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August 16, 2006

This Week's Finds in Mathematical Physics (Week 239)

John Baez

David Corfield, Urs Schreiber and I have started up a new blog!

David is a philosopher, Urs is a physicist, and I'm a mathematician,

but one thing we all share is a fondness for n-categories. We

also like to sit around and talk shop in a public place where our

friends can drop by. Hence the title of our blog:

1) The n-Category Cafe, http://golem.ph.utexas.edu/category/

Technologically speaking, the cool thing about this blog is that it

uses itex and MathML to let us (and you) write pretty equations in TeX.

For this we thank Jacques Distler, who pioneered the technology on

his own blog:

2) Jacques Distler, Musings, http://golem.ph.utexas.edu/~distler/blog/

Urs began by posting about 11d supergravity and higher gauge theory

(see "week237"). Now he's discussing Barrett and Connes' new work

on the Standard Model. Meanwhile, I've been obsessed with the

categorical semantics of quantum computation, and David has been

running discussions on categorifying Klein's Erlangen program (see

"week213"), the differences between mathematicians and historians

when it comes to writing histories of math, and so on.

And, it's all free.

Meanwhile, in the bad old world of extortionist math publishers,

we see a gleam of hope. The entire editorial board of the journal

Topology resigned to protest Reed-Elsevier's high prices!

3) Topology board of editors, letter of resignation,

http://math.ucr.edu/home/baez/topology-letter.pdf

The board includes some topologists I respect immensely. It takes

some guts for full-fledged memmbers of the math establishment to

do something like this, and I congratulate them for it. It'll be

fun to see what stooges Reed-Elsevier rounds up to form a new board

of editors. I can't imagine they'll just declare defeat and let the

journal fold.

This is part of trend where journal editors "declare independence"

from their publishers and move toward open access:

4) Open Access News, Journal declarations of independence,

http://www.earlham.edu/~peters/fos/lists.htm#declarations

Speaking of open access, you can now get the notes from the course

Freeman Dyson taught on quantum electrodynamics when he first

became a professor of physics at Cornell:

5) Freeman J. Dyson, 1951 Lectures on Advanced Quantum Mechanics,

second edition, available as quant-ph/0608140. For historical

context and original mimeographs, see

http://hrst.mit.edu/hrs/renormalization/dyson51-intro/

These notes are from an exciting period in physics, shortly after

the 1947 Shelter Island conference where Feynman and Schwinger

presented their approaches to quantum electrodynamics to an audience

of luminaries including Bohr, Oppenheimer, von Neumann, and Weisskopf.

Nobody understood Feynman's diagrams except Schwinger and maybe

Feynman's thesis advisor, John Wheeler.

Every true fan of physics loves reading about this heroic era and

its figures, especially Feynman. So, if you haven't read these yet,

run to the bookstore and buy them now!

6) James Gleick, Genius: the Life and Science of Richard Feynman,

Vintage Press, 1993.

7) Jagdish Mehra, The Beat of a Different Drum: the Life and Science

of Richard Feynman, Oxford U. Press, 1996.

8) Silvan S. Schweber, QED and the Men Who Made It, Princeton U.

Press, Princeton, 1994.

The first book is a barrel of fun but doesn't get into the nitty-gritty

details of Feynman's work. The second more scholarly treatment also

has lots of Feynman anecdotes - even some new ones! But, it covers

his work in enough detail to intimidate any non-physicist. The third

offers a broader panorama of the development of quantum electrodynamics.

Taken together, they add up to quite a nice story.

Of course, I'm *assuming* you've read these:

9) Richard P. Feynman, Surely You're Joking, Mr. Feynman! (Adventures

of a Curious Character), W. W. Norton and Company, New York, 1997.

10) Richard P. Feynman, What Do *You* Care What Other People Think?

(Further Adventures of a Curious Character), W. W. Norton and Company,

New York, 2001.

