Course on stacks and action functionals

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

  1. some differential higher geometry: the differential geometry of Lie groupoids/differentiable stacks and more generally of smooth ∞-groupoids/smooth ∞-stacks;

  2. 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 ((n1)(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.


Motivation – Natural Action functionals for QFTs

In recent years, the notion of (extended) topological quantum field theory has been fully formalized. An extended nn-dimensional TQFT is a functor Z:Bord n𝒞Z : 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 nBord_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 ZZ. 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 (n1)(n-1)-dimensional manifold Σ\Sigma.

The action functional

S:Conf S : Conf \to \mathbb{R}

will be a functional on a configuration space of fields

Conf=Fields(Σ×[0,1]) Conf = Fields(\Sigma \times [0,1])

The covariant phase space of the theory is the critical locus relative boundary,

PConf P \to Conf

If SS 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 ΣPE_\Sigma \to P, the prequantum line bundle. Find a polarization, a splitting into coordinates and momenta.

Finally, form the space Z ΣZ_\Sigma of polarized sections of this line bundle.

This is what the quantum field theory assigns to Σ\Sigma.

Z:ΣΓ P(E Σ) polarized. Z : \Sigma \mapsto \Gamma_P(E_\Sigma)_{polarized} \,.

Chern-Simons type functionals

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 𝔤\mathfrak{g} equipped with a binary non-degenerate invariant polynomial ,\langle -,- \rangle and a choice of “level” kk \in \mathbb{Z}.

Its configuration space over a 3-dimensional Σ×[0,1]\Sigma \times [0,1] is the groupoid of Lie-algebra valued forms

Conf=Ω 1(Σ×[0,1],𝔤). Conf = \Omega^1(\Sigma \times [0,1] , \mathfrak{g}) \,.

The Lagrangian is the Chern-Simons form

ACS(A)Ω 3(Σ×[0,1]) A \mapsto CS(A) \in \Omega^3(\Sigma \times [0,1])

and the action functional is the integral over that

Ak Σ12AdA+16A[AA]. A \mapsto k \int_\Sigma \frac{1}{2}\langle A \wedge d A\rangle + \frac{1}{6} \langle A \wedge [A \wedge A]\rangle \,.

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 kk-ary brackets

[,,] k: k𝔤𝔤 [-,\cdots, -]_k _: \wedge^k \mathfrak{g} \to \mathfrak{g}

and the action functional is of the form

S:Ψ k=1 1(k+1)!Ψ,[Ψ,,Ψ] k =12Ψ,DΨ+16Ψ,[Ψ,Ψ] 2+. \begin{aligned} S : \Psi & \mapsto \sum_{k = 1}^\infty \frac{1}{(k+1)!} \langle \Psi, [\Psi, \cdots, \Psi]_k\rangle \\ & = \frac{1}{2} \langle \Psi , D \Psi \rangle + \frac{1}{6} \langle \Psi, [\Psi, \Psi]_2\rangle + \cdots \end{aligned} \,.

where D:=[] 1D := [-]_1.

Wess-Zumino-Witten type functionals


1) Smooth groupoids and the 1d U(n)U(n)-Chern-Simons functional

We discuss some basics of Lie groupoids and smooth stacks, and then derive the action functional of U(n)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.

Lie groupoids

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 𝒢\mathcal{G} is

  • a set 𝒢 0\mathcal{G}_0 (think: of gauge field configurations);

  • a set 𝒢 1\mathcal{G}_1 (think: of gauge transformations between pairs of gauge field configurations)

  • equipped with

    • functions s,t:𝒢 1𝒢 0s, t : \mathcal{G}_1 \to \mathcal{G}_0 that project out the source and the target of a gauge transformation;

    • a map id:𝒢 0𝒢 1id : \mathcal{G}_0 \to \mathcal{G}_1 that assigns to every configuration the trivial (identity) gauge transformation;

  • 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… )

  • and such that under this composition every transformation has an inverse.

