Eric A. Forgy
Jul13-04, 03:37 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no,scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hello,\n\nUrs and I have been having lots of fun over at the weblog\n\nString Coffee Table\nhttp://golem.ph.utexas.edu/string/\n\ndiscussing all kinds of things that are in some way or other related\nto string theory, but typically the discussions focus on more general\naspects of differential geometry. We don\'t really intend it to be a\nprivate discussion between the two of us even though that is what\noften happens. Everyone is more than welcome to participate.\n\nA recent discussion\n\nhttp://golem.ph.utexas.edu/string/archives/000394.html#c001284\n\nhas gotten me fairly excited so I thought I\'d open it up and see what\nothers might think.\n\nFor some background, I have probably spent more time thinking about\nthe most basic aspects of differential geometry than anyone else on\nthe planet. It doesn\'t mean I understand things better because I\ndon\'t, but I am always REALLY trying to understand the basic "meaning"\nof it all and how it relates to nature. What I mean by "basic aspects\nof differential geometry" are simple questions like "What really is a\ntangent vector?", "What really is a differential form?", "What really\nis a Lie derivative?", etc.\n\nUrs has been doing some really amazing work lately on deformations and\nmost recently on connections in loop space and how it relates to\nhigher gauge theories. I accused him of becoming "assimilated" into\nthe habits of most theoretical physicists these days, i.e. speaking in\nsuch abstraction that none of them can understand one another :) In\nresponse, he agreed to try to present his work on differential\ngeometry of loop space in a manner that might actually be intelligible\nto mere mortals like myself. I am doing what I can to help out.\n\nThe first sticking point we came across was a bit of terminology. Urs\nlikes to refer to the operator\n\nd_X := d + i_X\n\nas a "deformation of the exterior derivative", where X is a vector\nfield and i_X is the interior product. I don\'t like this terminology\nbecause I have been involved in a long term love affair with Stokes\'\ntheorem and the exterior derivative is that thing that goes into\nStokes\' theorem. No one can deform my beloved Stokes\' theorem!\n\nThere are actually (at least) three geometrical operators that may be\nreferred to as a deformation of the exterior derivative. The three I\nhave in mind are\n\n1.) d + A\n2.) d + d^dag\n3.) d + i_X\n\nwhere A is a connection 1-form and d^dag is the adjoint of d. A very\ninteresting thing about each of these operators is that they square to\nbe something fundamental in differential geometry, i.e.\n\n(d + A)^2 --> "curvature"\n\n(d + d^dag)^2 --> "Laplace-Beltrami"\n\n(d + i_X)^2 --> "Lie derivative"\n\nThe first "deformation" d+A is, of course, the covariant exterior\nderivative, the second d+d^dag is the Dirac-Kaehler operator, and the\nthird is (apparently) referred to as a superconformal generator\n(phew!).\n\nIn this way, it makes a certain sense to think of d+A as the "square\nroot" of the curvature, d+d^dag is the "square root" of the\nLaplace-Beltrami operator, and d+i_X is the "square root" of the Lie\nderivative.\n\nI try to keep open minded about things and if\n\nd_X = d + i_X\n\nis to really be some kind of deformed exterior derivative, then is it\npossible that there is some operator\n\nh_X: C_p -> C_{p+1}\n\nthat takes a p-chain and returns a (p+1)-chain satisfying\n\nint_S i_X alpha = int_{h_X S} alpha\n\n? If this were possible, then we could define a deformed boundary map\n\n@_X = @ + h_X\n\nwhere @ is the usual boundary map, such that\n\nint_S d_X alpha = int_{@_X S} alpha\n\ncould be some kind of "super" Stokes\' theorem.\n\nGiven a vector field X, it does not take much imagination to come up\nwith a natural operator that takes a p-chain and returns a\n(p+1)-chain. The obvious candidate is something that "sweeps" the\np-chain along the flow generated by X.