When doing generalised linear algebra, we want to study transformations that preserve the canonical pairing from the last note (of signature ):
The group of linear transformations that preserve the inner product is the orthogonal group. What we’ll see is that the linear endomorphisms of preserving this bilinear form is the orthogonal group . The corresponding Lie algebra, which represents the algebra of infinitesimal transformations, is denoted by and, as we’ll see, . This is the Lie algebra of endomorphisms of which are skew adjoint with respect to the bilinear form.
Elements satisfy . Using the splitting of the direct sum we may write in matrix form
with , , , and . Note that these are all linear maps. So we find that the condition on R takes the form
For we have , which must hold for all . This implies , with the adjoint of (i.e., it is the pullback). Similarly, consider ; then . For the case , , it follows . It is from these conditions that satisfy (3) that we often read in the literature the Lie algebra consisting of matrices of the form
where, to be clear, is an arbitrary matrix generating . 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 generates what we call B-transformations. In the context of generalised geometry, are , with an element of . In the maths literature, and are noted as being skew such that
where the skewness implies alternativity.
Any endomorphisms of E that is skew adjoint with respect to the bilinear form can be viewed as a skew-symmetric bilinear form via . Under this identification we therefore have
Structure group generatorsLet’s now look at the three primary orthogonal transformations belonging to . (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 reduces to generated by the following elements:
- Diffeomorphisms: It is important that from (4), when we get for any the element
where . 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 takes the form
which yields a skew-symmetric action of on . Taking the exponential of (7) we get
In the most general case for any we have an action
- B-transformations: From (4) consider transformations parameterised by an anti-symmetric matrix with integer entries. The two-form corresponding to the orthogonal transformation
acts as an endomorphisms of the generalised vector space such that
and so its exponential then acts
with the two-form having components . In the maths literature this B-transformation is also known as a shear transformation as it shifts but keeps V invariant. Similar to (7), the determinant of is +1.
Remark 1 (Acting on generalised metric) The requirement that is anti-symmetric comes from the most general 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 and the 2-form in such a way that it is manifestly invariant under . When acting on the generalised metric, for the B-transformation we find
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 with a one-form on the worldsheet, these shifts are gauge transformations and they will be a topic of consideration in a later note.
- -transformations: Similar to B-transformations, from (4) for these are transformations parametrised by an anti-symmetric matrix of the form
The determinant of is also +1. It gives the following action on
with its exponential
where the action of the bi-vector field on forms is defined by way of contraction. So, for some one-form it follows with a local basis on the tangent space . This type of transformation shifts V and leaves invariant.
Using and the elements we can write
Patching and transition functions
Note that the non-trivial nature of the generalised tangent bundle discussed in Note #1 is encoded in transition functions between local patches .
Consider, moreover, the definition of a generalised vector also given in the same note. Given the generalised bundle , with and , we can write the generalised vector , in which denotes the sections on E, in component form
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
It will become clear in a later note that satisfies all the requirements as an -invariant metric on the extended space (for the reader familiar with double field theory, this is indeed the constant metric we know well).
If we now introduce notation for the components of the generalised vector , using coordinate bases on and on over the patch , it is quite plain to see we can write as expected, where . For the index that runs over the entire extended space, it is conventional to use capital Latin letters . Hence,
Therefore, it is possible to define a basis
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
which is a principle bundle with fibre that makes possible our previous discussion on the structure generators such that we were able to reduce the structure group to . The highest exterior power of E can be decomposed as
in which, given and , there exists a natural canonical pairing between both sides of this direct sum
It is conventional to make the identification 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 to another , in principle diffeomorphisms should relate the vectors and the one-forms . So is non-trivial. Likewise, additional transformations of the one-forms encode the way in which is fibred over .
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
where as the indices suggest, we are considering the overlap of two local patches .
As described in the previous section, if for now we interpret as an invertible matrix describing diffeomorphisms, a two-form, and denotes contraction with the vector field , then in component form for a generalised vector (27) can be expressed as
where the group is generated by elements denoted here as , . The two-form is typically restricted such that, in the case of a triple overlap, the one-forms must satisfy
As discussed more deeply later, this condition suggests that for the function , it is an element of given by . 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 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 , we will learn explicitly how the overall structure group is necessarily given by the semi-direct product and embeds as a subgroup of . 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