John Baez
May21-06, 04:00 AM
Also available at http://math.ucr.edu/home/baez/week233.html
May 20, 2006
This Week's Finds in Mathematical Physics (Week 233)
John Baez
On Tuesday I'm supposed to talk with Lee Smolin about an idea he's
been working on with Fotini Markopoulou and Sundance Bilson-Thompson.
This idea relates the elementary particles in one generation of the
Standard Model to certain 3-strand framed braids:
1) Sundance O. Bilson-Thompson, A topological model of composite preons,
available as hep-ph/0503213.
2) Sundance O. Bilson-Thompson, Fotini Markopoulou, and Lee Smolin,
Quantum gravity and the Standard Model, hep-th/0603022.
It's a very speculative idea: they've found some interesting relations,
but nobody knows if these are coincidental or not.
Luckily, one of my hobbies is collecting mysterious relationships between
basic mathematical objects and trying to figure out what's going on.
So, I already happen to know a bunch of weird facts about 3-strand braids.
I figure I'll tell Smolin about this stuff. But if you don't mind, I'll
practice on you!
So, today I'll try to tell a story connecting the 3-strand braid group,
the trefoil knot, rational tangles, the groups SL(2,Z) and PSL(2,Z),
conformal field theory, and Monstrous Moonshine.
I've talked about some of these things before, but now I'll introduce
some new puzzle pieces, which come from two places:
3) Imre Tuba and Hans Wenzl, Representations of the braid
group B_3 and of SL(2,Z), available as math.RT/9912013.
4) Terry Gannon, The algebraic meaning of genus-zero, available as
math.NT/0512248.
You could call it "a tale of two groups".
On the one hand, the 3-strand braid group has generators
| | |
\ / |
A = / |
/ \ |
| | |
and
| | |
| \ /
B = | /
| / \
| | |
and the only relation is
ABA = BAB
otherwise known as the "third Reidemeister move":
| | | | | |
\ / | | \ /
/ | | /
/ \ | | / \
| \ / \ / |
| / = / |
| / \ / \ |
\ / | | \ /
/ | | /
/ \ | | / \
On other hand, the group SL(2,Z) consists of 2x2 integer matrices
with determinant 1. It's important in number theory, complex analysis,
string theory and other branches of pure mathematics. I've described
some of its charms in "week125", "week229" and elsewhere.
These groups look pretty different at first. But, there's a
homomorphism from B_3 onto SL(2,Z)! It goes like this:
1 1
A |->
0 1
1 0
B |->
-1 1
Both these matrices describe "shears" in the plane. You may enjoy
drawing these shears and visualizing the equation ABA = BAB in these
terms. I did.
I would like to understand this better... and here are some clues.
The center of B_3 is generated by the element (AB)^3. This
element corresponds to a "full twist". In other words, it's
the braid you get by hanging 3 strings from the ceiling, grabbing
them all with one hand at the bottom, and giving them a full 360-degree
twist:
| | |
\ / |
/ | A
/ \ |
| \ /
| / B
| / \
\ / |
/ | A
/ \ |
| \ /
| / B
| / \
\ / |
/ | A
/ \ |
| \ /
| / B
| / \
| | |
This full twist gets sent to -1 in SL(2,Z):
-1 0
(AB)^3 |->
0 -1
So, the double twist gets sent to the identity:
1 0
(AB)^6 |->
0 1
In fact, Tuba and Wenzl say the double twist *generates* the kernel
of our homomorphism from B_3 to SL(2,Z). So, SL(2,Z) is isomorphic
to the group of 3-strand braids modulo double twists!
This reminds me of spinors... since you have to twist an electron
around *twice* to get its wavefunction back to where it started.
And indeed, SL(2,Z) is a subgroup of SL(2,C), which is the double
cover of the Lorentz group. So, 3-strand braids indeed act on the
state space of a spin-1/2 particle, with double twists acting
trivially!
(For more on this, check out Trautman's work on "Pythagorean spinors"
in "week196". There's also a version where we use integers mod 7,
described in "week219".)