\nThus, following Frankel, define\n\nS(t) = phi(t)_* S\n\nto be the p-chain obtained by carrying the p-chain S along the flow\nfor a time t. Now let\n\nH_X(t): C_p -> C_{p+1}\n\nbe a map defined by\n\nH_X(t) S = Union_{0 <= t\' <= t} S(t\'),\n\ni.e. H_X(t) consists of the point set obtained by "sweeping" the\np-chain S along the flow generated by X for a time t. Give it the\nrelative topology and orient the (p+1)-chain such that its boundary\nconsists of the three pieces\n\n@[H_X(t) S] = S(t) + "sides" - S,\n\ni.e. the original orientation of S gets carried along the flow leaving\na copy of S behind having opposite orientation. The orientation of the\nsides is assigned just to make things consistent. For example, here is\nsome ascii art\n\n0-chain\n-------\n\n"+" "-" H_X(t) c "+"\nc o ==> o------>-------o c(t)\n\n\n1-chain\n-------\n\no o------<-------o\n| | |\nc ^ ==> v H_X(t) c ^ c(t)\n| | |\no o------>-------o\n\n2-chain\n-------\n\nI wish :)\n\nThe definition of H_X(t) probably needs some work to make rigorous,\nbut I hope its meaning is clear. It basically "sweeps" a p-chain along\nresulting in a (p+1)-chain while keeping track of orientation.\n\nAlthough I failed to find a map h_X: C_p -> C_{p+1} satisfying\n\nint_S i_X alpha = int_{h_X S} alpha,\n\nI did manage to come pretty close. The interior product satisfies\n\nint_S i_X alpha = d/dt [int_{H_X(t) S} alpha] |_{t = 0}.\n\nThis is absolutely beautiful :) I have never seen this before, but it\nsays that the interior product i_X is essentially dual to (i.e. the\ntranspose of) the operation of sweeping out a p-chain :)\n\nI think that the picture this affords is priceless and that all\nauthors should, from here on out, use this as the definition of the\ninterior product :)\n\nNow, it is well known (e.g. see Frankel) that the Lie derivative\nsatisfies the related relation\n\nint_S L_X alpha = d/dt [int_{S(t)} alpha] |_{t = 0}\n\nNow, with\n\nL_X = (d + i_X)^2 = d i_X + i_X d,\n\nalthough I worked it out on the String Coffee Table, it is a fun\nexercise to show that if we define\n\n@_X(t) := @ + H_X(t),\n\nthen we have\n\n@_X(t)^2 = @ H_X(t) + H_X(t) @ = phi(t)_* - 1,\n\ni.e.\n\n@_X(t)^2 S = S(t) - S.\n\nTherefore,\n\nint_S L_X alpha\n= int_S d i_X alpha + int_S i_X alpha\n= d/dt [int_{@_X(t)^2 S} alpha] |_{t = 0}\n= d/dt [int_{S(t) - S} alpha] |_{t = 0}\n= d/dt [int_{S(t)} alpha] |_{t=0}\n\nas it should :) This gives a very cool way to look at the Lie\nderivative. Instead of considering it as a operator on forms, you can\nlook at its transpose as an operator on chains. The form that gets\nintegrated is unchanged, but the domain of integration gets swept\nalong under it :)\n\nThis approach to presenting the interior product is too beautiful and\ntoo simple to not have been written down ages ago. Has anyone seen\nthis before? I would love to see a reference discussing this.\n\nAnyway, I think this is beautiful and figured I would share it so that\nperhaps some students learning differential geometry for the first\ntime can have one more picture to aid in understanding these things. I\nnever did like the purely algebraic definition of i_X you see in most\ntexts and this provides an alternative.\n\nBest wishes,\nEric\n\nPS: Here is a look at\n\n@_X(t)^2 = @ H_X(t) + H_X(t) @\n\napplied to a 1-chain c\n\n@[H_X(t) c]\n-----------\n\no o------<-------o o o------<------o o\n| | | | |\nc ^ ==> v H_X(t) c ^ ==> v @[H_X(t) c] ^\n| | | | |\no o------>-------o o o------>------o o\n\nH_X(t) @c\n---------\n"+"\no o o------>------o\n|\nc ^ ==> @c ==> H_X(t) @c\n|\no o o------<------o\n"-"\n\n[@ H_X(t) + H_X(t) @]c\n----------------------\n\no o------<------o o o------>------o o o\n| | | |\nv @[H_X(t) c] ^ + H_X(t) @c ==> v ^\n| | | |\no o------>------o o o------<------o o o\n\nIn other words,\n\n[@ H_X(t) + H_X(t) @]c = [phi(t)_* - 1]c = c(t) - c.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hello,
Urs and I have been having lots of fun over at the weblog
String Coffee Table
http://golem.ph.utexas.edu/string/
discussing all kinds of things that are in some way or other related
to string theory, but typically the discussions focus on more general
aspects of differential geometry. We don't really intend it to be a
private discussion between the two of us even though that is what
often happens. Everyone is more than welcome to participate.
