Schreiber differential cohomology in a cohesive topos -- references

A commented list of references for differential cohomology in a cohesive topos .

Topos theory

A standard monograph on topos theory is

• Peter Johnstone, Sketches of an Elephant .

The standard text on (∞,1)-topos theory is

• Jacob Lurie, Higher Topos Theory

Cohesive toposes

The axioms for a cohesive topos originate in

• Bill Lawvere, Categories of spaces may not be generalized spaces, as exemplified by directed graphs , preprint, State University of New York at Buffalo, (1986) Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1?7 (pdf)

Under the name categories of cohesion these axioms, slightly refined, are presented in

• Bill Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41?49. (pdf)

The axioms on a cohesive (∞,1)-topos are an immediate generalization of these.

Cohomology

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

• Ken Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology (1973)

A discussion of principal ∞-bundles in topological ∞-groupoids and their geometric realization of simplicial topological spaces is in

• David Roberts, Danny Stevenson, Simplicial principal bundles in parametrized spaces

Homotopy

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

• Richard Williamson, private communication

A discussion of how this general abstract definition relates to various other construction in the literature is at

• nLab, geometric homotopy groups in an (∞,1)-topos

I profited from useful interaction with Mike Shulman when working on the related nLab pages.

Differential cohomology

The standard reference on (generalized) differential cohomology is

• Mike Hopkins, I. Singer, Quadratic Functions in Geometry, Topology,and M-Theory J. Differential Geom. Volume 70, Number 3 (2005), 329-452.

Deligne cohomology and differential characters

• Deligne cohomology

• differential character

Quantum anomalies

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

• Dan Freed, On determinant line bundles, Math. aspects of string theory, ed. S. T. Yau, World Sci. Publ. 1987, (revised pdf, dg-ga/9505002)

The fundamental article for the role of the determinant line bundle in understanding quantum anomalies is

• Michael Atiyah, I. M. Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. USA 81, 2597-2600 (1984)

A physicists’ account if the situation is in

• Reinhold Bertlmann, Anomalies in quantum field theory, Oxford Science Publ., 1996, 2000

An account that clearly identifies the mathematical nature of quantum anomalies for higher gauge theories is

• Dan Freed, Dirac charge quantization and generalized differential cohomology (arXiv)

The role of spin structures as the anomaly cancellation condition for the spinning particle is discussed in

• Edward Witten, Global anomalies in String theory in Symposium on anomalies, geometry, topology , World Scientific Publishing, Singapore (1985)

The string worldsheet Green-Schwarz mechanism which trivializes the worldsheet Pfaffian line bundle, and its relation to string structures goes back to

• Killingback, World-sheet anomalies and loop geometry Nuclear Physics B Volume 288, 1987, Pages 578-588

• Edward Witten, The Index of the Dirac Operator in Loop Space Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology, Princeton, N.J., Sep 1986.

• Edward Witten, Elliptic Genera and Quantum Field Theory

  Commun.Math.Phys.109:525,1987

and has been fully formalized in

• Ulrich Bunke, String structures and trivialisations of a Pfaffian line bundle (arXiv)

using the description of differential string structures as given in

• Konrad Waldorf, String Connections and Chern-Simons Theory (arXiv:0906.0117)

Connections

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.

In terms of parallel transport

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

• S. Kobayashi, Comptes Rendus, 238, 443-444. (1954). (web)

A more detailed and more general discussion has then been given in

• J. Milnor, Annals of Mathematics, 63, 272-284.(1956).

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

• J. W. Barrett, Holonomy and path structures in general relativity and Yang-Mills theory Int. J. of Theor. Phys., Iiol. 30, No. 9, 1991 (pdf)

who himself gives a proof. Barret implicitly uses the diffeological space structure on the space of loops.

In

• Jerzy Lewandowski Group of loops, holonomy maps, path bundle and path connection , Class. Quantum Grav. 10 (1993) 879-904 (pdf)

the statement of the equivalence is attributed to

• J. Anandan, Holonomy groups in gavity and gauge fields Pmc. Conf Differential Geometric Methods in Plystcs (fieste 1981) ed. G Denardo and H. D. Doebner (Singapore, World Scientific)

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

• A. Caetano, R.F. Picken, An axiomatic definition of holonomy , Int. Journ. Math. 5 (1994) 835 (scan)

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

• Dan Freed, Classical Chern-Simons theory, Part I (arXiv:hep-th/9206021)

A quick proof that bundles with connections are encoded in their parallel transport along paths was noted in

• Florin Dumitrescu, Connections and Parallel Transport (arXiv:0903.0121).

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:

• Roger Picken and Marco Mackaay, Holonomy and parallel transport for abelian gerbes (arXiv:0007053).

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.

• John Baez, Categorified gauge theory (arXiv)

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

• Joao Faria Martins, Roger Picken, The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module (arXiv)

On nonabelian gerbes

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.

Flat infinitesimal parallel transport

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

• Anders Kock,

  • Synthetic differential geometry , Cambridge University Press, London Math. Society Lecture NotesSeries No 333 (June 2006) (pdf)

  • Synthetic differential geometry of manifolds Cambridge Tracts in Mathematics 180 (2010) (pdf)

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

• Larry Breen, William Messing, Combinatorial differential forms (arXiv:0005087)

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

• Carlos Simpson, Constantin Teleman, de Rham?s theorem for ∞-stacks (pdf)

In

• Herman Stel, Master thesis (2010)

will appear some discussion of the infinitesimal path $\infty$-groupoid in ?LieGrpd?.