If instead we take 3-strand braids modulo full twists, we get the
so-called "modular group":
PSL(2,Z) = SL(2,Z)/{+-1}
Now, SL(2,Z) is famous for being the "mapping class group" of the torus -
that is, the group of orientation-preserving diffeomorphisms, modulo
diffeomorphisms connected to the identity. Similary, PSL(2,Z) is famous
for acting on the rational numbers together with a point at infinity
by means of fractional linear transformations:
az + b
z |-> -------
cz + d
where a,b,c,d are integers and ad-bc = 1. The group PSL(2,Z) also
acts on certain 2-stand tangles called "rational tangles". In
"week229", I told a nice story I heard from Michael Hutchings,
explaining how these three facts fit together in a neat package.
But now let's combine those facts with the stuff I just said!
Since PSL(2,Z) acts on rational tangles, and there's a homomorphism
from B_3 to PSL(2,Z), 3-strand braids must act on rational tangles.
How does that go?
There's an obvious guess, or two, or three, or four, but let's just
work it out.
I just said that the 3-strand braid A gets mapped to this shear:
1 1
A |->
0 1
In "week229" I said what this shear does to a rational tangle.
It gives it a 180 degree twist at the bottom, like this:
| | | |
| | | |
| | | |
------- -------
| T | |----> | T |
------- -------
| | \ /
| | /
| | / \
Next, Tuba and Wenzl point out that
0 1
ABA = BAB |->
-1 0
which is a quarter turn. From "week229" you can see how this quarter
turn acts on a rational tangle:
| | | |
| | ____ | |
| | / \ | |
------- | ------- |
| T | |----> | | T | |
------- | ------- |
| | | | \____/
| | | |
| | | |
It gives it a quarter turn!
From these facts, we can figure out what the braid B does to a
rational tangle. So, let me do the calculation.
Scribble, scribble, curse and scribble.... Eureka!
Since we know what A does, and what ABA does, we can figure out
what B must do. But, to make the answer look cute, I needed a
sneaky fact about rational tangles, which is that A *also* acts
like this:
| | \ /
| | /
| | / \
------- -------
| T | |----> | T |
------- -------
| | | |
| | | |
| | | |
This is proved in Goldman and Kauffman's paper cited in "week228".
With the help of this, I can show B acts like this:
| | | |
| | | ___ |
| | | / \ |
------- | / -------
| T | |----> \ | T |
------- / \ -------
| | | \___/ |
| | | |
| | | |
And this is *great*! It means our action of 3-strand braids on
rational tangles is really easy to describe. Just take your tangle
and let the upper left strand dangle down:
|
____ |
/ \ |
| -------
| | T |
| -------
| | |
| | |
| | |
To let a 3-strand braid act on this, just attach it to the bottom of
the picture!
(That's why there were *four* obvious guesses about this would work:
one can easily imagine four variations on this trick, depending on
which strand is the "odd man out" - here it's the upper right. It's
just a matter of convention which we use, but my conventions give this.)
In fact, even the group of 4-strand braids acts on rational tangles in
an obvious way, but the 3-strand braid group is enough for now.
Let me summarize. The 3-strand braid group B_3 acts on rational tangles
in an obvious way. The subgroup that acts trivially is precisely the
center of B_3, generated by the full twist. Using stuff from "week230",
it follows that the quotient of B_3 by its center acts on the projectivized
rational homology of the torus. We thus get a topological explanation
of why B_3 mod its center is PSL(2,Z).
But there's more.
For starters, the 3-strand braid group is also the fundamental group of
R^3 minus the trefoil knot!
And, R^3 minus the trefoil knot is secretly the same as SL(2,R)/SL(2,Z)!
In fact, Terry Gannon writes that the 3-strand braid group can be regarded
as "the universal central extension of the modular group, and the universal
symmetry of its modular forms". I'm not completely sure what that means,
but here's *part* of what it means.
Just as PSL(2,C) is the Lorentz group in 4d spacetime, PSL(2,R) is the
Lorentz group in 3d spacetime. This group has a double cover SL(2,R),
which shows up when you study spinors. But, it also has a universal
cover, which shows up when you study anyons. The universal cover has
infinitely many sheets. And up in this universal cover, sitting over
the subgroup SL(2,Z), we get... the 3-strand braid group!
In math jargon, we have this commutative diagram:
B_3 --------> SL(2,Z) --------> PSL(2,Z)
| | |
| | |
| | |
v v v
SL(2,R)~ -----> SL(2,R) ---------> SO_0(2,1)
Here SL(2,R)~ is the universal cover of SL(2,R). This universal cover
is related to something called the Maslov index.
Gannon believes that number theorists should think about all this stuff,
since he thinks it's lurking behind that weird network of ideas called
Monstrous Moonshine (see "week66").
