The n-Category Café

I want to talk about

$\;\;\bullet$ twisted $n$-bundles with connection over $d$-dimensional base space

coupled to

$\;\;\bullet$ (electrically) charged $n$-particles (($n-1$)-branes);

how they are

$\;\;\bullet$ sections with covariant derivatives of $(n+1)$-bundles with connection

which can be interpreted as

$\;\;\bullet$ obstructions to lifts through extensions of $n$-groups

or equivalently

$\;\;\bullet$ magnetic charges

of

$\;\;\bullet$ magnetically charged ($d-n-1$)-particles (($d-n-2$)-branes).

A crucial new ingredient compared to my former (I,II) discussion of sections of $n$-bundles is the method from groupoidification: think of an $n$-representation of an $n$-group not as an $n$-functor, but in terms of the corresponding action $n$-groupoid, as described more recently in $L_\infty$-associated bundles and sections.

Much of what I say is, in the language of generalized differential cohomology, in the great

D. S. Freed
Dirac Charge Quantization and Generalized Differential Cohomology
(arXiv)

only that what I describe in the language of $\infty$-parallel transport/$L_\infty$-connections (pdf, arXiv, blog, brief slides, detailed slides) is supposed to be a nonabelian generalization of the purely abelian treatment which allows to not just say things like

The Green-Schwarz mechanism says that the Kalb-Ramond 2-bundle coupling to the electric string is twisted by magnetic 5-brane charge.

(discussion around equation (3.5) in Freed’s article)

But also things like

The Green-Schwarz mechanism says that over the 10-dimensional boundary of 11-dimensional supergravity base space the supergravity Chern-Simons 3-bundle obstructing the lift of an $SO(10,1) \times E_8$-bundle to a $String(SO(10,1))\times E_8)$-2-bundle trivializes by admitting a global 2-section: the twisted Kalb-Ramond 2-bundle.

(previewed in section 3 of $L_\infty$-connections)

where the underlying $SO(10,1) \times E_8$-bundle as well as its String-2-bundle lift with connections represent nonabelian differential cohomology.

LaTeXified notes on what I will talk about are beginning to evolve as

Sections and covariant derivatives of $L_\infty$-algebra connections
(pdf, blog, Bruce Bartlett’s recollection)

and

Twisted $L_\infty$-connections
(pdf).

By the winter of 1897-8, the jobless philosopher Charles Peirce was financially crippled, continuing his studies despite the cold, unable to heat his house. Regretting this situation, William James, at Harvard, organised for Peirce to give a series of lectures for much needed remuneration. Peirce wanted to talk about formal logic, but James persuaded him that this would severely reduce his audience (“Now be a good boy and think a more popular plan out. I don’t want the audience to dwindle to 3 or 4…”), and that he would be better off discussing “topics of a vitally important character”.

He did adjust the content somewhat, as you can see in the published lectures Reasoning and the Logic of Things, (Kenneth Ketner ed., Harvard University Press, 1992), where you can read some of the correspondence between Peirce and James.

Over the years, Dan Freed, Michael Hopkins and I. M. Singer and others have been developing the theory of Generalized differential cohomology and applied it with great success to various problems appearing in the theory of charged $n$-particles, usually known as NS-branes and D-branes and M-branes.

The idea is:

Given a class $\omega$ in the (generalized) cohomology $\Gamma^\bullet(X)$ of a space $X$, regard it as classifying an $n$-bundle-like thing and then find a way to equip that with something like a connection $\nabla$, such that the curvature differential form $F_\nabla$ of that connection reproduces the image of $\omega$ in deRham cohomology (with coefficients in the ring $\Gamma(pt)$): $$[\omega_\mathbb{R}] = [F_\nabla] \,.$$

When $\Gamma^\bullet(-) = H^\bullet(-,\mathbb{Z})$ is ordinary integral cohomology, this reproduces the notion of Cheeger-Simons differential characters, which is a way of talking about equipping line $n$-bundles ((n-1) gerbes) with a connection.

