this entry is one chapter of geometry of physics
previous chapters: groups, principal connections
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We had seen that in higher differential geometry, then given any closed differential (p+2)-form $\omega \in \Omega^{p+2}_{cl}(X)$, it is natural to ask for a prequantization of it, namely for a circle (p+1)-bundle with connection $\nabla$ (equivalently: cocycle in degree-$(p+2)$-Deligne cohomology) on $X$ whose curvature is $F_\nabla = \omega$. In terms of moduli stacks this means asking for lifts of the form
in the homotopy theory of smooth homotopy types.
This immediately raises the question for natural classes of examples of such prequantizations.
One such class arises in infinity-Lie theory, where $\omega$ is a left invariant form on a smooth infinity-group given by a cocycle in L-∞ algebra cohomology. The prequantum n-bundles arising this way are the higher WZW terms discussed here.
In low degree of traditional Lie theory this appears as follows: On Lie groups $G$, those closed $(p+2)$-forms $\omega$ which are left invariant forms may be identified, via the general theory of Chevalley-Eilenberg algebras, with degree $(p+2)$-cocycles $\mu$ in the Lie algebra cohomology of the Lie algebra $\mathfrak{g}$ corresponding to $G$. We have $\omega = \mu(\theta)$where $\theta$ is the Maurer-Cartan form on $G$. These cocycles $\mu$ in turn may arise, via the van Est map, as the Lie differentiation of a degree-$(p+2)$-cocycle $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^{p+2}U(1)$ in the Lie group cohomology of $G$ itself, with coefficients in the circle group $U(1)$.
This happens to be the case notably for $G$ a simply connected compact semisimple Lie group such as SU or Spin, where $\mu = \langle -,[-,-]\rangle$, hence $\omega = \langle \theta , [\theta,\theta]\rangle$, is the Lie algebra 3-cocycle in transgression with the Killing form invariant polynomial $\langle -,-\rangle$. This is, up to normalization, a representative of the de Rham image of a generator $\mathbf{c}$ of $H^3(\mathbf{B}G, U(1)) \simeq H^4(B G, \mathbb{Z}) \simeq \mathbb{Z}$.
Generally, by the discussion at geometry of physics -- principal bundles, the cocycle $\mathbf{c}$ modulates an infinity-group extension which is a circle p-group-principal infinity-bundle
whose higher Dixmier-Douady class class $\int \Omega \mathbf{c} \in H^{p+2}(X,\mathbb{Z})$ is an integral lift of the real cohomology class encoded by $\omega$ under the de Rham isomorphism. In the example of Spin and $p = 1$ this extension is the string 2-group.
Such a Lie theoretic situation is concisely expressed by a diagram of smooth homotopy types of the form
where $\flat_{dR}\mathbf{B}^{p+2}\mathbb{R} \simeq \flat_{dR}\mathbf{B}^{p+2}U(1)$ is the de Rham coefficients (see also at geometry of physics -- de Rham coefficients) and where the homotopy filling the diagram is what exhibits $\omega$ as a de Rham representative of $\Omega \mathbf{c}$.
Now, by the very homotopy pullback-characterization of the Deligne complex $\mathbf{B}^{p+1}U(1)_{conn}$ (here), such a diagram is equivalently a prequantization of $\omega$:
For $\omega = \langle -,[-,-]\rangle$ as above, we have $p= 1$ and so $\nabla$ here is a circle 2-bundle with connection, often referred to as a bundle gerbe with connection. As such, this is also known as the WZW gerbe or similar.
This terminology arises as follows. In (Wess-Zumino 84) the sigma-model for a string propagating on the Lie group $G$ was considered, with only the standard kinetic action term. Then in (Witten 84) it was observed that for this action functional to give a conformal field theory after quantization, a certain higher gauge interaction term has to the added. The resulting sigma-model came to be known as the Wess-Zumino-Witten model or WZW model for short, and the term that Witten added became the WZW term. In terms of string theory it describes the propagation of the string on the group $G$ subject to a force of gravity given by the Killing form Riemannian metric and subject to a B-field higher gauge force whose field strength is $\omega$. In (Gawedzki 87) it was observed that when formulated properly and generally, this WZW term is the surface holonomy functional of a connection on a bundle gerbe $\nabla$ on $G$. This is equivalently the $\nabla$ that we just motivated above.
