### Seminar on 2-Vector Bundles and Elliptic Cohomology, I

#### Posted by Urs Schreiber

We currently have a series of seminars here on tensor categories and their application in CFT. After having heard talks about the basics of tensor categories and the way they appear in the FRS formalism of CFT, today Birgit Richter gave an introductory review of the work

N. Baas, B. Dundas & J. Rognes
**Two-vector bundles and forms of elliptic cohomology**

math.AT/0306027

which I surely have mentioned several times before here at the coffee table.

I’ll try to give a transcript of what was going on today. More meetings on this topic are planned and should go into deeper details.

Here, I’ll start with what, to me at least, is the bird’s eye perspective on the entire program.

**[Update:**

*Literature on elliptic cohomology*:

M. Ando, M. J. Hopkins, AND N. P. Strickland
*Elliptic Spectra, the Witten Genus and the theorem of the cube*

$\to $

P. Stolz & P. Teichner

What is an ellitpic object?

Jacob Lurie

A survey of Elliptic Cohomology

(*Thanks to A. J. Tolland for this link.*)

John Baez

TWF 197

Matthew Ando

Elliptic curves and algebraic topology

Fields Institute

Workshop on Forms of Homotopy Theory: Elliptic Cohomology and Loop Spaces

(2004)

(*Thanks to D. Corfield for this link.*)

Mark Hovey
*Major Problems in Algebraic Topology*,

Problem 3. and 4.

(*Thanks again to D. Corfield for this one.*)

Neil Strickland
*Survey talk given by Neil Strickland at the 2003 British Mathematical Colloquium in Birmingham*

slides

(*Thanks to David Roberts for this link.*)

*Contents:*

- First session, first part. 1) Bipermutative Categories

- First session, second part. 2) Algebraic K-Theory of Bipermutative Categories

- First session, third part. 3) $K(V)$ and Elliptic Cohomology

**]**

The basic motivating *idea* is that

Elliptic cohomology should be a categorification of K-theory.

or

Elliptic cohomology should be to K-theory like 2-bundles are to bundles.

This is based on a statement like

Elliptic cohomology is to K-theory like strings are to point particles.

This latter statement is the more well-known one, going back to Segal’s idea of realizing elliptic objects along the lines of his functorial conception of CFT. This is related to Witten’s approach which realizes the elliptic genus in terms of the index of the superstrings’s supercharge, which is a Dirac operator on loop space.

Note that this is so far just the idea. Baas, Dundas and Rognes (“BDR”, for short) made a first proposal for actually realizing this program, a proposal which is certainly successful, but also obviously only partially successful. So this is far from being a closed topic.

The way BDR approached this program is as follows:

**Step 1: Categorify the concept of K-theory.**

BD&R first of all pick one of the many equivalent definitions of K-theory, one that lends itself to categorification.

The one they use is the following:

*Definition:* If $R$ is a commutative ring (to be thought of as the ring of functions over some space) then the **algebraic K-theory** of $R$ is the space

Here $B$ is the operation of forming the classifying space, $\Omega $ is the operation of forming the space of based loops, and ${\mathrm{GL}}_{n}(R)$ is the group of invertible $n\times n$ matrices with values in $R$.

(The way how defining a space such as $K(R)$ defines a cohomology theory is nicely explained in John Baez’s TWFs, see week 149 and week 150 for an introduction to generalized cohomology. There is also something about elliptic cohomology in week 153 and week 197.)

Now, a group may be regarded as a category with a single object. Forming the classifying space of a group is a special case of taking the geometric realization $\mid C\mid $ of the nerve of a category $C$. This nerve is defined for any category and can also be defined for 2-categories.

For this reason, one could generalize the above formula tremendously by replacing ${\mathrm{GL}}_{n}(R)$ with some 1- or 2-categories ${C}_{n}$, and consider the space

But certainly, we don’t want the ${C}_{n}$ to be just anything. After all, ${\mathrm{GL}}_{n}(R)$ is some very special object, and we now want the ${C}_{n}$ to be appropriate categorifications of ${\mathrm{GL}}_{n}(R)$.

Proposing the “right” categorification for ${\mathrm{GL}}_{n}(R)$ here is the second step in the BDR proposal.

**Step 2: Categorify the concept of a vector space.**

Since ${\mathrm{GL}}_{n}(R)$ acts as automorphisms of vector spaces of the form ${R}^{n}$, one natural way to proceed in categorifying this is to construct a suitable notion of 2-vector spaces and of automorphisms between these.

Without further qualification, there is quite some freedom in how to do this. Before saying how BDR choose to realize this idea, I’ll say something about 2-vector spaces in general.

Categorification is a little bit like quantization in that it is a well-defined process that takes you from a “classical” structure to a “higher order” structure, but with the result possibly depending on which of several equivalent formulations of the classical structure one started with.

