A commented list of references for differential cohomology in a cohesive topos .
A standard monograph on topos theory is
The standard text on (∞,1)-topos theory is
The axioms for a cohesive topos originate in
Under the name categories of cohesion these axioms, slightly refined, are presented in
The axioms on a cohesive (∞,1)-topos are an immediate generalization of these.
See the references at cohomology.
The observation that cohomology is nothing but forming derived hom spaces in (∞,1)-toposes over a suitable site is essentially already given in
A discussion of principal ∞-bundles in topological ∞-groupoids and their geometric realization of simplicial topological spaces is in
The observation that geometric homotopy groups in an (∞,1)-topos are to be defined by the extra left adjoint of a locally ∞-connected (∞,1)-topos is due to
A discussion of how this general abstract definition relates to various other construction in the literature is at
I profited from useful interaction with Mike Shulman when working on the related nLab pages.
The standard reference on (generalized) differential cohomology is
We list references concerning quantum anomaly cancellation in the sense of trivializations of Pfaffian line bundles with connection on physical configurations spaces, whose sections constitute an action functional.
The general notion of Pfaffian line bundle is described for instance in section 3 of
The fundamental article for the role of the determinant line bundle in understanding quantum anomalies is
A physicists’ account if the situation is in
An account that clearly identifies the mathematical nature of quantum anomalies for higher gauge theories is
The role of spin structures as the anomaly cancellation condition for the spinning particle is discussed in
The string worldsheet Green-Schwarz mechanism which trivializes the worldsheet Pfaffian line bundle, and its relation to string structures goes back to
and has been fully formalized in
using the description of differential string structures as given in
The notion of connection on a bundle and its various generalizations and variants is fundamental and accordingly the relevant literature is vast, ranging from standard monographs to recent developments. We try to give commented lists of those references that in one way or other connect to our development.
There are many equivalent statements of the ordinary definition of a connection on a bundle. The following lists references related to the statement that the connection is equivalently encoded in terms of its parallel transport.
Apparently one of the oldest occurrences of the idea that a principal bundle $P \to X$ with connection $\nabla$ may be reconstructed from its holonomies around all smooth loops for any fixed base point in the connected base space $X$ appears in
A more detailed and more general discussion has then been given in
A detailed discussion of the differentiable case appears is
C. Teleman Généralisation du groupe fondamental , Annales Scientifiques de l’école Normale Supérieure 3, 77, 195-234. (1960) (web)
C. Teleman Annali di Matematica, Pura ed Applicata, LXII, 379-412. (1963).
This history is recollected in the introduction of
who himself gives a proof. Barret implicitly uses the diffeological space structure on the space of loops.
In
the statement of the equivalence is attributed to
Therein it is shown that smoothness of the parallel transport is a necessary condition for it to come from a smooth bundle with connection. Barrett also shows that this is sufficient.
Lewandowski adds to this a formulation of an equivalence of bundles with connections and the subset of loops around which the corresponding parallel transport is trivial.
Around the same time appeared
that generalizes these ideas from loops to general paths. These authors introduced the idea of sitting instants of paths and noticed that the most elegant way to (re)state the maximal equivalence relation on paths which is respected by parallel transport is in terms of thin homotopy.
Barrett originally had something very similar but slightly different. With Caetano and Picken’s relation, the space of thin homotopy classes of paths in $X$ becomes an groupoid $\mathbf{P}_1(X)$ – the path groupoid – internal to diffeological space.
(I am grateful to Christian Fleischhack and to Laurent Guillopé for help with tracking down some of the above links.)
A note on how a 1-form is encoded in the parallel transport that it induces along paths is also in appendix B of
A quick proof that bundles with connections are encoded in their parallel transport along paths was noted in
Motivated by the original results by Barrett et al. it was later observed that similarly an abelian gerbe with connection on a simply connected space is entirely encoded in the parallel surface transport that it induces on spheres:
John Baez noticed that these facts suggest that the proper formulation of bundles and higher bundles with connection should be in terms of smooth parallel transport n-functors $\mathbf{P}_n \to A$ that smoothly send the path n-groupoid of a space $X$ to a smooth n-groupoid.