Higher Lie theory

$\infty$-Lie algebra valued differential forms

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

• R. D’Auria, P. Fré Geometric supergravity in $D = 11$ and its hidden supergroup Geometric Supergravity in D=11 and its hidden supergroup

The standard textbook monograph on supergravity in general and this formalism is particular is

• Leonardo Castellani, Riccardo D'Auria, Pietro Fre, Supergravity and Superstrings - A Geometric Perspective

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

• Pietro Fre, Pietro Antonio Grassi, Free differential algebras, rheonomy and pure spinors (arXiv)

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)

Lie integration

The basic idea of identifying the Sullivan construction applied to Chevalley-Eilenberg algebras as Lie integration to discrete ∞-groupoids appears in

• Vladimir Hinich, Descent of Deligne groupoids, Internat. Math. Res. Notices, 5:223?239, 1997 (doi, preprint ddg.pdf)

and for general L-∞ algebras in

• Ezra Getzler, Lie theory for nilpotent $L_\infty$ algebras, Annals of Mathematics, 170 (2009), 271?301 (math.AT/0404003)

(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

• André Henriques, Integrating $L_\infty$ algebras, Compos. Math. 144 (2008), no. 4, 1017–1045 (doi,math.AT/0603563)

(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

• Pavol ?evera?, Some title containing the words “homotopy” and “symplectic”, e.g. this one (arXiv:math.SG/0105080).

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

• Marius Crainic, Rui Fernandes, Integrability of Lie brackets (arXiv:math.DG/0105033)

The description of Lie integration with values in smooth ∞-groupoids regarded as simplicial presheaves on CartSp is in

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech Cocycles for Differential characteristic Classes – An $\infty$-Lie theoretic construction

Essentially the same integration prescription is considered in

• Dmitry Roytenberg, Differential graded manifolds and associated stacks: an overview (pdf).

Higher Chern-Weil theory

Chern-Weil theory

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

• Jean-Luc Brylinski and Dennis McLaughlin, Cech cocycles for characteristic classes , Communications in Mathematical Physics, Volume 178, Number 1, (Springer)

The special case of the Cech-Deligne cocycle for the Chern-Simons circle 3-bundle also appears separately in

• Jean-Luc Brylinski and Dennis McLaughlin, A geometric construction of the first Pontryagin class (1993) (pdf)

Previous work by the author that is subsumed here

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.

Articles

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

• John Baez, U.S., Higher Gauge Theory ,in Categories in Algebra, Geometry and Mathematical Physics , eds. A. Davydov et al, Contemp. Math. 431, AMS, Providence, Rhode Island, 2007, pp. 7-30 (arXiv:math/0511710)

The description of ordinary bundles with connection as cocycles on the path n-groupoid in terms of higher parallel transport is given in

• U. S.Konrad Waldorf, Parallel transport and functors J. Homotopy Relat. Struct. 4, 187-244 (2009) (arXiv:0705.0452).

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

• U. S., David Roberts, The inner automorphism 3-group of a strict 2-group , Journal of Homotopy and Related Structures, vol. 3(1) ,(arXiv:0708.1741)

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

• David Roberts, Universal simplicial bundles and inner automorphism n-groups (2007),

published as

• David Roberts, The universal simplicial bundle is a simplicial group, New York Journal of Mathematics, Volume 19 (2013) 51-60, journal version, arXiv:1204.4886.

The observation that the String-group has a an incarnation as the string 2-group in ?LieGrpd as a Lie 2-group is from

• John Baez, Alissa Crans, U.S., Danny Stevenson, From loop groups to 2-groups , Homology, Homotopy and Applications, Vol. 9 (2007), No. 2 (arXiv).

The definition of connection on an ∞-bundle and aspects of the corresponding ∞-Chern-Weil theory appears in

• Hisham Sati, U.S., Jim Stasheff,

  $L_\infty$-algebra connections in Fauser (eds.) Recent Developments in QFT, Birkhäuser (arXiv:0801.3480)

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

• Domenico Fiorenza, U.S., Jim Stasheff, Cech cocycles for differential characteristic classes – An $\infty$-Lie theoretic construction (web page)

The application of this to the discussion of differential string structures and differential fivebrane structures is discussed in

• Hisham Sati, U.S., Jim Stasheff, Twisted Differential String and Fivebrane Structures (arXiv:0910.4001).

The general theory of twisted differential structures is developed in

• Domenico Fiorenza, Hisham Sati, U.S. , Twisted Differential Structures

The relevance of these structures in the anomaly cancellation for string theory backgrounds is discussd in

• Hisham Sati, U.S., Jim Stasheff, Fivebrane structures Reviews in mathematical physics, 10 (2009) 1197 (arXiv:0805.0564).

Aspects of the discussion of an $(\infty,1)$-topos with path ∞-groupoid functor as a context for differential cohomology appear in

• U. S., Zoran Skoda, Categorified Symmetries , 5th Summer School of Modern Mathematical Physics, SFIN, XXII Series A: Conferences, No A1, (2009), 397-424 (Editors: Branko Dragovich, Zoran Rakic) (arXiv:1004.2472)

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

• U.S., AQFT from n-Functorial QFT , Communications in Mathematical Physics, Volume 291, Issue 2 (2009), pp.357-401

Abstracts and introductions

Click on the links to the following articles to obtain hyperlinked abstracts and introductions

• differential cohomology in a cohesive topos

  • ∞-Chern-Weil theory

    • Principal ∞-bundles – models and general theory

    • Cech cocycles for differential characteristic classes

    • Twisted differential structures

      • Twisted Differential String and Fivebrane Structures

      • The moduli 3-stack of the C-field

  • ∞-Chern-Simons theory

    • Higher Chern-Weil Derivation of AKSZ Sigma-Models

    • 7d Chern-Simons theory and the 5-brane

  • ∞-Wess-Zumino-Witten theory

Talk notes

• 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