It is useful to depict (Lie) groupoids as

𝒢={ b f g a gf ca,b,c𝒢 0,f,g𝒢 1}. \mathcal{G} = \left\{ \array{ && b \\ & {}^{\mathllap{f}} \nearrow && \searrow^{\mathrlap{g}} \\ a &&\stackrel{g \circ f }{\to}&& c } \;\; | \;\; a,b,c \in \mathcal{G}_0 ,\; f,g \in \mathcal{G}_1 \right\} \,.

Here are some basic examples:

for GG a (Lie) group, its delooping groupoid is

BG={ * g 1 g 2 * g 2g 1 *g 1,g 2G}. \mathbf{B}G = \left\{ \array{ && * \\ & {}^{\mathllap{g_1}} \nearrow && \searrow^{\mathrlap{g_2}} \\ * &&\stackrel{g_2 \cdot g_1}{\to}&& * } \;\; | \;\; g_1, g_2 \in G \right\} \,.

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 𝒢 0\mathcal{G}_0.

for XX a smooth manifold it is naturally a Lie groupoid with only identity morphisms;

X={ y x zx,y,zX}. X = \left\{ \array{ && y \\ & && \\ x &&&& z } \;\; | \;\; x,y,z \in X \right\} \,.

For ρ:G×XX\rho : G \times X \to X an action of GG on XX, the orbifold X//GX /\!/G is

X//G={ y=ρ(g 1)(x) g 1 g 2 x g 2g 1 z=ρ(g 2)(y)x,y,zX,g 1,g 2G}. X/\!/G = \left\{ \array{ && y = \rho(g_1)(x) \\ & {}^{\mathllap{g_1}} \nearrow && \searrow^{\mathrlap{g_2}} \\ x &&\stackrel{g_2 \cdot g_1}{\to}&& z = \rho(g_2)(y) } \;\; | \;\; x,y,z \in X \;, g_1, g_2 \in G \right\} \,.

For {U iX}\{U_i \to X\} an open cover of a manifold, the Cech groupoid C({U i})C(\{U_i\})

C({U i})={ (x,j) (x,i) (x,k)xU iU jU k}. C(\{U_i\}) = \left\{ \array{ && (x,j) \\ & \nearrow && \searrow \\ (x,i) &&\to&& (x,k) } \;\; | \;\; x \in U_i \cap U_j \cap U_k \right\} \,.

Maps between Lie groupoids

Let’s then consider smooth functors C({U i})BGC(\{U_i\}) \to \mathbf{B}G. For {U iX}\{U_i \to X\} a good open cover, one finds that these are equivalent to Cech cocycle with values in GG, classifying GG-principal bundles on XX.

We really want to think of such functors as going out of XX itself. The canonical functor C({U i})XC(\{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.

Smooth groupoids / smooth stacks

For every Lie groupoid 𝒢\mathcal{G} and every smooth manifold UU, we may form the ordinary groupoid of UU-parameterized smooth families of objects and morphisms in 𝒢\mathcal{G}:

𝒢(U):=(Hom SmthMfd(U,𝒢 1)Hom SmthMfd(U,𝒢 0)). \mathcal{G}(U) := \left( Hom_{SmthMfd}(U, \mathcal{G}_1) \stackrel{\to}{\to} Hom_{SmthMfd}(U , \mathcal{G}_0) \right) \,.

This assignment is evidently contravariantly functorial. So 𝒢\mathcal{G} actually defined a presheaf on SmoothMfd with values in Grpd

𝒢:SmoothMfd opGrpd. \mathcal{G} : \mathrm{SmoothMfd}^{op} \to \mathrm{Grpd} \,.

Moreover, by the Yoneda lemma, all the information in 𝒢\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

LieGrpdPSh(SmoothMfd op,Grpd). LieGrpd \hookrightarrow PSh(SmoothMfd^{op}, Grpd) \,.

But this presheaf category is not quite what we need. Remember from the above example of a Cech nerve cover C({U i})XC(\{U_i\}) \to X of a smooth manifold XX, that there are morphisms of Lie groupoids that look like they ought to be invertible, but are not invertible in LieGrpdLieGrpd, nor, therefore, in Func(SmoothMfd op,Grpd)Func(SmoothMfd^{op}, Grpd).

