nLab
Bn-geometry

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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}{}& ʃ &\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)

String theory

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 nB_n-geometry (Baraglia). The T-duality interpretation is made explicit in Bouwknegt

Here “B nB_n” refers to the special orthogonal group of the form SO(n+1,n)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 nB_n-geometry in higher differential geometry.

For c conn,c conn:XBU(1)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

P× XP^ τ B 2U(1) * X fib() (BU(1))×(BU(1)) * B 3U(1). \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 τ\tau is the morphism that modulates the circle 2-bundle on the fiber product of the two circle bundles.

(…)

References

The term B nB_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 08:44:42. See the history of this page for a list of all contributions to it.