This is a page with notes, produced and being produced together with various collaborators, on “-Chern-Simons theory” in the context of differential cohomology in a cohesive topos.
We observe that in higher geometry and specifically in higher differential geometry the traditional notion of Chern-Simons theory generalizes naturally to a very general class of field theories (prequantum field theories) that on the one hand are of profound conceptual nature – they are precisely the universal characteristic classes in differential cohomology over higher moduli stacks – and on the other hand subsume a fairly encompassing list of fundamental field theories, not all of which are traditionally recognized as being of Chern-Simons type. This is true in particular if all boundary field theories and defects field theories for -Chern-Simons theories are also taken into account, which then subsumes also Wess-Zumino-Witten theory, Wilson line-theories and their higher and - analogs.
In fact, -Chern-Simons theories turn out to be themselves precisely the boundary field theories for a single prequantum field theory which we call the “universal higher topological Yang-Mills theory”. This is further discussed at Higher Chern-Simons local prequantum field theory.
More in technical detail:
The action functional of ordinary Chern-Simons theory for a simple Lie group may be understood as being the volume holonomy of the Chern-Simons 2-gerbe with connection that the refined Chern-Weil homomorphism assigns to any connection on a -principal bundle.
We observe that all the ingredients of this statement have their general abstract analogs in any cohesive (∞,1)-topos : for any cohesive ∞-group and any representative of a characteristic class for there is canonically the induced ∞-Chern-Weil homomorphism that sends intrinsic G-connections to cocycles in intrinsic differential cohomology with coefficients in . This may be thought of as the Lagrangian of the -Chern-Simons theory induced by .
In the cohesive (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids we give a natural general abstract procedure for integration of over an -dimensional parameter space . The resulting smooth function is the exponentiated action functional of -Chern-Simons theory on the smooth ∞-groupoid of field configurations. It may be regarded itself as a degree-0 characteristic class on the space of field configurations. As such, its differential refinement is the Euler-Lagrange equation of the theory. Its homotopy fiber is the smooth ∞-groupoid of classical solutions: the derived phase space.
We show that this construction subsumes the action functional of ordinary Chern-Simons theory, which is the one induced for the Lie integration of the canonical Lie algebra 3-cocycle on . We observe that there is a general abstract notion of ∞-Lie algebroid cohomology, each cocycle in which induces after Lie integration one type of -Chern-Simons theory, by the above. Among the action functionals obtained this way are those of BF-theory coupled to topological Yang-Mills theory, of all versions of AKSZ theory including the Poisson sigma-model and the Courant sigma model in lowest degree as well as Chern-Simons supergravity.
More on the general context of -Chern-Simons theory is at higher category theory and physics.
We give here a leisurely introduction to the ideas of -Chern-Simons theory and survey some of the constructions and results that are discussed in detail below.
A brief indication of what classical Chern-Weil theory fails to accomplish, as a motivation for the generalization to follow.
A survey of the general abstract formulation and generalization of Chern Weil theory in terms of cohesive (∞,1)-topos theory.
Concretely, the Chern-Weil homomorphism is presented by the following simple construction:
we get for each smooth manifold an assignment
of integral cohomology classes of base space to equivalence classes of -principal bundles by sending a bundle classified by a map to the class .
Let be the image of in real cohomology, induced by the evident inclusion of coefficients .
The first main statement of Chern-Weil theory is that there is an invariant polynomial
The second main statement is that this invariant polynomial serves to provide a differential (Lie integration) construction of :
The statement is that under the de Rham theorem-isomorphism this presents the class .
The third main statement, says that this construction may be refined by combining integral cohomology and de Rham cohomology to ordinary differential cohomology: the -form may be realized itself as the curvature -form of a circle n-bundle with connection – one speaks of a secondary characteristic class.
In summary this yields the following picture:
A central implication of the last step is that with the refinement from curvatures in de Rham cohomology to circle n-bundles with connection in differential cohomology is that these come with a notion of higher parallel transport and higher holonomy:
the corresponding higher parallel transport as an assignment
for the next higher invariant polynomial on a semisimple Lie algebra, is a Chern-Simons circle 7-bundle, and so on.
So the refined Chern-Weil homomorphism provides a large family of gauge quantum field theories of Chern-Simons type in odd dimensions whose field configurations are always connections on principal bundles and whose Lagrangians are Chern-Simons elements on a Lie algebra.