And here's the basic reason why. I'll try to get this right....
Any rational conformal field theory has a "chiral algebra" A which acts
on the left-moving states. Mathematicians call this sort of thing a
"vertex operator algebra". A representation of this on some vector
space V is a space of states for the circle in some "sector" of our
theory. Let's pick some state v in V. Then we can define a
"one-point function" where we take a Riemann surface with little
disk cut out and insert this state on the boundary. This is a number,
essentially the amplitude for a string in the give state to evolve like
this Riemann surface says.
In fact, instead of chopping out a little disk it's nice to just
remove a point - a "puncture", they call it. But, we get an ambiguous
answer unless we pick coordinates at this point, or in the lingo of
complex analysis, a choice of "uniforming parameter". Then our
one-point function becomes a function on the moduli space of Riemann
surfaces equipped with a puncture and a choice of uniformizing parameter.
If we didn't have this uniformizing parameter to worry about, we'd
just have the moduli space of tori equipped with a marked point,
which is nothing but the usual moduli space of elliptic curves,
H/PSL(2,Z)
where H is the complex upper halfplane. Then our one-point function
would have nice transformation properties under PSL(2,Z).
But, with this uniformizing parameter to worry about, our one-point
function only has nice transformation properties under B_3. This
is somehow supposed to be related to how B_3 is the "universal
central extension" of PSL(2,Z): in conformal field theory, all sorts
of naive symmetries hold only up to a phase, so you have to replace
various groups by central extensions thereof... and here that's what's
happening to PSL(2,Z)!
That last paragraph was pretty vague. If I'm going to understand this
better, either someone has to help me or I've got to read something
like this:
5) Y. Zhu, Modular invariance of characters of vertex operator algebras,
J. Amer. Math. Soc 9 (1996), 237-302. Also available at
http://www.ams.org/jams/1996-9-01/S0894-0347-96-00182-8/home.html
But I shouldn't need any conformal field theory to see how the moduli
space of punctured tori with uniformizing parameter is related to the
3-strand braid group! I bet this moduli space is X/B_3 for some space
X, or something like that. There's something simple at the bottom of
all this, I'm sure.
-----------------------------------------------------------------------
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
May 20, 2006
This Week's Finds in Mathematical Physics (Week 233)
John Baez
On Tuesday I'm supposed to talk with Lee Smolin about an idea he's
been working on with Fotini Markopoulou and Sundance Bilson-Thompson.
This idea relates the elementary particles in one generation of the
Standard Model to certain 3-strand framed braids:
1) Sundance O. Bilson-Thompson, A topological model of composite preons,
available as hep-ph/0503213.
2) Sundance O. Bilson-Thompson, Fotini Markopoulou, and Lee Smolin,
Quantum gravity and the Standard Model, hep-th/0603022.
It's a very speculative idea: they've found some interesting relations,
but nobody knows if these are coincidental or not.
Luckily, one of my hobbies is collecting mysterious relationships between
basic mathematical objects and trying to figure out what's going on.
So, I already happen to know a bunch of weird facts about 3-strand braids.
I figure I'll tell Smolin about this stuff. But if you don't mind, I'll
practice on you!
So, today I'll try to tell a story connecting the 3-strand braid group,
the trefoil knot, rational tangles, the groups SL(2,Z) and PSL(2,Z),
conformal field theory, and Monstrous Moonshine.
I've talked about some of these things before, but now I'll introduce
some new puzzle pieces, which come from two places:
3) Imre Tuba and Hans Wenzl, Representations of the braid
group B_3 and of SL(2,Z), available as math.RT/9912013.
4) Terry Gannon, The algebraic meaning of genus-zero, available as
math.NT/0512248.
You could call it "a tale of two groups".
On the one hand, the 3-strand braid group has generators
| | |
\ / |
A = / |
/ \ |
| | |
and
| | |
| \ /
B = | /
| / \
| | |
and the only relation is
ABA = BAB
otherwise known as the "third Reidemeister move":
| | | | | |
\ / | | \ /
/ | | /
/ \ | | / \
| \ / \ / |
| / = / |
| / \ / \ |
\ / | | \ /
/ | | /
/ \ | | / \
On other hand, the group SL(2,Z) consists of 2x2 integer matrices
with determinant 1. It's important in number theory, complex analysis,
string theory and other branches of pure mathematics. I've described
some of its charms in "week125", "week229" and elsewhere.
