Critical string models
Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
The field content of 11-dimensional supergravity contains a higher U(1)-gauge field called the supergravity C-field or M-theory 3-form , which is locally a 3-form and globally some variant of a circle 3-bundle with connection. There have been several suggestions for what precisely its correct global description must be.
Construction via gauge fields
In (DFM, section 3) the following definition is considered and argued to be a good model of the supergravity -field.
The homotopy groups of the classifying space of the Lie group E8 satisfy
Therefore for a manifold of dimension there is a canonical morphism
Let be a smooth manifold of dimension . For each . choose an E8-principal bundle which represents under the above isomorphism.
for the groupoid whose
objects are triples where
is one of the chosen -bundles,
is a connection on ;
is a degree-3 differential form on .
morphisms are parameterized by their source and target triples together with a closed 3-form with integral periods, subject to the condition that
where is the relative Chern-Simons form corresponding to the linear path of connections from to
the composition of morphisms
is given by
See (DFM, (3.22), (3.23)).
Here we think of as equipped with a pseudo Riemannian structure and spin connection and think of each object of as inducing an degree-4 cocycle in ordinary differential cohomology with curvature 4-form
Notice that with the normalization implicit here the second terms is one half of the image of something in integral cohomology. So this is not itself a differential character, but can be regarded as “shifted differential character”: a trivialization of the trivial 5-character with global connection 4-form given by . See below for more on this.
The above groupoid has homotopy groups
The first, the set of connected components (gauge equivalence classes of -fields) is isomorphic to the set of ordinary differential cohomology in degree 4 of . In fact is naturally a torsor over this abelian group: the torsor of -shifted differential characters.
The second, the fundamental group, is that of flat circle bundles.
Orientation and fractional classes
Ordinarily, given a -bundle with first fractional Pontryagin class
and second Chern class
the -field is supposed to have a curvature class in de Rham cohomology given by
Since in general is not further divisible in integral cohomology, this means that this cannot be the curvature of any differential character/bundle 2-gerbe/circle 3-bundle with connection, since these are necessarily the images in de Rham cohomology of their integral classes.
See also (DFM, section 12.1) where it is argued that this is related to boundaries and orientation double covers.
Restriction to the boundary
By DFM, section 12 on a manifold with boundary we are to impose .
See the discussion below for how this reproduces the Green-Schwarz mechanism for heterotic supergravity on the boundary.
Description in -Chern-Weil theory
Some remarks on ways to regard the -field from the point of view of ∞-Chern-Weil theory.
We shall consider the sum of two fields, whose curvature is the image in de Rham cohomology of the proper integral class
Recall from the discussion at circle n-bundle with connection that in the cohesive (∞,1)-topos Smooth∞Grpd the circle 3-bundles with local 3-form connection over an object (for instance a smooth manifold, or an orbifold) are objects in the 3-groupoid that is the (∞,1)-pullback
(Recall from the discussion there that if desired one may pass to the canonical presentation of this by the model structure on simplicial presheaves over CartSp and that in this explicit presentation we may replace with the more familiar . )
We consider now the analog of this definition for the universal curvature form on replaced by the difference of the differentially refined second Chern class of E8 and the first fractional Pontryagin class of the spin group. The resulting -pullback we tentatively call , though we shall have to discuss to which extend this faithfully models the -field, and which aspects of it.
For Smooth∞Grpd, let ∞Grpd be the (∞,1)-pullback
By its intrinsic definition we have that the differential characteristic class is the composite
of the smooth refinement of the second Chern class with the universal curvature form on . Similarly for .
Therefore we may either compute the (∞,1)-pullback in def. 2 directly, or in two consecutive steps. Both methods lead to their insights.
we consider general abstract consequences of the above definition, mainly making use of the factorization. In
we find a presentation by simplicial presheaves of the direct homotopy pullback.
In the first approach connections on the E8-principal bundles never appear explicitly. In the second approach they appear as pseudo-connections, or as genuine connections whose morphisms are however allowed to shift them arbitrarily. This means that these connections are purely auxiliary data that serve to present the required homotopies. They do not survive in cohomology. This is as in the DFM model above.
we comment how genuine -connections may appear inside the second presentation of the -model.
This implies by the pasting law for (∞,1)-pullbacks that the -pullback from def. 2 may be decomposed into two consecutive pullbacks of the form
where on the right we find the defining pullback for (the cocycle 3-groupoid of) ordinary differential cohomology.
This implies the following structure and properties.
By the above there exists canonically a morphism
that maps -field configurations to ordinary differential cohomology in degree 4, whose curvature is the image in de Rham cohomology of the second Chern-class of some -bundle.
The differential cocycle has all the general properties that make its higher parallel transport over membrane worldvolumes be well-defined. (Apart from the coefficient of , this is the only requirement from which DFM deduce their model.)
The following proposition describes the first two homotopy groups of the 3-groupoid .
Over a fixed -principal bundle we have a short exact sequence (of pointed sets)
is the group of pairs where is a smooth refinement under of the integral image of .