Later, such WZW terms, or at least their curvature forms $\omega$, were recognized all over the place in quantum field theory. For instance the Green-Schwarz sigma-models for super p-branes each have an action functional that is the sum of the standard kinetic action plus a WZW term of degree $p+2$.
In general WZW terms are “gauged” which means, as we will see, that they are not defined on the given smooth infinity-group $G$ itself, but on a bundle $\tilde G$ of differential moduli stacks over that group, such that a map $\Sigma \to \tilde G$ is a pair consisting of a map $\Sigma \to G$ and of a higher gauge field on $\Sigma$ (a “tensor multiplet” of fields).
Here we discuss the general construction and theory of such higher WZW terms.
We discuss how every cocycle $\mu \colon \mathfrak{g} \to b^{p+1} \mathbb{R}$ in L-∞ algebra cohomology has a Lie integration to a higher WZW term of the form
where $\tilde G$ is a differential extension of the smooth $p+1$-group that is the universal Lie integration of $\mathfrak{g}$, and where $\Gamma \hookrightarrow \mathbb{R}$ is the group of periods of $\mu$ on that group.
This construction is a differential refinement of Lie integration, so we start with recalling the relevant constructions and facts of Lie integration.
So let $\mathfrak{g}$ be an L-∞ algebra of finite type.
Write
$CE(\mathfrak{g})$ for the Chevalley-Eilenberg algebra of an L-∞ algebra $\mathfrak{g}$;
$\Delta^\bullet_{smth} \colon \Delta \to SmoothMfd$ for the cosimplicial smooth manifold with corners which is in degree $k$ the standard $k$-simpliex $\Delta^k \hookrightarrow \mathbb{R}^{k+1}$;
$\Omega^\bullet_{si}(\Delta_{smth}^k)$ for the de Rham complex of those differential forms on $\Delta_{smth}^k$ which have sitting instants, in that in an open neighbourhood of the boundary they are constant perpendicular to any face on their value at that face;
$\Omega^\bullet_{si}(U \times \Delta_{smth}^k)$ for $U \in SmoothMfd$ for the de Rham complex of differential forms on $U \times \Delta^k$ which when restricted to each point of $U$ have sitting instants on $\Delta^k$;
$\Omega^\bullet_{vert,si}(U \times \Delta_{smth}^k)$ for the subcomplex of forms that in addition are vertical differential forms with respect to the projection $U \times \Delta^k \to U$.
For $\mathfrak{g}$ an L-∞ algebra, write
$\exp(\mathfrak{g})_\bullet \in PreSmoothTypes = PSh(CartSp,sSet)$
for the simplicial presheaf
which is the universal Lie integration of $\mathfrak{g}$;
$\flat_{dR}\exp(\mathfrak{g})_\bullet \in PreSmoothTypes = PSh(CartSp,sSet)$
for the simplicial presheaf
of those differential forms on $U \times \Delta^\bullet$ with at least one leg along $U$;
$\Omega^1_{flat}(-,\mathfrak{g}) \coloneqq \flat_{dR}\exp(\mathfrak{g})_0 \longrightarrow \flat_{dR}\exp(\mathfrak{g})_\bullet$
for the canonical inclusion of the degree-0 piece, regarded as a simplicial constant simplicial presheaf.
From the discussion at Lie integration:
$\Omega^1_{flat}(-,b^{p+1}\mathbb{R}) = \mathbf{\Omega}^{p+2}_{cl}$;
for $\mathfrak{g}$ an ordinary Lie algebra, then for the 2-coskeleton
for $G$ the simply connected Lie group associated to $\mathfrak{g}$ by traditional Lie theory. If $\mathfrak{g}$ is furthermore a semisimple Lie algebra, then also
for $\mathfrak{g} = b^{p}\mathbb{R}$ the line Lie p+1-algebra, then
The constructions in def. 1 are clearly functorial: given a homomorphism of L-∞ algebras
it prolongs to a homomorphism of presheaves
and of simplicial presheaves
etc.
A degree-$(p+2)$-L-∞ cocycle $\mu$ on an L-∞ algebra $\mathfrak{g}$ is a homomorphism of the form
to the line Lie (p+2)-algebra $b^{p+1}\mathbb{R}$. The formal dual of this is the homomorphism of dg-algebras
which manifestly picks a $d_{CE(\mathfrak{g})}$-closed element in $CE^{p+2}(\mathfrak{g})$.
