Schreiber Higher Cartan Geometry

$\;\;\;\;\;\;\;\;\;\;\;$ Higher Cartan Geometry

$\;\;\;\;\;\;\;\;\;\;\;$ Theory and applications.

• Harmonic Analysis Seminar

  Charles University, Prague

  Spring 2015

Abstract

Classical Cartan geometry is the general theory of differential geometric structures, subsuming (pseudo-)Riemannian geometry, conformal geometry, … symplectic geometry, complex geometry, …, parabolic geometry, etc.

There are however applications which call for structures that are akin to Cartan geometries, but yet a bit richer. For instance:

• given a symplectic geometry one asks for its prequantization in which the symplectic form $\omega$ is refined to a line bundle with connection;

• given a left-invariant $(n+1)$ form $\omega$ on some coset space $G/H$, one asks for its refinement to a WZW term, a cocycle in ordinary differential cohomology, a line n-bundle with connection;

• given such a WZW term on $G/H$, one asks for its globalization over a $G/H$-Cartan geometry pre-quantizing a definite globalization of its curvature form $\omega$ as familiar form special holonomy;

• given a map of group stacks $H \to G$ (for instance $G$ being the String 2-group corresponding to a Kac-Moody loop group) one asks for an étale stack $X$ locally modeled on the homotopy quotient $G/H$.

All of these refinements involve higher differential geometry in the sense that they involve geometric homotopy n-types or n-stacks for $n \gt 0$.

This seminar 1) starts with a self-contained introduction to the elements of higher differential geometry, then 2) presents a theory of higher Cartan geometry in this context, and 3) discusses a collection of examples and applications of this theory.

As a running example, one application which involves all of the above ingredients is super p-brane geometry on higher dimensional supergravity super-spacetimes. Here, higher Cartan geometry serves to properly formulate and then solve problems such as the cancellation of classical anomalies of super $p$-branes, and the classification of BPS states.

Sessions

1. Feb 20 – Motivation – We informally survey motivation for higher Cartan geometry from phenomena and open problems visible in traditional geometry.

2. Feb 24$^\ast$ – Homogeneous bundles and higher geometry, guest talk by David Roberts on string 2-bundle extensions of the canonical $H$-principal bundles $G \to G/H$ over Klein geometries. Talk notes: pdf

3. Feb 27 – Deligne cohomology – We introduce ordinary differential cohomology in its incarnation as Cech cohomology with coefficients in the Deligne complex and discuss its main properties, notably the exact differential cohomology hexagon that it forms. In the course of this we introduce elements of homotopy theory via the structure of a category of fibrant objects on the category of chain complexes. Lecture notes at geometry of physics – principal connections.

4. Mar 6 – Smooth spaces – We introduce basics of regarding sheaves on the gros site of all smooth manifolds as generalized smooth spaces, which faithfully subsume smooth manifolds, infinite dimensional Fréchet manifolds, diffeological spaces as well as more exotic smooth moduli spaces, notably smooth spaces of differential forms. Lecture notes at geometry of physics – smooth sets.

5. Mar 12$^\ast$ – The (co-)reflective categories of supergravity – We discuss how the (higher) topos of super-stacks is stratified by a system of opposing (co-)reflective subcategories that serve to characterize its geometric content.

6. Mar 13$^\ast$ – Super-Cartan geometry – Cartan geometry naturally makes sense in the context of supergeometry. Much of the literature on supergravity is secretly about such super-Cartan geometry. In this talk I recall some basic concepts and then present in detail an interesting super-analog of the familiar geometry of definite/stable forms, with applications to the physics of super p-branes.

7. Mar 20 – Dold-Kan correspondence – In order to bring non-abelian principal connections into the context of Deligne cohomology we discuss the equivalence between chain complexes and simplicial abelian groups and use this to motivate Kan complexes and first elements of simplicial homotopy theory. Lecture notes at geometry of physics – homotopy types.

8. Mar 27 – ∞-Lie algebroids and their Lie integration – A key class of homotopy types/Kan complexes of interest in higher Lie theory are those arising via Lie integration of the homotopy-theoretic refinement of Lie algebras to L-∞ algebras and more generally of Lie algebroids to L-∞ algebroids.

9. Apr 3 – smooth ∞-groupoids – The previous example clearly suggests to equip homotopy types/Kan complexes with smooth structure. We introduce such smooth homotopy types in direct higher analogy with the smooth sets discussed before. Technically these are also known as smooth ∞-stacks, but we discuss how, despite this name, there are simple methods in a model structure on simplicial presheaves to handle these efficiently and in practice. We indicate the crucial abstract structure of the collection of smooth homotopy types: they form a cohesive ∞-topos. Lecture notes are at geometry of physics – smooth homotopy types

10. Apr 10 – principal ∞-bundles. We saw that morphisms between smooth groupoids neatly capture cocycles in Cech cohomology. Now we discuss that principal bundles are precisely the homotopy fibers of these cocycles. This fact immediately generalizes to principal infinity-bundles. Lecture notes at geometry of physics – principal bundles.

11. Apr 17 – ∞-groups, ∞-actions and associated ∞-bundles A key fact of geometric homotopy theory is that representation theory becomes neatly subsumes as simply the slice theory over deloopings of ∞-groups. We spell this out for ordinary groups (1-groups), their actions and their associated bundles. As before, once in this form the generalization to higher differential geometry is immediate. Lecture notes are at geometry of physics – groups and geometry of physics – representations and associated bundles.

12. Apr 23 – formal smooth ∞-groupoids – In order to efficiently speak about local diffeomorphisms (formally étale morphisms) and jet bundles in higher differential geometry it is convenient to further equip smooth homotopy types with “synthetic differential” structure by admitting infinitesimal spaces. This just amounts to passing from the site of smooth manifolds to that of formal smooth manifolds. We indicate the crucial abstract structure of the collection of formal formal smooth homotopy types: they form a differential cohesive ∞-topos. Lecture notes at geometry of physics – manifolds and orbifolds

13. May 15 – WZW terms – Now that we know what a manifold locally modeled on a cohesive $\infty$-group $G$ is, we will equip these with geometric structure given by “definite globalizations of WZW terms”. To that end here we first discuss WZW terms as such: higher prequantizations of L-∞ cocycles. Lecture notes at geometry of physics – WZW terms.

14. May 22 – BPS charges – Putting it all together, we consider definite globalizations of WZW terms over $V$-manifolds and find that these come with “reductions” of the higher structure group to the homotopy stabilizer group of the given WZW term. After globalization, there is then also the homotopy stabilizer of the globalized WZW terms under the isometry group of the induced G-structure. We discuss how when applied to the WZW terms for the super p-branes this yields BPS charge-extensions of super-isometry groups/Lie algebras, such as the M-theory super Lie algebra. Lecture notes at geometry of physics – BPS charges.

15. May 29${}^\ast$ infinitesimal symmetries of Deligne cocycles – We discuss in more detail the origin of the charge-extended higher current algebras given by the Poisson bracket Lie n-algebra. Lecture notes at geometry of physics – prequantum geometry

Further talk notes covering the material of the last three sessions:

• Structure Theory for Higher WZW Terms.

And further stuff in:

• From the Superpoint to T-Folds

Literature

The seminar is based on the material contained in the book-in-preparation

• Differential cohomology in a cohesive topos

and the references given there. Concretely, the course notes follow

• section 1.2, The geometry of physics

(which will be further expanded as we go along).

For more formal details see

• Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory, 2017