The String Coffee Table

As one can see on slides 9-12 one main idea is that which has been popularized in

Daniel S. Freed
Higher Algebraic Structures and Quantization
hep-th/9212115,

which says that number-valued partition function of an $n$-dimensional field theory is just the $(n-1)$-morphism part of something that takes values in an $(n-1)$-category of higher Hilbert spaces.

More concretely, if one thinks about it, one can realize that a field theory usually allows us to associate

$\bullet$ complex numbers to (closed) $n$-cobordimsms

$\bullet$ vector spaces (Hilbert spaces) to $(n-1)$-dimensional boundary parts

$\bullet$ something like 2-Hilbert spaces to $(n-2)$-dimensional objects.

John C. Baez
Higher-Dimensional Algebra II: 2-Hilbert Spaces
q-alg/9609018 .

The same idea appears for instance also in Simon Willerton’s work (which, recall, I planned to report about in detail once I find the time)

Simon Willerton
The twisted Drinfeld double of a finite group via gerbes and finite groupoids
math.QA/0503266 .

As I have indicated before, there is a nice way to understand this phenomenon systematically:

Where 1-dimensional quantum field theory (quantum mechanics) is a functor from 1-cobordisms to $\mathbb{C}$-modules, an $n$-dimensional quantum field theory should be an $n$-functor (“transport functor” $\to$) to $C$-module $n$-categories, where $C$ is some monoidal $n$-category.

This reproduces Freed’s prescription (after you account for the fact that it is one step more refined than what Freed has, in that it also assigns data at level 0), as you can be checked with a little knowledge of higher linear algebra ($\to$, $\to$).

For instance, for $n=2$ you’d assign module categories to the lowest dimensional objects - this are the 2-Hilbert spaces. You assign 1-morphisms of these module categories to the next higher dimensional objects - and in the examples one deals with these $\mathrm{Hom}$-spaces are vector spaces. Finally, 2-morphisms between these correspond to linear maps, which, when the manifolds in question have only one boundary component, define vectors and, after a choice of basis, numbers. That’s what the partition function (the “surface holonomy”) takes values in.

One can nicely map this 1-1 to a similar description involving categories of D-branes, defect lines and field insertions ($\to$), as indicated on slide number 12 of Gukov’s talk.

Gukov relates this to knot theory and the line operators ($\to$) that made a prominent appearance in the recent Kapustin-Witten work ($\to$).

This is described in detail in

Nathan M. Dunfield, Sergei Gukov, Jacob Rasmussen
The Superpolynomial for Knot Homologies
math.GT/0505662,

which is about the categorification of the HOMFLY polynomial.

Similarly, the Kauffman polynomial is categorified in

Sergei Gukov, Johannes Walcher
Matrix Factorizations and Kauffman Homology
hep-th/0512298.

There is an interesting remark on how all this categorified knot theory can be understood physics-wise in terms of membranes ending on 5-branes.

Hirosi Ooguri, Cumrun Vafa
Knot Invariants and Topological Strings
hep-th/9912123.

This is a situation which has long been suspected to lead to the sort of categorification (namely of strings ending on D-branes) that is going on here ($\to$, $\to$), so I should one day sit down and try to understand the above paper.