Precomposing this $\mu^\ast$ with a flat L-∞ algebra valued differential form
yields, by example 1, a plain closed $(p+2)$-form
Given an L-∞ cocycle
as in example 2, then its group of periods is the discrete additive subgroup $\Gamma \hookrightarrow \mathbb{R}$ of those real numbers which are integrations
of the value of $\mu$, as in example 2, on L-∞ algebra valued differential forms
over the boundary of the (p+3)-simplex (which are forms with sitting instants on the $(p+2)$-dimensional faces that glue together; without restriction of generality we may simply consider forms on the $(p+2)$-sphere $S^{p+2}$).
Given an L-∞ cocycle $\mu \colon \mathfrak{g} \to b^{p+1}\mathbb{R}$, as in example 2, then the universal Lie integration of $\mu$, via def. 1 and remark 1, descends to the $(p+2)$-coskeleton
up to quotienting the coefficients $\mathbb{R}$ by the group of periods $\Gamma$ of $\mu$, def. 2, to yield the bottom morphism in
This is fairly immediate from the definitions, detailed discussion is in (FSS 12).
Here and in the following we are freely using example 1 to identify $\exp(b^{p+1}\mathbb{R}) \simeq \mathbf{B}^{p+2}\mathbb{R}$. Establishing this is the only real work in prop. 1.
For $\mu \colon \mathfrak{g}\longrightarrow b^{p+1}\mathbb{R}$ an L-∞ cocycle, then there is the following canonical commuting diagram of simplicial presheaves
which is given
Moreover, this presents a refinement of the canonical Hodge filtration on $\mathbf{B}^{p}(\mathbb{R}/\Gamma)$, def. 6, along the cocycle $\mathbf{c}$ which Lie integrates $\mu$ via prop. 1.
Write
for the homotopy pullback of the left vertical morphism in prop. 2 along (the modulating morphism for) the Maurer-Cartan form $\theta_G$ of $G$, i.e. for the object sitting in a homotopy Cartesian square of the form
For the special case that $G$ is an ordinary Lie group, then $\flat_{dR}\mathbf{B}G \simeq \Omega^{1}_{flat}(-,\mathfrak{g})$, hence in this case the morphism being pulled back in def. 3 is an equivalence, and so in this case nothing new happens, we get $\tilde G \simeq G$.
On the other extreme, when $G = \mathbf{B}^{p}U(1)$ is the circle (p+1)-group, then def. 3 reduces to the homotopy pullback that characterizes the Deligne complex and hence yields
This shows that def. 3 is a certain non-abelian generalization of ordinary differential cohomology. We find further characterization of this below in corollary 2, see remark 6.
From example 3 one see the conceptial meaning of def. 3:
For $G$ a Lie group, then the de Rham coefficients are just globally defined differential forms, $\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})$ (by the discussion here), and in particular therefore the Maurer-Cartan form $\theta_G \colon G \to \flat_{dR}\mathbf{B}G$ is a globally defined differential form. This is no longer the case for general smooth ∞-groups $G$. In general, the Maurer-Cartan forms here is a cocycle in hypercohomology, given only locally by differential forms, that are glued nontrivially, in general, via gauge transformations and higher gauge transformations given by lower degree forms.
But the WZW terms that we are after are supposed to be prequantizations of globally defined Maurer-Cartan forms. The homotopy pullback in def. 3 is precisely the universal construction that enforces the existence of a globally defined Maurer-Cartan form for $G$, namely $\theta_{\tilde G} \colon \tilde G \to \Omega^1_{flat}(-,\mathfrak{g})$.
Given an L-∞ cocycle $\mu \colon \mathfrak{g}\longrightarrow b^{p+1}\mathbb{R}$, then via prop. 2, prop. 1 and using the naturality of the Maurer-Cartan form, we have a morphism of cospan diagrams of the form
By the homotopy fiber product characterization of the Deligne complex this yields a morphism of the form
which modulates a p+1-connection/Deligne cocycle on the differentially extended smooth $\infty$-group $\tilde G$ from def. 3.
