tr(F ^ ... ^ F)where F is the curvature of a connection A on some G-bundle P over the spacetime M. This Lagrangian vacuous equations of motion, which follow from the Bianchi identity
d_A F = 0Then we looked at the Yang-Mills Lagrangian
tr(F ^ *F)This gives the Yang-Mills equation
d_A *F = 0The quantized version of Yang-Mills theory describes all the forces *except* gravity. It relies on a background metric, which we need to define the Hodge * operator. To avoid this, we can introduce an extra field E, which like *F is an Ad(P)-valued (n-2)-form. This lets us write down the Lagrangian
tr(E ^ F)This gives the equations of "EF theory":
F = 0
d_A E = 0
In dimensions 2, 3 and 4 we can add extra terms to the EF Lagrangian,
making the field equations more interesting.
We talked about this already in Week 2!
What can we do in dimension 4?"
Miguel said, "We can add a term like tr(E ^ E)..."
"... or tr(F ^ F)!" added Jay.
"Right!" said the Wiz. "The Chern theory Lagrangian tr(F ^ F) gives vacuous equations of motion, so adding a term like this won't affect the classical equations of motion in EF theory. It can still be interesting in the quantized theory, but let's not worry about that -- let's just try
tr(E ^ F + Lambda E ^ E)What equations of motion do we get? How do we figure this out?"
"Well," said Jay, "We start with the action
S = \int tr(E ^ F + Lambda E ^ E)and then vary it:
DS = D \int tr(E ^ F + Lambda E ^ E)Pushing the D in and using the Leibniz law, we get
DS = \int tr(DE ^ F + E ^ DF + Lambda DE ^ E + Lambda E ^ DE)And then...." He hesitated.
"Those last two terms look awfully similar," commented the Wiz.
"Oh!" said Toby. "If E and DE were 2-forms, we'd have E ^ DE = DE ^ E. They're actually Ad(P)-valued 2-forms, but they still commute after we take the trace, so we get
DS = \int tr(DE ^ F + E ^ DF + 2 Lambda DE ^ E)or
DS = \int tr(E ^ DF + DE ^ (F + 2 Lambda E))after we simplify."
"Good!" said the Wiz. "Now what?"
"The magic formula!" cried John.
"Right! Never forget the magic formula DF = d_A DA! This gives
DS = \int tr(E ^ d_A DA + DE ^ (F + 2 Lambda E))Then what?"
"When in doubt," said Miguel, "integrate by parts. So:
DS = \int tr(d_A E ^ DA + DE ^ (F + 2 Lambda E))I'm not completely sure about the sign of the first term, but it doesn't matter: the whole thing can only be zero for all variations DA and DE if both these equations hold:
d_A E = 0
F + 2 Lambda E = 0
These are the equations of motion."
"Excellent!" said the Wizard. "Notice anything funny about them?"
The class stared intently at the equations. "Hmm," said Jay. "You can solve for E in the second equation and plug it into the first one...."
The Wizard growled and tossed a low-intensity fireball in Jay's general direction, singing him slightly.
"Ouch!" said Jay. "Why'd you do that?"
"It was a nice idea," said the Wiz, "but you forgot a crucial clause: *if Lambda is nonzero*, we can solve the second equation for E and get
E = F / 2 LambdaThen the first equation says
d_A F = 0But this is just a tautology: the Bianchi identity."
"Hey!" said Miguel. "That's weird! When Lambda is nonzero, the equations say: take any connection A you like, work out F, and let E = F / 2 Lambda. Just as in Chern theory, any connection gives a solution. But when Lambda is zero, they say: A is flat and d_A E = 0. The two cases are utterly different!"
"Yup!" said the Wiz. "But believe it or not, some people study the theory for arbitrary Lambda by doing perturbation about the Lambda = 0 theory."
The class gasped in horror.
"We'll talk more about that later. Now let's go down to 3 dimensions. What terms can we add to the EF theory Lagrangian here?"
"Just tr(E ^ E ^ E)," said Toby.
"Okay, so take this Lagrangian
tr(E ^ F + Lambda E ^ E ^ E)and work out the equations of motion. Any takers?"