The notion of generalized differential cohomology allows to go beyond that and equip any other kind of cohomology class with a corresponding notion of “connection and curvature”. This has notably been applied to the next interesting generalized cohomology theory after ordinary integral cohomology: K-theory. It turns out that the differential forms appearing in differential K-theory model the RR-fields appearing in string theory.

Here I try to review some basics, provide some links – and then start to relate all this to the theory of parallel $\infty$-transport and $L_\infty$-connections.

Yay!

Julia Bergner has accepted a tenure-track position in the math department here at U. C. Riverside. She’s just finishing her third year as a postdoc at the University of Kansas. She’s done a lot of important work on the ‘homotopy theory of homotopy theories’.

So, life here at UCR just got a lot more interesting for people who like $n$-categories.

While talking to people who will hold still and listen, like when I talk to Bruce Bartlett or to Danny Stevenson, I realized that while I talked about integration theory of Lie $\infty$-algebras here and there, I might not have gotten my point across succinctly.

So, as a kind of meta-response to some aspect of Bruce’s highly appreciated exegesis. I have now prepared a manifesto:

Impressions on $\infty$-Lie theory
pdf (6 pages)

Abstract. I chat about some of the known aspects of the categorified version of Lie theory – the relation between Lie $\infty$-algebras and Lie $\infty$-groups – indicate how I am thinking about it, talk about open problems to be solved and ideas for how to solve them.

This starts with very roughly reviewing the basic ideas underlying the work by Getzler, Henriques and Ševera, then exhibits the “shift in perspective” which I keep finding helpful and important, mentions applications like the strict integration of the String Lie 2-algebra using strict path 2-groupoids and ends by quickly sketching how that relates the “fundamental problem of Lie $\infty$-theory” to the relation between smooth spaces and “$C^\infty$DGCAs” which we are having a long discussion about with Todd Trimble and Andrew Stacey, scattered over various threads (notably here, and here).

At the end of the notes I indicate how I am currently trying to address this issue. But I need more time to work this out. Comments are appreciated.

Guest post by Bruce Bartlett

Dear reader,

I’m sure you’ll agree with me that there is a remarkable person on this blog : Urs. The rate at which he produces new posts and deep ideas is nothing short of a phenomenon. Indeed, he is so fast that perhaps many of you are like me and have been left in the dust long ago!

If so, this post is for you! I was lucky enough to have Urs visit me recently, and after much patience on his part I think I am finally beginning to see the first glimmers of daylight. Let me mention some of the things he explained to me; perhaps it will help some of you to understand what Urs has been going on about.

Points of a set, $X$, correspond to certain maps from the Boolean algebra of subsets, $P(X)$, to $2$, namely those corresponding to prime ideals of the algebra.

Points of a space, $Y$, correspond to certain functors from the topos of locally constant sheaves to Set, via evaluation at a point again. Is there a way to construe this by analogy to the prime ideal story? Is there a ‘spectrum’ around?

How does one characterise the fibre functors to Set which correspond to points? Is there something ‘ideal’ going on?

Cartier’s Mad Day’s Work paper seems to suggest there is such a story going on here. In the same paragraph (p. 404) as the description of the fundamental group as the automorphism group of a fibre functor, he speaks of the Galois group of a field extension in terms of the fields’ spectra.

Over at the Secret Blogging Seminar Noah Snyder is reporting on talks Jacob Lurie gave on extended TQFT, which is the theory of representations of $\infty$-categories of cobordisms (in contrast to ordinary TQFT, which is just representations of mere 1-categories of cobordisms):

Noah Snyder
Jacob Lurie on 2-d TQFT

From Noah’s notes, the talks mostly centered on the observation by John Baez and James Dolan

John C. Baez, James Dolan
Higher-dimensional Algebra and Topological Quantum Field Theory
arXiv:q-alg/9503002v2

that this should be essentially about representing the free stable $\infty$-groupoid and that hence such a representation is fixed by choosing just one object, the image of the point, with suitable dualities on it. See also our recent discussion about that here.

The extended TQFT would then be an assignment of this object to the point, and of the $n$-fold higher “trace” on this object to closed $n$-dimensional manifolds.

In his talk Lurie apparently talked about some new classification results on this. With a little luck, more details will percolate through to us eventually.

