# Schreiber differential cohomology in a cohesive topos

A book that I am writing:

• Urs Schreiber

Differential cohomology in a cohesive $\infty$-topos

v2: pdf

For exposition and survey see at Higher Prequantum Geometry.

# Contents

## Abstract

We formulate differential cohomology and Chern-Weil theory – the theory of connections on fiber bundles and of gauge fields – abstractly in homotopy toposes that we call cohesive. Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, complex-analytic, formal, supergeometric, etc.) and equipped with connections, hence higher gauge fields. Furthermore we formulate differential geometry abstractly in a homotopy toposes that we call differentially cohesive. The manifolds in this theory are higher étale stacks (orbifolds) equipped with higher Cartan geometry (higher Riemannian-, complex, symplectic, conformal-, geometry) together with partial differential equations on spaces of sections of higher bundles over them, and equipped with higher pre-quantization of the resulting covariant phase spaces. We also formulate supergeometry abstractly in homotopy toposes and lift all these constructions to include fermionic degrees of freedom. Finally we indicate an abstract formulation of non-perturbative quantization of prequantum local field theory by fiber integration in twisted generalized cohomology of spectral linearizations of higher prequantum bundles.

We then construct models of the abstract theory in which traditional differential super-geometry is recovered and promoted to higher (derived) differential super-geometry.

We show that the cohesive and differential refinement of universal characteristic cocycles constitutes a higher Chern-Weil homomorphism refined from secondary characteristic classes to morphisms of higher moduli stacks of higher gauge fields, and at the same time constitutes extended geometric prequantization – in the sense of extended/multi-tiered quantum field theory – of hierarchies of higher dimensional Chern-Simons-type field theories, their higher Wess-Zumino-Witten-type boundary field theories and all further higher codimension defect field theories.

We find that in the Whitehead tower of superpoints in higher supergeometry one finds god given such cocycles on higher supersymmetry-groups, reflecting the completed brane scan of string/M-theory. We show that the induced higher super Cartan geometry is higher dimensional supergravity with super $p$-brane charge corrections included. For the maximal case of 11-dimensional supergravity we find the Einstein equations of motion and the higher extension of their super-isometry groups by M2/M5-brane charges in twisted generalized cohomology.

We close with an outlook on the cohomological quantization of these higher boundary prequantum field theories by a kind of cohesive motives.

## Sub-projects

The discussion here builds on and subsumes various sub-projects that appear separately.

## Followup articles

In

the case of stable cohesion for smooth cohesion is discussed in more detail – smooth spectra – and the “differential cohomology diagram” is derived from stable cohesion.

In

global equivariant homotopy theory is shown to be cohesive.

In

the cohesive homotopy type theory of (Schreiber-Shulman 12) provides an example (p. 10) of homotopy type theory with higher modalities.

In

the (∞,n)-category of correspondences in a slice (∞,1)-topos as appearing in Local prequantum field theory is analyzed.

In

nonabelian Hodge theory is generalized to twisted bundles via the theory of cohesive principal infinity-bundles.

In

further aspects of the smooth shape modality of cohesion are worked out (the etale homotopy type operation in the context of smooth infinity-stacks) as applied to orbifolds and étale groupoids and generally étale ∞-groupoids.

In

the formalization in cohesive homotopy type theory is developed via adjoint logic.

## Lecture notes and talk notes

List of invited talks and lectures, in chronological order.

Revised on August 29, 2017 12:07:15 by Urs Schreiber (192.41.132.62)