One natural way to categorify vector spaces is to look at categories internal to the category of vector spaces. This leads to the notion of 2-vector spaces studied by Baez and Crans in math.QA/0307263. These 2-vectors have proven useful in several circumstances. For instance, by way of such 2-vector spaces which carry the structure of 2-Lie algebras one can construct structures closely related to elliptic cohomology, namely 2-group realizations of the string group. Still, at least as far as I can see currently, this is not the notion of 2-vectors which is the right thing to use in a categorification of K-theory. In fact, I have been told that BDR did try to plug in these Baez-Crans 2-vectors into their construction and found that the resulting categorified K-theory is somehow too trivial.

But there are other possibilities. Regard ordinary vector spaces as *modules* over some ring $R$. Categorify this.

The categorification of a (semi-)ring, which I sometimes like to call a (semi-)*2-ring*, is an (abelian) monoidal category, namely a category which has a product and a sum operation implemented functorially, such that some sort of distributivity holds.

(Semi-)2-rings are just (semi-)rings internal to $\mathrm{Cat}$. They can act on *2-modules*, which are modules internal to $\mathrm{Cat}$, known as *module categories*. (See this and this for more details on the idea of 2-rings.)

Therefore, the categorification of an ordinary vector space over a ring $R$ – an $R$-module – is a module category over a 2-ring ${R}_{2}$.

Even better, the categorification of the category of vector spaces over $R$ should be the 2-category of ${R}_{2}$-2-modules.

Many things are known about ${R}_{2}$-(2-)modules. A theorem by Victor Ostrik says that under nice conditions these are nothing but categories of internal modules. In fact, Victor Ostrik kindly confirmed my conjecture that under suitably nice conditions the entire 2-category of internal bimodules internal to ${R}_{2}$ is equivalent to that of ${R}_{2}$ module categories.

That should be relevant for the present discussion due to the relation of the 2-category of bimodules to the FRS description of conformal field theory. I’ll say more about this later.

Like $\mathbb{N}$ plays a special role among all semi-rings, there is a monoidal category which similarly plays a special role among all semi-2-rings. That’s the monoidal category ${\mathrm{Vect}}_{R}$ of all (finite dimensional) vector spaces over $R$, with morphisms being linear maps between them.

The addition operation in ${\mathrm{Vect}}_{R}$ is just the direct sum, and multiplication is just the tensor product of vector spaces and linear maps.

Since this is about as simple as it gets, one would like to understand the 2-category ${\mathrm{Vect}}_{R}-\mathrm{Mod}$ of module categories over ${\mathrm{Vect}}_{R}$, since the objects in there would be like 2-vector spaces over ${R}_{2}={\mathrm{Vect}}_{R}$.

But what do these module categories look like, in detail?

As John Baez only recently pointed out to me, there is a result by David Yetter which says that (under some conditions on the properties of morphisms)

The 2-category of 2-modules of the 2-ring ${R}_{2}={\mathrm{Vect}}_{K}$ is equivalent to the 2-category of 2-vector spaces defined by Kapranov and Voevodsky.

That’s good, because the Kapranov-Voevodsky (“KV” for short) description of 2-vector spaces is very concrete. It was introduced in the paper

M. Kapranov & V. Voevodsky
**2-categories and Zamolodchikov tetrahedra equations**

In *Proc. Symp. Pure Math.* **56**, Part 2, pages 177-260.

American Mathematical Society, 1994 .

I cannot find an electronic version of this paper, but the basic idea is very easy to describe (though some details are a little more involved).

Namely, if ${R}_{2}$ is a 2-ring, it naturally acts on the space of tuples $({R}_{2},{R}_{2},\dots ,{R}_{2})$. These tuples simply live in the category

where the product is just the cartesian product in $\mathrm{Cat}$. So that’s a module catgegory for ${R}_{2}$!

In particular, for the simple case ${R}_{2}={\mathrm{Vect}}_{K}$ that we decided to restrict attention to at the moment, the category $({\mathrm{Vect}}_{K}{)}^{n}$ has

- objects which are n-dimensional vectors whose entries are vector spaces over $K$

- morphisms which are $n$-dimensional vectors whose entries are linear transformations between vector spaces.

You should mentally decategorify everything here while reading this, in order to see how very simple this is. Ismomorphism classes of vector spaces are just given by the dimension of the vector space, so passing to isomorphism classes reduces a vector whose entries are vector spaces to a vector whose entries are natural numbers.

That’s what it means to say that ${\mathrm{Vect}}_{K}$ is a categorification of the natural numbers.

Once one gets used to it, this game can be continued. The 2-vector spaces $({\mathrm{Vect}}_{K}{)}^{n}$ are objects of a 2-category (the 2-category of ${R}_{2}$ module categories) whose

- objects are $({\mathrm{Vect}}_{K}{)}^{n}$ for all $n$

- 1-morphisms are matrices whose entries are $K$-vector spaces

- 2-morphisms are matrices whose entries are linear maps between the entries of the 1-morphisms.

These KV-2-vector spaces are what Baas, Dundas and Rognes proposed to use for their program of categorifying K-theory.

