∞-Lie theory (higher geometry)
Formal Lie groupoids
A Chern-Simons element on an L-∞ algebroid (named after Shiing-shen Chern and James Simons who considered this for semisimple Lie algebras) is an element of its Weil algebra that exhibits a transgression between an ∞-Lie algebroid cocycle and an invariant polynomial.
It is construct that arises in the presentation of the ∞-Chern-Weil homomorphism by an ∞-anafunctor of simplicial presheaves.
We discuss ∞-Lie algebras and ∞-Lie algebroids of finite type in terms of their Chevalley-Eilenberg algebras . For -Lie algebras these are objects in the category dgAlg of dg-algebras (over a given ground field). For -Lie algebroids these are dg-algebras equipped with a lift of the degree-0 algebra to an algebra over a given Fermat theory and such that the differential is a -derivation in this degree. (See ∞-Lie algebroid for details). We shall write in the following also for the category of dg-algebras with this extra structure and leave the Fermat theory implicit.
For an ∞-Lie algebra or more generally ∞-Lie algebroid, a ∞-Lie algebra cocycle (a closed element of the Chevalley-Eilenberg algebra) and an invariant polynomial, a Chern-Simons element exhibiting the transgression between the two is an element
where the restriction is along the canonical morphism .
A Chern-Simons element an witnessing the transgression of to is equvivalently a morphism
such that we have a commuting diagram in dgAlg
where the vertical morphisms are the canonical ones.
For a given transgressive cocycle and transgressing invariant polynomial the set of Chern-Simons elements witnessing the transgression is a torsor (based over the point and) over the additive group
of Chern-Simons elements for vanishing cocycle and vanishing invariant polynomial.
Canonical -Chern-Simons elements
Since the Weil algebra of an L-∞ algebra has trivial cohomolgy in positive degree, every invariant polynomial has a Chern-Simons element and there is a standard formula for it.
Let be an L-∞ algebra with -ary brackets and equipped with a quadratic invariant polynomial .
A Chern-Simons element for is given by the formula
where is any -valued form
There is a canonical contracting homotopy
and the above element is
To see this, let be a basis and the dual basis. Then the differential of the Chevalley-Eilenberg algebra can be written
is the corresponding -ary bracket.
for the components of the invariant polynomial in this basis.
Then the claim is that
where the coefficients are
Write for the free dg-algebra on the graded vector space . In terms of the above basis this is generated from . As discussed at Weil algebra, there is a dg-algebra isomorphism
given by sending and .
Let be the derivation which on generators is defined by
Notice that this is not the homotopy that exhibits the triviality of , rather that homotopy is , where is the word length operator for element in in terms of the generators .
Therefore the homotopy is the composite top morphism in the diagram
Unwinding this, we find
We consider the ordinary Chern-Simons element as an example of this formula: let be a semisimple Lie algebra and the Killing form invariant polynomial. Then the above computation gives
We discuss the general abstract structures of which Chern-Simons elements are presentations and how they are related to other structures.
The term Chern-Simons element alludes to the term Chern-Simons form and Chern-Simons theory. In the following we explain the relation.
As presentations for the -Chern-Weil homomorphism
We explain here briefly how Chern-Simons elements provide a presentation of a generalization of the Chern-Weil homomorphism – the ∞-Chern-Weil homomorphism in cohesive (∞,1)-topos theory – in the sense in which (∞,1)-toposes have presentations by a model structure on simplicial presheaves.
To warm up, we start with considering a traditional setup of Lie groupoid theory. Recall that for a Lie group, we may form its delooping Lie groupoid dnoted or . Then with any smooth manifold, we have that the groupoid of morphisms of Lie groupoids is equivalent to that of -principal bundles on :
Here we are thinking of Lie groupoids as differentiable stacks, hence as objects in the (2,1)-topos
of stacks/(2,1)-sheaves on the site SmoothMfd (equivalently on its small dense subsite CartSp of Cartesian spaces). (This is discussed in detail at principal bundle ).
