Notes for talks given at the 8th meeting of the Polish Category theory seminar and at the 4th Odense winter school on geometry and theoretical physics 2011.
under construction
Abstract These are lecture notes that are meant to expose and explain in parallel
some differential higher geometry: the differential geometry of Lie groupoids/differentiable stacks and more generally of smooth ∞-groupoids/smooth ∞-stacks;
action functionals of gauge theory and higher gauge theory of ∞-Chern-Simons theory-type and their associated ∞-Wess-Zumino-Witten theory defined on the full moduli ∞-stack of field configurations (to which the BRST complex is the infinitesimal approximation).
These two poitns are naturally related by a higher analog of Chern-Weil theory (“∞-Chern-Weil theory”), which naturally induces these action functionals from universal characteristic classes of smooth ∞-groups. Accordingly they enjoy various nice properties, among them is their “extended” nature: instead of inducing just a prequantum line bundle on covariant phase space, they induce prequantum circle n-bundles ($(n-1)$-bundle gerbes) on the moduli ∞-stack of field configurations over the point.
The exposition follows part of the material in section 1 – Introduction of differential cohomology in a cohesive topos.
In recent years, the notion of (extended) topological quantum field theory has been fully formalized. An extended $n$-dimensional TQFT is a functor $Z : Bord_n \to \mathcal{C}$ on an (∞,n)-category of cobordisms.
Moreover, this concept has been found to naturally originate in higher category theory – by a universal construction: $Bord_n$ is simply the free symmetric monoidal (∞,n)-category on a point. This is the statement of the celebrated cobordism hypothesis-theorem.
However, the quantum field theories found in nature are not random such $Z$. Most of them arise from quantization of action functionals, and these action functionals themselves are not random. There are certain recurring patterns in the action functionals that one actually cares about in physics.
We present a unified picture of higher geometry that turns out to generate such natural action functionals, again from a universal construction, now in higher topos theory – “cohesion”.
We start below by briefly surveying some of these natural action functionals in physics.
Then in 1) we pick the simplest non-trivial one among them, the action for 1-dimensional Chern-Simons theory, and use it as a motivating example for the development of some classical theory of Lie groupoids and differentiable stacks, but formulated in a fashion that paves the way to the higher theory to follow.
This we develop in 2). We discuss a notion of smooth ∞-groupoids and indicate a series of constructions on them that combine to produce natural “infinity-Chern-Simons theory”-action functionals: principal ∞-bundles, geometric realization, differential cohomology.
Finally we look into the construction of examples in 3). To that end, we introduce L-∞ algebroids and their Lie integration. Applying this to the corresponding notion of L-∞ algebra cohomology yields large classes of “universally connnected examples”. Truncating these finally yields classes of examples of natural action functionals on moduli stacks of gauge field configurations.
For completeness, we briefly recall aspects of how one can hope to obtain a quantum field theory from a action functional by geometric quantization.
Consider an $(n-1)$-dimensional manifold $\Sigma$.
will be a functional on a configuration space of fields
The covariant phase space of the theory is the critical locus relative boundary,
If $S$ is a local action functional then this is naturally a presymplectic smooth space – equipped with a closed differential 2-form.
Quotient out symmetries to get a symplectic manifold.
(This quotient will in general be very badly behaved. What one really does is form a derived critical locus and quotient out symmetries only up to homotopy. This is accomplished by BV-BRST formalism.)
Lift symplectic form to a line bundle with connection $E_\Sigma \to P$, the prequantum line bundle. Find a polarization, a splitting into coordinates and momenta.
Finally, form the space $Z_\Sigma$ of polarized sections of this line bundle.
This is what the quantum field theory assigns to $\Sigma$.
An archetypical example of a TQFT arising from an action functional is 3-dimensional Chern-Simons theory.
This is determined by a choice of Lie algebra $\mathfrak{g}$ equipped with a binary non-degenerate invariant polynomial $\langle -,- \rangle$ and a choice of “level” $k \in \mathbb{Z}$.
