The n-Category Café

Deformation Quantization of Surjective Submersions

Posted by Urs Schreiber

I am in Leipzig, attending the last day of Recent Developments in QFT in Leipzig.

First talk this morning was by Stefan Waldman on Deformation quantization of surjective submersions.

What he described is this:

Suppse over some manifold $M$ we have a surjective submersion $$p : P \to M \,.$$ In applications we will want to restrict to the case that $P$ is some principal $G$-bundle. But much of the following is independent of that restriction.

Next, suppose that we want to consider (noncommutative) deformations $$(C^\infty(M)[[\lambda]],\star)$$ of the algebra of functions on $M$. This is a popular desideratum in many approaches of quantum field theory that try to go beyond the standard model and classical gravity.

Stefan Waldman and his collaborators ask: given such a deformation of base space, what are suitably compatible deformations of the covering space $P$ over $M$?

The motivating idea is this:

There is that wide-spread idea in quantum field theory that theories which go beyond the energy scale currently observable by experimental means will involve noncommutative deformations of the algebra of functions on spacetime. But since physical fields are usually not just functions on spacetime, but either sections of some vector bundles, or connections on these, the question arises in which sense these bundles then have to be deformed, too.

Therefore Waldman is looking for deformations $$(C^\infty(P)[[\lambda]],\bullet)$$ of the algebra of functions on the total space of the bundle which are, in some sense to be determined, compatible with the deformation downstairs.

The first idea is to find deformations such that the pullback $p^*$ extends to an algebra homomorphism of the deformed algebras $$p^* : (C^\infty(M)[[\lambda]],\star) \to (C^\infty(P)[[\lambda]],\bullet) \,.$$ But they show that in many physically interesting cases, like the Hopf fibration which in physics corresponds to the “Dirac monopole”, there are obstructions for such an extension to exist at all. Hence they reject this idea.

The next idea is to realize $(C^\infty(M)[[\lambda]],\star)$ just as a bimodule for $(C^\infty(M)[[\lambda]],\star)$. But this, too, turns out to be too restrictive.

Finally, they settle for requiring that $p^*$ induces just a one-sided $(C^\infty(M)[[\lambda]],\star)$-module structure on $(C^\infty(P)[[\lambda]],\bullet)$. This turns out to be a tractable problem with interesting solutions.

They show that the module structure of $(C^\infty(P)[[\lambda]],\bullet)$, when computed order by order in the deformation parameter $\lambda$, is given by terms in Hochschild cohomology. Stefan Waldman emphasized the nice coincidence that these are actually explicitly solvable in the present situation.

So, in the end, they find the following nice result:

there is, up to equivalence, a unique deformation of the algebra of functions on the total space of the surjective submersion such that it becomes a right module for the given deformed algebra of functions on base space and extends the pullback $p^*$.

Moreover, if the surjective submersion happened to be a principal $G$-bundle, then, there is, up to equivalence, a unique $G$-equivariant such deformation.

Waldman and collaborators want to understand connections on these deformed bundles eventually, but they are not there yet. One application of the existing technology is this:

a section of a vector bundle associated to the principal bundle $P \to M$ can be understood as a $G$-equivariant function on $P$ with values in the representation space of the rep of $G$ inducing the association. This construction can be immediately fed into the above deformation procedure and hence yields a notion of sections of noncommutatively deformed vector bundles over deformed base spaces.