Arriving to Strings: Einstein Eqns, Riemann Tensor, Poliakov Action

[arivero]

When I was first told that strings were able to see Einstein equations I thought, "what is the deal?". The Riemann Curvature Tensor is a two dimensional object, nothing rare you capture it by using two dimensional objects. It is well known so it is not rare that high level books do not mention it. But now I am checking some of these "string for undergraduate" stuff and it is not mentiones neither, so perhaps it is not a valid argument. Has anybody seen some paper approaching to stringgy gravity from this perspective? The closest thing I have seen is Connes observation that the Poliakov action is a two dimensional object and that its Dixmier trace in n dimensions is always proportional to Einstein-Hilbert action.

 

[arivero]

related to this, a peregrine idea about rigid strings is that to any rigid open string one can associate the vector going from one extreme to the other. And to any rigid (circular) closed string one can associate the axial vector defining the plane of the string. So it seems that strings can be used to describe both covariant and contravariant vectors. One wonders if this is actually exploited in string theory.

 

[Hurkyl]

And to any rigid (circular) closed string one can associate the axial vector defining the plane of the string.

Only in three-dimensional space. The dual to a 2-dimensional thing is an (n-2)-dimensional thing, and n-2=1 only when n=3.

 

[arivero]

Only in three-dimensional space. The dual to a 2-dimensional thing is an (n-2)-dimensional thing, and n-2=1 only when n=3.

Sloppy I am. I should use the terminology of covariant and contravariant objects, but yes, only in three dimensional objects we have a (co?)vector. Still, for the open string it is always true we can build a vector.

 

[josh1]

When I was first told that strings were able to see Einstein equations I thought, "what is the deal?"

The deal is that conformal invariance on the world-sheet requires the vacuum einstein equations be satisfied, which is a really amazing way for the field equations of general relativity to show up in string theory. It's one of the reasons why most physicists in high energy theory are confident that strings can't be completely wrong. Meanwhile, lqg is supposed to be quantum general relativity, but nobody knows whether it produces anything like einsteins theory in the low energy limit.

 

[selfAdjoint]

The deal is that conformal invariance on the world-sheet requires the vacuum einstein equations be satisfied, which is a really amazing way for the field equations of general relativity to show up in string theory. It's one of the reasons why most physicists in high energy theory are confident that strings can't be completely wrong. Meanwhile, lqg is supposed to be quantum general relativity, but nobody knows whether it produces anything like einsteins theory in the low energy limit.

Josh, those Einstein equations only show up on the 2-D world sheet of the string (it is pretty nifty, I agree), NOT on spacetime, so yoiur comparison is quite lame.

 

[arivero]

Josh, those Einstein equations only show up on the 2-D world sheet of the string (it is pretty nifty, I agree), NOT on spacetime, so yoiur comparison is quite lame.

Regretly the relevant famous pages from Polchinsky book do not fit here as attachment, so I am putting them here:
http://dftuz.unizar.es/~rivero/polch.pdf (2 Mbytes download)

Myself, I will read them again while I am at the cafeteria and then I will take a side

(EDITED: as the section 3.7 refers to 3.4, I have also extracted a scan of it: http://dftuz.unizar.es/~rivero/polch3_4.pdf )

 

[arivero]

The deal is that conformal invariance on the world-sheet requires the vacuum einstein equations be satisfied, which is a really amazing way for the field equations of general relativity to show up in string theory.

Well, I assume you refer to the argument developed in Polchinski section 3.7 and particularly to equation 3.7.14a. Note that Einstein equation comes not exactly from conformal invariance of the Poliakov action, but from the requeriment of keeping this conformal invariance after quantisation, i.e. from the requeriment of cancellation of any produced anomaly. Thus we have already applied a variational principle (call it Schwinger if you want, or just think that quantisation is a way to regularise the variational principle of the action, using h discrete instead of h->0) and if we have a such principle it is not so surprising to extract einstein equations. I agree it is a very interesting technique, but also Connes is able to extract the Einstein-Hilbert action from the trace of a bidimensional object.

(EDITED: I am attaching here Connes's theorem from pg 492 of Gracia-Bondia, Varilly, Figueroa. "Elements of noncommutative geometry". Amusingly, these authors choosed a normalisation for Dixmier trace that makes evident the relationship of both results; I asked one of the authors and I was told they had not noticed the similarity. In fact if you follow the proof in the book you can see that the normalisation is choosen on purely classical arguments).

 

[josh1]

...those Einstein equations only show up on the 2-D world sheet...

Just to be sure we're on the same page, it's the target space metric - this being the metric of spacetime itself - and not the world-sheet metric that must satisfy einstein's equations for the world-sheet theory to be self-consistent. I'm unsure why you think my response is lame.

Well, I assume you refer to the argument developed in Polchinski section 3.7 and particularly to equation 3.7.14a. Note that Einstein equation comes not exactly from conformal invariance of the Poliakov action, but from the requeriment of keeping this conformal invariance after quantisation, i.e. from the requeriment of cancellation of any produced anomaly.

Yes, of course: I didn’t say that it’s required to ensure only classical conformal invariance.

