differential monad

Differential endofunctors and differential monads appear in the treatment of differential calculus on noncommutative spaces/schemes represented by their abelian categories "of quasicoherent sheaves"; the formalism is due [[Valery Lunts]] and [[Alexander Rosenberg]].

#### Filtrations associated to topologizing subcategories

Below, a "subscheme" of an abelian category is in the sense of [[coreflective subcategory|coreflective]] [[topologizing subcategory]]. A "subscheme" is Zariski closed if it is also reflective.

Let $B$ be an abelian category $B$ satisfying the Gabriel's [[property sup]]. For any coreflective topologizing subcategory ("subscheme") $\mathbb{T}$ of $B$ one defines the notion of __$\mathbb{T}$-filtration__ on any object $M$ of $B$ as an increasing $M_\cdot = \{M_i \,|\, i\gt -1\}$ on $M$ such that $M_{-1} = 0$ and $M_i/M_{i-1} \in Ob \mathbb{T}$. By coreflectiveness of $\mathbb{T}$ there is a canonical choice of a $\mathbb{T}$-filtration given by $M_i = \mathbb{T}^{(i+1)}M$ (see [[n-th neighborhood of a topologizing subcategory]]); this is also called $\mathbb{T}^{i+1}$-torsion (part) of $M$. By the property (sup), there is a well-defined subobject $\mathbb{T}^{(\infty)}(M) := sup\{\mathbb{T}^{(i)}M\}$ which is calle the __$\mathbb{T}$-part__ of $M$. An object is a $\mathbb{T}$-object if it equals its own $\mathbb{T}$-part. Denote by $\mathbb{T}^\infty$ (note no brackets in the notation) the full subcategory of $B$ whose objects are $\mathbb{T}$-objects.

__Proposition.__ (Lunts-Rosenberg MPI1996-53, Lemma 3.3.1) The subcategory $\mathbb{T}^\infty$ is a coreflective topologizing subcategory of $B$.

#### The case of the diagonal "subscheme": differential filtration

Fix an abelian category $A$; let $B = End A$ be the category of additive endofunctors $A\to A$ (essential smallness or some sort of accessibility assumption may be required on $A$). If $A$ has the Gabriel's [[property sup]] then $End A$ also has the property sup).

Let $\mathbb{T}$ be the diagonal subscheme of $B = End A$, i.e. the minimal "subscheme" of the $End A$ containing the identity functor $Id_A$.  For example, if $A$ is the category ${}_R Mod$ for $R$ a simple ring, then all "subschemes" of it are (Zariski) closed.

__Definition.__ The $\Delta$-part of any object $M$ of $B = End A$ is called the __differential part__ of $M$, and $\Delta$-objects in $B$ are called __differential endofunctors__. Differential endofunctors are (Lunts-Rosenberg MPI 1996-53, 4.2) closed under composition. A __differential monad__ is a monad whose underlying endofunctor is differential.

In these definitions it is often convenient to redefine $B$ as some other full subcategory $B \subset End A$ containing $Id_A$, and closed under composition and colimits in $End A$.

#### The case of (noncommutative) rings

If $A$ is the category ${}_R Mod$ and $B = End_c A$ of endofunctors having right adjoint for a (possibly noncommutative associative) $k$-algebra $R$, then this modification leads to the (noncommutative generalization of the) notion of [[differential bimodule]]s. In other words, one looks at $\Delta_R$-part of bimodules where $\Delta_R$ is (basically, by the [[Eilenberg-Watts theorem]]) equivalent to the full subcategory of the category of $R\otimes_k R^{op}$-modules (i.e. $R$-bimodules) generated by all $M$, such that the kernel of the multiplication is an ideal in $R\otimes_k R^{op}$ contained in $Ann(m)$.

### Literature

The general abstract nonsense is proposed in these 1996 MPI preprints:

* V. A. Lunts, [[Alexander Rosenberg|A. L. Rosenberg]], _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preblob/3894), _II. D-Calculus in the braided case. The localization of quantized enveloping algebras_, MPI 1996-76 [pdf](http://www.mpim-bonn.mpg.de/preblob/3916)

This is applied (without mentioning differential monads) to the case of noncommutative rings in

* V. A. Lunts, [[Alexander Rosenberg|A. L. Rosenberg]], _Differential operators on noncommutative rings_, Selecta Math. (N.S.)  __3__  (1997),  no. 3, 335--359 ([doi](http://dx.doi.org/10.1007/s000290050014)); sequel: _Localization for quantum groups_, Selecta Math. (N.S.) __5__ (1999), no. 1, pp. 123--159 ([doi](http://dx.doi.org/10.1007/s000290050044)).

[[!redirects differential endofunctor]]


differential nonabelian cohomology

For differential [[nonabelian cohomology]], see [[schreiber:Differential Nonabelian Cohomology|Differential Nonabelian Cohomology]] on [[Urs Schreiber]]\'s web.

[[!redirects Differential Nonabelian Cohomology]]


differential object

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## Definition

A **differential object** in a [[category with translation]] $T : C \to C$ is an [[object]] $V$ equipped with a [[morphism]] $d_V : V \to T V$.

## Remarks

* This says that a differential object is a [[coalgebra]] for the endofunctor $T$.

## Further constructions

* Usually, when addressing [[coalgebra]]s for $T$ as  _differential objects_ one considers these in [[additive category|additive categories]] and requires that they are _nilpotent_ in that
$V \stackrel{d_V }{\to} T V \stackrel{T d_V}{\to} T T V$
is the [[zero morphism]]. Such a differential object is called a [[chain complex]].

* In a differential object $d_V : V \to T V$ in an [[additive category]] $C$ the **shifted differential object** is $T V$ with differential given by
$
  d_{T V} = - T(d_V)
$. The minus sign here is crucial in many constructions such as that of the [[mapping cone]]. It is naturally understood in terms of [[fiber sequence]]s in [[stable infinity-category|stable infinity-categories]].

[[!redirects differential objects]]


differential operator

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## Definition
 {#Definition}

We define general non-linear differential operators.

### In differential geometry
 {#DefinitionInDifferentialGeometry}

Depending on which definition of differential operators one regards as fundamental, the following are either definitions or are propositions.

+-- {: .num_defn #DifferentialOperatorViaJetBundles}
###### Definition/Proposition

For $X$ a [[smooth manifold]] and $(E\to X)$ a smooth [[bundle]] over $X$, write $(Jet(E)\to X)$ for its [[jet bundle]].

For $(E_1 \to X)$, $(E_2 \to X)$
two [[bundles]] over $X$, then a _differential operator_

$$
  D \colon \Gamma_X(E_1) \to \Gamma_X(E_2)
$$

between their [[spaces of sections]] is equivalently a map of the form

$$
  \phi \mapsto \tilde D \circ j_\infty(\phi)
$$

where $j_\infty(\phi) \in \Gamma_X(Jet(E_1))$ is the jet prolongation of the section $\phi \in \Gamma_X(E_1)$, and where

$$
  \tilde D \colon Jet(E_1) \to E_2
$$

is a bundle morphism from the [[jet bundle]] of $E_1$ to the bundle $E_2$.

=--

In this form this appears for instance as ([Saunders 89, def. 6.2.22](#Saunders89)).  Discussion showing the equivalence of this definition with the maybe more traditional definition is in ([Krasil'shchikVerbovetsky 98, def. 1.1, prop. 1.1, prop. 1.9](#KrasilshchikVerbovetsky98), [Krasilshchik 99, theorem 10](#Krasilshchik99)).

+-- {: .num_remark }
###### Remark

The [[jet bundle]] construction $Jet \colon \mathbf{H}_{/X} \to \mathbf{H}_{/X}$ is (by the discussion there) a [[comonad]] on the category of bundles over $X$. In terms of this def. \ref{DifferentialOperatorViaJetBundles} says that a differential operators from a bundle $E_1$ to a bundle $E_2$ is a morphism from $E_1$ to $E_2$ in the [[co-Kleisli category]] of the jet comonad.

=--

Indeed, also the composition of differential operators is the composition in this [[co-Kleisli category]] (e.g. [Marvan 93, section 1.1](#Marvan93)):

+-- {: .num_prop #CompositionOfDifferentialOperatorsViaCoKleisli}
###### Proposition

The [[composition]] $D_2 \circ D_1 \colon \Gamma_X(E_1) \to \Gamma_X(E_3)$ of two differential operators $D_1 \colon \Gamma_X(E_1) \to \Gamma_X(E_2)$ and $D_2 \colon \Gamma_X(E_2)\to \Gamma_X(E_3)$ , def. \ref{DifferentialOperatorViaJetBundles}, is given,under the identification of def. \ref{DifferentialOperatorViaJetBundles}, by the composite

$$
  \widetilde{D_2 \circ D_1}
  \;\colon\;
  Jet(E_1)
  \longrightarrow
  Jet(Jet(E_1))
  \stackrel{Jet(\tilde D_1)}{\longrightarrow}
  Jet(E_2)
  \stackrel{\tilde D_2}{\longrightarrow}
  E_3
  \,,
$$

where the first morphism is the [[counit of a comonad|counit]] of the [[jet bundle]] [[comonad]].

=--

+-- {: .proof}
###### Proof

Abbreviating $P_i \coloneqq \Gamma_X(E_i)$ and $J^\infty(P_i) = \Gamma_X(Jet(E_i))$, consider the following pasting diagram:

$$
  \array{
    P_1 &\stackrel{id}{\longrightarrow}& P_1 &\stackrel{D_1}{\longrightarrow}& P_2 &\stackrel{D_2}{\longrightarrow}& P_3
    \\
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{\mathrlap{j_\infty}}
      &&
    \downarrow^{\mathrlap{j_\infty}}
      &&
    \downarrow^{\mathrlap{id}}
    \\
    P_1
     &\stackrel{j_\infty}{\longrightarrow}&
    J^\infty(P_1)
    &\stackrel{J^\infty(D_1)}{\longrightarrow}&
    J^\infty(P_2)
    \\
    \downarrow^{\mathrlap{j_\infty}}
      &&
    \downarrow^{\mathrlap{j_\infty}}
      &&
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{\mathrlap{id}}
    \\
    J^\infty (P_1)
      &\stackrel{c^{\infty,\infty}}{\longrightarrow}&
    J^\infty(J^\infty(P_1))
     &\stackrel{J^\infty (\tilde D_1)}{\longrightarrow}&
    J^\infty(P_2)
     &\stackrel{\tilde D_2}{\longrightarrow}&
    P_3
  }
  \,.
$$

Here all the nontrivial squares are as in ([Krasil'shchik-Verbovetsky 98, p. 12-13](#KrasilshchikVerbovetsky98)), with the bottom middle square being the image under $J^\infty$ of the square defining $\tilde D_1$. The bottom horizontal fillers of these squares are unique by ([Krasil'shchik 99, theorem 10](#Krasilshchik99)) (which is just our def/prop. \ref{DifferentialOperatorViaJetBundles}), hence the identification of the middle bottom morphism as displayed in the diagram.

With this, the morphism that our proposition claims is the correct composite is the total bottom morphism, and the differential operator that this defines by def. \ref{DifferentialOperatorViaJetBundles} is the further composite with the left vertical morphism. Therefore the commutativity of the total diagram gives that this is equal to the total top morphisms, which is the composite of the two differential operators as claimed.

=--

The co-Kleisli-like composition for finite order differential operators also appears in ([Kock 10, section 7.3](#Kock10)), from a perspective of [[synthetic differential geometry]].

### In differential cohesion
 {#DefinitionInDifferentialCohesion}

In view of the [above](#DefinitionInDifferentialGeometry) one may axiomatize the category of differential operators in any context $\mathbf{H}$ of [[differential cohesion]] with [[infinitesimal shape modality]] $\Im$ as being the [[co-Kleisli category]] of the [[jet comonad]]

$$
  Jet_X \coloneqq i^\ast i_\ast
$$

induced by [[base change]] along the [[unit of a monad|unit]] $i \colon X \to \Im X$, for any choice of base space $X \in \mathbf{H}$.

### In algebraic geometry / D-geometry

For the case of [[algebraic geometry]], where $\Im X$ is known as the [[de Rham stack]] of a [[scheme]] $X$, and the [[quasicoherent sheaves]] on $i^\ast i_\ast X$ are the [[D-modules]] over $X$ (see at _[[jet bundle]]_ for more on this), this statement is implicit in ([Saito 89, def. 1.3](#Saito89)).

## Properties

+-- {: .num_prop #RetainsOrShrinksWaveFrontSetDifferentialOperator}
###### Proposition
**([[differential operator]] preserves or shrinks [[wave front set]])**

Let $P$ be a [[differential operator]] (with [[smooth function|smooth]] [[coefficients]]). Then for $u \in \mathcal{D}'$ a [[distribution]], the [[wave front set]] of the [[derivative of distributions]] $P u$ is contained in the original wave front set of $u$:

$$
  WF(P u) \subset WF(u)
  \,.
$$

=--

([Hörmander 90, (8.1.11)](#Hoermander90))

## Types of differential operators

* [[elliptic differential operator]]

* [[regular differential operator]], [[regular differential operator in noncommutative geometry]]

* [[pseudodifferential operator]]

* [[chiral differential operator]]

* [[crystalline differential operator]]

## Related concepts

* [[formally adjoint differential operator]]

* [[symbol of a differential operator]]

* [[index of a differential operator]]

* [[formal adjoint differential operator]]

* [[D-module]]

* [[Spencer cohomology]]

## References

* {#Marvan86} [[Michal Marvan]], _A note on the category of partial differential equations_, in _Differential geometry and its applications_, Proceedings of the Conference August 24-30, 1986, Brno ([[MarvanJetComonad.pdf:file]])

  (notice that prop. 1.3 there holds only when the equalizer exists in the first place)

* {#Saunders89} [[David Saunders]], _The geometry of jet bundles_, London Mathematical Society Lecture Note Series __142__, Cambridge Univ. Press 1989.

* {#Saito89} [[Morihiko Saito]], _Induced D-modules and differential complexes_, Bull. Soc. Math. France 117 (1989), 361–387, [pdf](http://smf4.emath.fr/Publications/Bulletin/117/pdf/smf_bull_117_361-387.pdf)

* {#Hoermander90} [[Lars Hörmander]], _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

* {#Marvan93} [[Michal Marvan]], _On Zero-Curvature Representations of Partial Differential Equations_,  (1993) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.5631))

* {#KrasilshchikVerbovetsky98} [[Joseph Krasil'shchik]], [[Alexander Verbovetsky]], _Homological Methods in Equations of Mathematical Physics_,  Lectures given in August 1998 at the International Summer School in Levoca, Slovakia ([arXiv:math/9808130](http://arxiv.org/abs/math/9808130))

* {#Krasilshchik99} [[Joseph Krasil'shchik]] in collaboration with Barbara Prinari, _Lectures on Linear Differential Operators over Commutative Algebras_ ([pdf](http://diffiety.ac.ru/preprint/99/01_99.pdf))

Discussion from a point of view of [[synthetic differential geometry]] is in

* {#Kock10} [[Anders Kock]], _Synthetic geometry of manfiolds_, Cambridge Tracts in Mathematics 180 (2010). ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

[[!redirects differential operators]]

[[!redirects linear differential operator]]
[[!redirects linear differential operators]]


differential programming

(Mostly) a rebranding of [[automatic differentiation]] by the programming languages community.


differential propositional calculus

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview).


differential renormalization

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## Idea

In [[perturbative quantum field theory]], _differential renormalization_ ([Freedman-Johnson-Latorre 92](#FreedmanJohnsonLatorre92)) is a method of _[[renormalization|("re"-)normalization]]_ which makes choices for defining [[time-ordered products]]/[[Feynman amplitudes]] at coincident interaction [[vertices]].

Differential normalization follows directly ([Prange 97](#Prange97)) from [[Epstein-Glaser renormalization]] in the guise of [[extensions of distributions]] of [[time-ordered products]]/[[Feynman amplitude]] to coinciding interaction points
([this prop.](renormalization#RenormalizationIsInductivelyExtensionToDiagonal)):

In the proof of that construction ([this prop](renormalization#SpaceOfPointExtensions)) over [[Minkowski spacetime]] $\mathbb{R}^{p,1}$, a choice of [[renormalization|("re"-)normalization]] is embodied at each order $k \in \mathbb{N}$ (number of [[vertices]] in [[Feynman diagrams]]) by a choice of projection operator

$$
  p_{\rho_k}
   \;\colon\;
  \mathcal{D}(\mathbb{R}^{(p+1)(k-1)})
    \longrightarrow
  \mathcal{D}_{\rho_k}(\mathbb{R}^{(p+1)(k-1)})
$$

([this equation](renormalization#eq:ForExtensionOfDistributionsProjectionMaps)) from all test functions to those that vanish to order $\rho_k$ at the origin. (This projction is often denoted "$W$", see [this remark](renormalization#WExtensions)).

Restricted to the image of this projector the relevant [[time-ordered product]] $T_k$ has a _unique_ [[extension of distributions]] to the origin, and this defines the [[renormalization|("re"-)normalization]].

This defines a new [[distribution]] $T_{k,R}$ by

$$
  \left\langle T_{k,R}, b \right\rangle
  \;\coloneqq\;
  \left\langle T_k, p_{\rho_k}(b)\right\rangle
  \,.
$$

Differential renormalization focuses on manipulating theses expressions $T_{k,R}$ ([Prange 97, section 1.1](#Prange97)).

## References

The concept is due to

* {#FreedmanJohnsonLatorre92} D. Z. Freedman, K. Johnson, J. I. Latorre, _Differential regularization and renormalization: a new method of calculation in quantum field theory_, Nucl. Phys. B 371 (1992) 353-414

* J. I. Latorre, X. Vilasis-Cardona, _Systematic Differential Renormalization to All Orders_, Ann. Phys. (N.Y.) 231 (1994) 149

Discussion in [[causal perturbation theory]]/relation to [[Epstein-Glaser renormalization]] is due to

* {#Prange97} Dirk Prange, _Epstein-Glaser renormalization and differential renormalization_, J. Phys. A 32, 2225 (1999) ([arXiv:hep-th/9710225](http://xxx.lanl.gov/abs/hep-th/9710225))

* [[Jose Gracia-Bondia]], _Differential renormalization and Epstein-Glaser renormalization_, Mod. Phys. Lett. A, 16, 281 (2001).  ([spire](http://inspirehep.net/record/557491/))

* [[Michael Dütsch]], [[Klaus Fredenhagen]], _Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity_, Reviews in Mathematical Physics, Volume 16, Issue 10, November 2004 ([arXiv:hep-th/0403213](https://arxiv.org/abs/hep-th/0403213))

* [[Michael Dütsch]], [[Klaus Fredenhagen]], Kai Johannes Keller, [[Katarzyna Rejzner]], _Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization_, J. Math. Phy. 55(12), 122303 (2014) ([arXiv:1311.5424](https://arxiv.org/abs/1311.5424))

* [[José Gracia-Bondía]], Heidy Gutiérrez, Joseph C. Várilly, _Improved Epstein-Glaser renormalization in $x$-space versus differential renormalization_, Nuclear Physics B 886 (2014), 824-869 ([arXiv:1403.1785](https://arxiv.org/abs/1403.1785))

Discussion of application to [[anomalous magnetic moment]] and [[supergravity]]:

* F. del Aguila, A. Culatti, R. Munoz-Tapia, M. Perez-Victoria, _Supergravity corrections to $(g-2)_l$ in differential renormalization_, Nuclear Physics  B  504  (1997)  532-550 ([arXiv:hep-ph/9702342](https://arxiv.org/abs/hep-ph/9702342))


differential ring

[[!redirects differential rings]]

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## Idea ##

A differential ring is an $\mathbb{Z}$-[[differential algebra]].

## Definition ##

A **differential ring** is a [[ring]] $R$ with an [[abelian group]] [[homomorphism]] $d:R \to R$ such that $d(a \cdot b) = a \cdot d(b) + d(a) \cdot b$.

## See also ##

* [[ring]]

* [[tensor ring]]

* [[symmetric ring]]

* [[exterior ring]]

* [[Clifford ring]]

* [[graded ring]]

* [[super ring]]

* [[supercommutative ring]]

* [[differential ring groupoid]]

* [[differential algebra]]


differential ring groupoid

[[!redirects differential ring groupoid]]

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## Idea ##

The [[groupoidal categorification]] of a [[differential ring]]. Hypothetically, a [[1-truncation]] of some "differential [[A-infinity-ring|$A_\infty$-ring]]".

## Definition ##

A **differential ring groupoid** is a [[ring groupoid]] $R$ with a [[symmetric 2-group]] [[functor]] $d:R \to R$ which preserves the symmetric 2-group structure of $R$, with an [[isomorphism]] $i_{a, b}:d(a \otimes b) \cong (a \otimes d(b)) \oplus (d(a) \otimes b)$ and coherence conditions (if any) which have not been figured out yet by the creator of this article.

## See also ##

* [[ring groupoid]]

* [[differential ring]]


differential spin structure

> to be merged with [[spin connection]]

#Contents#
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{:toc}

## Definition

A [[spin structure]] is given by a [[spin group]]-[[principal bundle]]. A _differential_ spin structure is a [[principal connection]] on this bundle.

## Related concepts

* [[twisted differential c-structures|(twisted, differential) c-structures]]

  * [[orientation]]

  * [[spin structure]], [[twisted spin structure]], [[differential spin structure]]

    [[spin^c structure]], [[twisted spin^c structure]],

  * [[string structure]], [[twisted differential string structure]],

  * [[fivebrane structure]], [[twisted differential fivebrane structure]]


differential string structure

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## Idea

Where a [[string structure]] is a trivialization of a class in [[integral cohomology]], a _differential string structure_ or _geometric string structure_ is the trivialization of this class refined to [[ordinary differential cohomology]]:

the first fractional [[Pontryagin class]]

$$
  \frac{1}{2} p_1 : B Spin \to B^4 \mathbb{Z}
$$

in the [[(∞,1)-topos]] [[∞Grpd]] $\simeq$ [[Top]] has a refinement to $\mathbf{H} = $ [[Smooth∞Grpd]] of the form

$$
  \frac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin \to \mathbf{B}^3 U(1)
$$

-- the smooth first fractional Pontryagin class.

The induced morphism on [[cocycle]] [[∞-groupoid]]s

$$
  \frac{1}{2}\mathbf{p}_1
  :
  \mathbf{H}(X,\mathbf{B} Spin)
  \stackrel{}{\to}
  \mathbf{H}(X,\mathbf{B}^3 U(1))
$$

sends a [[spin group]]-[[principal bundle]] $P$ to its corresponding [[Chern-Simons circle 3-bundle]] $\frac{1}{2}\mathbf{p}_1(P)$.

A choice of trivialization of $\frac{1}{2}p_1(P)$ is a [[string structure]]. The [[2-groupoid]] of smooth string structures is the [[homotopy fiber]] of $\frac{1}{2}\mathbf{p}_1$ over the trivial [[circle n-bundle with connection|circle 3-bundle]].

By [[Chern-Weil theory in Smooth∞Grpd]] this morphism may be further refined to a [[differential characteristic class]] $\frac{1}{2}\hat \mathbf{p}_1$ that lands in the [[ordinary differential cohomology]]
$\mathbf{H}_{diff}(X, \mathbf{B}^3 U(1))$,
classifying [[circle n-bundle with connection|circle 3-bundles with connection]]

$$
  \frac{1}{2}\hat \mathbf{p}_1
  :
  \mathbf{H}_{conn}(X,\mathbf{B} Spin)
  \stackrel{}{\to}
  \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))
$$

The [[2-groupoid]] of **differential string structures** is the [[homotopy fiber]] of this refinement $\frac{1}{2}\hat \mathbf{p}_1$ over the [[circle n-bundle with connection|trivial circle 3-bundle with trivial connection]] or more generally over the trivial circle 3-bundles with possibly non-trivial connection

Such a differential string structure over a [[smooth manifold]] $X$ is characterized by a tuple consisting of

1. a [[connection on a bundle|connection]] $\nabla$ on a [[Spin group|Spin]]-[[principal bundle]] on $X$;

1. a choice of trivial [[circle n-bundle with connection|circle 3-bundle]] with connection $(0, H_3)$, hence a differential 3-form $H_3 \in \Omega^3(X)$;

1. a choice of [[equivalence in an (∞,1)-category|equivalence]] $\lambda$ of the [[Chern-Simons circle 3-bundle]] with connection $\frac{1}{2}\hat\mathbf{p}_1(\nabla)$ of $\nabla$ with this chosen 3-bundle

  $$
    \lambda : \frac{1}{2}\hat \mathbf{p}_1(\nabla) \stackrel{\simeq}{\to}
    (0,H_3)
    \,.
  $$

More generally, one can consider the [[homotopy fiber]]s of $\frac{1}{2}\hat \mathbf{p}_1$ over arbitrary circle 3-bundles with connection $\hat \mathcal{G}_4 \in \mathbf{H}_{diff}^4(X, \mathbf{B}^3 U(1))$ and hence replace $(0,H_3)$ in the above with $\hat \mathcal{G}_4$. According to the general notion of [[twisted cohomology]], these may be thought of as **twisted differential string structures**, where the class $[\mathcal{G}_4] \in H^4_{diff}(X)$ is the twist.

## Definition
  {#Definition}

We will assume that the reader is familiar with basics of the discussion at [[Smooth∞Grpd]]. We often write $\mathbf{H} := Smooth \infty Grpd$ for short.

Let $Spin(n) \in $ [[SmoothMfd]] $\hookrightarrow$ [[Smooth∞Grpd]] be the [[Spin group]], for some $n \in \mathbb{N}$, regarded as a [[Lie group]] and thus canonically as an [[∞-group]] object in [[Smooth∞Grpd]]. We shall notationally suppress the $n$ in the following. Write $\mathbf{B}Spin$ for the [[delooping]] of $Spin$ in [[Smooth∞Grpd]]. (See the discussion here).
Let moreover $\mathbf{B}^2 U(1) \in Smooth \infty Grpd$ be the circle Lie 3-group and $\mathbf{B}^3 U(1)$ its [[delooping]].

At [[Chern-Weil theory in Smooth∞Grpd]] the following statement is proven ([FSS](#FSS)).

+-- {: .num_prop #LieIntegrationToPOne}
###### Proposition

The image under [[Lie integration]] of the canonical [[Lie algebra cohomology|Lie algebra 3-cocycle]]

$$
  \mu = \langle -,[-,-]\rangle : \mathfrak{so} \to b^2 \mathbb{R}
$$

on the [[semisimple Lie algebra]] $\mathfrak{so}$ of the [[Spin group]] -- the [[special orthogonal Lie algebra]] -- is a morphism in [[Smooth∞Grpd]] of the form

$$
  \frac{1}{2} \mathbf{p}_1 :
  \mathbf{B}Spin
    \to
  \mathbf{B}^3 U(1)
$$

whose image under the the fundamental ∞-groupoid (∞,1)-functor/ geometric realization $\Pi : Smooth \infty Grpd \to $ [[∞Grpd]] is the ordinary fractional [[Pontryagin class]]

$$
  \frac{1}{2}p_1 : B Spin \to B^4 \mathbb{Z}
$$

in [[Top]]. Moreover, the corresponding refined differential characteristic class

$$
  \frac{1}{2}\hat \mathbf{p}_1 :
  \mathbf{H}_{conn}(-,\mathbf{B}Spin)
  \to
  \mathbf{H}_{diff}(-, \mathbf{B}^3 U(1))
$$

is in [[cohomology]] the corresponding refined [[Chern-Weil homomorphism]]

$$
  [\frac{1}{2}\hat \mathbf{p}_1] :
  H^1_{Smooth}(X,Spin) \to H_{diff}^4(X)
$$

with values in [[ordinary differential cohomology]] that corresponds to the [[Killing form]] [[invariant polynomial]] $\langle - , - \rangle $ on $\mathfrak{so}$.

=--

+-- {: .num_defn #DifferentialStringStructure}
###### Definition

For any $X \in $ [[Smooth∞Grpd]], the [[2-groupoid]]  of **differential string-structures** on $X$ -- $String_{diff}(X)$ -- is the [[homotopy fiber]] of  $\frac{1}{2}\hat \mathbf{p}_1(X)$ over the trivial differential cocycle.

More generally (see [[twisted cohomology]]) the 2-groupoid of **twisted differential string structures** is the [[(∞,1)-pullback]] $String_{diff,tw}(X)$ in

$$
  \array{
    String_{diff,tw}(X) &\to& H_{diff}^4(X)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}Spin)
      &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}&
    \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))
  }
  \,,
$$

where the right vertical morphism is a choice of (any) one point in each [[connected component]] (differential cohomology class) of the [[cocycle]] [[∞-groupoid]] $\mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))$ (the [[homotopy type]] of the [[(∞,1)-pullback]] is independent of this choice).

More specifically, a **geometric string structure** is a twisted differential string structure whose differential twist has underlying trivial class.

=--

+-- {: .num_note NoteOnLiterature}
###### Note

In terms of local [[∞-Lie algebra valued differential forms]] data this has been considered in ([SSSIII](#SSSIII)), as we shall discuss [below](#ChernWeilTheory).

For the case where the the underlying integral class of the twist is trivial -- geometric string structures -- something close to this definition, explicitly modeled on [[bundle 2-gerbe]]s, has been given in ([Waldorf](#Waldorf)). See the discussion [below](#Properties).

=--

## Properties
 {#Properties}

### General
 {#GeneralProperties}

+-- {: .num_prop #GroupoidOfStringStructures}
###### Observation

The [[∞-groupoid]] $String_{tw,diff}(X)$ of twisted differential string structures is [[2-truncated]], hence is a [[2-groupoid]].

=--

+-- {: .proof}
###### Proof

This follows from the [[long exact sequence of homotopy groups]] associated to the defining [[(∞,1)-pullback]], using that

* $\mathbf{H}_{diff}(X, \mathbf{B}^3 U(1))$ is a [[3-groupoid]];

* $\mathbf{H}(X, \mathbf{B}Spin)$ is a [[1-groupoid]];

* $H^4_{diff}(X)$ is a [[0-groupoid]].

=--

See also ([Waldorf, cor. 1.1.5](#Waldorf)).

+-- {: .num_prop #PropertiesOfTrivialIntegralTwistClass}
###### Observation

If the underlying [[integral cohomology]] class of the twist is trivial, $c(tw) = 0 \in H^3(X, \mathbb{Z})$, then a $tw$-twisted differential string structures on a $Spin$-[[connection on a bundle|connection]] $\nabla$ are characterized by a globally defined 3-form on $X$.

=--

This 3-form is the globally defined connection 3-form of an appropriate [[circle n-bundle with connection|circle 3-bundle with connection]] equivalent to the [[Chern-Simons circle 3-bundle]] $CS(\nabla)$ whose underlying 3-bundle is by assumption trivial: on a trivial circle $n$-bundle every connection may be represented by a globally defined $n$-form.

This statement appears as ([Waldorf, theorem 1.3.3](#Waldorf)), where circle 3-bundles are modeled as [[bundle 2-gerbe]]s. The explicit construction of the globally defined 3-form in this model is spelled out in lemma 3.2.4 there.

+-- {: .proof}
###### Proof

Elements in the defining [[homotopy pullback]] from def. \ref{DifferentialStringStructure} over a given [[connection on a bundle|connection]] $\nabla \in \mathbf{H}(X,\mathbf{B} Spin)_{conn}$ are chracterized by an element $[\alpha] \in H^4_{diff}(X)$ and an [[equivalence in an (infinity,1)-category|equivalence]]

$$
  \Omega : CS(\nabla) \stackrel{\simeq}{\to} \alpha
$$

between the corresponding [[Chern-Simons circle 3-bundle]] and the given [[circle n-bundle with connection|circle 3-bundle]] $\alpha$. In the case at hand, both have underlying trivial class $c(CS(\nabla)) = c(\alpha) = 0$.

By the [characteristic class exact sequence](http://nlab.mathforge.org/nlab/show/ordinary%20differential%20cohomology#AbstractProperties)

$$
  0 \to \Omega^3(X)/\Omega_{int}^{3}(X)
  \to
   H^4_{diff}(X) \stackrel{c}{\to} H^4(X, \mathbb{Z})
$$

any two classes in $\pi_0 \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1)) \simeq H^4_{diff}(X)$ that have trivial underlying class in $\pi_0 \mathbf{H}(X, \mathbf{B}^3 U(1)) \simeq H^4(X, \mathbb{Z})$ differ by a 3-form modulo a closed 3-form with integral periods.

Therefore both $[\alpha]$ as well as $[CS(\nabla)] \in H^4_{diff}(X)$ are given by a globally defined 3-form modulo an integral form: the global connection 3-form on these trivial [[circle n-bundle with connection|circle 3-bundles with connection]].

=--

### Construction in terms of $L_\infty$-Cech cocycles
  {#ChernWeilTheory}

We use the [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-topos]] [[Smooth∞Grpd]] (as described there) by the local [[model structure on simplicial presheaves]] $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ to give an explicit construction of twisted differential string structures in terms of [[Cech cohomology|Cech]]-cocycles with coefficients in [[∞-Lie algebra valued differential forms]].

Proofs not displayed here can be found at _[[differential string structure -- proofs]]_ .

Recall the following fact from [[Chern-Weil theory in Smooth∞Grpd]] ([FSS](#FSS)).

+-- {: .num_prop #LieIntegrationOfDifferentialPOne}
###### Proposition

The differential fractional Pontryagin class $\frac{1}{2} \hat \mathbf{p}_1$ is presented in $[CartSp_{smooth}^{op}, sSet]_{proj}$ by the top morphism of simplicial presheaves in

$$
  \array{
    \mathbf{cosk}_3\exp(\mathfrak{so})_{ChW,smp}
     &\stackrel{\exp(\mu, cs)}{\to}&
    \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{ChW,smp}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{cosk}_3\exp(\mathfrak{so})_{diff,smp}
     &\stackrel{\exp(\mu, cs)}{\to}&
    \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{smp}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}Spin_{c}
  }
  \,.
$$

=--

Here the middle morphism is the direct [[Lie integration]] of the [[infinity-Lie algebra cohomology|L-∞ algebra cocycle]] while the top morphisms is its restriction to coefficients for [[connection on an ∞-bundle|∞-connections]].

In order to compute the [[homotopy fiber]]s of
$\frac{1}{2}\hat \mathbf{p}_1$ we now find a [[resolution]] of this morphism $\exp(\mu,cs)$ by a fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$. By the fact that this is a [[simplicial model category]] then also the hom of any cofibrant object into this morphism, computing the cocycle $\infty$-groupoids, is a fibration, and therefore, by the general discussion at [[homotopy pullback]], we obtain the [[homotopy fiber]]s as the ordinary [[fiber]]s of this fibration.

#### Presentation of the differential class by a fibration
 {#PresentationOfClassByFibration}

In order to factor $\exp(\mu,cs)$ into a weak equivalence followed by a fibration, we start by considering such a factorization before differential refinement, on the underlying characteristic class $\exp(\mu)$.

To that end, we replace the [[Lie algebra]] $\mathfrak{g} = \mathfrak{so}$ by an equivalent but bigger [[∞-Lie algebra|Lie 3-algebra]] (following [SSSIII](#SSSIII)). We need the following notation:

* $\mathfrak{g} = \mathfrak{so}$, the [[special orthogonal Lie algebra]] (the Lie algebra of the [[spin group]]);

* $b^2 \mathbb{R}$ the [[line Lie n-algebra|line Lie 3-algebra]], the single generator in degee 3 of its [[Chevalley-Eilenberg algebra]] we denote $c \in CE(b^2 \mathbb{R})$, $d c = 0$.

* $\langle -,-\rangle \in W(\mathfrak{g})$ is the [[Killing form]] [[invariant polynomial]], regarded as an element of the [[Weil algebra]] of $\mathfrak{so}$;

* $\mu := \langle -,[-,-]\rangle \in CE(\mathfrak{g})$ the degree 3 [[Lie algebra cohomology|Lie algebra cocycle]], identified with a morphism

  $$
    CE(\mathfrak{g}) \leftarrow CE(b^2 \mathbb{R}) : \mu
  $$

  of [[Chevalley-Eilenberg algebra]]s; and normalized such that its continuation to a 3-form on $Spin$ is the image in
[[de Rham cohomology]] of $Spin$ of a generator of $H^3(Spin,\mathbb{Z}) \simeq \mathbb{Z}$;

* $cs \in W(\mathfrak{g})$ is a [[Chern-Simons element]] interpolating between the two; characterized by the fact that it fits into the commuting diagram

  $$
    \array{
       CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}&
       CE(b^2 \mathbb{R})
       \\
       \uparrow && \uparrow
       \\
       W(\mathfrak{g})
       &\stackrel{cs}{\leftarrow}&
       W(b^2 \mathbb{R})
       \\
       \uparrow && \uparrow
       \\
       inv(\mathfrak{g}) &\stackrel{\langle-,-\rangle}{\leftarrow}&
       inv(b^2 \mathbb{R})
       & = CE(b^3 \mathbb{R})
    }
  $$

* $\mathfrak{g}_\mu := \mathfrak{string}$ the [[string Lie 2-algebra]].

+-- {: .num_defn #ResolutionOfLieAlgebra }
###### Definition

Let $(b\mathbb{R} \to \mathfrak{g}_\mu)$ denote the
[[L-∞ algebra]] whose [[Chevalley-Eilenberg algebra]] is

$$
  CE(b\mathbb{R} \to \mathfrak{g}_\mu) =
  (\wedge^\bullet(  \mathfrak{g}^* \oplus \langle b\rangle \oplus \langle c \rangle ), d)
  \,,
$$

with $b$ a generator in degree 2, and $c$ a generator in degree 3, and with differential defined on generators by

$$
  \begin{aligned}
    d|_{\mathfrak{g}^*} & = [-,-]^*
    \\
    d  b & = - \mu + c
    \\
    d c & =  0
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop #FactorizationOfTheCocycle}
###### Proposition

The 3-cocycle $ CE(\mathfrak{g}) \stackrel{\mu}{\leftarrow} CE(b^2 \mathbb{R})$ factors as

$$
  CE(\mathfrak{g})
   \stackrel{(c \mapsto \mu, b \mapsto 0)}{\leftarrow}
  CE(b\mathbb{R} \to \mathfrak{g}_\mu)
   \stackrel{(c \mapsto c)}{\leftarrow}
  CE(b^2 \mathbb{R})
  :
  \mu
  \,,
$$

where the morphism on the left (which is the identity when restricted to $\mathfrak{g}^*$ and acts on the new generators as indicated) is a [[quasi-isomorphism]].

=--

The point of introducing the resolution $(b \mathbb{R} \to \mathfrak{g}_\mu)$ in the above way is that it naturally supports the obstruction theory of lifts from $\mathfrak{g}$-[[connection on a bundle|connections]] to [[string Lie 2-algebra]] [[connection on an infinity-bundle|2-connection]]

+-- {: .num_prop #LongFiberSequenceOnLieAlgebras}
###### Observation

The defining projection $\mathfrak{g}_\mu \to \mathfrak{g}$
factors through the above quasi-isomorphism $(b \mathbb{R} \to \mathfrak{g}_\mu) \to \mathfrak{g}$ by the canonical inclusion

$$
  \mathfrak{g}_\mu \to (b \mathbb{R} \to \mathfrak{g}_\mu)
  \,,
$$

which dually on $CE$-algebras is given by

$$
  t^a \mapsto t^a
$$

$$
  b \mapsto - b
$$

$$
  c \mapsto 0
  \,.
$$

In total we are looking at a convenient presentation of the long [[fiber sequence]] of the [[string Lie 2-algebra]] extension:

$$
  \array{
    && && (b \mathbb{R} \to \mathfrak{g}_\mu) &\to& b^2 \mathbb{R}
    \\
    && & \nearrow & \downarrow^{\mathrlap{\simeq}}
    \\
    b \mathbb{R} &\to& \mathfrak{g}_\mu &\to& \mathfrak{g}
  }
  \,.
$$

=--

(The signs appearing here are just unimportant convention made in order for some of the formulas below to come out nice.)

+-- {: .num_prop #BareFibration}
###### Proposition

The image under [[Lie integration]] of the above factorization is

$$
  \exp(\mu)
  :
   \mathbf{cosk}_3\exp(\mathfrak{g})
   \to
   \mathbf{cosk}_3\exp(b \mathbb{R} \to \mathfrak{g}_\mu)
   \to
   \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_c
$$

where the first morphism is a weak equivalence followed by a fibration in the [[model structure on simplicial presheaves]] $[CartSp_{smooth}^{op}, sSet]_{proj}$.

=--

+-- {: .proof}
###### Proof

To see that the left morphism is objectwise a [[weak homotopy equivalence]], notice that
a $[k]$-cell of $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)$ consists of a triple $(A,B,C)$, where $A$ is a vertical flat $\mathfrak{g}$-valued 1-form on $U\times\Delta^k$, $B$ is a vertical 2-form and $C$ a 3-form on $U\times\Delta^k$, such that $d B=C-\mu(A,A,A)$ and $d C=0$, since $A$ is flat. Therefore $C$ is uniquely determined by $A$ and $B$, and there are no conditions on $B$. This means that  a $[k]$-cell of $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)$ is identified with a pair consisting of a based [[smooth function]] $f : \Delta^k \to Spin$ and a [[vertical differential form|vertical 2-form]] $B \in \Omega^2_{si,vert}(U \times \Delta^k)$, (both suitably with sitting instants perpendicular to the boundary of the simplex). Since there is no further condition on the 2-form, it can always be extended from the boundary of the $k$-simplex to the interior (for instance simply by radially rescaling it smoothly to 0). Accordingly the [[simplicial homotopy group]]s of $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)(U)$ are the same as those of $\exp(\mathfrak{g})(U)$. The morphism between them is the identity in $f$ and picks $B = 0$ and is hence clearly an isomorphism on homotopy groups.

We turn now to discussing that the second morphism is a fibration. The nontrivial degrees of the lifting problem

$$
  \array{
    \Lambda[k]_i
     &\to&
     \mathbf{cosk}_3\exp(b\mathbb{R} \to \mathfrak{g}_\mu)(U)
    \\
    \downarrow && \downarrow
    \\
    \Delta[k] &\to& \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_c(U)
  }
$$

are $k = 3$ and $k = 4$.

Notice that a $3$-cell of $\mathbf{B}^3 \mathbb{R}/ \mathbb{Z}_c(U)$ is a [[smooth function]] $U \to \mathbb{R}/\mathbb{Z}$ and
that the morphism $\exp(b\mathbb{R} \to \mathfrak{g}_\mu) \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_c$ sends the pair $(f,B)$ to the [[fiber integration]] $\int_{\Delta^3}(f^* \langle \theta \wedge [\theta \wedge \theta]\rangle + d B) mod \mathbb{Z}$.

Our lifting problem in degree 3,
has given a function $c : U \times \Delta^3 \to \mathbb{R}/\mathbb{Z}$ and a smooth function (with sitting instants at the subfaces) $f : U \times \Lambda^3_i \to Spin$ together with a 2-form $B$ on the [[horn]] $U \times \Lambda^3_i$.

By pullback along the standard [[continuous function|continuous]] [[retract]] $\Delta^3 \to \Lambda^3_i$ which is non-smooth only where $f$ has sitting instants,  we can always extend $f$ to a smooth function $f' : U \times \Delta^3 \to Spin$ with the property that $\int_{\Delta^3} (f')^* \langle \theta \wedge [\theta \wedge \theta]\rangle = 0$. (Following the general discussion at [[Lie integration]].)

In order to find a horn filler for the 2-form component, consider any smooth 2-form with sitting instants and non-vanishing integeral on $\Delta^2$, regarded as the missing face of the [[horn]]. By multiplying it with a suitable smooth function on $U$ we can obtain an extension $\tilde B \in \Omega^3_{si,vert}(U \times \partial \Delta^3)$ of $B$ to all of $U \times \partial \Delta^3$ with the property that its integral over $\partial \Delta^3$ is the given $c$. By the [[Stokes theorem]] it remains to extend $\tilde B$ to the interior of $\Delta^3$ in any way, as long as it is smooth and has sitting instants.

To that end, we can find in a similar fashion a smooth $U$-parameterized family of closed 3-forms $H$ with sitting instants on $\Delta^3$, whose integral over $\Delta^3$ equals $c$. Since by sitting instants this 3-form vanishes in a neighbourhood of the boundary, the standard formula for the [[Poincare lemma]] applied to it produces a 2-form $B' \in \Omega^2_{si, vert}(U \times \Delta^3)$ with $d B' = C$ that itself is radially constant at the boundary. By construction the difference $\tilde B - B'|_{\partial \Delta^3}$ has vanishing surface integral. By the discussion at [[Lie integration]] it follows that the difference extends smoothly and with sitting instants to a closed 2-form $\hat B \in \Omega^2_{si,vert}(U \times \Delta^3)$. Therefore the sum

$$
  B' + \hat B \in \Omega^2_{si,vert}(U \times \Delta^3)
$$

equals $B$ when restricted to $\Lambda^k_i$ and has the property that its integral over $\Delta^3$ equals $c$.
Together with our extension $f'$, this constitutes a pair that solves the lifting problem.

The extension problem in degree 4 amounts to a similar construction: by coskeletalness the condition is that for a given $c : U \to \mathbb{R}/\mathbb{Z}$ and a given vertical 2-form on $U \times \partial \Delta^3$ such that its integral equals $c$, as well as a function $f : U \times \partial \Delta^3 \to Spin$, we can extend the 2-form and the function along $U \times \partial \Delta^3 \to U \times \Delta^3$. The latter follows from the fact that $\pi_2 Spin = 0$ which guarantees a continuous filler (with sitting instants), and using the [[Steenrod-Wockel approximation theorem]] to make this smooth. We are left with the problem of extending the 2-form, which is the same problem we discussed above after the choice of $\tilde B$.

=--

We now proceed to extend this factorization to the exponentiated differential coefficients (see [[connection on an ∞-bundle]]).

+-- {: .num_prop #PresentationByFibration}
###### Proposition
**(presentation of the differential class by a fibration)**

Under [[Lie integration]]
the [above factorization](#FactorizationOfTheCocycle)
of the Lie algebra cocycle
maps to the factorization

$$
  \exp(\mu, cs)
   :
  \mathbf{cosk}_3 \exp(\mathfrak{g})_{ChW}
   \stackrel{\simeq}{\to}
  \mathbf{cosk}_3 \exp((b \mathbb{R} \to \mathfrak{g}_\mu))_{ChW}
   \to
  \mathbf{B}^3 U(1)_{ChW,ch}
$$

of $\exp(\mu,cs)$ in $[CartSp^{op}, sSet]_{proj}$, where the first morphism is a weak equivalence and the second a fibration.

=--

+-- {: .proof}
###### Proof

The following proof makes use of details discussed at _[[differential string structure -- proofs]]_ .

We discuss that the first morphism is an equivalence. Clearly it is injective on homotopy groups: if a sphere of $A$-data cannot be filled, then also adding the $(B,C)$-data does not yield a filler. So we need to check that it is also surjective on homotopy groups: if the $A$-data can be filled, then also the corresponding $(B,C)$-data has a filler. Since the curvature $H$ is horizontal it is already extended. We may extend $B$ in any smooth way to $U \times \Delta^k$ (for instance by rescaling it smoothly to zero at the center of the $k$-simplex) and then take the equation $d B = - CS(A) + C + H$ to define the extension of $C$.

We now check that the second morphism is a fibration. It is itself the composite

$$
  \mathbf{cosk}_{3} \exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{ChW}
  \to
  \exp(b^2 \mathbb{R})_{ChW}/\mathbb{Z}
  \stackrel{\int_{\Delta^\bullet}}{\to}
  \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{ChW,ch}
  \,.
$$

Here the second morphism is a degreewise surjection of simplicial abelian groups, hence a degreewise surjection under the [[normalized chain complex]] functor, hence is itself already a projective fibration. Therefore it is sufficient to show that the first morphism here is a fibration.

In degree $k = 0$ to $k = 3$ the lifting problems

$$
  \array{
     \Lambda[k]_i
       &\to &
      \exp(b\mathbb{R} \to \mathfrak{g}_{\mu})_{smp,ChW}(U)
      \\
      \downarrow && \downarrow
      \\
      \Delta[k]
        &\to&
      \exp(b^2 \mathbb{R})_{smp,ChW}/\mathbb{Z}(U)
  }
$$

may all be equivalently reformulated as lifting against a [[cylinder]] $D^k \hookrightarrow D^k \times [0,1]$ by using the sitting instants of all forms.

We have then a 3-form $C \in \Omega^3_{si}(U \times D^{k-1}\times [0,1])$ with horizontal curvature $\mathcal{G} \in \Omega^4(U)$ and differential form data $(A,B)$ on $U \times D^{k-1}$ given. We may always extend $A$ along the cylinder direction $[0,1]$ (its vertical part is equivalently a based smooth function to $Spin$ which we may extend constantly). $H$  has to be horizontal so it is to be constantly extended along the cylinder.

We can then use the kind of formula that proves the [[Poincare lemma]] to extend $B$. Let $\Psi : (D^k \times [0,1]) \times [0,1] \to (D^k  \times [0,1])$ be a smooth contraction. Then while $d(H - CS(A) + C)$ may be non-vanishing, by horizonatlity of their [[curvature characteristic form]]s we still have that $\iota_{\partial_t} \Psi_t^* d(H - CS(A) + C)$ vanishes (since the contraction vanishes).

Therefore the 2-form

$$
  \tilde B := \int_{[0,1]} \iota_{\partial_t} \Psi_t^*(H - CS(A) + C)
$$

satisfies $d \tilde B = (H - CS(A) + C)$. It may however not coincide with our given $B$ at $t = 0$. But the difference $B - \tilde B|_{t = 0}$ is a closed form on the left boundary of the cylinder. We may find some closed 2-form on the other boundary such that the integral around the boundary vanishes. Then the argument from the proof of the [[Lie integration]] of the [[line Lie n-algebra]] applies and we find an extension $\lambda$ to a closed 2-form on the interior. The sum

$$
  \hat B := \tilde B + \lambda
$$

then still satisfies $d \hat B = H - CS(A) - C$ and it coincides with $B$ on the left boundary.

Notice that here $\tilde B$ indeed has sitting instants: since $H$, $CS(A)$ and $C$ have sitting instants they are constant on their value at the boundary in a neighbourhood perpendicular to the boundary, which means for these 3-forms in the degrees $\leq 3$ that they _vanish_ in a neighbourhood of the boundary, hence that the above integral is towards the boundary over a vanishing integrand.

In degree 4 the nature of the lifting problem

$$
  \array{
  \Lambda[4]_i &\to& \mathbf{cosk}_3\exp(b\mathbb{R} \to \mathfrak{g}_\mu)(U)
  \\
  \downarrow && \downarrow
  \\
  \Delta[4] &\to& \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{ChW,ch}
  }
$$

starts out differently, due to the presence of $\mathbf{cosk}_3$, but it then ends up amounting to the same kind of argument:

We have four functions $U \to \mathbb{R}/\mathbb{Z}$ which we may realize as the [[fiber integration]] of a 3-form $C$ on $U \times (\partial \Delta[4] \setminus \delta_i \Delta[3])$,
and we have a lift to $(A,B,C, H)$-data on $U \times (\partial \Delta[4]\setminus \delta_i(\Delta[3]))$ (the boundary of the 4-simplex minus one of its 3-simplex faces).

We observe that we can

* always extend $C$ smoothly to the remaining 3-face such that its [[fiber integration]] there reproduces the signed difference of the four given functions corresponding to the other faces (choose any smooth 3-form with sitting instants and with non-vanishing integral and rescale smoothly);

* fill the $A$-data horizonatlly due to the fact that $\pi_2 (Spin) = 0$.

* the $H$-form is already horizontal, hence already filled.

Moreover, by the fact that the 2-form $B$ already is defined on all of $\partial \Delta[4] \setminus \delta_i(\Delta[3])$ its fiber integral over the boundary $\partial \Delta[3]$ coincides with the fiber integral of $H - CS(A) + C$ over $\partial \Delta[4] \setminus \delta_i (\Delta[3])$). But by the fact that we have lifted $C$ and the fact that $\mu(A_{vert}) = CS(A)|_{\Delta^3}$ is an integral cocycle, it follows that this equals the fiber integral of $C - CS(A)$ over the remaining face.

Use then as above the vertical Poincare lemma-formula to find $\tilde B$ on $U \times \Delta^3$ with sitting instants that satisfies the equation
$d B = H - CS(A) + C$ there. Then extend the closed difference $B - \tilde B|_{0}$ to a closed smooth 2-form on $\Delta^3$. As before, the difference

$$
  \hat B := \tilde B  + \lambda
$$

is an extension of $B$ that constitutes a lift.

=--

#### Explicit Cech cocycles
  {#ExplicitCocycles}

+-- {: .num_cor #PresentationBySimplicialPresheaves}
###### Corollary

For any $X \in $ [[SmoothMfd]] $\hookrightarrow$ [[Smooth∞Grpd]], for any choice of differentiaby [[good open cover]] with corresponding cofibrant presentation $\hat X = C(\{C_i\})\in [CartSp_{smooth}^{op}, sSet]_{proj}$ we have that the [[2-groupoid]]s of [twisted different String structures](#DifferentialStringStructure)s are presented by the ordinary [[fiber]]s of the morphism of [[Kan complex]]es

$$
  [CartSp^{op}, sSet](\hat X,\exp(\mu,cs))
  :
  [CartSp^{op}, sSet](\hat X, \mathbf{cosk}_3 \exp(b \mathbb{R}
   \to \mathfrak{g}_\mu)_{ChW})
  \to
  [CartSp^{op}, sSet](\hat X, \mathbf{B}^3 U(1)_{ChW})
  \,.
$$

over any basepoints in the connected components of the [[Kan complex]] on the right, which correspond to the elements
$[\hat \mathbf{C}_3] \in H_{diff}^4(X)$ in the [[ordinary differential cohomology]]  of $X$.

=--

+-- {: .proof}
###### Proof

Since $[CartSp_{smooth}^{op}, sSet]_{proj}$ is a [[simplicial model category]] the morphism
$[CartSp^{op}, sSet](\hat X,\exp(\mu,cs))$
is a fibration because $\exp(\mu,cs)$ is and $\hat X$ is cofibrant.

It follows from the discussion at [[homotopy pullback]] that the ordinary [[pullback]] of [[simplicial presheaves]]

$$
  \array{
    String_{diff,tw}(X) &\to& H_{diff}^4(X)
    \\
    \downarrow && \downarrow
    \\
    [CartSp^{op}, sSet](\hat X, \mathbf{cosk}_3 \exp(b \mathbb{R}
     \to \mathfrak{g}_\mu)_{ChW})
    &\to&
  [CartSp^{op}, sSet](\hat X, \mathbf{B}^3 U(1)_{ChW})
  }
$$

is a presentation for the defining [[(∞,1)-pullback]] for $String_{diff,tw}(X)$, as defined [above](#DifferentialStringStructure).

=--

We unwind the explicit expression for a twisted differential string structure under this equivalence.

Any twisting cocycle is in the above presentation given by a [[Cech cohomology|Cech]] [[Deligne cohomology|Deligne]]-cocycle (as discussed at [[circle n-bundle with connection]])

$$
  \hat \mathbf{H}_3 = ((H_3)_i, \cdots)
$$

with local connection 3-form $(H_3)_i \in \Omega^3(U_i)$ and globally defined [[curvature]] 4-form $\mathcal{G}_4 \in \Omega^4(X)$.

+-- {: .num_note #UnwindingTheLocalData}
###### Note

A differential string structure on $X$ twisted by this cocycles is on patches $U_i$ a morphism

$$
  \Omega^\bullet(U_i)
  \leftarrow
  \tilde W(b\mathbb{R}\to \mathfrak{g}_\mu)
$$

in [[dgAlg]], subject to some horizontality constraints.
The components of this are over each $U_i$ a collection of differential forms of the following structure

$$
  \left(
    \array{
      F_\omega =& d \omega + \frac{1}{2}[\omega \wedge \omega]
      \\
      H_3 =& \nabla B := d B + CS(\omega) - C_3
      \\
      \mathcal{G}_4  =& d C_3
      \\
      d F_\omega =& - [\omega \wedge F_\omega]
      \\
      d H_3 =&  \langle F_\omega \wedge F_\omega\rangle - \mathcal{G}_4
      \\
      d \mathcal{G}_4 =& 0
    }
  \right)_i
  \;\;\;\;
  \stackrel{
    \array{
       t^a & \mapsto \omega^a
       \\
       r^a & \mapsto F^a_\omega
       \\
       b & \mapsto B
       \\
       c & \mapsto C_3
       \\
       h & \mapsto H_3
       \\
       g & \mapsto \mathcal{G}_4
    }
  }{\leftarrow}|
  \;\;\;\;
  \left(
    \array{
       r^a  =& d t^a + \frac{1}{2}C^a{}_{b c} t^b \wedge t^c
       \\
       h = & d b + cs - c
       \\
       g  =& d c
       \\
       d r^a  =&  - C^a{}_{b c} t^b \wedge r^a
       \\
       d h =& \langle -,-\rangle - g
       \\
       d g =&  0
    }
  \right)
  \,.
$$

=--

Here we are indicating on the right the generators and their relation in $\tilde W(b\mathbb{R} \to \mathfrak{g}_\mu)$ and on the left their images and the images of the relations in $\Omega^\bullet(U_i)$.   This are first the definitions of the [[curvature]]s themselves and then the [[Bianchi identities]] satisfied by these.

### Differential string structures and fermionic string quantum anomalies

The [[Pfaffian line bundle]] controlling the [[fermionic path integral]] of the [[heterotic superstring]] propagating on target $X$ trivializes precisely if the target has a (geometric) string structure.

One shows that the [[Pfaffian line bundle]] on the worldsheet is [[isomorphic]] as a bundle with connection with the [[transgression]] of the differential string structure on the target space to the mapping space $[\Sigma,X]$. So the target space having a (differential) string structure is a sufficient condition for the cancellation of the [[quantum anomaly]].

(First argued in [Killingback](#Killingback), later made precise in ([Bunke](#Bunke))).

### The Green-Schwarz mechanism in heterotic supergravity
 {#InheteroticSugra}

We discuss the application of twisted differential string structures in [[supergravity]] and [[string theory]].

Local differential form data as in note \ref{UnwindingTheLocalData} above is known in [[higher category theory and physics|theoretical physics]] in the context of the [[Green-Schwarz mechanism]] for 10-dimensional [[supergravity]].

In this context

* $\omega$ is called the [[spin connection]];

* the components $((H_3)_i, \cdots)$ of the above cocycle are known as the $\hat \mathcal{G}_4$-twisted [[Kalb-Ramond field]].

In this application the twisting cocycle $\hat \mathcal{G}_4 \in H^4_{diff}(X)$ is itself the [[Chern-Simons circle 3-bundle]] of a [[unitary group]]-[[principal bundle]] with local connection form $A \in \Omega^1(U, \mathfrak{u})$. Therefore in this case $C_3 = CS(A)$ and the above local form data becomes

$$
  H_3 = d B + CS(\omega) - CS(A)
$$

$$
  d H_3 = \langle F_\omega \wedge F_\omega \rangle
   -
   \langle F_A \wedge F_A \rangle
  \,.
$$

Since $H_3$ is the would-be [[curvature]] of a [[circle n-bundle with connection|circle 2-bundle with connection]], this is the first higher [[Maxwell equation]] that exhibits

$$
  j_{mag}
  :=
  \langle F_\omega \wedge F_\omega \rangle
   -
   \langle F_A \wedge F_A \rangle
$$

as the [[magnetic charge]] distribution that twists this 2-bundle. This may be interpreted as the magnetic charge density of a classical background density of magnetic [[fundamental brane|fivebranes]].  For more details on this see [[Green-Schwarz mechanism]].

More precisely, the twisted differential string structure of the [[Green-Schwarz mechanism]] in heterotic [[supergravity]] for fixed gauge bundles are therefore given by the [[(∞,1)-pullback]]

$$
  \array{
    GSBackground_{fixed gauge bundle}(X) &\to& \pi_0 \mathbf{H}_{conn}(X, \mathbf{B}U)
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{c}_2}}
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}Spin)
     &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^3 U(1))
  }
  \,.
$$

Clearly, if we take into account also gauge transformations of the gauge bundle, we should replace this by the full

$$
  \array{
    GSBackground(X) &\to& \mathbf{H}_{conn}(X, \mathbf{B}U)
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{c}_2}}
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}Spin)
     &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^3 U(1))
  }
  \,.
$$

The look of this diagram makes manifest how in this situation we are looking at the structures that homotopically cancel the differential classes $\frac{1}{2}\hat \mathbf{p}$ and $\hat \mathbf{c}_2$ against each other.

More discussion of this is in ([SSSIII](#SSSIII)).

Since $\mathbf{H}_{dR}(X, \mathbf{B}^3 U(1))$ is abelian, we may consider the corresponding Mayer-Vietoris sequence by realizing $GSBackground(X)$ equivalently as the [[homotopy fiber]] of the difference of differential cocycles $\frac{1}{2}\hat \mathbf{p}_1 - \hat \mathbf{c}_2$.

$$
  \array{
    GSBackground(X) &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}Spin \times \mathbf{B}U)
     &\stackrel{\frac{1}{2}\hat \mathbf{p}_1-\hat \mathbf{c}_2}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^4 U(1))
  }
  \,.
$$

Indeed, the above explicit presentation by simplicial presheaves generalizes immediately to describe this case, realizing $U$-twisted differential string structures equivalently as differential "untwisted $U$-twisted-string-structures".

We may usefully formalize this further by defining the $String^{\mathbf{c}_2}$-2-group to be the homotopy fiber

$$
  \array{
    \mathbf{B} String^{\mathbf{c}_2} &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{G}(Spin \times U)
    &\stackrel{\frac{1}{2}\mathbf{p}_1 - \mathbf{c}_2}{\to}&
    \mathbf{B}^3 U(1)
  }
  \,.
$$

We have then that $GSBackground(X)$ is the 3-groupoid of _untwisted_
differential $\mathbf{B}String^{\mathbf{c}_2}$-structures.

$$
  \array{
    GSBackground(X) &\to& *
    \\
    \downarrow && \downarrow^{0}
    \\
    \mathbf{H}_{conn}(\mathbf{B} (Spin \times U))
    &\stackrel{\frac{1}{2}\hat \mathbf{p}_1 - \mathbf{c}_2}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^4 U(1))
  }
  \,.
$$

More on this in ([FiSaSc](#FiSaSc)).

This is supposed to be (see section 12 of ([DFM](#DFM))) the restriction to the boundary of the [[supergravity C-field]], which is the $(\infty,1)$-pullback

$$
  \array{
    C Field(Y) &\to& H^4_{dR}(Y)
    \\
    \downarrow && \downarrow^{\mathrlap{d C_3}}
    \\
    \mathbf{H}(Y,\mathbf{B} (Spin \times U))
    &\stackrel{\frac{1}{2}\hat \mathbf{p}_1 - \mathbf{c}_2}{\to}&
    \mathbf{H}(Y, \mathbf{B}^4 U(1))
  }
  \,.
$$

where $Y$ is 11-dimensional with $\parital Y = X$. Notice that here in the bottom left we have bundles _without_ connection, or equivalently (when computing the [[homotopy pullback]] by an ordinary pullback along a fibration)  with [[pseudo-connection]]s.

By the discussion at [[supergravity C-field]] under a shift of connection $\nabla_1 \mapsto \nabla_2$ the $C$-field transforms as

$$
  C_2 = C_1 + CS(\nabla_1, \nabla_2)
  \,,
$$

where on the right we have the relative [[Chern-Simons form]]. This vanishes precisely on the genuine gauge transformations. Hence as we restrict from  11-dimensions to 10, two things happen:

1. the supergravity $C$-field vanishes,

1. the gauge bundles develop genuine connections.

### Relation to string 2-connections

By the discussion at _[[connection on an ∞-bundle]]_ we have that for $\mathfrak{g}$ an [[L-∞ algebra]] and

$$
  \mathbf{B}G := \mathbf{cosk}_{n+1} \exp(\mathfrak{g})
$$

the [[delooping]] of the [[smooth ∞-groupoid|smooth Lie n-group]] obtained from it by [[Lie integration]], the coefficient for [[connection on an ∞-bundle|∞-connections]] on $G$-[[principal ∞-bundle]]s is

$$
  \mathbf{B}G_{conn} := \mathbf{cosk}_{n+1} \exp(\mathfrak{g})_{conn}
  \,,
$$

where on the very right we have the [[simplicial presheaf]]

$$
  \exp(\mathfrak{g}) : (U,[k]) \mapsto
   \left\{
     A : CE(\mathfrak{g}) \to \Omega^\bullet(U \times \Delta^n)
     |
     A\;is\;vertically\;flat\; and\;F_A\;is \; horizontal
   \right\}
   \,.
$$

(See [[∞-Chern-Weil homomorphism]] for details).

+-- {: .num_prop #StringConnectionsFromDiffStringStructzres}
###### Proposition

The [[2-groupoid]] of entirely untwisted differential string structures on $X$ (the twist being $0 \in H^4_{diff}(X)$) is equivalent to that of [[string 2-group]] [[principal 2-bundle]]s with [[connection on an ∞-bundle|2-connection]]:

$$
  String_{diff, tw = 0}(X) \simeq String 2Bund_{\nabla}(X)
  \,.
$$

=--

+-- {: .proof}
###### Proof

By the above discussion of [Cech cocycles](#ExplicitCocycles) we compute $String_{diff, tw = 0}(X)$ as the ordinary fiber of the morphism of simplicial presheaves

$$
  [CartSp^{op}, sSet](
  C(\{U_i\}),  \mathbf{cosk}_3 \exp(b \mathbb{R} \to \mathfrak{g}_\mu))
  \to
  [CartSp^{op}, sSet](
  C(\{U_i\}), \mathbf{B}^3 U(1)_{diff})
$$

over the identically vanishing cocycle.

In terms of the component formulas spelled out in the [above discussion](#InheteroticSugra) of the GS-mechanism, this amounts to restricting to those cocyles for which n each degree the equations

$$
  C = 0
$$

$$
  G = 0
$$

holds.

Comparing this to the explicit formulas for $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)$ and $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{conn}$ in the [above](#PresentationOfClassByFibration) we see that these cocycles are exactly those that factor through the canonical inclusion

$$
  \mathfrak{g}_\mu \to (b \mathbb{R} \to \mathfrak{g}_\mu)
$$

from observation \ref{LongFiberSequenceOnLieAlgebras} of the [[string Lie 2-algebra]] into the [[mapping cone]] Lie 3-algebra of the extension $b \mathbb{R} \to \mathfrak{g}_\mu \to \mathfrak{g}$ that defines it.

=--

## Related concepts

* [[nonabelian differential cohomology]]

* [[Whitehead tower]] of the [[orthogonal group]]

  * [[orientation]]

  * [[spin structure]], [[twisted spin structure]], [[differential spin structure]]

    [[twisted spin^c structure]]

  * [[string structure]],  **differential string structure**

    * [[chiral Dolbeault complex]]

  * [[fivebrane structure]]

    * [[schreiber:Hypothesis H]]

  * [[twisted differential c-structure]]

    * **differential string structure**

    * [[supergravity C-field]]

    * [[twisted differential fivebrane structure]]

    * [[differential T-duality]]

## References

A discussion of differential string structures in terms of [[bundle 2-gerbe]]s is given in

* {#Waldorf} [[Konrad Waldorf]], _String Connections and Chern-Simons Theory_, Trans. Amer. Math. Soc. **365** (2013), 4393-4432, ([arXiv:0906.0117](http://arxiv.org/abs/0906.0117), [doi:10.1090/S0002-9947-2013-05816-3](https://doi.org/10.1090/S0002-9947-2013-05816-3))

The description of the [[gauge transformation]]s of the [[supergravity C-field]] is in section 3 of

* {#DFM} E. Diaconescu, [[nLab:Greg Moore]], [[nLab:Dan Freed]], _The $M$-theory 3-form and $E_8$-gauge theory_ ([arXiv:hep-th/0312069](http://arxiv.org/abs/hep-th/0312069))

The local data for the [[∞-Lie algebra valued differential forms]] for the description of twisted differential string structures as above was given in

* {#SSSIII} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]]:  *[[schreiber:Twisted Differential String and Fivebrane Structures]]*, Communications in Mathematical Physics **315** 1 (2012)  169-213 [[arXiv:0910.4001](https://arxiv.org/abs/0910.4001), [doi:10.1007/s00220-012-1510-3)]([arXiv:](https://link.springer.com/article/10.1007/s00220-012-1510-3))]

The full Cech-Deligne cocycles induced by this (but not yet the homotopy fibers over them) were discussed in

* {#FiorenzaSatiSchreiber12} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech cocycles for differential characteristic classes|Čech cocycles for differential characteristic classes]]_, Advances in Theoretical and Mathematical Physics, **16** 1 (2012) 149-250 [[arXiv:1011.4735](https://arxiv.org/abs/1011.4735), [doi:10.1007/BF02104916](https://doi.org/10.1007/BF02104916)]

A comprehensive discussion including all the formal background and the applications is attempted at

* {#dcct} [[Urs Schreiber]], section 4.2 of: _[[schreiber:differential cohomology in a cohesive topos]]_

The translation between the [[bundle 2-gerbe]]-picture of [Waldorf](#Waldorf) and the $String$-[[principal 2-bundle]]-piucture of [Sati et al.](#SSSIII), [Fiorenza et al.](#FiorenzaSatiSchreiber12) and [dcct](#dcct) is worked out in some detail in:

* Alessandra Capotosti, *[[From String structures to Spin structures on loop spaces]]*, Ph.D. thesis, Università degli Studi Roma Tre, Rome (April 2016) [[thesis pdf](http://www.matfis.uniroma3.it/Allegati/Dottorato/TESI/capotosti/PhD%20Thesis%202016%20A%20Capotosti.pdf), [[Capotosti-FromStringStructures.pdf:file]], talk slides: [[Capotosti-FromStringStruc-Slides.pdf:file]]]

The relation to [[quantum anomaly]] cancellation in [[heterotic string theory]] has been first discussed in

* {#Killingback} Killingback, _World-sheet anomalies and loop geometry_ Nuclear Physics B Volume 288, 1987, Pages 578-588

and given a rigorous treatment in

* {#Bunke} [[Ulrich Bunke]], _String structures and trivialisations of a Pfaffian line bundle_ ([arXiv](http://arxiv.org/abs/0909.0846))

More discussion on the relation to spin structures on smooth loop space is in

* Alessandra Capotosti, _[[From String structures to Spin structures on loop spaces]]_, Ph.D. thesis, Università degli Studi Roma Tre, Rome, April 2016

[[!redirects differential string structures]]

[[!redirects differential String structure]]
[[!redirects differential String structures]]
[[!redirects twisted differential String structure]]
[[!redirects twisted differential String structures]]

[[!redirects differential string-structure]]
[[!redirects differential string-structures]]

[[!redirects twisted differential string structure]]
[[!redirects twisted differential string structures]]
[[!redirects twisted differential string-structure]]
[[!redirects twisted differential string-structures]]

[[!redirects geometric string structure]]
[[!redirects geometric string structures]]
[[!redirects geometric string-structure]]
[[!redirects geometric string-structures]]

[[!redirects twisted string structure]]
[[!redirects twisted String structure]]
[[!redirects twisted string structures]]
[[!redirects twisted String structures]]

[[!redirects smooth first fractional Pontryagin class]]

[[!redirects string 2-connection]]
[[!redirects string 2-connections]]


differential string structure -- proofs

This page contains technical details to be used at the main page _[[differential string structure]]_ . See there for context.