Notice that we have the pasting diagram of (∞,1)-pullbacks
where the top right square is discussed at cohesive (∞,1)-topos – Differential cohomology. By the discussion at smooth ∞-groupoid – Flat cohomology we have that , where on the right we have ordinary cohomology (for instance realized as singular cohomology). Finally observe that , by the above remark. Therefore after passing to connected components by applying we get on cohomology
by reasoning as discussed at fiber sequence. In parallel to the familiar short exact sequence for ordinary differential cohomology
this therefore implies also the short exact sequence
Next we redo the entire discussion after applying the loop space object-construction to everything. Using that
on general grounds (see fiber sequence for details) and that also
(since and are right adjoint (∞,1)-functors – by the discussion at cohesive (∞,1)-topos – and hence commute with the (∞,1)-pullback that defines ), we have then the looped pasting diagram of (∞,1)-pullbacks
Observe that here is a smooth but 0-truncated object: so that
is the set of smooth functions (to be thought of as the the set of gauge transformations from the trivial -principal bundle on to itself).
Presentation by differential form data
In order to compute the -pullback more explicitly, we follow the discussion at differential string structure, where presentations of this pullback in terms of simplicial presheaves arising from Lie integration is given.
for the Lie algebra of and write
for the sum of the canonical Lie algebra cocycles in transgression with the respective Killing form invariant polynomials.
for the canonical diagonal embedding Write
for the corresponding smooth characteristic class. See ∞-Chern-Weil homomorphism for details. By the discussion there we present by
By the discussion at differential string structure we have that the top morphism is a fibration in the global projective model structure on simplicial presheaves (there it is shown that the analogous morphism out of is a fibration, but then so is this one, because the components on the left are the same but with fewer conditions on them, so that the lifts that existed before still exist here).
Over some CartSp and we have that is given by differential form data
on . Here, recall, takes values in , so that for instance the -curvature is in detail given by
where denotes the spin connection.
Let be a differentiably good open cover. We hit all connected components of by considering in
those cocycles that
Write therefore for such a cocycle.
For gauge transformations between two such pairs, parameterized by the above form data patchwise on , the fact that vanishes on implies the infinitesmal gauge transformation law
where is the shift of the 1-forms. This integrates to
is the relative Chern-Simons form corresponding to the shift of -connection.
Restriction to the boundary
We have seen that is the 3-goupoid of those Cech cocycles on with coefficients in such that the curvature 4-form has a fixed globally defined value.
Consider the subobject
of the simplicial presheaf on those objects and k-morphisms for which .
By the gauge transformation law (2)
this means that this picks those morphisms for which the Chern-Simons form vanishes
where is the 1-form datum (with the canonical coordinate on the 1-simplex ).
In the literature often the relative Chern-Simons form is considered for “ungauged” paths of connections: for in the above formula, hence for a -valued 1-form on with no leg along the simplex (only depending on the simplex coordinate). Here, however, it is crucially important that we consider the general “gauged” paths.
Notice that on the semisimple Lie algebra and compact Lie algebra the Killing form is non-degenerate and positive definite (or negative definite, depending on convention). The latter condition means that this integral vanishes precisely if
This is the case on paths for which , but this are exactly the paths that induce genuine gauge transformations between and , where
This means that cocycles with coefficients in this subobject for are cocycles as described at differential string structure, exhibiting the Green-Schwarz mechanism on the heterotic boundary, witnessed by the restriction of the curvature equation (1) to vanishing -field
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
|brane||in supergravity||charged under gauge field||has worldvolume theory|
|black brane||supergravity||higher gauge field||SCFT|
|D-brane||type II||RR-field||super Yang-Mills theory|
|D0-brane||BFSS matrix model|
|D4-brane||D=5 super Yang-Mills theory with Khovanov homology observables|
|D1-brane||2d CFT with BH entropy|
|D3-brane||N=4 D=4 super Yang-Mills theory|
|(D25-brane)||(bosonic string theory)|
|NS-brane||type I, II, heterotic||circle n-connection|
|NS5-brane||B6-field||little string theory|
|M-brane||11D SuGra/M-theory||circle n-connection|
|M2-brane||C3-field||ABJM theory, BLG model|
|M5-brane||C6-field||6d (2,0)-superconformal QFT|
|M9-brane/O9-plane||heterotic string theory|
|topological M2-brane||topological M-theory||C3-field on G2-manifold|
|topological M5-brane||C6-field on G2-manifold|
|solitons on M5-brane||6d (2,0)-superconformal QFT|
|self-dual string||self-dual B-field|
|3-brane in 6d|
The state-of-the-art in the literature concerning attempts to find the correct mathematical model for the supergravity C-field seems to be
A summary and rview of this is in
The discussion in twisted nonabelian differential cohomology is given in
See also section 4.3.4 of
A detailed discussion of the quantum anomaly of the supergravity C-field – and its cancellation – is in