There is a differential refinement of the Lie groupoid , to the smooth groupoid
where is the path groupoid of . This is the (2,1)-sheaf given by the (2,1)-sheafification of the assignment that sends a smooth manifold to the groupoid of Lie algebra-valued 1-forms on .
There is a corresponding natural equivalence
of morphisms into with the groupoid of -principal bundles with with connection on . (This is described in detail at connection on a bundle ).
In particular if is the circle group, a morphism is a circle bundle with connection. This
This allows already to consider a simple case of a characteristic class and its refinement to a differential characteristic class: Let be the unitary group. There is a canonical morphism of Lie groupoids given by the determinant. This – or rather its image in cohomology
is a smooth representative of the characteristic class called the first Chern class . Its differential refinement is the evident morphism
that sends a -valued differential form to the trace of its Lie algebra value. Postcomposition with this is the refined Chern-Weil homomorphism
with values in circle bundles with connection, hence in degree-2 ordinary differential cohomology.
It is this kind of construction on Lie groupoids that we now want to generalize to a notion of smooth ∞-groupoids, to see that Chern-Simons elements are a means to constructi morphisms akind to the differential first Chern-class .
A general abstract context for higher geometry equipped with differential cohomology is a cohesive (∞,1)-topos of ∞-groupoids equipped with cohesive structure , such as smooth cohesive structure.
An example for such is the ∞-stack-analog of the stack-(2,1)-topos over SmoothMfd: the ∞-stack (∞,1)-topos Smooth∞Grpd .
In that context we have for instance all the higher deloopings of : the circle Lie (n+1)-groups
This is such that the evident generalizations of the above classification statements hold: we have that morphisms form an n-groupoid
equivalent to that of circle n-bundles/-bundle gerbes on .
If here is again the delooping of a Lie group, this means that now also the higher characteristic classes are represented by morphisms
For instance for the spin group, the first fractional Pontryagin class has a smooth incarnation given by a morphism of the form
corresponding under the above equivalence to the ordinary Chern-Simons circle 3-bundle on .
Every cohesive (∞,1)-topos comes canonically and essentially uniquely equipped with all the intrinsic structure that we need for the discussion of a refinement of this to differential characteristic classes:
There is an endo-(∞,1)-adjunction
A morphism encodes the flat higher parallel transport of a flat circle n-bundle with connection, and we have that the n-groupoid of morphisms
is that of flat circle n-bundles with connection/ (n-1)-bundle gerbes with connection.
We observe that a trivial circle -bundle with connection is equivalently just a globally defined differential n-form. Therefore if we define the modified (∞,1)-adjunction
by forming the (∞,1)-pullback
which is the coefficient object for trivial principal -bundles equipped with flat -connection, one finds (discussed in detail here) that morphisms correspond to trivial circle bundle with connection, hence to cocycles in de Rham cohomology of ;
This now allows us to construct differential refinements:
one can show (detailed discussion is here) that there are canonical cocycles
in the degree -de Rham cohomology of : these are the universal curvature characteristic forms on .
Then for any smooth characteristic class, the corresponding (unrefined) differential characteristic class is simply the composite
The (unrefined) ∞-Chern-Weil homomorphism is postcomposition with this morphism:
This is finally where the Chern-Simons elements come in:
This presentation we describe in the next section.
(In fact a bit more is true: the serve to present the refinement of to a morphism with values in ordinary differential cohomology. This we come to further below.)
We explain now how Chern-Simons elements arise as a presentation of a differential characteristic class by a span of simplicial presheaves.
At the heart of the presentation of differenial characteristic classes by morphisms of simplicial presheaves is a differential refinement of the Lie integration of L-∞ algebras and ∞-Lie algebroid: for an ordinary Lie algebra, one finds that the 3-coskeleton of the simplicial presheaf that assigns flat vertical -Lie algebra valued 1-forms
is the delooping of the simply connected Lie group integrating
Similarly the Lie integration of the line Lie n-algebra
is the -fold delooping of :
Moreover, for a degree- cocycle in Lie algebra cohomology, simple postcomoposition gives its image under Lie integration
Under coskeletization on the left this carves out the periods of as a lattice in , which typically is the integers, so that this descends to degree -cocycle in Lie group cohomology with coefficients in
(See Lie group cohomology and smooth ∞-groupoid for discussion of the refined notion of Lie group cohomology arising here.)
