Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

String theory

Differential cohomology



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).


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.



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.