## March 11, 2004

### Introductory String Theory Seminar

#### Posted by Urs Schreiber

I have been asked by students if I would like to talk a little about introductory string theory. Since it is currently semester break, we decided to make an experiment (which is unusual for string theory) and try to do an informal and inofficial seminar.

The background of the people attending the semiar is very inhomogeneous and a basic knowledge of special relativity and quantum mechanics is maybe the greatest common divisor. Therefore we’ll start with elementary stuff and will try to acquaint ourselfs with the deeper mysteries of the universe (such as QFT, YM, GR, CFT, SUSY) as we go along.

If I were in my right mind I’d feel overwhelmed with the task of conducting such a seminar, but maybe at least I can be of help as a guide who has seen the inside of the labyrinth before. Hence I’d like to stress that

I can only show you the door. You’re the one that has to walk through it.

;-)

In this spirit, the very first thing I can and should do is prepare a commented list of introductory literature. Here it is:

Actually, the task of writing such a list has already been done:

and I won’t be able and won’t try to do better than that. But I can provide a couple of convenient hyperlinks and personal comments.

First of all, everybody must know that there are two canonical textbooks, the old and the new testament. The old one is

M. Green & J. Schwarz & E. Witten, Superstring Theory Vol.1 , Vol. 2, Cambridge University Press (1987)

and the new one is

J. Polchinski, String Theory Vol. 1, Vol. 2, Cambridge University Press (1998).

Both are to some degree complementary. Polchinski is more modern (no branes in GSW) and more concise. GSW is more more old-fashioned and more elementary.

Those who want to read textbooks should probably start with the first couple of chapters of GSW, first volume, and then begin reading volume 1 of Polchinski in parallel - and then see what happens to your neurons and decide on that basis how to proceed further.

There are also some non-canonical textbooks:

B. Hatfield, Quantum Field Theory of Point Particles and Strings, Perseus Publishing (1992)

(This one is very pedagogical but only covers very little string theory.)

B. Zwieback, A First Course in String Theory, Cambridge University Press (2004)

M. Kaku, Introduction to Superstrings and M-Theory, Springer (1998)

M. Kaku, Strings, Conformal Fields, and M-Theory, Springer (2000) .

(I haven’t read these last three books myself.)

More important for our purposes, there are a large number of very good lecture notes available online at the so called arXiv. This is a preprint server which is a way to make research papers publically available that have not yet went through the full process of peer-reviewed publishment in print journals.

Of interest for this seminar are mostly the sections hep-th (theoretical high energy physics) and maybe gr-qc (general relativity and quantum cosmology) of the arXiv archive.

Most notably in the fields covered by hep-th, there has been an ongoing process away from an emphasis of print journals towards an emphasis of online communication, and except for articles dating from before 1992 most every publication in high energy physics that one will ever want to see can be found here, online and for free!

In this context one should also mention the SPIRES HEP Literature Database that reaches all the way back to 1974 - which is incidentally the year in which it was realized that string theory is a theory of quantum gravity.

The most easily accessible introductory lecture on string theory that I know is

R. Szabo, BUSSTEPP Lectures on String Theory (2002)

In

J. Schwarz, Introduction to Superstring Theory (2000)

a brief elementary introduction of the basic ideas of string theory aimed at
experimentalists is given.

Another nice introduction is

T. Mohaupt, Introduction to String Theory (2002) .

The notes by E. Kiritsis

E. Kiritsis, Introduction to Superstring Theory (1998)

are a thorough introduction to the string with some emphasis on conformal field theory and a bit on branes and dualities.

I always find the lecture notes by M. Kreuzer extremely valuable as a second

M. Kreuzer, Einführung in die Superstring-Theorie (2001)

for the bosonic string and

M. Kreuzer, Einführung in die Superstring-Theorie II (2001)

for conformal field theory and a (tiny) little bit on the superstring. (The
text is in English, only the title is German.)

E. Alvarez & P. Meessen, String Primer (2001)

and

There is much more available, but this should give a first idea. The above list is basically taken from this post to the newgroup sci.physics.research, which can be a very valuable resource and place to ask and answer questions. Before participating please read this and this. Maybe there will be a similar newsgroup concerned exclusively with string theory soon. Of course, everybody is also invited to post any questions and comments to the String Coffee Table. See here for some tips and tricks.

If I find the time I may expand the above list in the future. Suggestions are very welcome.

Last not least, I cannot refrain from pointing to the fun little Java applet which visualizes the classical motion of string.