A recent discussion
http://golem.ph.utexas.edu/string/archives/000394.html#c001284
has gotten me fairly excited so I thought I'd open it up and see what
others might think.
For some background, I have probably spent more time thinking about
the most basic aspects of differential geometry than anyone else on
the planet. It doesn't mean I understand things better because I
don't, but I am always REALLY trying to understand the basic "meaning"
of it all and how it relates to nature. What I mean by "basic aspects
of differential geometry" are simple questions like "What really is a
tangent vector?", "What really is a differential form?", "What really
is a Lie derivative?", etc.
Urs has been doing some really amazing work lately on deformations and
most recently on connections in loop space and how it relates to
higher gauge theories. I accused him of becoming "assimilated" into
the habits of most theoretical physicists these days, i.e. speaking in
such abstraction that none of them can understand one another :) In
response, he agreed to try to present his work on differential
geometry of loop space in a manner that might actually be intelligible
to mere mortals like myself. I am doing what I can to help out.
The first sticking point we came across was a bit of terminology. Urs
likes to refer to the operator
d_X := d + i_X
as a "deformation of the exterior derivative", where X is a vector
field and i_X is the interior product. I don't like this terminology
because I have been involved in a long term love affair with Stokes'
theorem and the exterior derivative is that thing that goes into
Stokes' theorem. No one can deform my beloved Stokes' theorem!
There are actually (at least) three geometrical operators that may be
referred to as a deformation of the exterior derivative. The three I
have in mind are
1.) d + A
2.) d + d^{dag}
3.) d + i_X
where A is a connection 1-form and d^{dag} is the adjoint of d. A very
interesting thing about each of these operators is that they square to
be something fundamental in differential geometry, i.e.
(d + A)^2[/itex] --> "curvature"
(d + d^{dag})^2 --> "Laplace-Beltrami"
(d + i_X)^2 --> "Lie derivative"
The first "deformation" d+A is, of course, the covariant exterior
derivative, the second d+d^{dag} is the Dirac-Kaehler operator, and the
third is (apparently) referred to as a superconformal generator
(phew!).
In this way, it makes a certain sense to think of d+A as the "square
root" of the curvature, d+d^{dag} is the "square root" of the
Laplace-Beltrami operator, and d+i_X is the "square root" of the Lie
derivative.
I try to keep open minded about things and if
d_X = d + i_X
is to really be some kind of deformed exterior derivative, then is it
possible that there is some operator
h_X: C_p -> C_{p+1}
that takes a p-chain and returns a (p+1)-chain satisfying
\int_S i_X \alpha = \int_{h_X S} \alpha
? If this were possible, then we could define a deformed boundary map
@_X = @ + h_X
where @ is the usual boundary map, such that
\int_S d_X \alpha = \int_{@_X S} \alpha
could be some kind of "super" Stokes' theorem.
