higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Type II geometry is to Riemannian geometry as generalized complex geometry is to complex geometry.
Where the latter is the geometry induced by reduction of the structure group of the generalized tangent bundle of an even dimensional manifold along the inclusion $U(d,d) \to O(2d,2d)$ of the indefinite unitary group into the orthogonal group, type II geometry is the geometry induced by reduction along the inclusion of the product of orthogonal groups
which is the inclusion of the maximal compact subgroup into the Narain group.
This notion takes its name from the fact that it describes a good bit of the geometry of type II supergravity.
The definition of type II geometry proceeds in direct analogy with that of Riemannian geometry in terms of orthogonal structure/vielbein fields on the tangent bundle, generalized here to the generalized tangent bundle:
(…)
We discuss how a type II geometry is the reduction of the structure group of the generalized tangent bundle along the inclusion $O(d) \times O(d) \to O(d,d)$.
Consider the Lie group inclusion
of those orthogonal transformations, that preserve the positive definite part or the negative definite part of the bilinear form of signature $(d,d)$, respectively.
If $\mathrm{O}(d,d)$ is presented as the group of $2d \times 2d$-matrices that preserve the bilinear form given by the $2d \times 2d$-matrix
then this inclusion sends a pair $(A_+, A_-)$ of orthogonal $n \times n$-matrices to the matrix
This inclusion of Lie groups induces the corresponding morphism of smooth moduli stacks of principal bundles
There is a fiber sequence of smooth stacks
where the fiber on the left is the coset space of the action of $O(d) \times O(d)$ on $O(d,d)$.
There is a canonical embedding
of the general linear group.
In the above matrix presentation this is given by sending
where in the bottom right corner we have the transpose of the inverse matrix of the invertble matrix $a$.
Under inclusion of def. 1, the tangent bundle of a $d$-dimensional manifold $X$ defines an $\mathrm{O}(d,d)$-cocycle
The vector bundle canonically associated to this composite cocycles may canonically be identified with the direct sum vector bundle $T X \oplus T^* X$, and so we will refer to this cocycle by these symbols, as indicated. This is also called the generalized tangent bundle of $X$.
Therefore we may canonically consider the groupoid of $T X \oplus T^* X$-twisted $\mathbf{TypeII}$-structures, according to the general notion of twisted differential c-structures.
More generally, instead of $E = T X \oplus T^* X$ one considers bundle extensions $E$ of the form
These may have structure froups in $O(n,n)$ but not in the inclusion $GL(n) \hookrightarrow O(n,n)$. For more on this see the section Geometric and non-geometric type II geometries below. Accordingly, in all of the following $T X \oplus T^* X$ could be replaced by a more general extension $E$.
A type II generalized vielbein on a smooth manifold $X$ is a diagram
in $\mathbf{H} =$ Smooth∞Grpd, hence a cocycle in the smooth twisted cohomology
The groupoid $\mathbf{TypeII}\mathrm{Struc}(X)$ is that of “generalized vielbein fields” on $X$, as considered for instance around equation (2.24) of (GMPW) (there only locally, but the globalization is evident).
In particular, its set of equivalence classes is the set of type-II generalized geometry structures on $X$.
Over a local coordinate chart $\mathbb{R}^d \simeq U_i \hookrightarrow X$, the most general such generalized vielbein (hence the most general $\mathrm{O}(d,d)$-valued function) may be parameterized as
where $e_+, e_- \in C^\infty(U_i, \mathrm{O}(d))$ are thought of as two ordinary vielbein fields, and where $B$ is any smooth skew-symmetric $n \times n$-matrix valued function on $\mathbb{R}^d \simeq U_i$.
By an $\mathrm{O}(d) \times \mathrm{O}(d)$-gauge transformation this can always be brought into a form where $e_+ = e_- =: \tfrac{1}{2}e$ such that
The corresponding “generalized metric” over $U_i$ is
where
is the metric (over $\mathbb{R}^q \simeq U_i$ a smooth function with values in symmetric $n \times n$-matrices) given by the ordinary vielbein $e$.
An element in $O(d,d)$ which in the canonical matrix presentation is of the block form
is called a $B$-transform. An element of the block form
is called a $\beta$-transform. The subgroup
generated by $Gl(d) \hookrightarrow O(d,d)$ and the B-transforms, hence that of matrices with vaishing top right block is called the geometric subgroup (e.g. GMPW, p.5).
A type II background where the structure group of the generalized tangent bundle is not in the inclusion of the geometric subgroup is often called a non-geometric background (e.g. GMPW, section 5).
The target space geometry for type II superstrings in the NS-NS sector , type II supergravity, is naturally encoded by type II geometry.
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The appearance of type II geometry in type II supergravity/type II string theory is discussed for instance in
Ian Ellwood, NS-NS fluxes in Hitchin’s generalized geometry (arXiv:hep-th/0612100)
Mariana Graña, Ruben Minasian, Michela Petrini, Daniel Waldram, T-duality, Generalized Geometry and Non-Geometric Backgrounds (arXiv:0807.4527)
The genuine reformulation of type II supergravity as a $(O(d)\times O(d) \hookrightarrow O(d,d))$-gauge/gravity theory is in
In
the geometry of the reduction $O(d) \times O(d) \to O(d,d)$ was referred to as “type I geometry”, with “type II geometry” instead referring to further U-duality group extensions, discussed at exceptional generalized geometry.
The above formulation in terms of twisted smooth cohomology is discussed in section 5 of