The String Coffee Table

It seems helpful to me to roughly distinguish three different levels, or kinds, of application of category theory in mathematical physics.

1) Categorical methods in general mathematics arising in mathematical physics.

2) Functorial phenomena in quantum field theory.

3) Higher order structures (categorification) in string theory.

1)

Categories are a way to talk about algebra, hence they tend appear whenever algebra plays a role.

The most prominent example for an application of categories to mathematical physics in this general purpose sense is probably the description of topological D-branes by means of derived categories of coherent sheaves.

In the study of topological strings one encounters a lot of homological algebra. Hence it is not too surprising that at some point derived categories make an appearance ($\to$). There is by now a detailed understanding of derived categories applied to topological strings ($\to$) as well as to related systems, like Landau-Ginzburg models ($\to$). In particular, the string theoretic notion of mirror symmetry is encoded in the equivalence of certain derived categories of coherent sheaves with certain Fukaya categories.

There are other mathematical aspects of string theory which can be described without using categories, but which find their most natural description in categorical language.

For instance important special cases of 2-dimensional conformal field theories used in string theory are described in terms of sigma models of maps into orbifolds. As is well known, these orbifolds are in some cases best thought of as groupoids ($\to$) or, equivalently, as quotient stacks ($\to$). There are arguments that viewing orbifolds in string theory from this category theoretic perspective is more than a matter of language. Eric Sharpe for instance provides evidence that the notion of equivalence of stacks is related to renormalization group flows between gauged linear sigma models and conformal sigma models whose target spaces are described by the respective stacks ($\to$).

Like point particle quantum mechanics is closely related to the geometry of target space ($\to$) string theory leads to a more detailed analysis of the geometry of loop spaces over target space. This has lead to a field known as string topology. The above mentioned conception of orbifolds in terms of groupoids plays a crucial role in orbifold string topology ($\to$).

2)

In quantum field theory, two central concepts are (temporal) propagation and locality. As emphasized in particular by Segal, this leads to functorial structures.

From this point of view one tries to give a rigorous definition of $d$-dimensional quantum field theory in terms of a functor from $d$-dimensional cobordisms to some target category (usually vector spaces).

This is very well understood for the simplest nontrivial case, that of 2-dimensional topological field theory (topological strings) ($\to$, $\to$).

The next simplest case, conformal 2-dimensional field theory (“physical strings”), can be split into a complex-analytic aspect (providing “conformal blocks”, using vertex operator algebra ($\to$)) and a topological aspects (providing the “structure constants” and hence the correlators). At least for the rational case, the topological aspect is captured in FRS formalism ($\to$) by using modular tensor categories ($\to$) and 3-dimensional (functorial) topological field theory ($\to$).

While this FRS formalism offers a rigorous, detailed and inheretently categorical description and understanding of (correlators in) rational 2-dimensional conformal field theory, the 2-dimensional functorial transport is not manifest, but can possibly be made manifest ($\to$). Another approach to 2-D CFT which emphasizes the 2-dimensional functorial transport is that followed by Stolz&Teichner ($\to$, $\to$) in their attempt to describe elliptic cohomology as a categorification of K-theory ($\to$).

This leads seamlessly to point 3) below.

There are variations on how the details of a (topological) field theory can be described in terms of functors. Segal and Getzler have proposed what is now being called topological conformal field theory ($\to$). This has in particular been developed by Kevin Costello ($\to$) and it provides new insights into the category of topological D-branes and the appearance of $A_{\mathrm{\infty }}$-structures in open string theory.

Recently, there has been increased interest in the proper nature of superstrings. Recent developments suggest that these are best described by a model called the pure spinor superstring ($\to$). It turns out that this involves several structures related to gerbes, in particular gerbes of chiral differential operators ($\to$).

There are other phenomena, distinct from propagation/transport ($\to$), in quantum field theory which are inherently functorial.

In algebraic quantum field theory (AQFT) functoriality encodes the covariance of quantum fields with respect to diffeomorphisms of spacetime. The study of AQFT has famously lead to Doplicher-Roberts reconstruction ($\to$) of group representation categories, which, physically, is related to the study of gauge symmetries and superselection sectors.

Another place where functoriality and related category theoretic structures seem to play a cruciual role is renormalization in quantum field theory. Renormalization is based on a certain algebra structure on the space of Feynman graphs which, I think, should be thought of as the category algebra of a certain multicategory of graphs ($\to$).

3)

In string theory, 2-dimensional quantum field theory describing propagation of strings is regarded as a dimensional lift of 1-dimensional quantum field theory (“quantum mechanics”) describing propagation of point particles. This might remind one of the process of categorification, lifting ordinary categories to higher categories, in particular to 2-categories, bicategories.

That there is something to this analogy is currently best understood for the case of (higher) gauge theory ($\to$). Point particles are “charged” under fiber bundles. Strings are by now well known to be charged under (abelian) gerbes ($\to$).

Using the language of bundle gerbes it is in principle possible to reason about gerbes without using the word category, which is commonly done in the literature. But gerbes happen to be categorified versions of fiber bundles (roughly) and in the end morphisms and 2-morphisms of bundle gerbes (and hence the 2-category of bundle gerbes) are best understood using the proper categorical language ($\to$).

Infinitesimal concepts ($\to$) in the context of gerbes have more recently attracted a lot of attention in mathematical physics. These are related to algebroids and in particular to higher algebroids (e.g. Courant algebroids). These have been found to be relevant for certain topological 2-dimensional field theories like the Poisson sigma-model ($\to$) as well as to what Hitchin calls generalized geometry ($\to$), which has been found to be very useful for understanding string dynamics in Kalb-Ramond backgrounds.

Gerbes play a major role also in non-stringy quantum field theory, where they are mainly related to lifting problems which on the physical side are known as anomalies ($\to$).

Categorification is by now also known to explain the nature of the mysterious String-group ($\to$), which governs the spinorial nature of superstrings. It has been found that this group is the realization of the nerve of a 2-group ($\to$, $\to$) and that String(n)-bundles can be thought of as nonabelian gerbes (or 2-bundles ($\to$)) with this 2-group as structure 2-group ($\to$).

A nice overview over many applications of categories to mathematical physics can be found in Bruce Bartlett’s thesis ($\to$).

John Baez and Aaron Lauda have compiled a History of $n$-Categorical Physics ($\to$).