Categorification and Quantum Field Theories

Oxford Mathematician Elena Gal talks about her recently published research.

"Categorification is an area of pure mathematics that attempts to uncover additional structure hidden in existing mathematical objects. A simplest example is replacing a natural number $n$ by a set with $n$ elements: when we attempt to prove a numerical equation, we can think about it in terms of sets, subsets and one-to-one correspondences rather than just use algebraic manipulations. Indeed this is the natural way to think about the natural numbers - this is how children first grasp rules of arithmetic by counting stones, coins or sea shells. In modern mathematics we often encounter far more complicated objects which intuitively we suspect to be "shadows" of some richer structure - but we don't immediately know what this structure is or how to construct or describe it. Uncovering such a structure gives us access to hidden symmetries of known mathematical entities. There is reasonable hope that this will eventually enrich our understanding of the physical world. Indeed the object our present research is concerned with originates in quantum physics.

The term "categorification" was introduced about 15 years ago by Crane and Frenkel in an attempt to construct an example of 4-dimensional Topological Quantum Field Theory (TQFT for short). TQFTs are a fascinating example of the interaction between modern physics and mathematics. The concept of quantum field theory was gradually developed by theoretical physicists to describe particle behavior. It was later understood that it can also be used to classify intrinsic properties of geometric objects. The state of the art for now is that mathematicians have a way of constructing TQFTs in 3 dimensions, but not higher. The key to rigorous mathematical constructions of 4-dimensional TQFTs is thought to lay in categorification.

The kind of structure that sets (rather than numbers!) form is called a category. This concept was first introduced in 1945 by Eilenberg and Mac Lane. It is now ubiquitous in pure mathematics and appears to find its way into more "applied sciences", like computer science and biology. A category is a collection of objects and arrows between them, satisfying certain simple axioms. For example the objects of the category $\mathbf{Sets}$ are sets and the arrows are the maps between them. The arrows constitute the additional level of information that we obtain by considering $\mathbf{Sets}$ instead of numbers.

Suppose we are given two sets: $A$ with $n$ elements and $B$ with $m$ elements - what is the number of different ways to combine them into one set within the category Sets? We must take a set with $n+m$  elements $C$ and count the number of arrows $A \hookrightarrow C$ that realize $A$ as a subset of $C$. The number of such arrows is given by a binomial coefficient $n+m\choose n$. We can now consider a new operation $\diamond$ given by the rule $n\diamond m = {n+m\choose m}(n+m)$. In this way given a sufficiently nice category $\mathcal{C}$ we can define a mathematical object called the Hall algebra of $\mathcal{C}$. Remarkably, if instead of the category $\mathbf{Sets}$ we consider the category $\mathbf{Vect}$ of vector spaces, we obtain a quantization of this operation where the binomial coefficient $n+m\choose m$ is replaced by its deformation with a parameter $q$. The result is the simplest example of a mathematical object called a quantum group. Quantum groups were introduced by Drinfeld and Jimbo 30 years ago, and as it later turned out they (or more precisely the symmetries of vector spaces that they encode their categories of representations) are precisely what is needed to construct 3-dimensional TQFTs. The upshot is that to move one dimension up we need to categorify Hall algebras.

In our joint project Adam GalKobi Kremnitzer and I use a geometric approach to quantum groups pioneered by Lusztig. The categorification of quantum groups is given by performing the Hall algebra construction we saw above for mathematical objects called sheaves. Using sheaves allows us to remember more information about the category. In our recent work we define an algebra structure on the dg-category of constructible sheaves. This recovers the existing work on categorification of quantum groups by Khovanov, Lauda and Rouquier in a different language. The advantage of our approach is that it incorporates the categorification of the "dual" algebra structure that every Hall algebra has. This should lead to an understanding of the class of symmetries that it generates (in mathematical language, to the construction of the monoidal category of its representations). 

This is the key step to constructing 4-dimensional TQFTs. For more details and updates please click here."

This work is funded by an Engineering and Physical Sciences Research Council (EPSRC) grant.