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Obstructions to globalizing higher WZW terms over Cartan geometries

WZW-Terms on infinitesimal Klein geometries

Given a Lie algebra $\mathfrak{g}$.

and Lie algebra 3-cocycle $\omega_{WZW}^{\mathfrak{g}} \in \Omega^3_{LI}(\mathfrak{g})$

consider a potential 2-form $B\in \Omega^2(\mathfrak{g})$ (a B-field) i.e. $\mathbf{d} B = \omega_{WZW}^{\mathfrak{g}}$.

For $\Sigma$ a 2-dimensional manifold, consider on the space of smooth functions

$$\phi \colon \Sigma \longrightarrow \mathfrak{g}$$

the functional (“action functional”)

$$\phi \mapsto \exp(\tfrac{i}{\hbar} \int_{\Sigma}\phi^\ast B )$$

taking this as the interaction term defines a a 2-dimensional sigma-model quantum field theory called the Wess-Zumino-Witten model or WZW model for short.

At first sight this may seem exotic, but on second sight WZW models are ubiquitous.

First of all, they are the worldsheet theory of the NSR string propagating on a patch of a group manifold.

These are rational conformal field theories, that part of the field of 2d conformal field theory that has been mathematically understood und classified (see at FRS formalism).

Specifically the chiral part of the E8-WZW model gives the current algebra-sector of the heterotic string, i.e. that piece that gives the gauge group in grand unified heterotic model building).

But, in fact, as we pass from the NSR superstring (with worldsheet supersymmetry) to the Green-Schwarz superstring (with target space supersymmetry), then every model is a WZW model!

Here $\mathfrak{g}/\mathfrak{h} = \mathbb{R}^{d-1,1|N}$ super translation Lie algebra, i.e. super Minkowski spacetime and $\omega_{WZW}$ is a super 3-form.

Now for any target spacetime any curved super-spacetime, the Green-Schwarz superstring, remaiins a super-WZW model – to first order around every point. This is what we come to below.

But of course the general idea of WZW models works in other dimensions, too. Let $\omega_{WZW}$ be a $(p+2)$-form, then it defines a sigma-model for p-branes.

Curious: For $p = 0$ there is a 2-cocycle on Galilei group which induces a 1-d WZW model which gives the jet-space Lagrangian of the free non-relativistic massive particle.

More relevant: there are the Green-Schwarz super p-brane sigma models:

whenever there is a super p-brane in string theory/M-theory on a target spacetime of dimension $d$ with supersymmetry “number” $N$ (i.e. real spin representation $N$), then there is a WZW curvature $\omega_{WZW}$ of degree $p+2$ on the super-Minkowski spacetime $\mathbb{R}^{d-1,1|N}$.

The viable combinations $(p,d,N)$ are the content of the brane scan, or rather, the The brane bouquet. Quite a bit of the usual lore of M-theory is captured this way in super Lie algebra cohomology (super L-infinity algebra L-infinity algebra cohomology) this way.

Finally: often WZW curvature invariant under sub-Lie algebra $\mathfrak{h} \hookrightarrow \mathfrak{g}$. Then defined on quotient $\mathfrak{g}/\mathfrak{h}$.

Indeed, this is precisely what happens for the super-$p$-branes where $\mathfrak{g} =$ super Poincaré Lie algebra $\mathfrak{iso}(\mathbb{R}^{d-1,1|N})$ and $\mathfrak{h}$ is the Lorentz Lie algebra $\mathfrak{o}(\mathbb{R}^{d-1,1|N})$ with

$$\mathbb{R}^{d-1,1|N} \simeq \mathfrak{iso}(\mathbb{R}^{d-1,1|N})/\mathfrak{o}(\mathbb{R}^{d-1,1|N})$$

WZW-terms on Klein geometry

For reasonable applications WZW models need to be generalized from target spaces of the form $\mathfrak{g}/\mathfrak{h}$ to curved manifolds only whose tangent spaces look like this.

These are first of all the Klein geometries given by the coset manifolds $G/H$, for $H \to G$ Lie group Lie integrating $\mathfrak{h} \to \mathfrak{g}$.

There is a very popular trick to define the globalization:

for $\phi \colon \Sigma \longrightarrow G/H$ a sigma-model configuration, find – if possible – an extension $\widehat \phi$ to a 3-manifold $\Sigma_3$ with boundary $\Sigma$ and then take the WZW term to be

$$\phi \mapsto \exp(\tfrac{i}{\hbar} \int_{\Sigma_3} \widehat \phi^\ast \omega_{WZW}) \,.$$

As with many simple solutions, this is deceptively simple. First of all, $\Sigma_3$ might not exist (particularly if one requires extra structure on $\Sigma$ and $\Sigma_3$).

But worse: this hack does not reflect the higher gauge symmetry of the WZW term: it has

1. gauge transformations $B \mapsto B + \mathbf{d} A$ (ghosts);

2. higher gauge transformations $A \mapsto A + \mathbf{d}f$ (ghosts-of-ghosts)

But as we globalize, WZW term need to be glued together by gauge transformations, and so it will be all-important to keep these around.