But the notion of invariant polynomials and Chern-Simons elements naturally exists much more generally for L-∞ algebras, and even more generally for L-∞ algebroids. We claim here that in this fully general case there is still a natural analog of the Chern-Weil homomorphism – which we call the ∞-Chern-Weil homomorphism . Accordingly this gives rise to a wide class of action functionals for gauge quantum field theories, which we call -Chern-Simons theories.
But there are applications where one needs differential refinements for instance of the higher connected covers in the Whitehead tower of Lie groups, which without further work are just topological groups. Central motivating examples are differential characteristic classes for the string 2-group and the fivebrane 6-group.
A refinement of the Chern-Weil homomorphism from cohomology sets to cocycle ∞-groupoids gives access to its homotopy fibers: these are ∞-groupoids that encode the obstruction theory of the given differential characteristic class: they encode how trivializations of differential characteristic classes are related, how these relations are related, and so on.
In the next paragraph we indicate how these deficiencies of the classical Chern-Weil homomorphism are removed by constructing it in the context of cohesive (∞,1)-toposes.
Chern-Weil theory is traditionally discussed in terms of smooth universal connections on the universal principal bundles over the classifying space of , where the topological spaces and are both equipped in a clever way with smooth structure of sorts.
However, even when equipped with a smooth structure, the classifying space is far from being a faithful representative of the theory of smooth -principal bundles: all it remembers (in its topological homotopy type) is their equivalence classes, but it fails – with or without smooth structure – to carry any information about the morphisms of smooth -principal bundles: their gauge transformations.
This is not just a question of cosmetics: Chern-Weil theory produces (differential) characteristic classes and with every characteristic map representing a characteristic class, there is the associated extension theory that extracts the information encoded in the characteristic map: but these extensions are homotopy fibers and these homotopy fibers only come out right when the morphisms of bundles are faithfully taken into account.
The correct object to replace the smooth classifying space with is the corresponding moduli stack, which we shall write (where here and in all of the following the boldface type is to indicate (good) smooth refinement of the topological structure that goes by the non-boldface symbols). This is nothing but the delooping Lie groupoid with a single object and worth of morphisms. For a smooth manifold, regarded as a Lie groupoid with only trivial morphisms, the groupoid of -principal bundles and smooth gauge transformations between them is
Lie groupoids are certain concrete stacks on smooth manifolds. If we allow more generally also non-concrete such stacks – which we shall call just smooth groupoids – then there is also the differential refinement , such that
If is the circle group, there is not just the Lie groupoid but also the Lie 2-groupoid with a single object, a single morphism and worth of 2-morphisms. This is the beginning of a pattern: for each there is a smooth n-groupoid which is a smooth refinement of the topological Eilenberg-MacLane space
is the n-groupoid whose objects are smooth circle n-bundles -bundle gerbes on , whose morphisms are smooth gauge transformations, whose 2-morphisms are smooth gauge-of-gauge transformations, and so on.
Ordinary Lie groupoids and the Morita morphisms are to be regarded as living in a context called a (2,1)-topos . The more general context needed for the above statement is an (∞,1)-topos of smooth ∞-groupoids. We write Smooth∞Grpd to indicate this.
For all smooth -groupoids , , , etc. in there is a smooth path ∞-groupoid , , , etc.;
If the underlying -principal ∞-bundle of such a flat -connection is trivial,
For this is a closed smooth differential -form and the smooth -groupoid
is the coefficient object for de Rham cohomology on smooth ∞-groupoids in degree .
There is a canonical curvature characteristic form
on each .
A general connection on a principal ∞-bundle is an object that universally lifts the unrefined -Chern-Weil homomorphism from de Rham cohomology to circle -bundles with connection.
such that the full -Chern-Weil homomorphism is the operation that maps a -principal -bundle with connection to the composite .
is the -Chern-Weil homomorphism.
of the -Chern-Simons theory associated with the given smooth cocycle .
In this fashion the Chern-Weil homomorphism and its refinements and generalizations exists on general abstract grounds, and many of its general properties follow on these abstract grounds. In the next survey section we indicate how this general abstract construction translates into concrete Lie theoretic expressions.
At least in good cases the general abstract ∞-Chern-Simons functional indicated above arises from Lie integration of infinitesimal data. We indicate the relevant notions of higher Lie theory and how Chern-Simons elements on L-∞ algebras serve to present Lagrangians for the general abstract -Chern-Simons action functional.
To start with, notice that to every Lie algebra (here taken to be finite dimensional over ) is associated its Chevalley-Eilenberg algebra , a dg-algebra refinement of the Grassmann algebra . This may usefully be thought of as the function algebra on the infinitesimal neighbourhood of the neutral element in the Lie group integrating (this is discussed in detail here).
