nLab
type II geometry

Idea

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

$$O(n)\times O(n)\to O(n,n),$$O(n) \times O(n) \to O(n,n) \,,

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.

Definition

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:

1. As a fiberwise metric on the generalized tangent bundle

2. By reduction of the structure group of the generalized tangent bundle

By fiberwise metric on the generalized tangent bundle

(…)

By reduction of 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 $O(d)\times O(d)\to O(d,d)$.

Definition

Consider the Lie group inclusion

$$\mathrm{O}(d)\times \mathrm{O}(d)\to \mathrm{O}(d,d)$$\mathrm{O}(d) \times \mathrm{O}(d) \to \mathrm{O}(d,d)

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

$$\eta ≔\left(\begin{array}{cc}0& {\mathrm{id}}_d\\ {\mathrm{id}}_d& 0\end{array}\right)$$\eta \coloneqq \left( \array{ 0 & \mathrm{id}_d \\ \mathrm{id}_d & 0 } \right)

then this inclusion sends a pair $(A_{+},A_{-})$ of orthogonal $n\times n$-matrices to the matrix

$$(A_{+},A_{-})\mapsto \frac{1}{\sqrt{2}}\left(\begin{array}{cc}{A}_{+}+A_{-}& A_{+}-A_{-}\\ A_{+}-A_{-}& A_{+}+A_{-}\end{array}\right).$$(A_+ , A_-) \mapsto \frac{1}{\sqrt{2}} \left( \array{ A_+ + A_- & A_+ - A_- \\ A_+ - A_- & A_+ + A_- } \right) \,.

This inclusion of Lie groups induces the corresponding morphism of smooth moduli stacks of principal bundles

$$\mathrm{TypeII}:B(\mathrm{O}(d)\times \mathrm{O}(d))\to B\mathrm{O}(d,d).$$\mathbf{TypeII} : \mathbf{B}(\mathrm{O}(d) \times \mathrm{O}(d)) \to \mathbf{B} \mathrm{O}(d,d) \,.

Proposition

There is a fiber sequence of smooth stacks

$$O(d)\backslash O(d,d)/O(d)\to B(\mathrm{O}(d)\times \mathrm{O}(d))\stackrel{\mathrm{TypeII}}{\to }B\mathrm{O}(d,d),$$O(d) \backslash O(d,d) / O(d) \to \mathbf{B}(\mathrm{O}(d) \times \mathrm{O}(d)) \stackrel{\mathbf{TypeII}}{\to} \mathbf{B} \mathrm{O}(d,d) \,,

where the fiber on the left is the coset space of the action of $O(d)\times O(d)$ on $O(d,d)$.

Definition

There is a canonical embedding

$$\mathrm{GL}(d)\hookrightarrow \mathrm{O}(d,d)$$\mathrm{GL}(d) \hookrightarrow \mathrm{O}(d,d)

of the general linear group.

In the above matrix presentation this is given by sending

$$a\mapsto \left(\begin{array}{cc}a& 0\\ 0& a^{-T}\end{array}\right),$$a \mapsto \left( \array{ a & 0 \\ 0 & a^{-T} } \right) \,,

where in the bottom right corner we have the transpose of the inverse matrix of the invertble matrix $a$.

Definition

Under inclusion of def. 1, the tangent bundle of a $d$-dimensional manifold $X$ defines an $\mathrm{O}(d,d)$-cocycle

$$TX\oplus T^{*}X:X\stackrel{TX}{\to }B\mathrm{GL}(d)\stackrel{}{\to }B\mathrm{O}(d,d).$$T X \oplus T^* X : X \stackrel{T X}{\to} \mathbf{B}\mathrm{GL}(d) \stackrel{}{\to} \mathbf{B} \mathrm{O}(d,d) \,.

The vector bundle canonically associated to this composite cocycles may canonically be identified with the direct sum vector bundle $TX\oplus T^{*}X$, 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 $TX\oplus T^{*}X$-twisted $\mathrm{TypeII}$-structures, according to the general notion of twisted differential c-structures.

More generally, instead of $E=TX\oplus T^{*}X$ one considers bundle extensions $E$ of the form

$$T^{*}X\to E\to TX.$$T^* X \to E \to T X \,.

These may have structure froups in $O(n,n)$ but not in the inclusion $\mathrm{GL}(n)\hookrightarrow O(n,n)$. For more on this see the section Geometric and non-geometric type II geometries below. Accordingly, in all of the following $TX\oplus T^{*}X$ could be replaced by a more general extension $E$.