The solution is familar in one dimension down: for $\omega$ a closed 2-form, then the solution to the problem of assigning $U(1)$-elements to curves $\Sigma_1$ such that whenever the curve bounds a disk $\Sigma_2$ the element is $\exp(\tfrac{i}{\hbar}\int_{\Sigma_2}\omega)$ is to find a line bundle with connection $\mathbf{L}$ with curvature $\omega$ and have the assignment be its holonomy (i.e. to find a prequantization of $\omega$).

This has an analog in all higher dimensions: given a closed $(p+2)$-form $\omega$, one may ask for a line (p+1)-bundle with connection such that $\omega$ is its curvature. This defines a volume holonomy for $(p+1)$-dimensional volumes. This is a higher prequantization.

These line (p+1)-bundles are equivalently cocycles in ordinary differential cohomology of degree $(p+2)$. For $p = 1$ these are also knwon as bundle gerbes with connection.

So then a proper WZW term is a line (p+1)-bundle with connection $\mathbf{L}_{WZW}$ on $G/H$ whose curvature is $\omega_{WZW}$. Then the WZW interaction functional is simply the higher holonomy map

$$\phi \mapsto Hol_{\mathbf{L}_{WZW}}(\phi) \,.$$

For $H \to G$ ordinary Lie groups and $p =1$, the possible WZW terms in this sense have been classified in great detail in the literature. Importantly, to a given curvature, there may be none, one, or (in general) many inequivalent full WZW terms. Under quantization these in general induce different 2d CFTs. So the extra information here crucially matters!

We are after a more subtle classification which hasn’t found attention yet:

WZW-Terms on Cartan geometry

The Klein geometries $G/H$ are the canonical globalizations of $\mathfrak{g}/\mathfrak{h}$.

The general globalization are $(H \to G)$-Cartan geometries.

Indeed, again this is just what happens for the super $p$-branes: one really wants to them to propagate not on super-Minkowski spacetime, but on surved super-spacetimes that are solutions to the relevant supergravity equations of motion. But such a curved spacetime with super-vielbein is precisely a $(O(\mathbb{R}^{d-1,1|N}) \to Iso(\mathbb{R}^{d-1,1|N}))$-Cartan geometry.

The Cartan-geometry globalization of the curvatures $\omega_{WZW}$ is well known. it follows the pattern of G2-manifolds etc.

If $(\omega_{a_1 a_2 \cdots a_{p+1}})$ denote the components of $\omega_{WZW}$ on $G/H$ in the basis of left invariant 1-forms, then the globalization over a superspacetime has to satisfy

1. definiteness: $\omega_{WZW} = \omega_{a_1 a_2 \cdots a_{p+1}} E^{a_1} \wedge E^{a_2} \wedge \cdots \wedge E^{a_{p+1}}$ for $(E^a)$ the vielbein field (super-vielbein) which encodes the field of gravity.

2. covariantly constancy: $\nabla \omega_{WZW} = 0$ ;

3. torsion constraint: on first-order infinitesimal neighbourhoods $\mathfrak{g}/\mathfrak{h} \hookrightarrow X$, vielbein must restric to left-invariant 1-forms.

open question: how do the two globalizations combine? Need to find full differential cocycles on Cartan geometry such that on the local model $\mathfrak{g}/\mathfrak{h}$ they restrict to presribed system.

or a little easier: extend WZW term on $G$ to $G$-principal bundle:

Example/Theorem

structure needed to globalize canonical WZW term on semisimple G to G-bundle is string structure

observe: String 2-group is the Heisenberg 2-group of the canonical 2-cocycle, its stabilizer 2-group under the $G$-action on itself.

Let’s understand this one dimension down:

for $p = 0$ then

• a WZW term is a line bundle with connection

• refining curvature 2-form to WZW is prequantization;

the action functional is the holonomy

the stabilizer group under the diffeomorphism group action is the quantomorphism group, the thing the Lie integrates the Poisson algebra

if $X$ is symplectic vector space, hence abelian group with Hamiltonian action on itself, then restriction of quantomorphisms to those translations is Heisenberg group $Heis(nabla)$

More generally, if a group $G$ acts by Hamiltonian action then – for lack of established term – we call the part of the quantomorphism group covering this the Heisenberg group $\mathbf{Heis}_G(\nabla)$

These concepts generalize to WZW terms in all dimensions Higher geometric prequantum theory

Using this we may state the general theorem:

The obstruction to globalizing a higher WZW term on $\mathfrak{g}/\mathfrak{h}$ over an $(H \to G)$-Cartan geometry is a first-order integrable $\mathbf{Heis}_{O(\mathfrak{g}/\mathfrak{h})}(\mathbf{L}_{WZW}^{inf})$-structure.

Notice: first order integrable but not torsion free: instead vielbein restricted to be first-order left-invariant. This means torsion for the sugra case! torsion constraints of supergravity

Notice: this statement holds in higher differential geometry. $X$ may be orbifold or worse. And $\mathbf{Heis}_{O(\mathfrak{g}/\mathfrak{h})}(\mathbf{L}_{WZW}^{inf})$ is a smooth infinity-group In fact for the M5-brane sigma model with its B-field, then $X$ is itself modeled on the supergravity Lie 3-algebra!

Thanks.