The (2,1)-category of smooth groupoids or stacks on the site of smooth manifolds is.

H=L WFunct(SmthMfd op,Grpd) \mathbf{H} = L_W \mathrm{Funct}(\mathrm{SmthMfd}^{op}, Grpd)

This is the (2,1)-category of presheaves with values in Grpd where we formally invert (“localization”) the morphisms in

W:={stalkwiseequivalences} W := \{stalkwise equivalences\}

hence those natural transformations η:XY\eta : X \to Y of functors such that for each manifold UU, and for each point uUu \in U, the stalk of the transformation at xx is an equivalence of groupoids u *η:u *Xu *Yu^* \eta : u^* X \stackrel{\simeq}{\to} u^* Y.

In H\mathbf{H} a morphism is presented by a zig-zag of morphisms of presheaves of groupoids, where all the left-pointing morphisms are in WW.

In particular the Cech cocycle functors above are now functors out of XX in H\mathbf{H}

XC({U i})gBG. X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \stackrel{g}{\to} \mathbf{B}G \,.

Smooth moduli stacks of gauge fields

We now consider smooth stacks that do not arise from Lie groupoids, namely differential refinements of Lie groupoids.

BG conn:U{AgA gAΩ 1(U,𝔤)gC (U,G)}. \mathbf{B}G_{conn} : U \mapsto \left\{ A \stackrel{g}{\to} A^g | A \in \Omega^1(U, \mathfrak{g}) \; g \in C^\infty(U, G) \right\} \,.

This object has the property that maps (in H\mathbf{H}) into it classify (Yang-Mills) gauge fields.

For XX a smooth manifold, we have an equivalence of groupoids

GBund conn(X)H(X,BG conn). G Bund_conn(X) \simeq \mathbf{H}(X, \mathbf{B}G_{conn}) \,.

Therefore one may say that BG conn\mathbf{B}G_{conn} is the “classifying space” or moduli stack for GG-gauge fields.

The first Chern-class and its smooth and differential refinement

The determinant function is a (Lie) group homomorphism

det:U(n)U(1) det : U(n) \to U(1)

from the unitary group in dimension nn to the circle group.

This induces the evident map between the corresponding delooping Lie groupoids

c 1:=Bdet:BU(n)BU(1). \mathbf{c}_1 := \mathbf{B}det \; : \; \mathbf{B}U(n) \to \mathbf{B}U(1) \,.

Composition with this sends U(n)U(n)-principal bundles to circle bundles. Under the classification of circle bundles by second integral cohomology this refines the first Chern class.

H(X,BU(n)) H(X,c 1) H(X,BU(1)) π 0 π 0 VectBund rnkn(X)/ [c 1] H 2(X,). \array{ \mathbf{H}(X,\mathbf{B} U(n)) &\stackrel{\mathbf{H}(X, \mathbf{c}_1)}{\to}& \mathbf{H}(X, \mathbf{B}U(1)) \\ \downarrow^{\mathrlap{\pi_0}} && \downarrow^{\mathrlap{\pi_0}} \\ VectBund_{rnk n}(X)/_\sim &\stackrel{[c_1]}{\to}& H^2(X, \mathbb{Z}) } \,.

There is an evident refinement of the smooth first Chern class c 1\mathbf{c}_1 further to a differential first Chern class c^ 1\hat {\mathbf{c}}_1

c^ 1:BU(n) connBU(1) conn \hat \mathbf{c}_1 : \mathbf{B}U(n)_{conn} \to \mathbf{B}U(1)_{conn}

given by the morphism of groupoid-valued presheaves which over a test space UU \in SmoothMfd sends

c^ 1:Atr(A) \hat {\mathbf{c}}_1 : A \mapsto tr(A)

and hence

c^ 1:(AgA g)tr(A)detgtr(A g)=tr(A) det(g). \hat {\mathbf{c}}_1 : (A \stackrel{g}{\to} A^g) \mapsto tr(A) \stackrel{det g}{\to} tr(A^g) = tr(A)^{det(g)} \,.