Its configuration space over a 3-dimensional $\Sigma \times [0,1]$ is the groupoid of Lie-algebra valued forms
The Lagrangian is the Chern-Simons form
and the action functional is the integral over that
It turns out that many action functionals for quantum field theories are of a “similar form”, in a way that we will make precise. Notice that while many actions functionals appeating in physics are not topological – such as that of electromagnetism and Yang-Mills theory, these become topological once coupled to gravity.
An example of this is closed string field theory.
Here the field configuration form a higher generalization of a Lie algebra called an L-∞ algebra with $k$-ary brackets
and the action functional is of the form
where $D := [-]_1$.
(…)
We discuss some basics of Lie groupoids and smooth stacks, and then derive the action functional of $U(n)$-1-dimensional Chern-Simons theory in this context. A similar but more detailed exposition is in the first two sections at infinity-Chern-Weil theory introduction. Various aspects mentioned in the talks are to be found there and not stated here.
The notion of groupoid was – not under this name – invented in physics when gauge theory was understood in the middle of the 20th century. There one deals with objects – gauge field configurations – that may be equivalent to other such objects – by gauge transformations – but where it is crucial not to identitfy them, equivalent as they may be. Instead, it is crucial to keep track of how exactly they are equivalent, by remembering which gauge transformations take them into each other. For instance for a fixed field configuration it is of interest to remember the group of gauge transformations that take it to itself, which mathematically is the group of automorphisms of the object or first homotopy group of the corresponding groupoid.
So, a groupoid $\mathcal{G}$ is
a set $\mathcal{G}_0$ (think: of gauge field configurations);
a set $\mathcal{G}_1$ (think: of gauge transformations between pairs of gauge field configurations)
equipped with
a composition operation (associative and unital in the evident sense) that says how to compose two consecutive gauge transformations;
(so far this defines a category, this becomes a groupoid by adding the further condition… )
It is useful to depict (Lie) groupoids as
Here are some basic examples:
for $G$ a (Lie) group, its delooping groupoid is
This example is the reason for the name “groupoid”. A general groupoid is like this example of (a delooped) group, but with possibly more than just a single object in $\mathcal{G}_0$.
for $X$ a smooth manifold it is naturally a Lie groupoid with only identity morphisms;
For $\rho : G \times X \to X$ an action of $G$ on $X$, the orbifold $X /\!/G$ is
For $\{U_i \to X\}$ an open cover of a manifold, the Cech groupoid $C(\{U_i\})$
Let’s then consider smooth functors $C(\{U_i\}) \to \mathbf{B}G$. For $\{U_i \to X\}$ a good open cover, one finds that these are equivalent to Cech cocycle with values in $G$, classifying $G$-principal bundles on $X$.
We really want to think of such functors as going out of $X$ itself. The canonical functor $C(\{U_i\}) \to X$ is an equivalence of bare groupoids, but has no inverse as a smooth functor of Lie groupoids. Therefore, in order to proceed, we need a somewhat more general perspective on Lie groupoids.
For every Lie groupoid $\mathcal{G}$ and every smooth manifold $U$, we may form the ordinary groupoid of $U$-parameterized smooth families of objects and morphisms in $\mathcal{G}$:
This assignment is evidently contravariantly functorial. So $\mathcal{G}$ actually defined a presheaf on SmoothMfd with values in Grpd
Moreover, by the Yoneda lemma, all the information in $\mathcal{G}$ (certainly the underlying bare groupoid, but also the smooth structure on it) is still available in this induced presheaf. We have an embedding
But this presheaf category is not quite what we need. Remember from the above example of a Cech nerve cover $C(\{U_i\}) \to X$ of a smooth manifold $X$, that there are morphisms of Lie groupoids that look like they ought to be invertible, but are not invertible in $LieGrpd$, nor, therefore, in $Func(SmoothMfd^{op}, Grpd)$.
The (2,1)-category of smooth groupoids or stacks on the site of smooth manifolds is.
