### “Area Metric” Manifolds

#### Posted by Urs Schreiber

David Roberts made me have a look at the recent preprint

F. Schuller & M. Wohlfarth
**Canonical differential structure of string backgrounds**

hep-th/0511157.

People are trying to find the right mathematical formalisms to capture ‘higher’ stringified structures.

The best example is Hitchin’s currently very popular *generalized geometry*, a mathematical framework motivated by and modeled after the fact that the metric and the Kalb-Ramond 2-form appear on equal footing in string theory. The authors of the above paper propose *another* way to unify a symmetric and an antisymmetric rank-2- tensor into some generalized geometric structure.

An ordinary metric $g$ can be regarded as a linear map from vectors to covectors

In this spirit, Schuller and Wohlfarth define an **area metric** $G$ to be a linear map from 2-vectors (no, not those 2-vectors :-) to 2-forms:

An **area metric manifold** is then a manifold $M$ equipped with such an area metric.

Every ordinary metric $g$ gives rise to an area metric ${G}_{g}$ in the obvious way. Similarly every nondegenerate 2-form $B$ on $M$ gives rise to an area metric ${G}_{B}$.

Consider a ‘worldsheet’ $\Sigma :[\mathrm{0,1}{]}^{2}\to M$ with tangent 2-vector

It is a well known and widely used fact that studying the Polyakov action in a background with gravity $g$ and Kalb-Ramond fields $B$ leads one to use the ‘generalized metric’ tensor $g\pm B$.

Using the above language this fact appears in form of the statement that the Polyakov action in such a background is classically equivalent to an action given by the term

formally analogous to the Nambu-Goto point partcicle action.

This is not too surprising, it seems, being essentially a new coordinate-independent formulation of well-known formulas, I think. But it could point us to useful new concepts.

The authors of the above paper propose that such a useful new concept is that of **area derivative** and **area parallel transport**.

They discuss in detail various generalizations of the concept of a covariant derivative to the context of manifolds with area metric. For the applications to string actions the most important one is the **canonical area derivative** described in section IV. In the case where this is *area metric compatible* (in analogy to how the Levi-Civita connection is metric compatible) this a gadget that maps two infinitesimal surfaces $\Sigma ,\Omega \in {\Lambda}^{2}TM$ to the 1-form ${D}_{\Sigma}\Omega $ given by

for $v\in TM$.

Using this object, it turns out that the classical equations of motion of the Polyakov string in metric and KR-background simply read

and hence look completely similar to the geodesic equation of motion of a free point particle.

I would like to better understand to what extent this operator ${D}_{\Omega}$ is the right thing to consider, generally. For instance, what does it mean that ${D}_{\Sigma}\Omega $ is a 1-form, something I find somewhat counterintuitive. Naïvely I would have expect it to be an element in ${\Lambda}^{2}TM$.

Hence what I would like to understand is to what extent area metric geometry and in partiular the ‘canonical area derivative’ follow from some general abstract reasoning, possibly along the lines of our recent discussion on higher Poisson geometry and higher geometric quantization.

That would help understand in which sense ${D}_{\Omega}\Omega =0$ is just a rewriting of known things or a concept leading to something genuinely new.

## Re: “Area Metric” Manifolds

I haven’t absorbed this yet, but it seems fascinating to me, because in the “BF theory” approach to general relativity that spin foam people like, the basic field is not the metric but a Lorentz-Lie-algebra-valued 2-form, the B field, which plays the role of an “area frame field” - assigning internal bivectors to tiny parallelograms, instead of assigning internal vectors to tiny vectors, as a (co)frame field does.

Does string theory really involve an area metric?

Of course the other reason this “Lie-algebra-valued 2-form” is so fascinating, is that such things naturally show up in 2-connections.

Which of course is part of my sneaky plan to unify string theory, spin foam models and n-categories. :-)

By the way, Urs: why don’t you put up an entry about our new paper? And while you’re at it, you can advertise this even shorter version:

Higher Gauge Theory: 2-Connections

… my talk at the Union College Mathematics Conference this weekend.