Incidentally, while typing this I am on my way back from Edinburgh to Sheffield to meet Bruce Bartlett again. Bruce is writing his PhD thesis, advised by Simon Willerton, on a beautiful description of (higher) Dijkgraaf-Witten theory – a finite group version of Chern-Simons theory – in this extended sense. He finds plenty of fascinating relations between these “higher traces” and finite group representation theory, providing a useful blueprint for and nice insights into what fully extended Chern-Simons theory has to eventually look like.

Mike Stay and I are writing a paper for a book Bob Coecke is editing: New Structures for Physics. The deadline is coming up soon, and we need your help!

• John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone. (Draft version: rose.pdf.)

We’d really love comments on the ‘logic’ section, because neither of us are professional logicians. I haven’t included the ‘computation’ section in this draft, since it’s embarrassingly far from finished… but Mike knows computation.

At Riverside we recently heard a fascinating talk:

• Bertram Kostant, On Some Mathematics in Garrett Lisi’s ‘E8 Theory of Everything’, U. C. Riverside, February 12th.
  
  Abstract: A physicist, Garrett Lisi, has published a highly controversial, but fascinating, paper purporting to go beyond the Standard Model in that it unifies all 4 forces of nature by using as gauge group the exceptional Lie group $E_8$. My talk, strictly mathematical, will be about an elaboration of the mathematics of $E_8$ which Lisi relies on to construct his theory.

Luckily we had a video camera on hand. So, at the above link you can see streaming or downloadable videos of Kostant’s talk, as well as lecture notes.

Andreas Döring writes:

Dear all,

we hereby wish to invite you to participate in the second workshop on “Categories, Logic and Foundations of Physics”, which will take place at Imperial College on

Wednesday, 14th May 2008, 11:00 - 18:45, Lecture Theatre 3, Blackett Laboratory.

Our workshop series is aimed at nourishing research in the fields named in the title and at bringing together scientists from the different fields involved. The success of the first workshop already showed that there is substantial interest in these topics and a great potential for collaborations. With the second workshop, we hope to keep and increase the momentum.

Please also have a look at the website.

The videos and slides of the January workshop are online now.

Last Tuesday, the Harvard Faculty of Arts and Sciences voted overwhelmingly to make their research papers freely available online!

Under the new system, faculty will deposit finished papers in an open-access repository run by the library. The papers will instantly become available for free on the Internet. Authors will still retain their copyright. So, they can still publish anywhere they want — as long as the journal is okay with this. (Most are.)

Sounds like the arXiv, right? One difference is that this covers all faculty in the arts and sciences, including departments where free online access is not a given. And, instead of an “opt-in” system, Harvard is adopting an “opt-out” system: all papers will be included unless the author specifically requests them not to be.

Let’s get our universities to do the same thing! I know the UCR librarians are interested.

Next week I’ll be travelling through Great Britain.

Ezra Getzler has kindly invited me to speak at the London Geometry and Topology Seminar on Monday Feb 18.

After that I will take a train to Sheffield where I’ll meet Bruce Bartlett (and possibly Simon Willerton and Eugenia Cheng and …?) to chat about extended Chern-Simons TQFT (I, II, III).

Then on Wednesday Feb 20 I am invited to speak at the Edinburgh Mathematical Physics Group Seminar.

After that I go back to Sheffield to continue chatting with Bruce (and possibly Simon and Eugenia and …?).

I suppose I will give blackboard talks, but I like to prepare slides anyway.

$L_\infty$-connections and applications to String- and Chern-Simons $n$-transport
(slides, pdf article, arXiv, blog)

(The slides contain plenty of hyperlinks which allow you to navigate through them – and in particular to skip over various details. )

Of course there also are the closely related very extensive slides I prepared last time:

On String- and Chern-Simons $n$-Transport
(pdf slides (long, don’t try to print!) , blog)

Next month, March 17, Gregory Ginot has kindly invited me to give a talk in Paris.

Then in April I will be travelling to bridge some time, first visiting Jens Fjelstad # then Stephan Stolz # and finally Jim Stasheff, before the summer starts, with its Trimester Program in Geometry and Physics in Bonn.