This we call the WZW term obtained by universal Lie integration from $\mu$.
Essentially this construction originates in (FSS 13).
The WZW term of def. 4 is a prequantization of
More generally, one has a sequence of L-∞ cocycles, each defined on the extension that is classified by the previous one – a bouquet of cocycles.
In each stage, for $\mu_1 \colon \mathfrak{g}\to b^{p_1+1}\mathbb{R}$ a cocycle and $\hat {\mathfrak{g}} \to \mathfrak{g}$ the extension that it classifies (its homotopy fiber), then the next cocycle is of the form $\mu_2 \colon \hat \mathfrak{g} \to b^{p_2+1}\mathbb{R}$
The homotopy fiber $\hat \mathfrak{g} \to \mathfrak{g}$ of $\mu_1$ is given by the ordinary pullback
where $e b^{p_1}\mathbb{R}$ is defined by its Chevalley-Eilenberg algebra $CE(e b^{p_1}\mathbb{R})$ being the Weil algebra of $b^{p_1}\mathbb{R}$, which is the free differential graded algebra on a generator in degree $p_1$, and where the right vertical map takes that generator to 0 and takes its free image under the differential to the generator of $CE(b^{p_1+1}\mathbb{R})$.
This follows with the recognition principle for L-∞ homotopy fibers.
A homotopy fiber sequence of L-∞ algebras $\hat \mathfrak{g} \to \mathfrak{g}\stackrel{\mu}{\longrightarrow} b^{p+1}\mathbb{R}$ induces a homotopy pullback diagram of the the associated objects of L-∞ algebra valued differential forms, def. 1, of the form
(hence an ordinary pullback of presheaves, since these are all simplicially constant).
The construction $\mathfrak{g} \mapsto Hom_{dgAlg}(CE(\mathfrak{g}), \Omega^\bullet(-))$ preserves pullbacks ($CE$ is an anti-equivalence onto its image, pullbacks of (pre-)-sheaves are computed objectwise, the hom-functor preserves pullbacks in the covariant argument).
Observe then (see the discussion at L-∞ algebra valued differential forms), that while
we have
With this the statement follows by lemma 1.
We say that a pair of L-∞ cocycles $(\mu_1, \mu_2)$ is consecutive if the domain of the second is the extension (homotopy fiber) defined by the first
and if the truncated Lie integrations of these cocycles via prop. 1 preserves the extension property in that also
The issue of the second clause in def. 5 is to do with the truncation degrees: the universal untruncated Lie integration $\exp(-)$ preserves homotopy fiber sequences, but if there are non-trivial cocycles on $\mathfrak{g}$ in between $\mu_1$ and $\mu_2$, for $p_2 \gt p_1$, then these will remain as nontrivial homotopy groups in the higher-degree truncation $\mathbf{B}G_{2} \coloneqq \tau_{p_2}\exp(\hat\mathfrak{g})$ (see Henriques 06, theorem 6.4) but they will be truncated away in $\mathbf{B}G_1 \coloneqq \tau_{p_1}\exp(\mathfrak{g})$ and will hence spoil the preservation of the homotopy fibers through Lie integration.
Notice that extending along consecutive cocycles is like the extension stages in a Whitehead tower.
Given two consecutive L-∞ cocycles $(\mu_1,\mu_2)$, def. 5, let
and
be the WZW terms obtained from the two cocycles via def. 4.
There is a homotopy pullback square in smooth homotopy types of the form
Consider the following pasting composite
where
the top left square is the evident homotopy;
the bottom left square is from prop. 2
the right square is the naturality of the Maurer-Cartan form construction.
Under forming homotopy limits over the horizontal cospan diagrams here, this turns into
by prop. 4. On the other hand, forming homotopy limits vertically this turns into
(on the left by corollary 1, on the right by the second clause in def. 5).
The homotopy limit over that last cospan, in turn, is $\widetilde{\hat G}$. This implies the claim by the fact that homotopy limits commute with each other.
Prop. 3 says how consecutive pairs of $L_\infty$-cocycles Lie integrate suitably to consecutive pairs of WZW terms.