Thus we have already applied a variational principle (call it Schwinger if you want, or just think that quantisation is a way to regularise the variational principle of the action, using h discrete instead of h->0) and if we have a such principle it is not so surprising to extract einstein equations

What on Earth are you talking about?

 

[arivero]

What on Earth are you talking about?

Just recalling my first statement: that Einstein equations are usually got by using a variational procedure upon the action functional associated to a bidimensional object, the curvature. Eq 3.7.14a of Polchinski had been surprising if it were not a variation, or not a bidimensional object. But it is a quantisation of an action, so it is variational procedure in some sense.

The extant surprise of the string approach, the need of D=26, get diminished when you llok at 3.7.14c (as explained in the text) and then one sees that the only exceptional thing about non critical strings is that they ask for non-null sources of the gravitational field.

 

[josh1]

...The extant surprise of the string approach, the need of D=26, get diminished when you llok at 3.7.14c...

D=26 is required by diffeo x Weyl-invariance:

In light-cone gauge, the mass m of a bosonic string state is given as the sum of it's excitation level and the zero-point energy obtained by replacing the divergent factor (1+2+...) in the naive expression m = (1+2+...)(D-2)/2 with the more general expression (1^-s+2^-s+3^-s+...).

For Re(s) > 1 this converges to the Riemann-zeta function which one continues to the point s = -1 (a process commonly referred to as "zeta function regularization") where it equals -1/12.

This value's physical origin is however obscured in this approach by the way that the underlying world-sheet symmetries - whose preservation is what ultimately requires D=26 - are hidden. In fact, the same value may be obtained by regularizing the divergent sum by performing a sort of discrete path-integral regularization using a cutoff wherein the addition of a counterterm to the Polyakov action before one goes to light-cone gauge to cancel the resulting cutoff-dependent term is implicitly required by Weyl-invariance.

The constraint on the dimension of spacetime comes about because the level one excitations have mass (D-26)/24 and form an SO(D-2) vector, which by Lorentz-invariance must be massless.

In the path-integral approach based on the Polyakov action, one eliminates the overcounting of states due to the diffeo x Weyl symmetry by fixing the gauge using the Fadeev-Popov method familiar from quantum field theory.

For the X CFT (conformal field theory) this leads to the introduction of the pure ghost bc CFT which is necessarily of central charge -26. The central charge of the combined Xbc CFT is then D-26. To see that this must vanish, one observes that Weyl-invariance is respected only when the trace of the energy-momentum tensor is zero - the condition in a CFT that energy-momentum be conserved.

However, because for a curved world-sheet there is no fully gauge-invariant way to regulate the integral, the trace acquires an anomalous value, known as a "Weyl" or "conformal" anomaly, that breaks this invariance. Such anomalies render different gauges inequivalent, resulting in pathologies of one kind or another.

Diffeo x Poincare-invariance of the trace requires that the anomalous correction be proportional to the Ricci curvature scalar. The constant of proportionality is determined using diffeo-invariance to be the coefficient '- (central charge)/12' of the energy-momentum tensor's infinitesimal variation under conformal transformations.

If the CFT has holomorphic and antiholomorphic parts, since the same argument must apply to both, the two central charges must be equal, again requiring diffeo-invariance.

So in general, diffeo x Weyl-invariance dictates that the central charge vanish, which for the Xbc CFT means that D=26.

Note that the light-cone result for the zero-point energy may be reproduced in the Xbc CFT by defining |0> = |X> + |bc>. Here, |X> and |bc> are the ground states of the X and bc CFTs and satisfy L|X> = 0, L|bc> = -|bc> with L the zero-mode Virasoro generator. The ground state energy is then <0|(L-c/24)|0> =-1+(26-D)/24 = (2-D)/24, as one would expect.

 

[arivero]

Diffeo x Poincare-invariance of the trace requires that the anomalous correction be proportional to the Ricci curvature scalar. The constant of proportionality is determined using diffeo-invariance to be the coefficient '- (central charge)/12' of the energy-momentum tensor's infinitesimal variation under conformal transformations.

Yep, the point was that in curved backgrounds the Ricci curvature scalar gets also contributions from fields that can be absorbed as sources in Einstein equation, so when we get general relativity we need to be careful about the claim D=26.

A Lorentz invariant background is a flat space.

diffeo x Weyl-invariance...

I do not feel sure about this wording. I guess it suppossed to hint that there are two different invariances in string theory, the one of the background and the one of the worldsheet.

 

[arivero]

The Riemann Curvature Tensor is a two dimensional object,

Hmm how it is that nobody has disputed this affirmation ??

 

[selfAdjoint]

Just to be sure we're on the same page, it's the target space metric - this being the metric of spacetime itself - and not the world-sheet metric that must satisfy einstein's equations for the world-sheet theory to be self-consistent. I'm unsure why you think my response is lame.

Sorry for the long delay in answering.

But I see from Polchinski section 3.7, if that's what you are referring to, that this is the famous graviton. So if this makes the coordinates of the target space subject to the Einstein equations, and as Polchinski says "in string theory the target space is spacetime", then why is string theory done on a fixed, nondynamic background space? Never mind "curved", that's a nit; Einstein's equations define a spacetime that changes with the energy in it. But there ain't no dynamic background in SST, that I ever heard. Am I wrong?