There, they still don’t seem to have a list of participants, and it was only yesterday that I learned that my co-author Hisham Sati will be at the program, too! That should give us the opportunity to work more intensely on Fivebrane structures, the followup to our $L_\infty$-connections…

Let’s boldly venture on with our ascent of Mount 2-Logic.

Todd told us about Galois connections via relations

any relation $R \subseteq A \times B$ whatsoever can be used to set up a Galois connection between subsets $A'$ of $A$ and subsets $B'$ of $B$: $$\frac{A' \subseteq \{a \in A: \forall_{b: B'} (a, b) \in R\} }{B' \subseteq \{b \in B: \forall_{a: A'} (a, b) \in R\} } iff$$

And he also told us that the Galois connection relating theories to symmetry groups works by way of a relation:

In our case, $A$ is the group of permutations $g$ on $X$, $B$ is the set of finitary relations $p$ on $X$, and the relation $R$ is the set of $(g, p)$ such that $p(x_1, \ldots, x_n) = p(g x_1, \ldots, g x_n)$ [for all $n$-tuples $(x_1, \ldots, x_n)$ if $p$ is $n$-ary].

Anyone who wants to keep up with the latest research on $n$-categories needs to know what the Australians are doing. It helps to be on the Australian Category Seminar mailing list… which is free if you visit. (Maybe it’s free otherwise, too — I don’t know.)

Dominic Verity has been doing wonderful things lately, and here’s what he’s talking about tomorrow…

Just as I blog about Albert Lautman, a conference dedicated to his work comes along. Plenty of interesting looking talks there, including Fernando Zalamea’s “Transferts et obstructions en mathématiques, de Galois à Grothendieck: un parcours conceptuel minimal suivant le filtre de l’oeuvre de Lautman”.

Fernando wrote a favourable review of my book, and has himself written a book – Filosofia Sintetica de las
Matematicas Contemporaneas – to appear this year. If only my Spanish were better, or a translation were available.

This time in the Geometric Representation Theory seminar, we showed how to categorify the commutation relations between annihilation and creation operators for the harmonic oscillator:

$$a a^* = a^* a + 1$$

obtaining an isomorphism of spans of groupoids:

$$A A^* \cong A^* A + 1$$

This reduces a basic ingredient of quantum field theory to pure combinatorics, not involving the continuum in any form.

Even better, we did an in-class experiment demonstrating these commutation relations!

In the old article

Jean-Luc Brylinksi and Dennis McLaughlin
A geometric construction of the first Pontryagin class
Quantum Topology, 209-220
(review)

a concrete construction of a Čech cocycle (= local transition data) for a line 3-bundle (aka 2-gerbe) classified by the first Pontrjagin class $p_1(P) \in H^4(X,\mathbb{Z})$ of a given principal $G$-bundle $P \to X$ is discussed, following

Jean-Luc Brylinksi and Dennis McLaughlin
Čech cocycles for characteristic classes
Comm. Math. Phys. 178 (1996)
(pdf, review).

The point of the first article is to interpret the construction of the second in terms of the holonomy of the gerbe on $G$.

Here I want to discuss the following:

- how the fundamental 2-groupoid $\Pi_2(X_{CE(g_\mu)})$ of the smooth space obtained from the small version $g_\mu$ of the String Lie 2-algebra (section 6.4.1) as described here is another version of the strict String 2-group

$$\mathbf{B} String'(G) = \Pi_2(X_{CE}(g_\mu))$$

sitting inside the sequence (proposition 20)

$$1 \to \mathbf{B} U(1) \to String'(G) \to G \to 1$$

- how the construction described by Brylinski and McLaughlin is a special case of the construction of obstructing $n$-bundles for the attempted lift through the above sequence, using weak cokernels as described in the integral way used in the following from slide 518 on and as corresponds differentially to the construction in section 8 of $L_\infty$-connections and applications (pdf, blog, arXiv).