However, there is an obvious problem here. As I have emphasized above, ${R}_{2}={\mathrm{Vect}}_{K}$ is really like a categorification of the *natural numbers*. Hence, far from being a categorified field, it is not even a categorified ring, but just a semi-ring. This means that there are very few 2-linear operators of KV 2-vetor spaces which are (weakly) invertible. This again means that the (2-)group of automorphisms of 2-vector spaces which we are after (which we want to insert in the above formula for categorified K-theory) is pretty small. Too small.

This is a problem that has to be dealt with. There are several ways one could remedy this. Choosing one of these ways constitutes the third step in the BDR approach

**Step 3: Equip the 2-category of KV-2-vector spaces in a suitable way with with additional inverses.**

Before saying anything about how BDR deal with this, I would like to emphasize how this question looks like from the broader perspective of module categories.

The role of categorified linear operators is played by the 1-morphsims in the 2-category of ${R}_{2}$ module categories. The subcollection of these morphisms which are weakly invertible form the *Picard group* (think of the relation to bimodules mentioned above). Even though in the literature this group this is usually treated in form of its decategorification, it is actually a weak 2-group. It is this 2-group, the “Picard 2-group” which we would probably want to insert, in general, into the definition of categorified K-theory given above in part 1).

The Picard group of KV-2-vector spaces is very small. For other choices of module categories, however, one gets interesting Picard groups. One could for instance consider looking not just at internal bimodules, but at *derived* categories of internal bimodules.

Adopting some string theory imagery, we might think of (bundles of) vector spaces as stacks of D-branes. The fact that there are no additive inverses in ${\mathrm{Vect}}_{K}$ is, in this picture, related to the fact that you may stack D-branes on top of each other, but not annihilate them. Not, that is, unless you introduce anti-D-branes. Mathematically, the introduction of anti-D-branes is modeled by passing to derived categories. An ordinary vector bundle (or sheaf, if you like) is a complex of vector bundles concentrated in degree zero. An “anti” vector bundle (a bundle of categorified *negative* numbers) is a complex of bundles concentrated in degree -1. By using $\pi $-stability in triangulated categories these may mutually “annihilate” – the categorification of subtraction.

So there are very natural notions of 2-vector spaces in terms of module categories which do have highly nontrivial Picard groups, known as the *derived Picard group* in the case of derived categories. For instance, it is known that the derived Picard group of bimodules of path algebras of certain quivers encodes Seiberg duality on supersymmetric gauge theories defined by these quivers. Hence it might be tempting to study 2-vector bundles in this sense. Some comments along these lines are given in section 4.4 of this.

This is mainly to point out that there seem to be many different possibly interesting ways to proceed along the lines sketched by BDR.

Of course, there is also a much more straightforward approach to deal with the lack of invertibles in KV-2-vector space morphisms. That’s to do group completions by hand. And this is essentially what BDR do.

More precisely, while few $n\times n$-matrices with entries in $N$ are invertible, there are many more invertible $n\times n$-matrices if we allow the entries to take values in $\mathbb{Z}$. Moreover, many matrices with only non-negative entries do have inverses which contain negative inverses.

So, there is a category of matrices with values in vector spaces such that the isomorphism classes of these matrices, which are labelled by matrices with values in $\mathbb{N}$, are invertible in ${\mathbb{Z}}^{n\times n}$. Call the category of such matrices ${\mathrm{GL}}_{n}({\mathrm{Vect}}_{K})$. *This* is the category which BDR insert into the above formula for categorified K-theory.

**Step 4: What does this have to do with 2-Bundles??**

One can show that, similar to how K-theory is related to the classification of bundles, BDR’s categorified K-theory is related to the classification of KV-2-vector 2-bundles. This are bundles whose fibers are KV-2-vector spaces.

I’ll say more about this in a future entry.

## Re: Seminar on 2-Vector Bundles and Elliptic Cohomology, I

Re the title of the Kapranov-Voevodsky paper: has all this formalism led to any interesting solutions to the tetrahedron equation? To find such solutions was really my motivation. After all, the tetrahedron equation is nothing but vanishing 4-curvature in 3-gauge theory. Needless to say, I failed to find any such solution.

AFAIK, no satisfactory solution of the tetrahedron equation has been found at all. Zamolodchikov’s original solution was not unitary (Baxter reformulated it as a lattice model with some Boltzmann weights negative). There are also layered solutions, which can be expressed entirely in terms of Yang-Baxter matrices. Several people have published solutions of this type, including myself, but I have forgotten the details.

One problem is that there seems to be no interesting solution close to unity. If we into the tetrahedron equation

R_123 R_145 R_246 R_356 = R_356 R_246 R_145 R_123

plug the Ansatz R_ijk = 1 + r_ijk + …

we find to lowest order

[r_123, r_145] + 5 more terms = 0.

Of these 6 terms, only the first can act non-trivially on all of the first five factors. Hence one must indeed assume that

r_123 = r_12 + r_13 + r_23 + …