The differential refinement of these construction is based on the following fact (discussed in detail here)
the object Smooth∞Grpd is equivalently presented by a quotient of the presheaf of Kan complexes given by
where on the right we have the set of horizontal morphisms in dgAlg that make a commuting diagram with the canonical vertical morphisms as indicated.
(We may think of a morphism of simplicial presheaves as a circle n-bundle/-bundle gerbe equipped with a pseudo-connection . )
Notice that the bottom morphism here encodes precisely a degree- differential form on ,
The morphism is presented on this by the ∞-anafunctor
by the map that sends such a form to its curvature . If the pseudo-connections that we are dealing with are genuine connections the curvature is a basic form down on and this means diagrammatically that it forms the pasting composite
and then picks out the bottom horizontal morphism.
Therefore our task of presenting amounts to computing the composition of ∞-anafunctors
To do se we need to complete componentwise to commuting diagrams. To this end we first complete the assignment of to a diagram
Here in the middle row an unrestricted -∞-Lie algebra valued differential form appears, which is the local -∞-connection. And in the lower row all its curvature characteristic forms appear, obtained by evaluating the curvature in the invariant polynomials on .
This is such that a choice of Chern-Simons element witnessing the transgression of an invariant polynomial to allows to refine to
Here now the middle row is the evaluationn of the connection form inside the Chern-Simons element. This is the corresponding Chern-Simons form
of the -connection evaluated in the given Chern-Simons element. Its curvature is the curvature characteristic form appearing in the bottom line of the diagram, which is obtained by evaluating the -valued curvature in the given invariant polynomial.
A more comprehensive account of this is at Chern-Weil homomorphism in Smooth∞Grpd.
Chern-Simons action functionals
By the above construction, every Chern-Simons element of degree on an ∞-Lie algebroid induces an action functional on the space of ∞-Lie algebroid valued forms on over a -dimensional smooth manifold
This generalizes the action functional of ordinary Chern-Simons theory to general Chern-Simons elements. In the examples below is a list of various quantum field theories that arise as generalized Chern-Simons theories this way.
For more details see infinity-Chern-Simons theory.
On semisimple Lie algebra– Standard Chern-Simons action functional
Let be a semisimple Lie algebra. For the following computations, choose a basis of and let denotes the corresponding degree-shifted basis of .
Notice that in terms of this the differential of the CE-algebra is
and that of the Weil algebra
Let be the Killing form invariant polynomial. This being invariant
is equivalent to the fact that the coefficients
are skew-symmetric in and , and therefore skew in all three indices.
A Chern-Simons element for the Killing form invariant polynomial is
In particular the Killing form is in transgression with the degree 3-cocycle
Under a Lie algebra-valued form
this Chern-Simons element is sent to
If is a matrix Lie algebra then the Killing form is the trace and this is equivalently
This is a familiar form of the standard Chern-Simons form in degree 3.
For a 3-dimensional smooth manifold the corresponding action functional
is the standard action functional of Chern-Simons theory.
Higher CS-forms on semisimple Lie algebras
For any higher order cocycle, is the corresponding higher order Chern-Simons form.
Action functional for Chern-Simons (super-)gravity
Higher Chern-Simons elements on the Poincare Lie algebra or the super Poincare Lie algebra yield action functionals for gravity and supergravity.
Fractional secondary Pontryagin classes
For instance for the 7-cocycle on a semisimple Lie algebra, is the corresponding Chern-Simons 7-form, corresponding to the second Pontryagin class.
Notice that this we may also think of as a 7-cocycle on the corresponding string Lie 2-algebra. As such it is the one that classifies the extension to the fivebrane Lie 6-algebra. The corresponding Chern-Simons 7-form appears as the local conneciton data in the Chern-Simons circle 7-bundle with connection that obstructions the lift from a differential string structure to a differential fivebrane structure.
