Infinite dimensional 2-Hilbert spaces

Question

Is there a definition of an infinite dimensional 2-Hilbert space?

Finite dimensional 2-Hilbert spaces have been discussed by Baez in http://arxiv.org/abs/q-alg/9609018 In the more recent paper by Baez, Baratin, Freidel and Wise http://arxiv.org/abs/0812.4969 a notion of infinite dimensional 2-vector space is discussed, building on work by Crane, Sheppeard and Yetter. They also have a few proposals for what an infinite dimensional 2-Hilbert space should be, but "the details still need to be worked out". Is that the current state of knowledge?

Another way of asking this question is the following. If I want to see a field theory that has an infinite dimensional Hilbert space of states as an extended field theory, what kind of objects should I assign to codimension 2 manifolds?

This post imported from StackExchange MathOverflow at 2014-12-13 14:17 (UTC), posted by SE-user Samuel Monnier

Answer 1

This may be more of a comment or opinion.

A 2-Hilbert space is a Hilbert space on two levels: It's enriched in Hilbert spaces (morphisms) and it's a categorification of a Hilbert space in the sense of having biproducts for addition, Deligne-product with $\mathrm{Vect}$ for scalar multiplication and dual structure for the sesquilinear form. Which of those levels do you want to have infinite dimensional? The first one shouldn't be a big deal, you just allow for infinite dimensional Hilbert spaces as $\mathrm{hom}$-spaces.

The second one is the harder one. I personally think that it is no big deal "formally", it just amounts to dropping the requirement that the 2-Hilbert space be finitely semisimple. The question is, will you find interesting examples that are actually any use? You have the same in vector spaces. I can cook up tons of infinite dimensional vector spaces, but how do they become interesting and tractable at the same time? By adding some extra structure, say as a Banach space, a $C^*$-algebra, an $\mathrm{L}^2$-space over some measurable space or so. In that sense, I'd say that there already is a theory of "dull" infinite dimensional 2-Hilbert spaces, but people are thinking about interesting ones with extra structure, such as having a measure on objects.

As for your final question, I'd guess it will depend on the context of what you're doing. Are you considering some $n$-dimensional moduli space ($0 < n < \infty$)? Smooth groupoids? Then the approach with measurable categories may suffice. If your additional structure is different, a different variant might be needed.

This post imported from StackExchange MathOverflow at 2014-12-13 14:17 (UTC), posted by SE-user Turion

Comments

Thanks... I already have a vague idea along these lines of what a 2-Hilbert space should be, and actually there is even a concrete proposal in the paper by Baez, Baratin, Freidel and Wise. I was asking if there is a precise and well-studied definition people agree upon (the answer seems to be no...).

This post imported from StackExchange MathOverflow at 2014-12-13 14:17 (UTC), posted by SE-user Samuel Monnier

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For my final question, when I said "field theory", I meant it in the down-to-earth physical sense. Say consider the free boson in dimension 2 or higher. This is a non-topological theory with an infinite dimensional Hilbert space. What kind of object would we associate to codimension 2 manifolds? I would expect the answer to this question to be valid for any reasonable, non-anomalous field theory a physicist could be interested in.

This post imported from StackExchange MathOverflow at 2014-12-13 14:17 (UTC), posted by SE-user Samuel Monnier

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@SamuelMonnier, for any precise definition someone can probably come up with an example where the definition is unsuitable. As for well-studiedness, these guys have already worked out their example in depth a lot (and found some nice insights as I find). I don't know about your example. If you say it's a non-topological theory, then you must assume some background field like a metric. I have only met higher vector spaces in extended TQFTs, so I have no idea how one would use a 2-vector space here.

This post imported from StackExchange MathOverflow at 2014-12-13 14:17 (UTC), posted by SE-user Turion

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@SamuelMonnier, it just came to my mind, if you were interested in TQFTs, you might search for "non-semisimple" TQFTs and maybe find something useful there?

This post imported from StackExchange MathOverflow at 2014-12-13 14:17 (UTC), posted by SE-user Turion