The String Coffee Table

The detailed program of the workshop will be announced on Sunday. Each of the following participants is going to give a talk:

Aschieri, Paolo (Alessandria)
Bouwknegt, Peter (Canberra)
Braun, Volker (Philadelphia)
Carey, Alan (Canberra)
Evslin, Jarah (Bruxelles)
Gawedzki, Krzysztof (Lyon)
Husemöller, Dale (Bonn)
Jurčo, Branislav (München)
Mickelsson, Jouko (Helsinki)
Murray, Michael (Adelaide)
Pfeiffer, Hendryk (Cambridge)
Sati, Hisham (Canberra)
Schafer-Nameki, Sakura (Hamburg)
Schreiber, Urs (Essen)
Schweigert, Christoph (Hamburg)
Stevenson, Danny (Riverside)
Varghese, Mathai (Adelaide)

I was originally hoping to meet also Igor Baković and Zoran Skoda there, but they couldn’t come. Igor is currently finishing his math PhD (advised by Branislav Jurčo in Munich) in which he embeds 2-bundles in the wider context of bisites, which are bicategories (= weak coherent 2-categories) equipped with a (higher version of) Grothedieck topology.

I haven’t seen the whole thesis yet, but this sounds all extremely interesting. Among other things, Igor defines the notion of a torsor for a bigroupoid (= weak coherent 2-groupoid) which is certainly the right gadget to generalize 2-connections from structure 2-groups to ‘structure 2-groupoids’, thus providing the ‘integral’ version of the 2-algebroid Yang-Mills theories proposed by Thomas Strobl (as discussed here).

Due to the nice complementarity of our work we started thinking about a joint project which currently carries the working title

‘Higher Gauge Theory and its application to Strings and Elliptic Cohomology’.

This proposal was initially intended to be submitted as application for the Tandem-Programm of the Volkswagen-Stiftung, but even though it is not clear from its charta, it turned out that this program does not accept math/physics collaborations in general and no math/string collaborations in particular, but only very exotic combinations of subjects - in case any one here ever considers applying for this program.

So right now the proposal is more like a personal manifesto for future research. It begins like this:

Abstract

The study of geometric realizations of elliptic cohomology has lead to intriguing relations between relativistic string physics and higher dimensional algebra. These naturally find their place in what is beginning to be called higher gauge theory, where the stringification of point particle dynamics is understood in terms of categorified algebraic structures. Here we propose merging the approach of a physically motivated category theorist with that of a mathematically motiviated string theorist in order to study the higher K-theory of categorified vector bundles and the parallel transport of nonabelian strings, in an effort to understand (enriched) elliptic objects in the context of higher quantum theory.

Introduction

In modern physics a fruitful method for obtaining interesting higher order structures is the process of stringification, whereby theories of point particles are conceived as limits of theories of linearly extended strings that stretch between their endpoints.

In modern mathematics a fruitful method for obtaining interesting higher order structures is known as categorification, where algebraic structures are stringified by replacing objects by morphisms.

Recent developments taking place in the overlap of formal high energy physics and abstract higher dimensional algebra are indicative of a synthesis of these two. In particular, in what is beginning to be called higher gauge theory the dynamics of stringified point particles is studied using category theoretic insights.

The structures emerging thereby are beginning both to elucidate aspects of string theory as well as to become helpful tools in pure mathematics, as concerns notably the study of geometric realizations of elliptic cohomology.

Here we propose a project whose aim is to combine the approach of a physically motivated category theorist with that of mathematically motivated string theorist in order to develop a couple of existing partial constructions in elliptic cohomology into a coherent picture within higher gauge theory.

In particular, a central but still somewhat elusive concept in this context has turned out to be that of a categorified vector bundle. We shall begin by presenting and investigating several possible definitions of such structures and study their categorified K-theory, building on recent progress in the understanding of the categorification of principal fibre bundles and their relation to nonabelian gerbes, in which both applicants have been involved in their Ph.D. research. In particular, we propose a certain string-motivated refinement of one definition of categorified vector bundle which has already proven useful in the context of elliptic cohomology.

Categorified vector bundles ought to be just one puzzle piece in a grander scheme where supersymmetric quantum mechanics is systematically categorified to yield superstring physics. Judging from the relation of ordinary Dirac operators to K-theory it is reasonable to expect that what is needed for elliptic cohomology is a notion of categorified Dirac operator related to string supercharges, or in fact a categorification of the entire concept of a spectral triple.