This is by Igor Nikitin and the theory behind it is explained in

I. Nikitin, Introduction to String Theory.

So much for now. Summaries, links and background information concerning our Seminar meetings will be given in the comments.

Posted at March 11, 2004 12:29 PM UTC

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### Meeting 1: Nambu-Goto, Polyakov and back

For the convenience of those who had to decipher my handwriting on the blackboard while keeping track of my signs (which tend to pick up a stochastic dynamics) as well as of the number of dimensions I was talking about, here is a list of references where the material that I presented can be found in print.

(At the end there is also a little exercise. Please post proposed solutions here to the Coffee Table, so that everybody can benefit.)

First I made some historical remarks concerning the inception and development of what today is called ‘string theory’ or maybe ‘M-theory’. I didn’t even go to the level of detail found in R. Szabo’s lectures pp. 4-9. More on this can be found in GSW I, section 1 and a much shorter equivalent is section 1.1 of Polchinski. Since giving a reasonable glimpse of the Big Picture is beyond what I should try when standing with my back to the blackboard, I won’t say much more about this until maybe much later.

Instead there are some elementary but interesting calculations that one can get one’s hands on in order to get started:

First of all one should recall some basic facts about the relativistic point particle, like how its square-root form of the action looks like (Nambu-Goto-like action) and how the corresponding square form looks like (Polyakov-like action). This can be found for instance on pp. 293-295 of this text.

There is a (maybe surprisingly) obvious and straightforward generalization of this to the case where the object under consideration is not 0 but $p$-dimensional. One can write down the general Nambu-Goto-like action for $p$-branes and find the associated Polyakov-like action. For instance by varying the latter with respect to the auxiliary metric on the world-volume one can check that both are classically equivalent.

This is demonstrated in detail on pp. 171-179 of the above mentioned text.

Anyone who feels like he wants to read a more pedagogical discussion of these issues is invited to have a look at this.

We have also talked a lot about the basics of gauge theory after the seminar. I hope to come to that later, but if anybody feels like reading more on this he or she might want to have a look at chapter 20 of the very recommendable book

T. Frankel, The Geometry of Physics Cambridge (1997)

or of course pick up a book on field theory, like

M. Peskin & D. Schroer, An Introduction to Quantum Field Theory,

where it is chapter 15.

That wouldn’t hurt, because my evil plan is to eventually discuss the IIB Matrix Model in the seminar, which is a surprisingly elementary way to have a look into the

$\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}$Total Perspective Vortex.

But, as I said, we’ll come to that later.

Finally here is a little exercise concerning the material discussed in the first meeting:

I had demonstrated how the mass shell constraint

(1)${p}_{\mu }{p}^{\mu }=-{m}^{2}$

follows from the Nambu-Goto-like action of the point particle.

1) Derive the analogous constraint for the Nambu-Goto action of the string. Interpret it physically.

2) The action of the point particle coupled to an electromagnetic field with vector potential ${A}_{\mu }$ is

(2)$S=-m\int \left(\sqrt{-{\stackrel{˙}{x}}^{\mu }{\stackrel{˙}{x}}_{\mu }}+{A}_{\mu }\left(x\right){\stackrel{˙}{x}}^{\mu }\right)d\tau \phantom{\rule{thinmathspace}{0ex}}.$

How does the mass-shell constraint look now?

3) The generalization of the above action to the string is obviously

(3)$S=-T\int {d}^{2}\sigma \left(\sqrt{-h}+{B}_{\mu \nu }{ϵ}^{\alpha \beta }\left({\partial }_{\alpha }{x}^{\mu }\right)\left({\partial }_{\beta }{x}^{\nu }\right)\right)$

where $\alpha ,\beta \in \left\{0,1\right\}$ are the indices on the worldsheet, ${h}_{\alpha \beta }=\left({\partial }_{\alpha }{x}^{\mu }\right)\left({\partial }_{\beta }{x}^{\nu }\right){g}_{\mu \nu }$, is the induced metric on the worldsheet and $h=\mathrm{det}{h}_{\alpha \beta }$ is its determinant. ${ϵ}^{\alpha \beta }$ is the antisymmetric symbol and ${B}_{\mu \nu }=-{B}_{\nu \mu }$ is an antisymmetric tensor (i.e. a 2-form) on spacetime.

Derive the mass-shell constraint for the string for non-vanishing ${B}_{\mu \nu }$. Interpret the result by comparison with the point particle case.

The next meeting will be

Friday, 19 Mar 2004, 15:00, in S05 V07 E04.

(We cannot meet next Wednesday because I’ll be in Ulm).