"I bet I can do it," said Jay, striding confidently towards the board. "We write down the action
S = \int tr(E ^ F + Lambda E ^ E ^ E)and vary it:
DS = \int tr(DE ^ F + E ^ DF + Lambda (DE ^ E ^ E + E ^ DE ^ E + E ^ E ^ DE))Then, hmm... are those last 3 terms all equal?"
"Sure!" said Toby. "Use the cyclic property of the trace, and the fact that cyclic permutations are even, so we don't get nasty minus signs from switching the 1-form parts...."
"Okay, so it's just
DS = \int tr(DE ^ F + E ^ DF + 3 Lambda DE ^ E ^ E)Then I'll collect like terms and use the magic formula:
DS = \int tr(E ^ d_A DA + DE ^ (F + 3 Lambda E ^ E))To complete my performance, I'll integrate by parts:
DS = \int tr(-d_A E ^ DA + DE ^ (F + 3 Lambda E ^ E))I'm not sure about the sign of the first term, but it doesn't matter: this mess can only vanish for all variations of A and E if both these equations hold:
d_A E = 0
F + 3 Lambda E ^ E = 0
These are the equations of motion!"
The class erupted in applause; Jay took a bow and walked back to his desk.
"Excellent!" said the Wiz. "And when G = SO(2,1), they're the equations for Lorentzian 3d gravity with cosmological constant Lambda. By the way... do you notice anything funny about these equations?"
The Acolytes tried various tricks to derive one of the equations from the other, but failed miserably.
"Oh well," said the Wiz, looking down and flipping through his notes for the next topic. "It never hurts to try."
"... unless you get hit with a fireball!" someone quipped. The Wiz glanced up and glowered, but couldn't spot the culprit. Was Jay still smarting from that last one?
"Okay. To finish things off, I'll work out the equations of motion for Lorentzian *4d* gravity with cosmological constant. Here our basic fields are an \so(3,1) connection A and an \R^4-valued 1-form e, and the Lagrangian is
tr(e ^ e ^ F + Lambda e ^ e ^ e ^ e)I explained what this means in Week 2. The same Lagrangian works for the Riemannian theory if we use \so(4) instead of \so(3,1), but let's talk about the Lorentzian theory, since that's more like real physics. It's easy to work out the equations of motion. The action is
S = \int tr(e ^ e ^ F + Lambda e ^ e ^ e ^ e)so we get
DS = \int tr(2 De ^ e ^ F + e ^ e ^ DF + 4 Lambda De ^ e ^ e ^ e)where I've used the Leibniz law, the cyclic property of the trace, and the fact that
De ^ e = e ^ DeGrouping like terms and using the magic formula,
DS = \int tr(e ^ e ^ d_A DA + De ^ (2 e ^ F + 4 Lambda e ^ e ^ e))Integrating by parts,
DS = \int tr(d_A(e ^ e) ^ DA + De ^ (2 e ^ F + 4 Lambda e ^ e ^ e))so the equations of motion are
d_A(e ^ e) = 0
e ^ F + 2 Lambda e ^ e ^ e = 0
Believe it or not, these are the usual vacuum Einstein equations
in slight disguise, at least when the metric g(v,w) = tr((ev) (ew)
is nondegenerate. I'll prove this some other time;
it works like the 3d case I discussed last week.
For now, just compare these equations
to those for 4d EF theory with cosmological constant:
d_A E = 0
F + 2 Lambda E = 0
Notice anything funny?"
"Yeah," said Miguel, "e ^ e acts a lot like E."
"Right! More precisely, given any solution of the EF equations for which E is of the form e ^ e, we get a solution of the equations of general relativity! We only get a special class of solutions this way, but it's still pretty interesting. It hints at a relation between gravity and EF theory, and this relation is the key to "spin foam models". But we can talk about that over dinner. I'm famished!"
The class closed their notebooks and went down the winding stairs of the castle, out into the cool moonlit night. They hiked over to a nearby tavern, where they had dinner, laughing and talking into the wee hours of the morning. The conversation was very interesting -- but unfortunately, too secret for you.