Several aspects of such a point of view have already emerged in the literature, but its coherent development seems to be hampered by its intrinsic interdisciplinary nature, where crucial guidance is apparently to be found in string physics while crucial tools are provided only by category theory, two fields which are only beginning to be commonly recognized as closely related.

Recent developments in the context higher gauge theory have provided a bridge between these two fields which shall serve as the fundament for the project proposed here: the understanding of higher holonomy in categorified principal bundles (‘2-bundles’) has allowed to recognize the category theoretic structure underlying the well-known coupling of the string to the Kalb-Ramond field, and has for the first time provided a candidate formalism for describing nonabelian strings as they arise in configurations of M2-branes attached to M5-branes, thus possibly shedding light on the fundamental degrees of freedom in M-theory.

Urs Schreiber, working in string theory, has studied in his thesis the description of the superstring in terms of supersymmetric quantum mechanics on loop space, and, motivated by this, studied the theory of 2-holonomy in principal 2-bundles which describes the gauge theory of (non)abelian strings.

Igor Baković, working in category theory, has embedded in his thesis the 2-category of principal 2-bundles in the larger context of Grothendieck topologies for bicategories, 2-sheaves and 2-gerbes, thus illuminating crucial formal structures behind the concept of a 2-bundle.

Since elliptic cohomology can justly be expected to be in large parts about the classification of (vector) 2-bundles, describing global phenomena in string theory, we feel that our collaboration opens the interesting opportunity of merging expertise in string theory and in category theory to tackle this intriguing subject, which is possibly at the heart of M-theory as well as of much of modern geometry.

Background

There has for quite a while now been some cross-fertilization between string theory and category theory.

On the one hand the discovery of the description of open strings on D-branes in terms of derived categories of coherent sheaves and of Fukaya categories has shown that category theoretic language is inevitable for making progress with understanding the deeper structure of string theory’s still somewhat elusive nature. Much of recent progress on mirror symmetry and Seiberg-duality would have been hardly conceivable without both the basic point of view provided by category theory, which for instance allows to identify a functor between D-brane states as such, as well as its sophisticated results, as for instance concerning $\Pi$-stability in triangulated derived categories of topological branes.

On the other hand, string theory has motivated a rich web of category-theoretic constructions in abstract mathematical physics and in pure mathematics. String theory’s influence on the study of mirror symmetry is probably only the most commonly known of these. More archetypical and far-reaching is the description of topological, conformal or other field theories in terms of functors on cobordism categories. This originates in the simple picture of a quantum string (or $p$-brane) sweeping out its worldvolume, but is at the heart of fascinating results like the TFT construction of CFT correlators, and has lead to the conception of new structures like homotopy quantum field theories, which are being studied as rich mathematical objects in their own right.

But even in the face of this rather impressive list of currently known links between strings and categories, there are hints that a deeper, more systematic relation between stringification and categorification remains still to be revealed. Several of these hints have arisen in attempts of physicists and mathematicians to come to grips with geometric realizations if elliptic cohomology.

Elliptic cohomology is a generalized cohomology theory based on the concept of abstract genus. Several striking but mystifying results about the properties of the elliptic genus were dramatically clarified when it was noted by Witten that the elliptic genus is essentially nothing but the index of the superstring’s supercharge, which can be regarded as a Dirac operator on loop space. This lifts the relation between K-theory and the index of a supercharge in supersymmetric quantum mechanics from points to strings.

While Witten’s result was regarded as a breakthrough, there were issues with making his argument rigorous and with using this insight for the study of elliptic cohomology proper. In a reaction to that situation Segal famously suggested the concept of an elliptic object. In generalization of how an ordinary vector bundle with connection can be regarded as a functor from paths to the category of vector spaces, an elliptic object should be an assigment of vecor spaces to loops and of linear operators to cobordisms between loops, just as in the functorial description of conformal field theory. In this spirit, it was later proposed by Baas, Dundas and Rognes that it should be possible to go from K-theory to elliptic cohomology by categorifying the concept of a vector bundle and studying equivalence classes of the resulting `2-vector bundles’, thus essentially stringifying the study of ordinary fibre bundles, which describe charged point particles, to that of ‘2-bundles’, which describe charged strings. This was checked to be essentially true, even though the definition of categorified vector bundle seems to admit further refinement, to be discussed in more detail below.

