is also known as a shear transformation as it shifts ${V^{\star}}$ but keeps V invariant. Similar to (7), the determinant of ${h_B}$ is +1.

Remark 1 (Acting on generalised metric) The requirement that ${B}$ is anti-symmetric comes from the most general ${O(d,d;\mathbb{Z})}$ relations, which we first find in an analysis of the spectrum of the closed string. As we’ll see, we can define what is called a generalised metric (precisely the same as that in double field theory), which is a generalisation of the Riemannian metric that unifies $g$ and the 2-form $b$ in such a way that it is manifestly invariant under $O(d,d)$. When acting on the generalised metric, for the B-transformation we find

$\displaystyle h^{-T}_B \mathcal{H}(g,b) h^{-1}_B = \mathcal{H}(g,b + x B). \ \ \ \ \ (14)$

In general, this correlates with a shift of the B-field not as a symmetry of the action but strictly as a duality transformation. In the case ${B = d\Lambda}$ with ${\Lambda}$ a one-form on the worldsheet, these shifts are gauge transformations and they will be a topic of consideration in a later note.

3. ${\blacksquare}$ ${\beta}$-transformations: Similar to B-transformations, from (4) for ${\beta \in \wedge^2 V}$ these are transformations parametrised by an anti-symmetric ${d \times d}$ matrix of the form

$\displaystyle h_{\beta} = \begin{pmatrix} 1 & \beta \\ 0 & 1 \\ \end{pmatrix}. \ \ \ \ \ (15)$

The determinant of ${h_{\beta}}$ is also +1. It gives the following action on ${V \oplus V^{star}}$

$\displaystyle \beta (x + \xi) = \iota_{\xi} \beta \ \ \ \ \ (16)$

with its exponential

$\displaystyle e^{\beta} (x + \xi) = x + \iota_{\xi} \beta + \xi, \ \ \ \ \ (17)$

where the action of the bi-vector field ${\beta = 1/2 \beta^{ij} \partial_i \wedge \partial_j}$ on forms is defined by way of contraction. So, for some one-form ${\xi}$ it follows ${\beta \xi = -\xi_i \beta^{ij}\partial_j}$ with ${\partial_j}$ a local basis on the tangent space ${TM}$. This type of transformation shifts V and leaves ${V^{\star}}$ invariant.

Using ${h_B}$ and the elements ${O(d,d;\mathbb{Z})}$ we can write

$\displaystyle h_{\pm} = \begin{pmatrix} 0 & \pm \delta^{-1} \\ \pm \delta & 0 \\ \end{pmatrix}, \ h_{\pm} = \prod \limits^D_{i=1} h_{\pm i}, \ \ \ \ \ (18)$

as

$\displaystyle h_{\beta} = h_{\pm} h_B h_{\pm}, \ \text{where} \ \beta^{ij} = \delta^{ip}B_{pq} \delta^{qj}. \ \ \ \ \ (19)$

Patching and transition functions

Note that the non-trivial nature of the generalised tangent bundle ${E = TM \oplus T^{\star}M}$ discussed in Note #1 is encoded in transition functions between local patches ${U_a \subset M}$.

Consider, moreover, the definition of a generalised vector ${X = x + \xi}$ also given in the same note. Given the generalised bundle ${E \simeq TM \oplus T^{\star}M}$, with ${x \in TM}$ and ${\xi \in T^{\star}M}$, we can write the generalised vector ${X \in \Gamma (E)}$, in which ${\Gamma (E)}$ denotes the sections on E, in component form

$\displaystyle X \Gamma = \begin{pmatrix} x \\ \xi \\ \end{pmatrix}. \ \ \ \ \ (20)$

We discussed in the previous note linked above how this space comes equipped with an inner product and a metric. Let us now denote this metric as

$\displaystyle \eta (X, X) = \langle X, X \rangle = i_x \xi. \ \ \ \ \ (21)$

It will become clear in a later note that ${\eta}$ satisfies all the requirements as an ${o(d,d)}$-invariant metric on the extended space (for the reader familiar with double field theory, this is indeed the ${o(d,d)}$ constant metric we know well).

If we now introduce notation for the components of the generalised vector ${X}$, using coordinate bases ${\{\partial / \partial x^{\mu} \}}$ on ${TU_{a}}$ and ${\{ d x^{\mu} \}}$ on ${T^{\star}U_a}$ over the patch ${U_a}$, it is quite plain to see we can write ${X = x^{\mu} \partial / \partial x^{\mu} + \xi_{\mu}dx^{\mu}}$ as expected, where ${\mu = 1,...,d}$. For the index that runs over the entire extended space, it is conventional to use capital Latin letters ${M = 1,..., 2d}$. Hence,

$\displaystyle X^M = \begin{cases} x^{\mu} \ \text{for} \ M = \mu \\ \xi_{\mu} \ \text{for} \ M = \mu + d \end{cases}. \ \ \ \ \ (22)$

Therefore, it is possible to define a basis

$\displaystyle \langle E_A, E_B \rangle = \eta_{AB}. \ \ \ \ \ (23)$

We’ll speak more of this metric later. Simply note for now that, as it is therefore possible to define a natural o(d,d) action on the generalised bundle E, it follows we can define a generalised structure bundle

