# nLab Bn-geometry

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

## Phenomenology

#### Differential cohomology

differential cohomology

# Contents

## Idea

In the context of T-duality and in particular of differential T-duality one considers (as discussed in detail there) fiber products of two torus-fiber bundles together with a circle 2-bundle on this, (with connection).

In some disguise, this has been called $B_n$-geometry (Baraglia). The T-duality interpretation is made explicit in Bouwknegt

Here “$B_n$” refers to the special orthogonal group of the form $SO(n+1,n)$, which appears as the structure group of a generalized tangent bundle tensored with a line bundle (the Poincare line bundle of the T-duality correspondence).

## Properties

### Interpretation in higher differential geometry

We give the interpretation of $B_n$-geometry in higher differential geometry.

For $c_{conn},c'_{conn} \colon X \to \mathbf{B}U(1)$ modulating two circle principal bundles with conection, a differential T-duality structure is a choice of trivialization of their cup product class. From this we get the pasting diagram of homotopy pullbacks of smooth $\infty$-stacks

$\array{ P \times_X \hat P&\stackrel{\tau}{\to}& \mathbf{B}^2 U(1) &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\to& fib(\cup) &\to& (\mathbf{B}U(1)) \times (\mathbf{B}U(1)) \\ &\searrow & \downarrow && \downarrow^{\mathrlap{\cup}} \\ && * &\to& \mathbf{B}^3 U(1) } \,.$

Here $\tau$ is the morphism that modulates the circle 2-bundle on the fiber product of the two circle bundles.

(…)

## References

The term $B_n$-geometry was introduced in

A review is in

The relation to T-duality is made clear around slide 80 of

• Peter Bouwknegt, Courant Algebroids and Generalizations of Geometry, talk at StringMath2011 (pdf)

A discussion of the higher Lie theoretic aspects is in

Last revised on March 17, 2018 at 12:44:42. See the history of this page for a list of all contributions to it.