At that point the motivation for this very natural definition was mainly formal, while the relevance of higher nonabelian parallel transport for physics was felt to be compelling but remained somewhat unclarified:
John Baez, Higher Yang-Mills theory (arXiv:hep-th/0206130)
Florian Girelli, Hendryk Pfeiffer, Higher gauge theory – differential versus integral formulation (arXiv:hep-th/0309173)
The author’s work – joint with John Baez and Konrad Waldorf – was motivated by these considerations and gives the full description of smooth higher parallel transport on the path 2-groupoid $\mathbf{P}_2(-)$.
The next categorical step, the description of smooth 3-functors out of a path 3-groupoid $\mathbf{P}_3(X)$ was discussed in
A proposal for a Cech-de Rham cocycle description of connections on $H$-gerbes for $H$ an arbitrary (possibly nonabelian) group was given in
Larry Breen, William Messing, Differential Geometry of Gerbes (arXiv:0106083)
Larry Breen, Differential Geometry of Gerbes and Differential Forms (arXiv:0802.1833).
See the discussion at differential cohomology in an (∞,1)-topos – survey – connections on 2-bundles.
We list some references related to the notion of the infinitesima path ∞-groupoid.
The idea of considering the infinitesimal singular simplicial complex as a simplicial object in a smooth topos seems to go back at least to Andre Joyal. Its usefulness for synthetic differential geometry, in particular for the definition of differential forms in synthetic differential geometry has been particularly worked out in
Its concrete realization on schemes was already prominently considered by Alexander Grothendieck in the context of Grothendieck connections/ deRham descent and in that of deRham spaces. The full realization of the infinitesimal singular simplicial complex on spaces formally dual to algebras was spelled out in some detail in
Relevant references on this are collected at infinitesimal singular simplicial complex.
The explicit interpretation of the infinitesimal singular simplicial complex of a manifold or scheme as an object presenting an ∞-Lie groupoid by use of the model structure on simplicial sheaves must have been obvious to Andre Joyal, who found this model structure, but I am not aware of explicit statements to that extent in the literature, along the lines followed at path ∞-groupoid. Similarly, the construction considered here of Yoneda-extending to a Quillen functor on all ∞-Lie groupoids that sits inside the finite path ∞-groupoid-functor, seems to have not yet been conceived explicitly in the literature before.
At the level of the homotopy category, however, the $(\infty,1)$-version of the de Rham space is considered in the unfinished notes
In
will appear some discussion of the infinitesimal path $\infty$-groupoid in ∞LieGrpd.
The (simple but important) observation that Lie-algebra valued 1-forms are encoded in terms of morphisms out of the Chevalley-Eilenberg algebra and Weil algebra was promoted by Cartan in
The evident generalization of this to ∞-Lie algebras is the key point of the D'Auria-Fre formulation of supergravity. The original article that introduced th D’Auria-Fré-formalism is
The standard textbook monograph on supergravity in general and this formalism is particular is
These articles to not use the terms “∞-Lie algebra” nor “Weil algebra”. These articles speak of the “FDA formalism” (and mean really semi-free dgas) but what they call a soft group manifold and the way higher curvature forms and higher Bianchi identities is read off from these is evidently precisely what (SSSI) formalize as ∞-Lie algebra valued differential forms. A vague hunch that the “FDA-formalism” is about higher Lie theory is mentioned on page 2 of
Some more references on the D’Auria-Fre formalism are
Pietro Fré, M-theory FDA, twisted tori and Chevalley cohomology (arXiv)
Pietro Fré and Pietro Antonio Grassi, Pure spinors, free differential algebras, and the supermembrane (arXiv)
Pietro Fré and Pietro Antonio Grassi, Free differential algebras, rheonomy, and pure spinors (arXiv)
The basic idea of identifying the Sullivan construction applied to Chevalley-Eilenberg algebras as Lie integration to discrete ∞-groupoids appears in
and for general L-∞ algebras in
(whose main point is the discussion of a gauge condition applicable for nilpotent $L_\infty$-algebras that cuts down the result of the Sullivan construction to a much smaller but equivalent model) .
This was refined from integration to discrete ∞-groupoids to an integration to internal ∞-groupoids in Banach manifolds in
(whose origin possibly preceeds that of Getzler’s article).