Given a vector field X, it does not take much imagination to come up
with a natural operator that takes a p-chain and returns a
(p+1)-chain. The obvious candidate is something that "sweeps" the
p-chain along the flow generated by X.
Thus, following Frankel, define
S(t) = \phi(t)_* S
to be the p-chain obtained by carrying the p-chain S along the flow
for a time t. Now let
H_X(t): C_p -> C_{p+1}
be a map defined by
H_X(t) S = Union_{0 <= t' <= t} S(t'),
i.e. H_X(t) consists of the point set obtained by "sweeping" the
p-chain S along the flow generated by X for a time t. Give it the
relative topology and orient the (p+1)-chain such that its boundary
consists of the three pieces
@[H_X(t) S] = S(t) + "sides" - S,
i.e. the original orientation of S gets carried along the flow leaving
a copy of S behind having opposite orientation. The orientation of the
sides is assigned just to make things consistent. For example, here is
some ascii art
0-chain
-------
"+" "-" H_X(t) c "+"
c o ==> o------>-------o c(t)
1-chain
-------
o o------<-------o
| | |c ^ ==> v H_X(t) c ^ c(t)| | |
o o------>-------o
2-chain
-------
I wish :)
The definition of H_X(t) probably needs some work to make rigorous,
but I hope its meaning is clear. It basically "sweeps" a p-chain along
resulting in a (p+1)-chain while keeping track of orientation.
Although I failed to find a map h_X: C_p -> C_{p+1} satisfying
\int_S i_X \alpha = \int_{h_X S} \alpha,
I did manage to come pretty close. The interior product satisfies
\int_S i_X \alpha = d/dt [\int_{H_X(t) S} \alpha] |_{t = 0}.
This is absolutely beautiful :) I have never seen this before, but it
says that the interior product i_X is essentially dual to (i.e. the
transpose of) the operation of sweeping out a p-chain :)
I think that the picture this affords is priceless and that all
authors should, from here on out, use this as the definition of the
interior product :)
Now, it is well known (e.g. see Frankel) that the Lie derivative
satisfies the related relation
\int_S L_X \alpha = d/dt [\int_{S(t)} \alpha] |_{t = 0}
Now, with
L_X = (d + i_X)^2 = d i_X + i_X d,
although I worked it out on the String Coffee Table, it is a fun
exercise to show that if we define
@_X(t) := @ + H_X(t),
then we have
@_X(t)^2 = @ H_X(t) + H_X(t) @ = \phi(t)_* - 1,
i.e.
@_X(t)^2 S = S(t) - S.
Therefore,
[itex]\int_S L_X \alpha= \int_S d i_X \alpha + \int_S i_X \alpha= d/dt [\int_{@_X(t)^2 S} \alpha] |_{t = 0}= d/dt [\int_{S(t) - S} \alpha] |_{t = 0}= d/dt [\int_{S(t)} \alpha] |_{t=0}
as it should :) This gives a very cool way to look at the Lie
derivative. Instead of considering it as a operator on forms, you can
look at its transpose as an operator on chains. The form that gets
integrated is unchanged, but the domain of integration gets swept
along under it :)
This approach to presenting the interior product is too beautiful and
too simple to not have been written down ages ago. Has anyone seen
this before? I would love to see a reference discussing this.
Anyway, I think this is beautiful and figured I would share it so that
perhaps some students learning differential geometry for the first
time can have one more picture to aid in understanding these things. I
never did like the purely algebraic definition of i_X you see in most
texts and this provides an alternative.
Best wishes,
Eric
PS: Here is a look at
@_X(t)^2 = @ H_X(t) + H_X(t) @
applied to a 1-chain c
@[H_X(t) c]
-----------
o o------<-------o o o------<------o o
| | | | |c ^ ==> v H_X(t) c ^ ==> v @[H_X(t) c] ^| | | | |
o o------>-------o o o------>------o o
H_X(t) @c
---------
"+"
o o o------>------o
|
c ^ ==> @c ==> H_X(t) @c
|
o o o------<------o
"-"
[@ H_X(t) + H_X(t) @]c
----------------------
o o------<------o o o------>------o o o
| | | |v @[H_X(t) c] ^ + H_X(t) @c ==> v ^| | | |
o o------>------o o o------<------o o o
In other words,
[@ H_X(t) + H_X(t) @]c = [\phi(t)_* - 1]c = c(t) - c.