They're more fun than everything else I've ever recommended on This

Week's Finds, combined. If you haven't read them, don't just *run* to

the nearest bookstore - get in a time machine, go back, and make sure

you *did* read them.

Today I'd like to wrap up the discussion of Koszul duality which I

began last Week. As we'll see, this gives a really efficient way

of categorifying the theory of Lie algebras and defining "Lie

n-algebras". And, as Urs Schreiber notes, these seem to be just

what we need to understand 11-dimensional supergravity in a nice

geometric way.

But before I dive into this heavy stuff, something fun. Thanks to

Christine Dantas' blog, I just saw a webpage on the origins of math

and writing in Mesopotamia:

11) Duncan J. Melville, Tokens: the origin of mathematics,

from his website Mesopotamian Mathematics,

http://it.stlawu.edu/~dmelvill/mesomath/

Before people in the Near East wrote on clay tablets, there were "tokens":

12) The Schoyen Collection, MS 5067/1-8, Neolithic plain counting

tokens possibly representing 1 measure of grain, 1 animal and 1 man or

1 day's labour, respectively, http://www.nb.no/baser/schoyen/5/5.11/index.html [Broken]

These are little geometric clay figures that represented things like

sheep, jars of oil, and various amounts of grain. They are found

throughout the Near East starting with the agricultural revolution in

about 8000 BC. Apparently they were used for contracts! Eventually

groups of them were sealed in clay envelopes, so any attempt to tamper

with them would be visible.

But, it's annoying to have to break a clay envelope just to see what's

in it. So, after a while, they started marking the envelopes to say

what was inside.

Later, these marks were simply drawn on tablets. Eventually they gave

up on the tokens - a triumph of convenience over security. The marks

on tablets then developed into the Babylonian number system! The

transformation was complete by 3000 BC.

So, five millennia of gradual abstraction led to the writing of numbers!

From three tokens representing jars of oil, we eventually reach the

abstract number "3" applicable to anything.

Of course, all history is detective work. The story I just told is

an interpretation of archaeological evidence. It could be wrong.

This particular interpretation is due to Denise Schmandt-Besserat.

It seems to be fairly well accepted in broad outline, but scholars

are still arguing about it.

For more on her ideas, try this:

13) Denise Schmandt-Besserat, Accounting with tokens in the

ancient Near East,

http://www.utexas.edu/cola/centers/lrc/numerals/dsb/dsb.html [Broken]

For a bibliography of her many papers, try:

14) Denise Schmandt-Besserat, Publications,

http://www.utexas.edu/cola/centers/lrc/iedocctr/ie-pubs/dsb-pubs.html [Broken]

For more work on this subject - I want to read more! - try:

15) Eleanor Robson, Bibliography of Mesopotamian Mathematics,

http://it.stlawu.edu/~dmelvill/mesomath/erbiblio.html

From the distant past, let's now shoot straight into the 20th

century. Last week I gave three examples of Koszul duality:

Making the free graded-commutative algebra on SL* into a differential

graded-commutative algebra is the same as making L into a Lie algebra.

Making the free graded Lie algebra on SL* into a differential

graded Lie algebra is the same as making L into a commutative algebra.

Making the free graded associative algebra on SL* into a differential

graded associative algebra is the same as making L into an associative

algebra.

Here L is a vector space, which we think of as a graded vector space

concentrated in degree zero. L* is its dual, and SL* is the "shifted"

or "suspended" version of L*, where we add one to the degree of

everything.

Now, what if we replace L by a graded vector space that can have stuff

of any degree? We get a fancier version of Koszul duality, which goes

like this:

Making the free graded-commutative algebra on SL* into a differential

graded-commutative algebra is the same as making L into an L-infinity

algebra.

Making the free graded Lie algebra on SL* into a differential

graded Lie algebra is the same as making L into a C-infinity algebra.

Making the free graded associative algebra on SL* into a differential

graded associative algebra is the same as making L into an A-infinity

algebra.