In the above situation there is a homotopy fiber sequence of the form
By prop. 3 and the pasting law, the homotopy fiber of $\widetilde {\hat G} \to \tilde G$ is equivalently the homotopy fiber of $\mathbf{\Omega}^{p_1+1}\to \mathbf{\Omega}^{p_1+2}_{cl}$
Corollary 2 says that $\widetilde {\hat G}$ is a bundle of moduli stacks for differential cohomology over $\tilde G$. This means that maps
(which are the fields of the higher WZW model with WZW term $\mathbf{L}_2$) are pairs of plain maps $\phi \colon \Sigma \to \tilde G$ together with a differential cocycle on $\Sigma$, i.e. a $p_1$-form connection on $\Sigma$, which is twisted by $\phi$ in a certain way.
This oocurs for the (properly globalized) Green-Schwarz super p-brane sigma models of all the D-branes and of the M5-brane. For the D-branes $p_1 = 1$ and so there is a 1-form connection on their worldvolume, the Chan-Paton gauge field. For the M5-brane $p_1 = 2$ and so there is a 2-form connection on its worldvolume, the self-dual higher gauge field in 6d.
We discuss the general abstract formulation of WZW terms in a cohesive (infinity,1)-topos.
Throughout, let
$\mathbf{H}$ an cohesive (∞,1)-topos;
$\mathbb{G} \in \mathbf{H}$ be an object equipped with the structure of a braided ∞-group, i.e. with specified double delooping $\mathbf{B}^2 \mathbb{G}$.
$\Omega^2_{cl}(-,\mathbb{G}) \to \cdots \to \flat_{dR}\mathbf{B}\mathbb{G}$ a chosen Hodge filtration;
$G \in \mathbf{H}$ be any object equipped with ∞-group structure, i.e. with specified delooping $\mathbf{B}G$;
$\mathbf{c} \;\colon\; \mathbf{B}G \longrightarrow \mathbf{B}^2 \mathbb{G}$ a morphism, hence a cocycle in the group cohomology of $G$ with coefficients in $\mathbb{G}$.
Write
$\mathbf{B}\mathbb{G}_{conn} \coloneqq \mathbf{B}\mathbb{G} \underset{\flat_{dR}\mathbf{B}^2 \mathbb{G}}{\times} \Omega^2_{cl}(-,\mathbb{G})$
for the induced coefficients for $\mathbb{G}$-differential cohomology, as discussed at geometry of physics -- principal connections;
$\hat G \to G$ for the infinity-group extension classified by $\mathbf{c}$.
A refinement of the Hodge filtration of $\mathbb{G}$ along the cocycle $\mathbf{c}$ is a choice of 0-truncated object $\Omega^1_{flat}(-,G) \in \mathbf{H}$ and a completion to a diagram
We write $\tilde G$ for the homotopy pullback of this refinement along the Maurer-Cartan form $\theta_G$ of $G$
Let $\mathbf{H} =$ Smooth∞Grpd and $\mathbb{G} = \mathbf{B}^p U(1)$ the circle (p+1)-group.
For $G$ an ordinary Lie group, then $\mu$ may be taken to be the Lie algebra cocycle corresponding to $\mathbf{c}$ and then $\tilde G \simeq G$.
On the opposite extreme, for $G = \mathbf{B}^p U(1)$ itself with $\mathbf{c}$ the identity, then $\tilde G = \mathbf{B}^pU (1)_{conn}$ is the coefficients for ordinary differential cohomology (the Deligne complex under Dold-Kan correspondence and infinity-stackification).
Hence a more general case is a fibered product of these two, where $\tilde G$ is such that a map $\Sigma \longrightarrow \tilde G$ is equivalently a pair consisting of a map $\Sigma \to G$ and of differential $p$-form data on $\Sigma$. This is the case of relevance for WZW models of super p-branes with “tensor multiplet” fields on them, such as the D-branes and the M5-brane.
In the situation of def. 6 there is an essentially unique prequantization
of the closed differential form
whose underlying $\mathbb{G}$-principal ∞-bundle is modulated by the looping $\Omega \mathbf{c}$ of the original cocycle.
This we call the WZW term of $\mathbf{c}$ with respect to the chosen refinement of the Hodge structure.
The morphism in question is the image under forming homotopy limits of the morphism of cospan diagrams
where the top square is from def. 6 and where the bottom square is the naturality square of the homotopy fiber sequence that defines the Maurer-Cartan forms (see here).
…definite globalization of WZW terms…