In what might be regarded as a synthesis of these two approaches, Stolz and Teichner have begun approaching this issue by refining Segal’s concept of an elliptic object. They note that supersymmetric quantum mechanics, regarded as a functor from (super-)`worldline’ intervals to the category of (super-)vector spaces, encodes K-theory data, and then proceed to categorify this situation, obtaining a 2-functor on a 2-category of bounded surfaces. The resulting enriched elliptic objects map surfaces to 2-morphisms of a 2-vector space and can, in a currently controllable special case, be successfully related to elliptic cohomology.

While all three of these approaches currently fall short of completely capturing elliptic cohomology, they seem reasonably symptomatic of a grander underlying scheme, which points to the need for a more thorough understanding both of a systematic categorification of (supersymmetric) quantum mechanics of point particles, as well as of the concept of a vector bundle.

It should be noteworthy that these need not be two different tasks. One way to characterize an ordinary vector bundle is as a projectively generated module of the algebra of functions on its base space, a definition which is, incidentally, suitable for the generalization to noncommutative geometry. Such an algebra of functions, however, is already part of the data coming with a system of supersymmetric quantum mechanics with this space as its configuration space, and hence categorifying the latter is likely to yield a good categorification of the former.

Indeed, as has been particularly emphasized by Froehlich et al., the data defining supersymmetric quantum mechanics is precisely that describing spectral (not necessarily commutative) geometry. Froehlich as well as Chamseddine have taken this point of view seriously and have discussed possible ways to lift it from point particles to strings, in a way that is quite in resonance with the philosophy of the approaches to elliptic cohomology mentioned above. Further investigations along the lines of these proposals for spectral stringy geometry seem to have been thwarted at their time by the immense attention that was being paid to the newly discovered ubiquity of the noncommutative aspect of noncummutative spectral geometry in string physics, as it appears in Matrix Theory and in the study open strings in Kalb-Ramond backgrounds.

But meanwhile, undisturbed by the winds of fashion in string theory, necessary ingredients for a categorification of quantum mechanics were being pondered by Baez et al., who investigated the notion of a categorified Hilbert space. While motivated by purely formal considerations and the idea that concepts in physics eventually ought to be categorified, such categorified vector spaces turn out to reappear in Baas, Dundas and Rognes’ work on elliptic cohomology, as well as in the context of quiver gauge theories in string theory. This is further discussed below.

The same school of thought has recently lead to the first systematic approach towards 2-bundles and, building on that, to the study of the relation of 2-bundles to nonabelian gerbes and of connections and holonomy in such 2-bundles.

In this context it was found that a certain strict 2-group called ${𝒫}_1\mathrm{Spin}(n)$ has all the right properties to be the structure 2-group for 2-bundles that capture the parallel transport of spinning strings. A direct relation of this result to the approach by Stolz and Teichner seems inevitable, but remains to be investigated. This is further discussed below.

On the other hand, nonabelian 2-bundles with 2-connection are an interesting candidate for the description of M2-branes ending on M5-branes, which was rather recently speculated to be related to elliptic cohomology by Kriz and Sati in a series of papers. While the developments mentioned so far all pertain to the worldsheet description of strings, Kriz and Sati’s work gives a new twist to the relation between string physics and elliptic cohomology by placing elliptic cohomology at the heart of the broader understanding of string theory which has emerged after the so-called second superstring revolution and which is famously known under the working title `M-theory’.

When the process of stringification is applied twice, strings turn into membranes and their endpoints turn into endstrings. Like the endpoint of an open string may couple to a nonabelian bundle, the endstring of an open membrane ending on a collection of 5-branes is expected to couple to a nonabelian 2-bundle, or gerbe. Kriz and Sati argue that the worldsheet of such an `endstring’ should be closely related to the elliptic curves featuring in elliptic cohomology.

An argument for a special role played by elliptic cohomology in the context of full M-theory is given in another paper, where the authors note that due to common wisdom the RR-fields are objects in (twisted) K-theory, while the corresponding $\mathrm{NSNS}$-fields are ordinary $p$-forms. But for IIB string theory, which is self-S-dual with these fields interchanging their roles, this is inconsistent. Consequently, they argue, both the RR fields as well as the NSNS fields must be regarded as objects in a higher coholomoly theory, namely in elliptic cohomology.

There is more to say here, but at this point the exposition currently breaks off. All the ‘see below’ in the above refer to text that can be found at the above mentioned link.