There must be some standard textbook reference for the following statement:

The functor from smooth manifolds to algebras which sends each smooth manifold to the algebra of smooth functions on it, and which sends morphisms of smooth manifolds to the algebra homomorphisms obtained by the corresponding pullbacks of functions $$(X \stackrel{\phi}{\to} Y) \mapsto ( C^\infty(X) \stackrel{\phi^*}{\leftarrow}C^\infty(Y))$$ is full: every algebra homomorphism between algebras of smooth functions comes from a pullback along a smooth map of the underlying smooth manifolds.

This is of course closely akin to Gel’fand’s equivalence (for instace recalled as theorem 1 in Spaceoids), although a little different.

A proof should consist of two steps:

a) every homomorphism of function algebras comes from pullback along a map of the underlying sets.

b) only pullback along smooth maps will take all smooth functions to smooth functions.

What’s a good reference for this?

guest post by Tom Leinster

While the world goes wild for categorification (and the part that doesn’t, should!), I seem to be fixated on decategorification — or counting as it’s popularly known. And right now I’m obsessed with counting metric spaces.

We’ve talked about how you can count a category. There, counting went under other names still, ‘Euler characteristic’ and ‘cardinality’. With your mind on autopilot, you can extend this definition to enriched categories. And a metric space is a special kind of enriched category! So we obtain a definition of the cardinality of a metric space.

I’ve known for a while that this can be done, but it’s only in the last few weeks that I’ve started to see the point. Below, I explain where this idea leads: to the geometry of convex sets, to fractal dimension, and to how the world looks when you take off your glasses.

This is also a cry for help! It seems that there’s a really nice story to be told, but I’m frustratingly stuck on a couple of elementary lemmas. They’re probably the sort of thing that a bright schoolkid could settle — but they’ve defeated me.

We have now a detailed description and proof of the relation between

- strict smooth 2-functors on 2-paths in a manifold with values in a strict Lie 2-group $G_{(2)}$

and

- differential forms with values in the corresponding Lie 2-algebra $Lie(G_{(2)})$:

U.S. and Konrad Waldorf
Smooth Functors vs. Differential Forms
arXiv:0802.0663v1

Abstract: We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This way we set up a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in several fields, namely as connections on (non-abelian) gerbes, as the curvature of smooth functors and as critical points in BF theory. We demonstrate further that our dictionary provides a powerful tool to discuss the transgression of geometric objects to loop spaces.

I am continuing to look at the structure of $(4-d)$-states of Chern-Simons theory over $d$-dimensional manifolds by using the $L_\infty$-algebraic model of the Chern-Simons 3-bundle ($\sim$ 2-gerbe) with connection over $B G$ discussed in section 7.3 of $L_\infty$-Connections and Applications (pdf, blog, arXiv).

The program is: transgress this 3-bundle as in section 9.2. to spaces of maps from a $d$-dimensional parameter space, and then compute the collection of $(4-d)$-sections following the $L_\infty$-algebraic description for computing $\infty$-sections of the general concept of $\infty$-sections.

Last time I looked at the sections over 2-dimensional surfaces, that computation remaining somewhat inconclusive, as the holomorphic structure on the transgressed 1-bundle does not appear yet.

Today I instead looked at the states over 1-dimensional manifolds: over circles.

I find in section 4.1 of

Sections and covariant derivatives of $L_\infty$-algebra connections
(pdf, blog)

that the sections of the 2-bundle obtained from transgression of the Chern-Simons 3-bundle to the configuration space over the circle come from bundles of representations of the Kac-Moody central extension of the Lie algebra of loops over the space of $g$-holonomies over the circle.

And I think this is what the result should be, though a couple of details deserve a closer look.

One nice byproduct is this:

the computation explicitly gives a derivation in the present context of the fact that the 2-cocycle on the loop group

$$tg_{S^1} \mu(f,g) = \int_{S^1} \langle f \wedge d g \rangle$$

comes from the very transgression we are talking about of the 3-cocycle

$$\mu(x,y,z) = \langle x, [y,z]\rangle$$

on the semisiple Lie algebra $g$ that the Chern-Simons 3-bundle is governed by.

This crucially depends on some gymnastics with that “almost-internal-hom”

$$(A, B) \mapsto \Omega^\bullet(\mathrm{maps}(A,B))$$ of dg-algebras which we talked about in Transgression of $n$-Transport (section 5.1).