These groups look pretty different at first. But, there's a
homomorphism from B_3 onto SL(2,Z)! It goes like this:
1 1
A |->
0 1
1 0
B |->
-1 1
Both these matrices describe "shears" in the plane. You may enjoy
drawing these shears and visualizing the equation ABA = BAB in these
terms. I did.
I would like to understand this better... and here are some clues.
The center of B_3 is generated by the element (AB)^3. This
element corresponds to a "full twist". In other words, it's
the braid you get by hanging 3 strings from the ceiling, grabbing
them all with one hand at the bottom, and giving them a full 360-degree
twist:
| | |
\ / |
/ | A
/ \ |
| \ /
| / B
| / \
\ / |
/ | A
/ \ |
| \ /
| / B
| / \
\ / |
/ | A
/ \ |
| \ /
| / B
| / \
| | |
This full twist gets sent to -1 in SL(2,Z):
-1 0
(AB)^3 |->
0 -1
So, the double twist gets sent to the identity:
1 0
(AB)^6 |->
0 1
In fact, Tuba and Wenzl say the double twist *generates* the kernel
of our homomorphism from B_3 to SL(2,Z). So, SL(2,Z) is isomorphic
to the group of 3-strand braids modulo double twists!
This reminds me of spinors... since you have to twist an electron
around *twice* to get its wavefunction back to where it started.
And indeed, SL(2,Z) is a subgroup of SL(2,C), which is the double
cover of the Lorentz group. So, 3-strand braids indeed act on the
state space of a spin-1/2 particle, with double twists acting
trivially!
(For more on this, check out Trautman's work on "Pythagorean spinors"
in "week196". There's also a version where we use integers mod 7,
described in "week219".)
If instead we take 3-strand braids modulo full twists, we get the
so-called "modular group":
PSL(2,Z) = SL(2,Z)/{+-1}
Now, SL(2,Z) is famous for being the "mapping class group" of the torus -
that is, the group of orientation-preserving diffeomorphisms, modulo
diffeomorphisms connected to the identity. Similary, PSL(2,Z) is famous
for acting on the rational numbers together with a point at infinity
by means of fractional linear transformations:
az + b
z |-> -------
cz + d
where a,b,c,d are integers and ad-bc = 1. The group PSL(2,Z) also
acts on certain 2-stand tangles called "rational tangles". In
"week229", I told a nice story I heard from Michael Hutchings,
explaining how these three facts fit together in a neat package.
But now let's combine those facts with the stuff I just said!
Since PSL(2,Z) acts on rational tangles, and there's a homomorphism
from B_3 to PSL(2,Z), 3-strand braids must act on rational tangles.
How does that go?
There's an obvious guess, or two, or three, or four, but let's just
work it out.
I just said that the 3-strand braid A gets mapped to this shear:
1 1
A |->
0 1
In "week229" I said what this shear does to a rational tangle.
It gives it a 180 degree twist at the bottom, like this:
| | | |
| | | |
| | | |
------- -------
| T | |----> | T |
------- -------
| | \ /
| | /
| | / \
Next, Tuba and Wenzl point out that
0 1
ABA = BAB |->
-1 0
which is a quarter turn. From "week229" you can see how this quarter
turn acts on a rational tangle:
| | | |
| | ____ | |
| | / \ | |
------- | ------- |
| T | |----> | | T | |
------- | ------- |
| | | | \____/
| | | |
| | | |
It gives it a quarter turn!
From these facts, we can figure out what the braid B does to a
rational tangle. So, let me do the calculation.
Scribble, scribble, curse and scribble.... Eureka!
Since we know what A does, and what ABA does, we can figure out
what B must do. But, to make the answer look cute, I needed a
sneaky fact about rational tangles, which is that A *also* acts
like this:
| | \ /
| | /
| | / \
------- -------
| T | |----> | T |
------- -------
| | | |
| | | |
| | | |
This is proved in Goldman and Kauffman's paper cited in "week228".
With the help of this, I can show B acts like this:
| | | |
| | | ___ |
| | | / \ |
------- | / -------
| T | |----> \ | T |
------- / \ -------
| | | \___/ |
| | | |
| | | |
And this is *great*! It means our action of 3-strand braids on
rational tangles is really easy to describe. Just take your tangle
and let the upper left strand dangle down:
|
____ |
/ \ |
| -------
| | T |
| -------
| | |
| | |
| | |
To let a 3-strand braid act on this, just attach it to the bottom of
the picture!