For general ∞-Lie algebroids the general idea of the integration process by “$d$-paths” had been indicated in
A detailed review of how the traditional Lie integration of Lie algebras and Lie algebroids to Lie groups and Lie groupoids (including the smooth structure) is reproduced in terms of $d$-pathis is given in
The description of Lie integration with values in smooth ∞-groupoids regarded as simplicial presheaves on CartSp is in
Essentially the same integration prescription is considered in
The Cech-Deligne cohomology description of the refinement of the Chern-Weil homomorphism to ordinary differential cohomology that we reproduce in ∞-Chern-Weil theory has been discussed in
The special case of the Cech-Deligne cocycle for the Chern-Simons circle 3-bundle also appears separately in
The general theory of differential cohomology in an (∞,1)-topos that is presented here has grown out of, subsumes and generalizes the following earlier work by the author.
The idea of realizing higher nonabelian differential cocycles in terms of higher dimensional parallel transport, hence in terms of the cohomology of a higher path groupoid originates in
The description of ordinary bundles with connection as cocycles on the path n-groupoid in terms of higher parallel transport is given in
A detailed account of the description of (abelian and nonabelian) gerbes/bundle gerbes/principal 2-bundles in terms of their higher parallel transport $n$-functors is in
U.S., Konrad Waldorf, Smooth Functors vs. Differential Forms , Homology, Homotopy and Applications, Vol. 13 (2011), No. 1, pp.143-203. (HHA, arXiv:0802.0663)
U.S., Konrad Waldorf, Connections on non-abelian gerbes and their holonomy, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. (TAC, arXiv:0808.1923)
In order to have globally defined parallel 2-transport, this considers connections on a 2-bundle with vanishing curvature 2-form and otherwise arbitrary curvature 3-form as cocycles $\mathbf{P}_2(X) \to \mathbf{B}$ on the path 2-groupoid with coefficients in the delooping of a Lie 2-group $G$. As we discuss, for describing general nonabelian $\infty$-connections one needs a groupal model for universal principal ∞-bundles. For strict 2-groups such was constructed in
The fully general statement for ∞-groups modeled as simplicial groups – which here we use for the definition of the universal Ehresmann ∞-connection – was later given in
published as
The observation that the String-group has a an incarnation as the string 2-group in ∞LieGrpd as a Lie 2-group is from
The definition of connection on an ∞-bundle and aspects of the corresponding ∞-Chern-Weil theory appears in
The Lie integration of the $\infty$-connection data to $\infty$-stacks of $\infty$-connections and the construction of the Chern-Simons circle 3-bundle and Chern-Simons circle 7-bundle with connection as examples of $\infty$-Chern-Weil homomorphism is in
The application of this to the discussion of differential string structures and differential fivebrane structures is discussed in
The general theory of twisted differential structures is developed in
The relevance of these structures in the anomaly cancellation for string theory backgrounds is discussd in
Aspects of the discussion of an $(\infty,1)$-topos with path ∞-groupoid functor as a context for differential cohomology appear in
The general discussion of $\infty$-Chern-Simons functionals following from this $\infty$-Chern-Weil theory is in
Domenico Fiorenza, Chris Rogers, U.S., Higher Chern-Weil Derivation of AKSZ Sigma-Models (arXiv:1108.4378)
Domenico Fiorenza, Chris Rogers, U.S., infinity-Chern-Simons theory
The discussion of the local net of endomorphisms induced from a cocycle on the path 2-groupoid is in
Click on the links to the following articles to obtain hyperlinked abstracts and introductions
WZW terms in a cohesive $\infty$-topos, talk in Erlangen (2011) (pdf)
Chern-Simons terms on higher moduli stacks, talk at Hausdorff Institute Bonn (2011) (pdf)
$\infty$-Chern-Simons functionals, Talk at Higher Structures Göttingen (2011) (pdf)
Stacks, differential geometry and action functionals
course notes for the 8th meeting of the Polish Category theory seminar as well as the 4th Odense winter school on geometry and theoretical physics (2011)
On differential cohomology in an $(\infty, 1)$-topos at Higher Structures IV, Göttingen in June 2010.
handwritten notes by Bruce Bartlett: talk 1, talk 2
audio recordings: audio talk 1, audio talk 2 (quality not great)
Background fields in twisted differential nonabelian cohomology at Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology