Critical string models
Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
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
in the (∞,1)-topos ∞Grpd Top has a refinement to Smooth∞Grpd of the form
– the smooth first fractional Pontryagin class.
The induced morphism on cocycle ∞-groupoids
sends a spin group-principal bundle to its corresponding Chern-Simons circle 3-bundle .
A choice of trivialization of is a string structure. The 2-groupoid of smooth string structures is the homotopy fiber of over the trivial circle 3-bundle.
By Chern-Weil theory in Smooth∞Grpd this morphism may be further refined to a differential characteristic class that lands in the ordinary differential cohomology , classifying circle 3-bundles with connection
The 2-groupoid of differential string structures is the homotopy fiber of this refinement over the 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 is characterized by a tuple consisting of
a connection on a Spin-principal bundle on ;
a choice of trivial circle 3-bundle with connection , hence a differential 3-form ;
a choice of equivalence of the Chern-Simons circle 3-bundle with connection of with this chosen 3-bundle
More generally, one can consider the homotopy fibers of over arbitrary circle 3-bundles with connection and hence replace in the above with . According to the general notion of twisted cohomology, these may be thought of as twisted differential string structures, where the class is the twist.
We will assume that the reader is familiar with basics of the discussion at Smooth∞Grpd. We often write for short.
Let SmoothMfd Smooth∞Grpd be the Spin group, for some , regarded as a Lie group and thus canonically as an ∞-group object in Smooth∞Grpd. We shall notationally suppress the in the following. Write for the delooping of in Smooth∞Grpd. (See the discussion here). Let moreover be the circle Lie 3-group and its delooping.
At Chern-Weil theory in Smooth∞Grpd the following statement is proven (FSS).
The image under Lie integration of the canonical Lie algebra 3-cocycle
on the semisimple Lie algebra of the Spin group – the special orthogonal Lie algebra – is a morphism in Smooth∞Grpd of the form
whose image under the the fundamental ∞-groupoid (∞,1)-functor/ geometric realization ∞Grpd is the ordinary fractional Pontryagin class
in Top. Moreover, the corresponding refined differential characteristic class
is in cohomology the corresponding refined Chern-Weil homomorphism
with values in ordinary differential cohomology that corresponds to the Killing form invariant polynomial on .
For any Smooth∞Grpd, the 2-groupoid of differential string-structures on – – is the homotopy fiber of over the trivial differential cocycle.
More generally (see twisted cohomology) the 2-groupoid of twisted differential string structures is the (∞,1)-pullback in
where the right vertical morphism is a choice of (any) one point in each connected component (differential cohomology class) of the cocycle ∞-groupoid (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.
See also (Waldorf, cor. 1.1.5).
If the underlying integral cohomology class of the twist is trivial, , then a -twisted differential string structures on a -connection are characterized by a globally defined 3-form on .
This 3-form is the globally defined connection 3-form of an appropriate circle 3-bundle with connection equivalent to the Chern-Simons circle 3-bundle whose underlying 3-bundle is by assumption trivial: on a trivial circle -bundle every connection may be represented by a globally defined -form.
This statement appears as (Waldorf, theorem 1.3.3), where circle 3-bundles are modeled as bundle 2-gerbes. The explicit construction of the globally defined 3-form in this model is spelled out in lemma 3.2.4 there.
Elements in the defining homotopy pullback from def. 1 over a given connection are chracterized by an element and an equivalence
between the corresponding Chern-Simons circle 3-bundle and the given circle 3-bundle . In the case at hand, both have underlying trivial class .
By the characteristic class exact sequence
any two classes in that have trivial underlying class in differ by a 3-form modulo a closed 3-form with integral periods.
Therefore both as well as are given by a globally defined 3-form modulo an integral form: the global connection 3-form on these trivial circle 3-bundles with connection.
Construction in terms of -Cech cocycles
We use the presentation of the (∞,1)-topos Smooth∞Grpd (as described there) by the local model structure on simplicial presheaves to give an explicit construction of twisted differential string structures in terms of 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).
The differential fractional Pontryagin class is presented in by the top morphism of simplicial presheaves in
Here the middle morphism is the direct Lie integration of the L-∞ algebra cocycle while the top morphisms is its restriction to coefficients for ∞-connections.
In order to compute the homotopy fibers of we now find a resolution of this morphism by a fibration in . By the fact that this is a simplicial model category then also the hom of any cofibrant object into this morphism, computing the cocycle -groupoids, is a fibration, and therefore, by the general discussion at homotopy pullback, we obtain the homotopy fibers as the ordinary fibers of this fibration.
Presentation of the differential class by a fibration
In order to factor into a weak equivalence followed by a fibration, we start by considering such a factorization before differential refinement, on the underlying characteristic class .
To that end, we replace the Lie algebra by an equivalent but bigger Lie 3-algebra (following SSSIII). We need the following notation:
, the special orthogonal Lie algebra (the Lie algebra of the spin group);
the line Lie 3-algebra, the single generator in degee 3 of its Chevalley-Eilenberg algebra we denote , .
is the Killing form invariant polynomial, regarded as an element of the Weil algebra of ;
the degree 3 Lie algebra cocycle, identified with a morphism
of Chevalley-Eilenberg algebras; and normalized such that its continuation to a 3-form on is the image in de Rham cohomology of of a generator of ;
is a Chern-Simons element interpolating between the two; characterized by the fact that it fits into the commuting diagram
the string Lie 2-algebra.
Let denote the L-∞ algebra whose Chevalley-Eilenberg algebra is
with a generator in degree 2, and a generator in degree 3, and with differential defined on generators by
The 3-cocycle factors as
where the morphism on the left (which is the identity when restricted to and acts on the new generators as indicated) is a quasi-isomorphism.
The point of introducing the resolution in the above way is that it naturally supports the obstruction theory of lifts from -connections to string Lie 2-algebra 2-connection
The defining projection factors through the above quasi-isomorphism by the canonical inclusion
which dually on -algebras is given by
In total we are looking at a convenient presentation of the long fiber sequence of the string Lie 2-algebra extension:
(The signs appearing here are just unimportant convention made in order for some of the formulas below to come out nice.)
The image under Lie integration of the above factorization is
where the first morphism is a weak equivalence followed by a fibration in the model structure on simplicial presheaves .
To see that the left morphism is objectwise a weak homotopy equivalence, notice that a -cell of consists of a triple , where is a vertical flat -valued 1-form on , is a vertical 2-form and a 3-form on , such that and , since is flat. Therefore is uniquely determined by and , and there are no conditions on . This means that a -cell of is identified with a pair consisting of a based smooth function and a vertical 2-form , (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 -simplex to the interior (for instance simply by radially rescaling it smoothly to 0). Accordingly the simplicial homotopy groups of are the same as those of . The morphism between them is the identity in and picks 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
are and .
Notice that a -cell of is a smooth function and that the morphism sends the pair to the fiber integration .
Our lifting problem in degree 3, has given a function and a smooth function (with sitting instants at the subfaces) together with a 2-form on the horn .
By pullback along the standard continuous retract which is non-smooth only where has sitting instants, we can always extend to a smooth function with the property that . (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 , regarded as the missing face of the horn. By multiplying it with a suitable smooth function on we can obtain an extension of to all of with the property that its integral over is the given . By the Stokes theorem it remains to extend to the interior of 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 -parameterized family of closed 3-forms with sitting instants on , whose integral over equals . 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 with that itself is radially constant at the boundary. By construction the difference 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 . Therefore the sum
equals when restricted to and has the property that its integral over equals . Together with our extension , 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 and a given vertical 2-form on such that its integral equals , as well as a function , we can extend the 2-form and the function along . The latter follows from the fact that 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 .
We now proceed to extend this factorization to the exponentiated differential coefficients (see connection on an ∞-bundle).
(presentation of the differential class by a fibration)
Under Lie integration the above factorization of the Lie algebra cocycle
maps to the factorization
of in , where the first morphism is a weak equivalence and the second a fibration.
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 -data cannot be filled, then also adding the -data does not yield a filler. So we need to check that it is also surjective on homotopy groups: if the -data can be filled, then also the corresponding -data has a filler. Since the curvature is horizontal it is already extended. We may extend in any smooth way to (for instance by rescaling it smoothly to zero at the center of the -simplex) and then take the equation to define the extension of .
We now check that the second morphism is a fibration. It is itself the composite
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 to the lifting problems
may all be equivalently reformulated as lifting against a cylinder by using the sitting instants of all forms.
We have then a 3-form with horizontal curvature and differential form data on given. We may always extend along the cylinder direction (its vertical part is equivalently a based smooth function to which we may extend constantly). 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 . Let be a smooth contraction. Then while may be non-vanishing, by horizonatlity of their curvature characteristic forms we still have that vanishes (since the contraction vanishes).
Therefore the 2-form
satisfies . It may however not coincide with our given at . But the difference 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 to a closed 2-form on the interior. The sum
then still satisfies and it coincides with on the left boundary.
Notice that here indeed has sitting instants: since , and 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 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
starts out differently, due to the presence of , but it then ends up amounting to the same kind of argument:
We have four functions which we may realize as the fiber integration of a 3-form on , and we have a lift to -data on (the boundary of the 4-simplex minus one of its 3-simplex faces).
We observe that we can
always extend 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 -data horizonatlly due to the fact that .
the -form is already horizontal, hence already filled.
Moreover, by the fact that the 2-form already is defined on all of its fiber integral over the boundary coincides with the fiber integral of over ). But by the fact that we have lifted and the fact that is an integral cocycle, it follows that this equals the fiber integral of over the remaining face.
Use then as above the vertical Poincare lemma-formula to find on with sitting instants that satisfies the equation there. Then extend the closed difference to a closed smooth 2-form on . As before, the difference
is an extension of that constitutes a lift.
Explicit Cech cocycles
For any SmoothMfd Smooth∞Grpd, for any choice of differentiaby good open cover with corresponding cofibrant presentation we have that the 2-groupoids of twisted different String structuress are presented by the ordinary fibers of the morphism of Kan complexes
over any basepoints in the connected components of the Kan complex on the right, which correspond to the elements
in the ordinary differential cohomology of .
Since is a simplicial model category the morphism is a fibration because is and is cofibrant.
It follows from the discussion at homotopy pullback that the ordinary pullback of simplicial presheaves
is a presentation for the defining (∞,1)-pullback for , as defined above.
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 Deligne-cocycle (as discussed at circle n-bundle with connection)
with local connection 3-form and globally defined curvature 4-form .
A differential string structure on twisted by this cocycles is on patches a morphism
in dgAlg, subject to some horizontality constraints. The components of this are over each a collection of differential forms of the following structure
Here we are indicating on the right the generators and their relation in and on the left their images and the images of the relations in . This are first the definitions of the curvatures 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 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 . So the target space having a (differential) string structure is a sufficient condition for the cancellation of the quantum anomaly.
(First argued in Killingback, later made precise in (Bunke)).
The Green-Schwarz mechanism in heterotic supergravity
We discuss the application of twisted differential string structures in supergravity and string theory.
Local differential form data as in note 2 above is known in theoretical physics in the context of the Green-Schwarz mechanism for 10-dimensional supergravity.
In this context
is called the spin connection;
the components of the above cocycle are known as the -twisted Kalb-Ramond field.
In this application the twisting cocycle is itself the Chern-Simons circle 3-bundle of a unitary group-principal bundle with local connection form . Therefore in this case and the above local form data becomes
Since is the would-be curvature of a circle 2-bundle with connection, this is the first higher Maxwell equation that exhibits
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 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
Clearly, if we take into account also gauge transformations of the gauge bundle, we should replace this by the full
The look of this diagram makes manifest how in this situation we are looking at the structures that homotopically cancel the differential classes and against each other.
More discussion of this is in (SSSIII).
Since is abelian, we may consider the corresponding Mayer-Vietoris sequence by realizing equivalently as the homotopy fiber of the difference of differential cocycles .
Indeed, the above explicit presentation by simplicial presheaves generalizes immediately to describe this case, realizing -twisted differential string structures equivalently as differential “untwisted -twisted-string-structures”.
We may usefully formalize this further by defining the -2-group to be the homotopy fiber
We have then that is the 3-groupoid of untwisted differential -structures.
More on this in (FiSaSc).
This is supposed to be (see section 12 of (DFM)) the restriction to the boundary of the supergravity C-field, which is the -pullback
where is 11-dimensional with . 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-connections.
By the discussion at supergravity C-field under a shift of connection the -field transforms as
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:
the supergravity -field vanishes,
the gauge bundles develop genuine connections.
Relation to string 2-connections
By the discussion at connection on an ∞-bundle we have that for an L-∞ algebra and
the delooping of the smooth Lie n-group obtained from it by Lie integration, the coefficient for ∞-connections on -principal ∞-bundles is
where on the very right we have the simplicial presheaf
(See ∞-Chern-Weil homomorphism for details).
The 2-groupoid of entirely untwisted differential string structures on (the twist being ) is equivalent to that of string 2-group principal 2-bundles with 2-connection:
By the above discussion of Cech cocycles we compute as the ordinary fiber of the morphism of simplicial presheaves
over the identically vanishing cocycle.
In terms of the component formulas spelled out in the above discussion of the GS-mechanism, this amounts to restricting to those cocyles for which n each degree the equations
Comparing this to the explicit formulas for and in the above we see that these cocycles are exactly those that factor through the canonical inclusion
from observation 6 of the string Lie 2-algebra into the mapping cone Lie 3-algebra of the extension that defines it.
Whitehead tower of the orthogonal group
A discussion of differential string structures in terms of bundle 2-gerbes is given in
The description of the gauge transformations of the supergravity C-field is in section 3 of
E. Diaconescu, Greg Moore, Dan Freed, The -theory 3-form and -gauge theory (arXiv: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
The full Cech-Deligne cocycles induced by this (but not yet the homotopy fibers over them) were discussed in
The 2-group and its differential structurs, etc. are discussed in
A comprehensive discussion including all the formal background and the applications is attempted at
in section 4.2.
The relation to quantum anomaly cancellation in heterotic string theory has been first discussed in
- Killingback, World-sheet anomalies and loop geometry Nuclear Physics B Volume 288, 1987, Pages 578-588
and given a rigorous treatment in
- Ulrich Bunke, String structures and trivialisations of a Pfaffian line bundle (arXiv)
More discussion on the relation to spin structures on smooth loop space is in