Sifting through some old papers yesterday, I was reminded of my first forays into philosophy of mathematics while living in Paris. One very important influence on me was Albert Lautman (1908-1944). (The French Wikipedia has a little more on him than the English.) He’s often paired with Jean Cavaillès, understandably as they were friends, philosophers of mathematics, and both died as members of the French Resistance. Somehow I never got on with Cavaillès, even though he’s found greater favour in the Anglophone world. In Lautman, on the other hand, I recognised the kind of aesthetic sensibility I was later to enjoy in Mac Lane’s Mathematics: Form and Function and in This Week’s Finds.

I haven’t read Lautman in around 18 years, aside from a stolen hour spent in the Bodleian in Oxford, so went to see what was available on the Web. Not much it appears, but there is at least a translation of the introduction to Essai sur les notions de structure et d’existence en mathématiques: Les schémas de structure. Here are a couple of excerpts:

The structural design and the dynamic design of mathematics first of all seem to oppose themselves: the one indeed tends to regard a mathematical theory as a completed whole, independent of time, the other on the contrary does not separate the temporal stages from its development; for the former, the theories are like beings qualitatively distinct from each other, while the latter sees in each one an infinite power of expansion beyond its limits and connection with the others, because it affirms itself as the unity of the intelligence. We would however like, in the following pages, to try to develop a design of mathematical reality where the fixity of logical notions is combined with the movement that lives through these theories.

Jim Dolan and I are trying to learn about modular forms and the Modularity Theorem, once known as the Taniyama–Shimura–Weil Conjecture. It’s an irresistible challenge. After all, this result implies Fermat’s Last Theorem, but it’s much more conceptual, and closely related to the Hecke operators we’re always talking about.

I think we’re making decent progress — Diamond and Shurman’s book is very helpful. But, I’m still lacking in intuition in many ways.

For example: I think I have a decent intuition for level-1 modular forms. It took me about 10 years. Now I want to understand higher levels equally well. I’m hoping it won’t take another decade.

In the thread $L_\infty$ associated Bundles and Sections I started talking about how to compute the space of states of Chern-Simons theory using the $L_\infty$-algebraic model for the Chern-Simons 2-gerbe with connection on $B G$ that we describe in $L_\infty$-connections and applications (pdf, blog, arXiv).

Prompted by a request for more references on this question which I received, I shall try to collect some (incomplete) list of literature here, with some comments.

The general setup is as follows, and the various approaches to it may differ in terms of which concrete models are used to make sense of the various objects mentioned now:

For $G$ any suitably well behaved Lie group, there is supposed to be a canonical (family of) line 3-bundles (= 2-gerbes) $CS$ with connection (“and curving”) $\nabla$ over the space $B G$.

$$\array{ CS_\nabla \\ \downarrow \\ B G } \,.$$

For any 2-dimensional manifold $\Sigma$ we should be able to transgress this to a line bundle with connection $tg_\Sigma CS_\nabla$ on a suitable space $Maps(\Sigma, B G)$ of maps from $\Sigma$ to $B G$

$$\array{ tg_\Sigma CS_\nabla &&& \Sigma \times Maps(\Sigma, B G) &&& CS_\nabla \\ & \searrow & {}^{p_2} \swarrow && \searrow^{ev} & \swarrow \\ && Maps(\Sigma, B G) && B G } \,.$$

There happens to be a complex structure appearing which makes this a holomorphic line bundle. The space of states of Chern-Simons theory over $\Sigma$ is supposed to be the space of holomorphic sections $$Z(\Sigma) := \Gamma_{hol}(tg_\Sigma CS)$$ of $tg_\Sigma CS_\nabla$.

This space, in turn, has a “holographically” related interpretation in terms of the space of “pre-correlators” of another theory which comes from a line 2-bundle (= gerbe) on $G$: Wess-Zumino-Witten theory.

While much of the literature addresses both the Chern-Simons as well as the Wess-Zumino-Witten aspect, here I will concentrate mostly on the Chern-Simons aspect.