--

#Contents#
* table of contents
{:toc}

## Factorization of the $L_\infty$-cocycle
  {#FactorizationOfTheLInfintiyCocycle}

+-- {: .num_prop #FactorizationOfTheCocycle}
###### Proposition

The $L_\infty$-algebra cocycle

$$
  \mu : \mathfrak{so} \to b^2 \mathbb{R}
$$

factors as

$$
  \mathfrak{so} \stackrel{}{\to} (b \mathbb{R} \to \mathfrak{string})
  \stackrel{}{\to}
  b^2 \mathbb{R}
$$

given dually on CE-algebras by

$$
  CE(\mathfrak{so})
     \stackrel{
       \left(
       \array{
        t^a \mapsto t^a
        \\
        b \mapsto 0
        \\
        c \mapsto \mu
     }
     \right)
     }{\leftarrow}
  CE(b \mathbb{R} \to \mathfrak{string})
   \stackrel{
     \left(
     \array{
        c \mapsto c
     }
     \right)
   }{\leftarrow}
  CE(b^2 \mathbb{R})
  \,.
$$

The left morphism is a [[quasi-isomorphism]].

=--

+-- {: .proof}
###### Proof

To see that we have a quasi-isomorphism, notice that the dg-algebra $CE(b \mathbb{R} \to \mathfrak{string})$ is [[isomorphic]] to the one with generators $\{t^a, b, c'\}$ and differentials

$$
  \begin{aligned}
    d|_{\mathfrak{g}^*} & = [-,-]^*
    \\
    d  b & = c'
    \\
    d c' & =  0
  \end{aligned}
  \,,
$$

where the isomorphism is given by the identity on the $t^a$s and on $b$ and by

$$
  c \mapsto c' + \mu
  \,.
$$

The primed dg-algebra is the [[tensor product]] $CE(\mathfrak{g}) \otimes CE( inn(b \mathbb{R}))$, where the second factor is manifestly cohomologically trivial.

=--

## Factorization of the differential $L_\infty$-cocycle
  {#FactorizationOfTheLInfintiyCocycle}

We now give a concrete construction showing

+-- {: .num_prop #DifferentialLInfinityLift}
###### Proposition

The factorization

$$
  CE(\mathfrak{so})
     \stackrel{
     \left(
     \array{
        t^a \mapsto t^a
        \\
        b \mapsto 0
        \\
        c \mapsto \mu
     }
     \right)
     }{\leftarrow}
  CE(b \mathbb{R} \to \mathfrak{string})
   \stackrel{
     \left(
     \array{
        c \mapsto c
     }
     \right)
   }{\leftarrow}
  CE(b^2 \mathbb{R})
$$

from [above](#FactorizationOfTheLInfintiyCocycle) lifts to a factorization of differential $L_\infty$-algebraic cocycles

$$
  \array{
    CE(\mathfrak{so})
     &
       \underoverset{\simeq}{
       \left(
       \array{
          t^a \mapsto t^a
          \\
          b \mapsto 0
          \\
          c \mapsto \mu
       }
       \right)
       }{\leftarrow}
     &
    CE(b \mathbb{R} \to \mathfrak{string})
     &
     \stackrel{
       \left(
       \array{
          c \mapsto c
       }
       \right)
     }{\leftarrow}
     &
    CE(b^2 \mathbb{R})
    \\
    \uparrow^\mathrlap{}
     &&
    \uparrow^\mathrlap{}
     &&
    \uparrow^\mathrlap{}
    \\
    W(\mathfrak{so})
      &
       \underoverset{\simeq}{
       }{\leftarrow}
      &
    W(b \mathbb{R} \to \mathfrak{string})
      &
     \stackrel{
     }{\leftarrow}
     &
    W(b^2 \mathbb{R})
    \\
    \uparrow^\mathrlap{}
     &&
    \uparrow^\mathrlap{}
     &&
    \uparrow^\mathrlap{}
    \\
    inv(\mathfrak{so})
      &
       \underoverset{\simeq}{
       }{\leftarrow}
      &
     \tilde inv(b \mathbb{R} \to \mathfrak{string})
      &
     \stackrel{
     }{\leftarrow}
     &
    inv(b^2 \mathbb{R})
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof

This is at its heart trivial, but potentially a bit tedious. We proceed in two steps:

1. consider a "modified Weil algebra" of the twisted string Lie 3-algebra $(b \mathbb{R} \to \mathfrak{string})$

   in [The modified Weil algebra](#ModifiedWeilAlgebra);

1. construct the desired factorization by factoring itself through two fairly evident morphisms into and out of the modified Weil algebra,

   in [The differential lift](TheLInftyDifferentialLift).

=--

### The modified Weil algebra
 {#ModifiedWeilAlgebra}

Our factorization [below]() makes use of an isomorphic copy of the Weil algebra $W(b\mathbb{R} \to \mathfrak{g}_\mu)$.

+-- {: .num_prop #ShiftedWeilAlgebra}
###### Proposition

The [[Weil algebra]] $W(b\mathbb{R} \to \mathfrak{g}_\mu)$ of $(b^2 \mathbb{R} \to \mathfrak{g})$ is given on the extra shifted generators
$\{r^a := \sigma t^a, h := \sigma b, g := \sigma c\}$  (where $\sigma$ is the shift operator extended as a graded derivation, see [[Weil algebra]]) by

* $d t^a = -\frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a$;

* $d b = - \mu + c + h$;

* $d c = g$,

with [[Bianchi identities]]

* $d r^a = - C^a{}_{b c} t^b \wedge r^c$

* $d h = \sigma \mu - g$;

* $d g = 0$.

Let $\tilde W(b\mathbb{R} \to \mathfrak{g}_\mu)$ be the dg-algebra
with the same underlying graded algebra as $W(b\mathbb{R} \to \mathfrak{g}_\mu)$
but with the differential modified as follows

* $d t^a = -\frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a$;

* $d r^a  = - C^a{}_{b c} t^b \wedge r^a$;

* $d b = - cs + c + \tilde h$;

* $d \tilde h = \langle -,-\rangle - g$;

* $d c = g$ .

* $d g = 0$,

where "$\tilde h$" is the new name for the generator that used to be called "$h$"

There is an [[isomorphism]]

$$
  W(b\mathbb{R} \to \mathfrak{g}_\mu)
  \to
  \tilde W(b\mathbb{R} \to \mathfrak{g}_\mu)
$$

in [[dgAlg]] that is the identity on all generators except on $h$, where it is given by

$$
  h \mapsto \tilde h + (\mu - cs)
  \,.
$$

=--

+-- {: .num_note #RemarkOnShiftedWeilAlgebra}
###### Note

Where the formula for the differential of $W(b\mathbb{R}\to \mathfrak{g}_\mu)$ has the 3-cocycle $\mu$ that for $\tilde W(b\mathbb{R}\to \mathfrak{g}_\mu)$ has the [[Chern-Simons element]] $cs$.  The shift by $cs-\mu$ is precisely what shifts the curvature characteristic $d_{W(\mathfrak{g})}\mu$ into the shifted copy of $\mathfrak{g}^*$ in the Weil algebra, thus exhibiting the modified $h$ as an [[invariant polynomial]].

=--

+-- {: .num_cor #TheInvariantPolynomials}
###### Corollary

The [[invariant polynomial]]s on $(b \mathbb{R} \to \mathfrak{g}_\mu)$ are generated from those of $\mathfrak{g}_\mu$ together with $\tilde h$ and $g$:

$$
  \tilde inv(b \mathbb{R} \to \mathfrak{string})
  =
  (inv(\mathfrak{so})\otimes \langle \tilde h, g\rangle)/(d \tilde h = \langle -,-\rangle - g)
  \,.
$$

=--

### The differential lift
 {#TheLInftyDifferentialLift}

We now use the isomorphism

$$
  W(b \mathbb{R} \to \mathfrak{string}) \stackrel{\simeq}{\to}
  \tilde W(b \mathbb{R} \to \mathfrak{string})
$$

from prop. \ref{ShiftedWeilAlgebra} and obtain the desired factorization,  as the composite

$$
  \array{
    CE(\mathfrak{so})
     &
       \underoverset{\simeq}{
       \left(
       \array{
          t^a \mapsto t^a
          \\
          b \mapsto 0
          \\
          c \mapsto \mu
       }
       \right)
       }{\leftarrow}
     &
    CE(b \mathbb{R} \to \mathfrak{string})
     &
     =
     &
    CE(b \mathbb{R} \to \mathfrak{string})
     &
     \stackrel{
       \left(
       \array{
          c \mapsto c
       }
       \right)
     }{\leftarrow}
     &
    CE(b^2 \mathbb{R})
    \\
    \uparrow^\mathrlap{i^*_{\mathfrak{so}}}
     &&
    \uparrow^\mathrlap{i^*_{(b\mathbb{R} \to \mathfrak{string})}}
     &&
    \uparrow^\mathrlap{}
     &&
    \uparrow^\mathrlap{i^*_{b^2 \mathbb{R}}}
    \\
    W(\mathfrak{so})
      &
       \underoverset{\simeq}{
       \left(
       \array{
         t^a \mapsto t^a
         \\
         b \mapsto 0
         \\
         c \mapsto cs
         \\
         r^a \mapsto r^a
         \\
         \tilde h \mapsto 0
         \\
         g \mapsto \langle-,-\rangle
       }
       \right)
       }{\leftarrow}
      &
    W(b \mathbb{R} \to \mathfrak{string})
     &
     \underoverset{\simeq}{
      \left(
        \array{
          \tilde h \mapsto h + (cs - \mu)
        }
      \right)
     }
     {\leftarrow}
     &
    \tilde W(b \mathbb{R} \to \mathfrak{string})
      &
     \stackrel{
       \left(
       \array{
         c \mapsto c
         \\
         g \mapsto g
       }
       \right)
     }{\leftarrow}
     &
    W(b^2 \mathbb{R})
    \\
    \uparrow^\mathrlap{p^*_{\mathfrak{so}}}
     &&
    \uparrow^\mathrlap{}
     &&
    \uparrow^\mathrlap{p^*_{(b \mathbb{R} \to \mathfrak{string})}}
     &&
    \uparrow^\mathrlap{p^*_{b^2 \mathbb{R}}}
    \\
    inv(\mathfrak{so})
      &
       \underoverset{\simeq}{
       \left(
       \array{
         \tilde h \mapsto 0
         \\
         g \mapsto \langle -,-\rangle
         \\
         \langle \cdots \rangle
         \mapsto
         \langle \cdots \rangle
       }
       \right)
       }{\leftarrow}
      &
    \tilde inv(b \mathbb{R} \to \mathfrak{string})
      &
       =
      &
    \tilde inv(b \mathbb{R} \to \mathfrak{string})
      &
     \stackrel{
       \left(
       \array{
         g \mapsto g
       }
       \right)
     }{\leftarrow}
     &
    inv(b^2 \mathbb{R})
  }
  \,.
$$

Here

* the unlabelled vertical morphisms are defined as the unique ones that make the respective square commute;

* the notation $\langle \cdots \rangle$ stands for all the [[invariant polynomial]]s of $\mathfrak{so}$ and $\langle-,-\rangle$ specifically for the [[Killing form]].


differential T-duality

+-- {: .rightHandSide}
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###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Analogous to how [[topological T-duality]] describes the aspects of [[T-duality]] that are not related to the [[pseudo-Riemannian metric]] and the [[background gauge field]], _differential T-duality_ still ignores the metric, but takes the background [[Kalb-Ramond field]] and its effect on the [[RR-field]]s into account: it is a refinement of topological T-duality from [[cohomology]] to [[differential cohomology]].