Urs and I have been having lots of fun over at the weblog
String Coffee Table
http://golem.ph.utexas.edu/string/
discussing all kinds of things that are in some way or other related
to string theory, but typically the discussions focus on more general
aspects of differential geometry. We don't really intend it to be a
private discussion between the two of us even though that is what
often happens. Everyone is more than welcome to participate.
A recent discussion
http://golem.ph.utexas.edu/string/archives/000394.html#c001284
has gotten me fairly excited so I thought I'd open it up and see what
others might think.
For some background, I have probably spent more time thinking about
the most basic aspects of differential geometry than anyone else on
the planet. It doesn't mean I understand things better because I
don't, but I am always REALLY trying to understand the basic "meaning"
of it all and how it relates to nature. What I mean by "basic aspects
of differential geometry" are simple questions like "What really is a
tangent vector?", "What really is a differential form?", "What really
is a Lie derivative?", etc.
Urs has been doing some really amazing work lately on deformations and
most recently on connections in loop space and how it relates to
higher gauge theories. I accused him of becoming "assimilated" into
the habits of most theoretical physicists these days, i.e. speaking in
such abstraction that none of them can understand one another :) In
response, he agreed to try to present his work on differential
geometry of loop space in a manner that might actually be intelligible
to mere mortals like myself. I am doing what I can to help out.
The first sticking point we came across was a bit of terminology. Urs
likes to refer to the operator
d_X := d + i_X
as a "deformation of the exterior derivative", where X is a vector
field and i_X is the interior product. I don't like this terminology
because I have been involved in a long term love affair with Stokes'
theorem and the exterior derivative is that thing that goes into
Stokes' theorem. No one can deform my beloved Stokes' theorem!
There are actually (at least) three geometrical operators that may be
referred to as a deformation of the exterior derivative. The three I
have in mind are
1.) d + A
2.) d + d^{dag}
3.) d + i_X
where A is a connection 1-form and d^{dag} is the adjoint of d. A very
interesting thing about each of these operators is that they square to
be something fundamental in differential geometry, i.e.
(d + A)^2[/itex] --> "curvature"
(d + d^{dag})^2 --> "Laplace-Beltrami"
(d + i_X)^2 --> "Lie derivative"
The first "deformation" d+A is, of course, the covariant exterior
derivative, the second d+d^{dag} is the Dirac-Kaehler operator, and the
third is (apparently) referred to as a superconformal generator
(phew!).
In this way, it makes a certain sense to think of d+A as the "square
root" of the curvature, d+d^{dag} is the "square root" of the
Laplace-Beltrami operator, and d+i_X is the "square root" of the Lie
derivative.
I try to keep open minded about things and if
d_X = d + i_X
is to really be some kind of deformed exterior derivative, then is it
possible that there is some operator
h_X: C_p -> C_{p+1}
that takes a p-chain and returns a (p+1)-chain satisfying
\int_S i_X \alpha = \int_{h_X S} \alpha
? If this were possible, then we could define a deformed boundary map
@_X = @ + h_X
where @ is the usual boundary map, such that
\int_S d_X \alpha = \int_{@_X S} \alpha
could be some kind of "super" Stokes' theorem.
Given a vector field X, it does not take much imagination to come up
with a natural operator that takes a p-chain and returns a
(p+1)-chain. The obvious candidate is something that "sweeps" the
p-chain along the flow generated by X.