This is the (2,1)-category of presheaves with values in Grpd where we formally invert (“localization”) the morphisms in
hence those natural transformations $\eta : X \to Y$ of functors such that for each manifold $U$, and for each point $u \in U$, the stalk of the transformation at $x$ is an equivalence of groupoids $u^* \eta : u^* X \stackrel{\simeq}{\to} u^* Y$.
In $\mathbf{H}$ a morphism is presented by a zig-zag of morphisms of presheaves of groupoids, where all the left-pointing morphisms are in $W$.
In particular the Cech cocycle functors above are now functors out of $X$ in $\mathbf{H}$
We now consider smooth stacks that do not arise from Lie groupoids, namely differential refinements of Lie groupoids.
This object has the property that maps (in $\mathbf{H}$) into it classify (Yang-Mills) gauge fields.
For $X$ a smooth manifold, we have an equivalence of groupoids
Therefore one may say that $\mathbf{B}G_{conn}$ is the “classifying space” or moduli stack for $G$-gauge fields.
The determinant function is a (Lie) group homomorphism
from the unitary group in dimension $n$ to the circle group.
This induces the evident map between the corresponding delooping Lie groupoids
Composition with this sends $U(n)$-principal bundles to circle bundles. Under the classification of circle bundles by second integral cohomology this refines the first Chern class.
There is an evident refinement of the smooth first Chern class $\mathbf{c}_1$ further to a differential first Chern class $\hat {\mathbf{c}}_1$
given by the morphism of groupoid-valued presheaves which over a test space $U \in$ SmoothMfd sends
and hence
This is a local (“stacky”) refinement of the differential first Chern class as given by Chern-Weil theory
Combining now the differential refinement of the first Chern class with the holonomy operation we obtain an action functional for $U(n)$-gauge fields over 1-dimensional compact smooth manifolds $\Sigma$
This is the action function of $U(n)$-1-dimensional Chern-Simons theory
defined here on the groupoid of gauge fields. After Lie differentiation this is the BRST-invariant action functional on the corresponding BRST complex.
We now indicate the generalization of the above to a discussion of higher smooth groupoids/smooth ∞-stacks and the canonically induced higher Chern-Simons functionals such as, of course, ordinary 3-dimensional Chern-Simons theory. More exposition of this material is at infinity-Chern-Weil theory introduction, some of which was referred to in the actuals talks.
A little while after physicists discovered the notion of groupoids in gauge theory, they next discovered the notion of higher groupoids in higher gauge theory.
For instance the B-field in string theory is locally (on some open $U_i \hookrightarrow X$) not given by a connection 1-form, but by a connection 2-form.
Instead, a gauge transformation between two such $A_i : B_i \to \tilde B_i$ now given by a 1-form $A_i \in \Omega^1(U_i)$, subject to the relation
But now there is more: there is now also a notion of gauge transformations between gauge transformations which are labeled, locally, by $U(1)$-valued functions $\lambda_i \in C^\infty(U_i, U(1))$
subject to the relation
And it continues this way. Next the supergravity C-field is given locally by a 3-form $C_i \in \Omega^3(U_i)$, and there are third-order gauge transformations
(…)
As the diagrams suggest, where Yang-Mills gauge fields gave rise to the notion of groupoids, the B-field should give rise to a higher notion, called 2-groupoids, the supergravity C-field to 3-groupoids, and so on.
In general one speaks of ∞-groupoids.
(…)
The notion of equivalence for ∞-groupoids is the standard notion of weak homotopy equivalence.
(…)
(…)
where
Examples
$\mathbf{B}^n U(1) =$
moduli $n$-stack for circle n-bundles with connection
$\mathbf{B}^n U(1)_{conn} =$
As an application: due to the local smooth refinement we have access to the homotopy fibers of the characteristic class. These gives twisted differential string structures that control the Green-Schwarz mechanism.
7d String-Chern-Simons theory
BF-theory coupled to topological Yang-Mills theory
(…)
See differential cohomology in a cohesive topos.
Related survey talks are at
Chern-Simons terms on higher moduli stacks, Talk at Hausdorff Institute, Bonn (2011) (pdf)
$\infty$-Chern-Simons functionals, Talk at Higher Structures 2011 (pdf)