Here an "L-infinity algebra" is a chain complex that's like a Lie

algebra, except the Jacobi identity holds up to a chain homotopy called

the "Jacobiator", which in turn satisfies its own identity up to a

chain homotopy called the "Jacobiatorator", and so on ad infinitum.

Keeping track of all these higher homotopies is quite a chore. Well,

it's sort of fun when you get into it, but the great thing about

Koszul duality is that you don't need to remember any fancy formulas:

all the higher homotopies are packed into the *differential* on SL*.

Similarly, a "C-infinity algebra" is a chain complex that's like a

graded-commutative algebra up to homotopy, ad infinitum.

Similarly, an "A-infinity algebra" is a chain complex that's like an

associative algebra up to homotopy, ad infinitum. Here you can read off

all the higher homotopies from the Stasheff associahedra, which you

know and love from "week144" - but again, Koszul duality means you

don't have to!

As mentioned last week, all this stuff generalizes to any kind of

algebraic gadget in Vect - the category of vector spaces - which is

defined by a "quadratic operad" O. Any such operad has a "Koszul

dual" operad O* such that:

Making the free graded O-algebra on SL* into a differential

graded O-algebra is the same as making L into an O-infinity algebra.

Here O-infinity is an operad in the category of chain complexes

defined by "weakening" O in a systematic way - replacing all the

laws by chain homotopies, ad infinitum. We can define O-infinity

using the "bar construction", as nicely described here:

16) Todd Trimble, Combinatorics of polyhedra for n-categories,

http://math.ucr.edu/home/baez/trimble/polyhedra.html

or in the book by Markl, Schnider and Stasheff:

17) Martin Markl, Steve Schnider and Jim Stasheff, Operads in

Algebra, Topology and Physics, AMS, Providence, Rhode Island, 2002.

See "week191" for more on this book, and what the heck an "operad"

is.

Anyway, I don't have much intuition for how Koszul duality lets

us magically sidestep the bar construction of O-infinity - someday

I hope I'll understand this.

But, once we have the concept of "L-infinity algebra", we can

restrict ourselves to chain complexes that vanish except for their

first n terms - that is, degrees 0, 1, ..., n-1 - and get the

concept of "Lie n-algebra".

In fact, a Lie n-algebra is like a hybrid of a Lie algebra and an

n-category! The definition I just gave says a Lie n-algebra is

an L-infinity algebra which as a chain complex vanishes above

degree n-1. But, such chain complexes are equivalent to strict

n-category objects in Vect! So, we can think of Lie n-algebras as

strict n-categories that do their best to act like Lie algebras, but

with all the laws holding up to isomorphism, with these isomorphisms

satisfying their own laws up to isomorphism, etcetera.

But, the really cool part is that we can do *gauge theory* using

Lie n-algebras instead of Lie algebra, and taking n = 3 we get an

example that seems to explain the geometry of 11d supergravity...

that is, the classical limit of that mysterious thing called M-theory.

For this, you really need to read Urs Schreiber's stuff:

18) Urs Schreiber, Castellani on free differential algebras in

supergravity: gauge 3-group of M-theory,

http://golem.ph.utexas.edu/string/archives/000840.html

19) Urs Schreiber, SuGra 3-connection reloaded,

http://golem.ph.utexas.edu/category/2006/08/sugra_3connection_reloaded.html

and many other things he's been writing on the n-Category Cafe lately.

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Quote of the Week:

I never once doubted that I would eventually succeed in getting to the

bottom of things. - Alexander Grothendieck

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Previous issues of "This Week's Finds" and other expository articles on

mathematics and physics, as well as some of my research papers, can be

obtained at

http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try

http://math.ucr.edu/home/baez/twfcontents.html

A simple jumping-off point to the old issues is available at

http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

http://math.ucr.edu/home/baez/this.week.html

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# This Week's Finds in Mathematical Physics (Week 239)

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