Thus, following Frankel, define
S(t) = \phi(t)_* S
to be the p-chain obtained by carrying the p-chain S along the flow
for a time t. Now let
H_X(t): C_p -> C_{p+1}
be a map defined by
H_X(t) S = Union_{0 <= t' <= t} S(t'),
i.e. H_X(t) consists of the point set obtained by "sweeping" the
p-chain S along the flow generated by X for a time t. Give it the
relative topology and orient the (p+1)-chain such that its boundary
consists of the three pieces
@[H_X(t) S] = S(t) + "sides" - S,
i.e. the original orientation of S gets carried along the flow leaving
a copy of S behind having opposite orientation. The orientation of the
sides is assigned just to make things consistent. For example, here is
some ascii art
0-chain
-------
"+" "-" H_X(t) c "+"
c o ==> o------>-------o c(t)
1-chain
-------
o o------<-------o
| | |c ^ ==> v H_X(t) c ^ c(t)| | |
o o------>-------o
2-chain
-------
I wish :)
The definition of H_X(t) probably needs some work to make rigorous,
but I hope its meaning is clear. It basically "sweeps" a p-chain along
resulting in a (p+1)-chain while keeping track of orientation.
Although I failed to find a map h_X: C_p -> C_{p+1} satisfying
\int_S i_X \alpha = \int_{h_X S} \alpha,
I did manage to come pretty close. The interior product satisfies
\int_S i_X \alpha = d/dt [\int_{H_X(t) S} \alpha] |_{t = 0}.
This is absolutely beautiful :) I have never seen this before, but it
says that the interior product i_X is essentially dual to (i.e. the
transpose of) the operation of sweeping out a p-chain :)
I think that the picture this affords is priceless and that all
authors should, from here on out, use this as the definition of the
interior product :)
Now, it is well known (e.g. see Frankel) that the Lie derivative
satisfies the related relation
\int_S L_X \alpha = d/dt [\int_{S(t)} \alpha] |_{t = 0}
Now, with
L_X = (d + i_X)^2 = d i_X + i_X d,
although I worked it out on the String Coffee Table, it is a fun
exercise to show that if we define
@_X(t) := @ + H_X(t),
then we have
@_X(t)^2 = @ H_X(t) + H_X(t) @ = \phi(t)_* - 1,
i.e.
@_X(t)^2 S = S(t) - S.
Therefore,
[itex]\int_S L_X \alpha= \int_S d i_X \alpha + \int_S i_X \alpha= d/dt [\int_{@_X(t)^2 S} \alpha] |_{t = 0}= d/dt [\int_{S(t) - S} \alpha] |_{t = 0}= d/dt [\int_{S(t)} \alpha] |_{t=0}
as it should :) This gives a very cool way to look at the Lie
derivative. Instead of considering it as a operator on forms, you can
look at its transpose as an operator on chains. The form that gets
integrated is unchanged, but the domain of integration gets swept
along under it :)
This approach to presenting the interior product is too beautiful and
too simple to not have been written down ages ago. Has anyone seen
this before? I would love to see a reference discussing this.
Anyway, I think this is beautiful and figured I would share it so that
perhaps some students learning differential geometry for the first
time can have one more picture to aid in understanding these things. I
never did like the purely algebraic definition of i_X you see in most
texts and this provides an alternative.
Best wishes,
Eric
PS: Here is a look at
@_X(t)^2 = @ H_X(t) + H_X(t) @
applied to a 1-chain c
@[H_X(t) c]
-----------
o o------<-------o o o------<------o o
| | | | |c ^ ==> v H_X(t) c ^ ==> v @[H_X(t) c] ^| | | | |
o o------>-------o o o------>------o o
H_X(t) @c
---------
"+"
o o o------>------o
|
c ^ ==> @c ==> H_X(t) @c
|
o o o------<------o
"-"
[@ H_X(t) + H_X(t) @]c
----------------------
o o------<------o o o------>------o o o
| | | |v @[H_X(t) c] ^ + H_X(t) @c ==> v ^| | | |
o o------>------o o o------<------o o o
In other words,
[@ H_X(t) + H_X(t) @]c = [\phi(t)_* - 1]c = c(t) - c.