Traditionally this is taken as the definition of Lie algebra cohomology, but if one more thinks of deloopings of Lie algebras properly as being infinitesimal Lie groupoids , then this becomes a theorem (discussed here):
In fact captures the full information of a moduli stack: not only does encode the above cohomology groups and hence certain equivalence classes of Lie algebras, but it yields an equivalence of categories
As before, a class in the cochain cohomology of the Chevalley-Eilenberg algebra is a class in the L-∞ algebra cohomology of . In fact, using -algebras Lie algebra cohomology and L-∞ algebra cohomology becomes representable:
we write for the line Lie n-algebra: the -algebra defined by the fact that has a single generator in degree and trivial differential. Using this, a cocycle in L-∞ algebra cohomology is a morphism of L-∞ algebras
So there is an accurate infinitesimal analog of the general abstract cohomology of smooth ∞-groupoids, and this continues: there is a simple infinitesimal analog of groupal models for universal principal ∞-bundles: the Weil algebra
Using the Weil algebra, we obtain the following notions:
The curvature forms of such are the components
of the shifted generators. Precisely if these all vanish does factor through and we say it is flat .
An invariant polynomial is in transgression with an -algebra cocycle if there is an element with
The element we call the Chern-Simons element exhibiting the transgression.
For -valued form data , the composite
With all these -algebraic structures in place, the main statement is:
for a transgressive L-∞ algebra cocycle of degree and
integrates to a smooth characteristic class
(where is some lattice inside );
the differential refinement of this according to the above general abstract theory produces an -Chern-Weil homomorphism that sends -principal ∞-bundles with ∞-connection to circle n-bundles with connection whose connection -form is locally gauge equivalent to the Chern-Simons form (1).
It follows that the (exponentiated) action functional of the -Chern-Simons theory induced by is given by
for the induced Chern-Simons element .
Apart from producing general abstract statements about -Chern-Simons theory, it is of interest to scan the space of all those higher gauge theory Lagrangians that are, maybe secretly, instances of -Chern-Simons action functionals. Due to the additional generality of invariant polynomials and Chern-Simons element as we generalize from Lie algebras to L-∞ algebras and then further to L-∞ algebroids, this is a larger class than familiarity with ordinary Chern-Simons theory might suggest. We briefly list some examples and some of their properties. A detailed discussion is below in the section Examples.
ordinary Chern-Simons theory
The circle n-bundle with connection given by ordinary Chern-Simons theory is known as the Chern-Simons circle 3-bundle . The non-triviality of its underlying class is the obstruction to lifting the -principal bundle to a string structure. The non-triviality of its connection is the obstruction to having a differential string structure. In general it defines a twisted differential string structure.
Let be a discrete group, and
Due to discreteness, there cannot be any way to obtain this by Lie integration of infinitesimal data. Nevertheless, the general abstract story from above still applies and produces an action functional: that of Dijkgraaf-Witten theory.
magnetic dual Chern-Simons theory
A -valued field configuration is a pair
Then let be the next nontrivial invariant polynomial. The corresponding -Chern-Simons Lagrangian (in 7 dimension) is
The circle n-bundle with connection given by 7-dimensional Chern-Simons theory is the Chern-Simons circle 7-bundle . The non-triviality of its underlying class is the obstruction to lifting the string 2-group-principal 2-bundle to a fivebrane structure. The non-triviality of its 7-connection is the obstruction to having a differential fivebrane structure. In general it defines a twisted differential fivebrane structure.
BF-theory and topological Yang-Mills
(see below for the moment).
where is the Euler vector field.
For this is the Poisson sigma-model.
For this is the Courant sigma-model.
This concludes our introduction and survey. We now turn to the detailed discussion.
General abstract -Chern-Simons theory is defined in any cohesive (∞,1)-topos, as described in the section
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We discuss examples of -Chern-Simons functionals.
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Expositions, surveys and lecture notes on -Chern-Simons theory include the following
Chern-Simons terms on higher moduli stacks, Talk at Hausdorff Institute, Bonn (2011) (pdf)
The notion of Chern-Simons elements for -algebras and the associated -Chern-Simons Lagrangians is due to
In the general context of cohesive (∞,1)-toposes -Chern-Simons theory is discussed in section 4.3 of
The special case of the AKSZ sigma-model is discussed in
Discussion of symplectic Lie n-algebroids is in
On the structure of graded symplectic supermanifolds and Courant algebroids (arXiv)