This is a local (“stacky”) refinement of the differential first Chern class as given by Chern-Weil theory

VectBund conn(X) c^ 1 LineBund conn(X) VectBund conn(X)/ [c^ 1] H diff 2(X). \array{ VectBund_{conn}(X) &\stackrel{\hat \mathbf{c}_1}{\to}& LineBund_{conn}(X) \\ \downarrow && \downarrow \\ VectBund_{conn}(X)/_\sim &\stackrel{[\hat c_1]}{\to}& H^2_{diff}(X) } \,.

1-dimensional U(n)U(n)-Chern-Simons theory

Combining now the differential refinement of the first Chern class with the holonomy operation we obtain an action functional for U(n)U(n)-gauge fields over 1-dimensional compact smooth manifolds Σ\Sigma

exp(iS c 1):VectBund conn(Σ)c^ 1LineBund conn(Σ)exp(i Σ())U(1). \exp(i S_{\mathbf{c}_1}) : VectBund_conn(\Sigma) \stackrel{\hat \mathbf{c}_1}{\to} LineBund_{conn}(\Sigma) \stackrel{\exp (i \int_\Sigma(-)) }{\to} U(1) \,.

This is the action function of U(n)U(n)-1-dimensional Chern-Simons theory

Aexp(i Σtr(A)) A \mapsto \exp(i \int_\Sigma tr(A))

defined here on the groupoid of gauge fields. After Lie differentiation this is the BRST-invariant action functional on the corresponding BRST complex.

2) Smooth \infty-groupoids and higher Chern-Simons functionals

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.

Higher groupoids

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 iXU_i \hookrightarrow X) not given by a connection 1-form, but by a connection 2-form.

B iΩ 2(U i). B_i \in \Omega^2(U_i) \,.

Instead, a gauge transformation between two such A i:B iB˜ iA_i : B_i \to \tilde B_i now given by a 1-form A iΩ 1(U i)A_i \in \Omega^1(U_i), subject to the relation

B˜ i=B i+d dRA i. \tilde B_i = B_i + d_{dR} A_i \,.

But now there is more: there is now also a notion of gauge transformations between gauge transformations which are labeled, locally, by U(1)U(1)-valued functions λ iC (U i,U(1))\lambda_i \in C^\infty(U_i, U(1))

B˜ i A i λ id B i A˜ i B˜ i \array{ && \tilde B_i \\ & {}^{\mathllap{A_i}}\nearrow &\Downarrow^{\lambda}& \searrow^{\mathrlap{id}} \\ B_i &&\stackrel{\tilde A_i}{\to}&& \tilde B_i }

subject to the relation

A˜ i=A i+dlogλ. \tilde A_i = A_i + d log \lambda \,.

And it continues this way. Next the supergravity C-field is given locally by a 3-form C iΩ 3(U i)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.


Smooth higher groupoids


smooth ∞-groupoid

H:=L WFunct(SmoothMfd op,sSet) \mathbf{H} := L_W Funct(SmoothMfd^{op}, sSet)


W={stalkwiseweakhomotopyequivalence} W = \{stalkwise weak homotopy equivalence\}


circle n-group

B nU(1)=\mathbf{B}^n U(1) =

moduli nn-stack for circle n-bundles with connection

B nU(1) conn=\mathbf{B}^n U(1)_{conn} =

The refined second Chern class and 3d Chern-Simons theory

c^ 2:BSU conn(n)B 3U(1) conn \hat \mathbf{c}_2 : \mathbf{B} SU_{conn}(n) \to \mathbf{B}^3 U(1)_{conn}
exp(iS c 2):VectBund conn(Σ 3)c 2^LineBund conn(Σ 3)exp(i Σ())U(1) \exp(i S_{\mathbf{c}_2}) : VectBund_{conn}(\Sigma_3) \stackrel{\hat {\mathbf{c}_2}}{\to} LineBund_{conn}(\Sigma_3) \stackrel{\exp(i \int_\Sigma(-))}{\to} U(1)

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.

A list of higher Chern-Simons theories



See differential cohomology in a cohesive topos.

Related survey talks are at

Revised on December 20, 2011 17:12:43 by Urs Schreiber (