On strict Lie 2-algebras – BF-theory action functional
Let be a strict Lie 2-algebra.
every invariant polynomial on is a Chern-Simons element on , restricting to the trivial ∞-Lie algebra cocycle;
for a semisimple Lie algebra and the Killing form, the corresponding Chern-Simons action functional on ∞-Lie algebra valued forms
is the sum of the action functionals of topological Yang-Mills theory with BF-theory with cosmological constant (in the sense of gravity as a BF-theory):
where is the ordinary curvature 2-form of .
This is from (SSSI).
For a basis of and a basis of we have
Therefore with we have
The right hand is a polynomial in the shifted generators of , and hence an invariant polynomial on . Therefore is a Chern-Simons element for it.
Now for an ∞-Lie algebra-valued form, we have that the 2-form curvature is
On symplectic -Lie algebroids – The AKSZ Lagrangian
A symplectic Lie n-algebroid is a Lie n-algebroid equipped with a binary non-degenerate invariant polynomial of degree .
The corresponding Chern-Simons elements of are the integrands for the action functionals of various TQFT sigma-models.
With the graded Poisson bracket induced by we have (see Roytenberg) that there exists a ∞-Lie algebra cocycle such that
So in particular being a cocycle means that
The cocycle is in transgression with the invariant polynomial via the Chern-Simons element
where is the Euler vector field (Roytenberg).
Here is the shift derivation in the Weil algebra, in that .
To safe typing signs, we write as if all functions were even graded. By standard reasoning the computation holds true then also for arbitrary grading.
on unshifted generators we have
we have graded commutators
- (the degree operator)
(as one checks on generators).
where in the first line we used that by definition of invariant polynomial . Similarly, using that by definition we have
So in total we have
Higher phase space —Hamiltonian and Lagrangian mechanics
The symplectic Lie -algebroid may be thought of as an n-symplectic manifold that models the phase space of a physical system.
This means for a symplectic Lie -algebroid, the general diagram (1) exhibiting the transgression between cocycles and invariant polynomials via Chern-Simons elements may be labeled in terms of Hamiltonian mechanics, Lagrangian mechanics and symplectic geometry as follows
See Hamiltonian, Lagrangian, symplectic structure.
On a symplectic manifold – The topological particle
For a smooth manifold we may regard its cotangent bundle as a Lie 0-algebroid and the canonical 2-form as a binary invariant polynomial in degree 2.
The Chern-Simons element is the canonical 1-form which in local coordinates is .
The corresponding action functional on the line
is the familiar term for the action functional of the particle (missing the kinetic term, which makes it “topological”).
On a Poisson Lie algebroid – The Poisson -model
Let by a Poisson Lie algebroid. This comes with the canonical invariant polynomial .
The corresponding ∞-Lie algebroid cocycle is
and a Chern-Simons element for this is
For a 2-dimensional smooth manifold the corresponding action functional on ∞-Lie algebroid-valued forms is the actional functional of the Poisson sigma-model
We compute in a local coordinte patch:
On higher extensions of the super Poincare Lie algebra – supergravity
See D'Auria-Fre formulation of supergravity for the moment.
A classical reference on transgression of differential forms from the fiber to the base of a fiber bundle is section 9 of.
- Armand Borel, Topology of Lie groups and characteristic classes Bull. Amer. Math. Soc. Volume 61, Number 5 (1955), 397-432. (EUCLID)
The general definition of Chern-Simons element on -Lie algebras and -Lie algebroids is definition 21 in
The examples of the BF-theory invariant polynomials and Chern-Simons elements are in prop. 18 and def. 26 and the BF-action functional itself is extracted below proposition 28.
Dedicated discussion of -Chern-Simons theory is at
A comprehensive account is in
A survey of higher Chern-Simons elements and their action functionals as applied to gravity and supergravity is in
- Jorge Zanelli, Lecture notes on Chern-Simons (super-)gravities arXiv:0502193
Symplectic Lie -algebroids are discussed in
Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis (arXiv)
On the structure of graded symplectic supermanifolds and Courant algebroids (arXiv)
A talk about the historical origins of the standard Chern-Simons forms see
- Jim Simons, Origin of Chern-Simons talk at Simons Center for Geometry and Physics (2011) (video)