(That's why there were *four* obvious guesses about this would work:
one can easily imagine four variations on this trick, depending on
which strand is the "odd man out" - here it's the upper right. It's
just a matter of convention which we use, but my conventions give this.)
In fact, even the group of 4-strand braids acts on rational tangles in
an obvious way, but the 3-strand braid group is enough for now.
Let me summarize. The 3-strand braid group B_3 acts on rational tangles
in an obvious way. The subgroup that acts trivially is precisely the
center of B_3, generated by the full twist. Using stuff from "week230",
it follows that the quotient of B_3 by its center acts on the projectivized
rational homology of the torus. We thus get a topological explanation
of why B_3 mod its center is PSL(2,Z).
But there's more.
For starters, the 3-strand braid group is also the fundamental group of
R^3 minus the trefoil knot!
And, R^3 minus the trefoil knot is secretly the same as SL(2,R)/SL(2,Z)!
In fact, Terry Gannon writes that the 3-strand braid group can be regarded
as "the universal central extension of the modular group, and the universal
symmetry of its modular forms". I'm not completely sure what that means,
but here's *part* of what it means.
Just as PSL(2,C) is the Lorentz group in 4d spacetime, PSL(2,R) is the
Lorentz group in 3d spacetime. This group has a double cover SL(2,R),
which shows up when you study spinors. But, it also has a universal
cover, which shows up when you study anyons. The universal cover has
infinitely many sheets. And up in this universal cover, sitting over
the subgroup SL(2,Z), we get... the 3-strand braid group!
In math jargon, we have this commutative diagram:
B_3 --------> SL(2,Z) --------> PSL(2,Z)
| | |
| | |
| | |
v v v
SL(2,R)~ -----> SL(2,R) ---------> SO_0(2,1)
Here SL(2,R)~ is the universal cover of SL(2,R). This universal cover
is related to something called the Maslov index.
Gannon believes that number theorists should think about all this stuff,
since he thinks it's lurking behind that weird network of ideas called
Monstrous Moonshine (see "week66").
And here's the basic reason why. I'll try to get this right....
Any rational conformal field theory has a "chiral algebra" A which acts
on the left-moving states. Mathematicians call this sort of thing a
"vertex operator algebra". A representation of this on some vector
space V is a space of states for the circle in some "sector" of our
theory. Let's pick some state v in V. Then we can define a
"one-point function" where we take a Riemann surface with little
disk cut out and insert this state on the boundary. This is a number,
essentially the amplitude for a string in the give state to evolve like
this Riemann surface says.
In fact, instead of chopping out a little disk it's nice to just
remove a point - a "puncture", they call it. But, we get an ambiguous
answer unless we pick coordinates at this point, or in the lingo of
complex analysis, a choice of "uniforming parameter". Then our
one-point function becomes a function on the moduli space of Riemann
surfaces equipped with a puncture and a choice of uniformizing parameter.
If we didn't have this uniformizing parameter to worry about, we'd
just have the moduli space of tori equipped with a marked point,
which is nothing but the usual moduli space of elliptic curves,
H/PSL(2,Z)
where H is the complex upper halfplane. Then our one-point function
would have nice transformation properties under PSL(2,Z).
But, with this uniformizing parameter to worry about, our one-point
function only has nice transformation properties under B_3. This
is somehow supposed to be related to how B_3 is the "universal
central extension" of PSL(2,Z): in conformal field theory, all sorts
of naive symmetries hold only up to a phase, so you have to replace
various groups by central extensions thereof... and here that's what's
happening to PSL(2,Z)!
That last paragraph was pretty vague. If I'm going to understand this
better, either someone has to help me or I've got to read something
like this:
5) Y. Zhu, Modular invariance of characters of vertex operator algebras,
J. Amer. Math. Soc 9 (1996), 237-302. Also available at
http://www.ams.org/jams/1996-9-01/S0894-0347-96-00182-8/home.html
But I shouldn't need any conformal field theory to see how the moduli
space of punctured tori with uniformizing parameter is related to the
3-strand braid group! I bet this moduli space is X/B_3 for some space
X, or something like that. There's something simple at the bottom of
all this, I'm sure.
-----------------------------------------------------------------------
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