This is also called _geometric T-duality_ ([KahleValentino](#KahleValentino)), though this is a slight misnomer, since present technnology for the differential refinement does not take the metric into account.

## Definition

([KahleValentino](#KahleValentino)=.

## Formulation by twisted differential $\mathbf{c}$-structures

We discuss here how the notion of differential T-duality pairs
is a special case of the notion of [[twisted differential c-structures]].

(...)

> for the moment see the remarks included in [[T-Duality and Differential K-Theory]]

## Related concepts

* [[T-duality]]

  * [[topological T-duality]]

  * **differential T-duality**

* [[Bn-geometry]], [[generalized complex geometry]]

## References

The first proposal for a description of differential T-duality is

* [[Alexander Kahle]], [[Alessandro Valentino]], _[[T-Duality and Differential K-Theory]]_
 {#KahleValentino}

A review is in section 7.4 of

* [[Ulrich Bunke]], [[Thomas Schick]], _Differential K-theory. A survey_ ([arXiv:1011.6663](http://arxiv.org/abs/1011.6663)).
 {#BunkeSchick}

[[!redirects geometric T-duality]]


differential topology

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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

__Differential topology__ is the subject devoted to the study of [[algebraic topology|algebro]]-[[topology|topological]] and [[homotopy theory|homotopy-theoretic]] properties of [[differentiable manifolds]], [[smooth manifolds]] and related [[differential geometry|differential geometric]] [[spaces]] such as [[stratifolds]], [[orbifolds]] and more generally [[differentiable stacks]].

A key part of differential topology is [[cobordism theory]], where the [[Pontryagin theorem|Pontryagin-]][[Pontryagin-Thom theorem|Thom theorem]] relates the [[stable homotopy theory|stable]] [[homotopy theory]] of [[Thom spectra]] to [[cobordism classes]] of [[smooth manifold|smooth]] ([[submanifold|sub]]-)[[smooth manifold|manifolds]] (for instance [[cohomotopy]] to [[normal framing|normally framed]] [[cobordism]]).

Differential topology is also concerned with the problem of finding out which topological (or PL) manifolds allow a  [[smooth structure|differentiable structure]] and the degree of nonuniqueness of that structure if they do (e.g. [[exotic smooth structures]]). It is also concerned with concrete constructions of [[cohomology|(co)]][[homology]] classes (e.g. [[characteristic class]]es) for differentiable manifolds and of [[differential cohomology|differential refinements of cohomology theories]].

More recently, the *[[smooth Oka principle]]* reveals a deep structure in differential topology which is visible in the full generality of [[higher differential geometry]] ([[smooth infinity-groupoid|smooth $\infty$-stacks]]).

## Examples

Many considerations, and classification problems, depend crucially on [[dimension]], and the case of high-dimensional manifolds (the notion of 'high' depends on the problem) is often very different from the situation in each of the low dimensions; thus there are specialists' subjects like $3$-(dimensional) topology and $4$-topology. There are restrictions on an underlying topology which is allowed for some sorts of additional structures on a differentiable manifold.

For example, only some even-dimensional differentiable manifolds allow for [[symplectic manifold|symplectic structure]] and only some odd-dimensional one allow for a [[contact manifold|contact structure]]; in these cases moreover special constructions of topological invariants like [[Floer homology]] and [[symplectic field theory]] exist.

This yields the relatively young subjects of symplectic and contact topologies, with the first significant results coming from Gromov. Any (Hausdorff paracompact finite-dimensional) differentiable manifold allows for [[Riemannian manifold|riemannian structure]] however; therefore there is no special subject of 'riemannian topology'.

## Entries in differential topology

* [[Hopf degree theorem]], [[equivariant Hopf degree theorem]]

* [[Poincaré–Hopf theorem]]

* [Pontrjagin-Thom theorem](cohomotopy#RelationToCobordismGroup)

* [[Sard's theorem]], [[transversality]], [[Thom's transversality theorem]]

* [[Reeb sphere theorem]]

* [[cobordism]]

* ...

## Related entries

* [[low dimensional topology]]

* [[synthetic differential topology]]

* [[equivariant differential topology]]

## References

Though some of the basic results, methods and conjectures of differential topology go back to [Poincaré](http://en.wikipedia.org/wiki/Henri_Poincar%C3%A9), Whitney, Morse and Pontrjagin, it became an independent field only in the late 1950s and early 1960s with the seminal works of Smale, Thom, Milnor and Hirsch. Soon after the initial effort on foundations, mainly in the American school, a strong activity started in Soviet Union (Albert Schwarz, A. S. Mishchenko, S. Novikov, V. A. Rokhlin, M. Gromov...).

Introductions and monographs:

* {#Milnor64} [[John Milnor]], _Differential topology_, chapter 6 in T. L. Saaty (ed.) _Lectures On Modern Mathematic II_ 1964 ([[Milnor_DifferentialTopology.pdf:file]])

* [[John Milnor]], _Lectures on the h-cobordism theorem_, 1965 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/surgery/hcobord.pdf))

* [[James R. Munkres]], _Elementary Differential Topology_, Annals of Mathematics Studies 54 (1966), Princeton University Press ([doi:10.1515/9781400882656](http://dx.doi.org/10.1515/9781400882656)).

* Andrew H. Wallace, _Differential topology: first steps_, Benjamin 1968.

* [[Victor Guillemin]], Alan Pollack, _Differential topology_, Prentice-Hall 1974

* [[Morris Hirsch]], _Differential topology_, Springer Graduate Texts in Mathematics **33** (1976) [[doi:10.1007/978-1-4684-9449-5](https://link.springer.com/book/10.1007/978-1-4684-9449-5), [gBooks](http://books.google.com/books/about/?id=iSvnvOodWl8C)]

* [[Theodor Bröcker]], [[Klaus Jänich]], *Introduction to differentiable topology* (1982) [[ISBN:9780521284707](https://www.cambridge.org/ae/universitypress/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/introduction-differential-topology?format=PB&isbn=9780521284707)]

  > (translated from the German 1973 edition)

* [[Raoul Bott]], [[Loring Tu]], _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Math. __82__, Springer 1982. xiv+331 pp.

* [[John Milnor]], _Topology from the differential viewpoint_, Princeton University Press, 1997. ([ISBN:9780691048338](https://press.princeton.edu/books/paperback/9780691048338/topology-from-the-differentiable-viewpoint), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnortop.pdf))

* [[Mladen Bestvina]] (notes by [[Adam Keenan]]), _Differentiable Topology and Geometry_, 2002 ([pdf](http://www.math.utah.edu/~keenan/manifoldsnotes.pdf), [[BestvinaKeenanDifferentialTopology.pdf:file]])

* [[C. T. C. Wall]], _Differential topology_, Cambridge Studies in Advanced Mathematics 154, 2016

* [[Joel W. Robbin]], [[Dietmar Salamon]], _Introduction to differential topology_, 294 pp, webdraft 2018 [pdf](https://people.math.ethz.ch/~salamon/PREPRINTS/difftop.pdf)

* {#Benedetti19} [[Riccardo Benedetti]], _Lectures on Differential Topology_, Graduate Studies in Mathematics **218**, AMS 2021  ([arXiv:1907.10297](https://arxiv.org/abs/1907.10297), [ISBN: 978-1-4704-6674-9](https://bookstore.ams.org/cdn-1631100821985/gsm-218/))

Survey with connections to [[algebraic topology]]:

* [[Sergei Novikov]], _Topology I -- General survey_, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 ([doi:10.1007/978-3-662-10579-5](https://link.springer.com/book/10.1007/978-3-662-10579-5), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/novikovsurv.pdf))

* [[Jean Dieudonné]], _A History of Algebraic and Differential Topology, 1900 - 1960_, Modern Birkhäuser Classics 2009 ([ISBN:978-0-8176-4907-4](https://www.springer.com/de/book/9780817649067))

See also

* Wikipedia, _[Differential topology](https://en.wikipedia.org/wiki/Differential_topology)_

Generalization to [[equivariant differential topology]]:

* {#Wasserman69} [[Arthur Wasserman]], _Equivariant differential topology_, Topology Vol. 8, pp. 127-150, 1969 ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf))


differential topology of mapping spaces

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

1. [Introduction](#intro)
3. Manifold Structure
  2. [[linear mapping spaces|Linear Mapping Spaces]]
  3. [[manifold structure of mapping spaces|Manifold Structure of Mapping Spaces]]
  4. [[tangent spaces of mapping spaces|Tangent Spaces]]
  5. [[topology of mapping spaces|The Topology of the Loop Space]]
  6. [[smooth maps of mapping spaces|Some Useful Smooth Maps]]
4. Bundles
  1. [[tangent bundle of mapping spaces|The Tangent Bundle]]
  2. [[vector bundles on mapping spaces|Vector Bundles]]
  3. [[Principal and Gauge Bundles]]
  4. [[Connections]]
  5. [[The Enemy of my Enemy is not my Friend]]
5. Submanifolds and Tubular Neighbourhoods
  1. [[evaluation fibration of mapping spaces|The Fundamental Fibration]]
  2. [[tubular neighbourhood of a mapping space|Tubular Neighbourhoods]]
  3. [[equivariant tubular neighbourhoods|Equivariant Tubular Neighbourhoods]]
  4. [[A Not-So-Nice Submanifold]]
6. [[A Miscellany]]
  1. [[Weak Riemannian Manifolds]]
  2. [[More Fun with Based Loops]]
7. [References](#refs)

## Introduction {#intro}

This series of pages started out as an accompaniment to a series of seminars given at NTNU, and subsequently at Sheffield University, entitled:

* "The [[differential topology|Differential Topology]] of the [[loop space|Loop Space]]".

The original purpose of those seminars was to present an introduction to this topic leading up to the work contained in [[Andrew Stacey|my]] preprint on the construction of a [[Dirac operator]] on [[loop space]]s, [Sta](#Sta).

This was intended to be accessible to anyone with knowledge of basic finite
dimensional [[differential topology]]. Thus there was a reasonable amount of
background material to be explained before the subject of Dirac operators was
broached.  This could be divided into three broad areas: differential topology of loop spaces, [[spinors]] in arbitrary dimension, and the [[Atiyah-Singer index theorem]] (in finite dimensions).

The latter two of these topics are already superbly covered by books accessible to any
differential topologist, perhaps with a little [[functional analysis]]. The book [LM89](#LM89) is an excellent introduction to both topics in finite dimensions whilst [PR94](#PR94) deals with spinors in arbitrary dimension. A [[group]]-centric viewpoint is presented in [PS86](#PS86), which is required reading for anyone seriously thinking about loop spaces.

The main reference for the first topic is [KM97](#KM97) in which is developed a theory of analysis in infinite dimensions in arbitrary [[topological vector space]]s.  However, this means that whilst being an excellent book, its subject matter is perhaps too broad and too deep for someone who just wants to know about the differential topology of loop spaces.

This led to the writing of [arXiv:math.DG/0510097](#0510097) with the intention was to provide a more gentle
introduction than [KM97](#KM97) but still including the required detail to understand the differential topology of the space of smooth loops. It subsequently grew
beyond that remit as more topics were added.

That article is now in the process of being imported in to the nLab.  In the transfer, its scope is being extended to more general mapping spaces as many of the basic structure of loop spaces holds in this more general setting.

## References ## {#refs}

* [[Andreas Kriegl]], [[Peter Michor]], _The convenient setting of global analysis_ ,   1997 [MR1471480](http://www.ams.org/mathscinet-getitem?mr=1471480)
{#KM97}

* Spinors in Hilbert space, Plymen, R. J.  and Robinson, P. L., 1994 [MR1312612](http://www.ams.org/mathscinet-getitem?mr=1312612)
{#PR94}

* Andrew Pressley, [[Graeme Segal]], _Loop groups_   1986 [MR900587](http://www.ams.org/mathscinet-getitem?mr=900587)
{#PS86}

* Spin geometry, Lawson, H. Blaine, Jr. and Michelsohn, Marie-Louise, 1989, [MR1031992](http://www.ams.org/mathscinet-getitem?mr=1031992)
{#LM89}