$\displaystyle K = \{ (x, E_A) : x \in M, \ \text{and} \ \{E_A \} \ \text{is an O(d,d) basis of} \ E_x, \ \ \ \ \ (24)$

which is a principle bundle with fibre ${o(d,d)}$ that makes possible our previous discussion on the structure generators such that we were able to reduce the structure group to ${SO(d,d)}$. The highest exterior power of E can be decomposed as

$\displaystyle \Lambda^{2d} TM \oplus T^{\star}M = \lambda^d TM \oplus \Lambda^d T^{\star}M \ \ \ \ \ (25)$

in which, given ${x \in \Lambda^d TM}$ and ${y \in \lambda^d T^{\star}M}$, there exists a natural canonical pairing between both sides of this direct sum

$\displaystyle (x,y) = \det (y_i (x_i)). \ \ \ \ \ (26)$

It is conventional to make the identification ${\Lambda^{2d} TM \oplus T^{\star}M = \mathbb{R}}$ so that the canonical orientation on E is defined by a real number.

Finally, returning to the discussion on patching, what this all means is that when studying the symmetries of this extended space we see that when transitioning from one patch ${U_a}$ to another ${U_b}$, in principle diffeomorphisms should relate the vectors ${x}$ and the one-forms ${\xi}$. So ${TM}$ is non-trivial. Likewise, additional transformations of the one-forms encode the way in which ${T^{\star}M}$ is fibred over ${TM}$.

Remark 2 (T-folds and non-geometric backgrounds) In more advanced settings, such as when studying T-folds, we study transition functions that involve T-duality. In [Alf21], the global geometry of a T-fold is studied by dimensionally reducing a bundle gerbe on a torus bundle spacetime. Interestingly, it is then shown how the geometric structure underlying the T-fold can be interpreted as a particular case of a global tensor hierarchy. This is again something we’ll consider later, particularly in the context of double field theory.

More precisely, the transition between local patches and the diffeomorphism takes the form

$\displaystyle X_a = x_a + \xi_a = A^{-1}_{ab}x_b + [A^T_{ab} \xi_b - \iota_{A^{-1}_{ab} \alpha_b} B_{ab}], \ \ \ \ \ (27)$

where as the indices ${a,b}$ suggest, we are considering the overlap of two local patches ${U_a \cap U_b}$.

As described in the previous section, if for now we interpret ${A_{ab} \in GL(d, \mathbb{R})}$ as an invertible matrix describing diffeomorphisms, ${B_{ab}}$ a two-form, and ${\iota_x}$ denotes contraction with the vector field ${x}$, then in component form for a generalised vector (27) can be expressed as

$\displaystyle X_a = \begin{pmatrix} x_a \\ \xi_a \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ B_{ab} & 1 \\ \end{pmatrix} \begin{pmatrix} A^{-1}_{ab} & 0 \\ 0 & A^{T}_{ab} \end{pmatrix} \begin{pmatrix} x_b \\ \xi_b \end{pmatrix} = h_{B_{ab}}h_{A_{ab}}X_b, \ \ \ \ \ (28)$

where the group ${O(d,d; \mathbb{Z})}$ is generated by elements denoted here as ${h_A}$, ${h_B}$. The two-form is typically restricted ${B_{ab} = d\Lambda_{ab}}$ such that, in the case of a triple overlap, ${U_a \cap U_b \cap U_c}$ the one-forms ${\Lambda_{ab}}$ must satisfy

$\displaystyle \Lambda_{ab} + \Lambda_{bc} + \lambda_{ca} = g^{-1}_{abc}(dg_{abc}). \ \ \ \ \ (29)$

As discussed more deeply later, this condition suggests that for the function ${g_{abc}}$, it is an element of ${U(1)}$ given by ${g_{abc} = \exp(i\lambda_{abc})}$. In the literature, this is interpreted to describe the structure of a gerbe, which at its most basic represents the analogue of a fibre bundle where the fibre is the classifying stack of a group. What it means is that ${E}$ geometrises diffeomorphisms and B-field gauge transformations [Alf21]. The role of gerbes in extended field theory and its geometry proves both interesting and important. It is a topic that is very much still in its infancy, with research in development.

What also becomes clear as one advances deeper in the study of generalised geometry, is that there is a natural correspondence between the patching 1-forms and the B-transformation. Indeed, as will become clear, the role of this B-transformation is no coincidence and, from view of string theory, we see that corresponds directly to the 2-form NS-NS potential. So already, and certainly as we progress in these notes, we’re starting to see some very nice features that given an interesting geometric perspective on the target space of the string. Acting as a generator of the subgroup ${h_B \subset SO(d,d)}$, we will learn explicitly how the overall structure group ${G}$ is necessarily given by the semi-direct product ${G = h_b \ltimes GL(d)}$ and embeds as a subgroup of ${SO(d, d)}$. This structure is actually quite interesting from the perspective of double field theory, but there is still much more to uncover before such considerations.

References

[Alf21] L. Alfonsi. Doubled Geometry of Double Field Theory [PhD thesis]. url: https://researchprofiles.herts.ac.uk/portal/files/26014618/2108.10297v1.pdf. [Gual04] M. Gualtieri. Generalized complex geometry [PhD thesis]. arXiv: 0401221[math.DG]. [Hitc10] N. Hitchin. Lectures on generalized geometry. arXiv: 1008.0973 [math.DG]. [Rub18] R. Rubio. Generalised geometry: An introduction [lecture notes]. url: https://mat.uab.cat/~rubio/gengeo/Rubio-GenGeo.pdf