Definition

A type II generalized vielbein on a smooth manifold $X$ is a diagram

$$\begin{array}{ccccc}X& & \stackrel{\widetilde{(}TX\oplus T^{*}X)}{\to }& & B(O(n)\times O(n))\\ & {}_{TX\oplus T^{*}X}\searrow & {⇙}_E& {\swarrow }_{\mathrm{TypeII}}\\ & & BO(n,n)\end{array}$$\array{ X &&\stackrel{\widetilde(T X \oplus T^* X)}{\to}&& \mathbf{B}(O(n) \times O(n)) \\ & {}_{\mathllap{T X \oplus T^* X}}\searrow &\swArrow_{E}& \swarrow_{\mathrlap{\mathbf{TypeII}}} \\ && \mathbf{B} O(n,n) }

in $H=$ Smooth∞Grpd, hence a cocycle in the smooth twisted cohomology

$$E\in \mathrm{TypeII}\mathrm{Struc}(X)≔H_{/BO(n,n)}(TX\oplus T^{*}X,\mathrm{TypeII}).$$E \in \mathbf{TypeII}Struc(X) \coloneqq \mathbf{H}_{/\mathbf{B} O(n,n)}(T X \oplus T^* X, \mathbf{TypeII}) \,.

Proposition / Definition

The groupoid $\mathrm{TypeII}\mathrm{Struc}(X)$ 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 $X$.

Proof

Over a local coordinate chart ${ℝ}^d\simeq U_i\hookrightarrow X$, the most general such generalized vielbein (hence the most general $\mathrm{O}(d,d)$-valued function) may be parameterized as

$$E=\frac{1}{2}\left(\begin{array}{cc}(e_{+}+e_{-})+(e_{+}^{-T}-e_{-}^{-T})B& (e_{+}^{-T}-e_{-}^{-T})\\ (e_{+}-e_{-})-(e_{+}^{-T}+e_{-}^{-T})B& (e_{+}^{-T}+e_{-}^{-T})\end{array}\right),$$E = \frac{1}{2} \left( \array{ (e_+ + e_-) + (e_+^{-T} - e_-^{-T})B & (e_+^{-T} - e_-^{-T}) \\ (e_+ - e_-) - (e_+^{-T} + e_-^{-T})B & (e_+^{-T} + e_-^{-T}) } \right) \,,

where $e_{+},e_{-}\in C^{\mathrm{\infty }}(U_i,\mathrm{O}(d))$ are thought of as two ordinary vielbein fields, and where $B$ is any smooth skew-symmetric $n\times n$-matrix valued function on ${ℝ}^d\simeq U_i$.

By an $\mathrm{O}(d)\times \mathrm{O}(d)$-gauge transformation this can always be brought into a form where $e_{+}=e_{-}=:\frac{1}{2}e$ such that

$$E=\left(\begin{array}{cc}e& 0\\ -e^{-T}B& e^{-T}\end{array}\right).$$E = \left( \array{ e & 0 \\ - e^{-T}B & e^{-T} } \right) \,.

The corresponding “generalized metric” over $U_i$ is

$$E^{T}E=\left(\begin{array}{cc}{e}^T& B{e}^{-1}\\ 0& e^{-1}\end{array}\right)\left(\begin{array}{cc}e& 0\\ -e^{-T}B& e^{-T}\end{array}\right)=\left(\begin{array}{cc}g-B{g}^{-1}B& B{g}^{-1}\\ -g^{-1}B& g^{-1}\end{array}\right),$$E^T E = \left( \array{ e^T & B e^{-1} \\ 0 & e^{-1} } \right) \left( \array{ e & 0 \\ - e^{-T}B & e^{-T} } \right) = \left( \array{ g - B g^{-1} B & B g^{-1} \\ - g^{-1} B & g^{-1} } \right) \,,

where

$$g≔e^{T}e$$g \coloneqq e^T e

is the metric (over ${ℝ}^q\simeq U_i$ a smooth function with values in symmetric $n\times n$-matrices) given by the ordinary vielbein $e$.

Geometric and “non-geometric” type II geometries

Definition

An element in $O(d,d)$ which in the canonical matrix presentation is of the block form

$$e^{\omega }≔\left(\begin{array}{cc}{1}_d& 0\\ \omega & 1_d\end{array}\right)$$e^\omega \coloneqq \left( \array{ 1_d & 0 \\ \omega & 1_d } \right)

is called a $B$-transform. An element of the block form

$$e^{\beta }≔\left(\begin{array}{cc}{1}_d& \beta \\ 0& 1_d\end{array}\right)$$e^\beta \coloneqq \left( \array{ 1_d & \beta \\ 0 & 1_d } \right)

is called a $\beta$-transform. The subgroup

$$G_{\mathrm{geom}}(d)\hookrightarrow O(d,d)$$G_{geom}(d) \hookrightarrow O(d,d)

generated by $\mathrm{Gl}(d)\hookrightarrow O(d,d)$ 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).

Application in type II supergrabity

The target space geometry for type II superstrings in the NS-NS sector , type II supergravity, is naturally encoded by type II geometry.

…

Related concepts

• exceptional generalized geometry

• twisted smooth cohomology in string theory

References

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

• André Coimbra, Charles Strickland-Constable, Daniel Waldram, Supergravity as Generalised Geometry I: Type II Theories (arXiv:1107.1733)

In

• Chris Hull, Generalised Geometry for M-Theory (arXiv:hep-th/0701203)

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

• Urs Schreiber, differential cohomology in a cohesive topos