Posted by: Urs Schreiber on March 11, 2004 5:16 PM | Permalink | PGP Sig | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Urs,

When I saw you stating you were giving a “String Seminar”, my first thought went to Baez’s “Quantum Gravity Seminar” where he kept us updated on what was discussed in the seminar. It was almost as if we were there ourselves :) Now, reality has sunk in and I see you meant a usual seminar. Bummer! :) Do you think you might be able to get one of your students to volunteer to write up an informal expository overview of the lecture. It would be great if they could mimic something like what Toby Bartels and Baez did for the QG Seminar. Just a thought! :)

On a different note…

One big deterrant for me getting very far in the string literature is the insane notation. I feel like the index-ridden notation is something best left in the 20th century :)

If you look at Maxwell’s original expression for the equations, they extend over several pages. With the use of vector calculus notation they were reduced to

(1)$\nabla ×E+\frac{\partial B}{\partial t}=0,\phantom{\rule{1em}{0ex}}\nabla \cdot B=0,$
(2)$\nabla ×H-\frac{\partial D}{\partial t}=J,\phantom{\rule{1em}{0ex}}\nabla \cdot D=\varrho$

with constitutive relations

(3)$D\left(E\right)\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}H\left(B\right),$

but we soon learned that this was STILL coordinate dependent because of the appearance of $t$ and the realization that spacetime was a 4d manifold. Finally we end up with

(4)$\mathrm{dF}=0\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}{d}^{†}F=j.$

Voila! When you express the equations in their natural coordinate independent form, then a great simplification occurs. Not only that, we get some really nice geometrical pictures arising. The first equation says simply that the flux of F through any closed surface in spacetime is zero. This manifests itself as the first line of equations using the vector calculus notation. This beautiful interpretation is hardly obvious just by staring at the vector calculus versions (and hopeless looking at the originals of Maxwell :)).

Now, when I look at the string literature I can’t help but think that things are in as sad a shape as the original form of Maxwell’s equations. It is a mess of indices and different fields. Even worse, the indices are a contiuum :) My question is, in the list of literature you provided, is there anything that remotely resembles the string analog of

(5)$\mathrm{dF}=0,{d}^{†}F=j?$

Could it be that the various fields in the string expressions (I don’t have a particular example in mind) are really just different components of a single field in analogy to how E and B are just different components of F? I guess I am looking for an index free formulation of string theory. Does such a thing exist?

Best of luck with your seminar!

Eric

Posted by: Eric on March 12, 2004 2:43 PM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Eric -

you wrote:

my first thought went to Baez’s ‘Quantum Gravity Seminar’

Gulp. My aim is orders of magnitude more humble. I can only run this with a relatively low task priority and really have to learn many thinks myself.

Do you think you might be able to get one of your students to volunteer to write up an informal expository overview of the lecture.

Yes, I have thought about that, too. For our first meeting I have tried to provide the equivalent of a write up of what we did in my previous comment by pointing to the page numbers of texts from which I took the material. (For instance all the material necessary to solve the exercise is given… :-)

I would like to make the content of our meetings available here in general, not the least because not all of the local participants will be able to attend every week. But there is a limit to the amount of energy and time that I have, so it would be really good if one of the participants would volunteer to supply his notes to the Coffee Table. I’ll see if I find somebody.

One big deterrant for me getting very far in the string literature is the insane notation.

That’s too bad. Once you are used to it it does not seem that insane anymore.

Of course you can rewrite most anything that you encounter in a coordinate independent way. That usually involves making further definitions. For instance, as you know, you can define the worldline volume form

(1)$\mathrm{vol}:=\sqrt{-h}d\tau$

and the pull-back of ${A}_{\mu }{\mathrm{dx}}^{\mu }$ to the worldline

(2)${}^{*}\phantom{\rule{-0.1667 em}{0ex}}A:={A}_{\mu }\frac{{\mathrm{dx}}^{\mu }}{d\tau }d\tau$

and then rewrite the first equation in the above exercise as

(3)$S=-m\int \left(\mathrm{vol}+{}^{*}\phantom{\rule{-0.1667 em}{0ex}}A\right)\phantom{\rule{thinmathspace}{0ex}}.$

You can then invent notations that allow you to vary this action without ever writing down an index. But I think this will not make things easier, necessarily. But if you find a nice way, let me know! :-)

Now, when I look at the string literature I can’t help but think that things are in as sad a shape as the original form of Maxwell’s equations.

I don’t think that that’s fair. The notation usually used is more like the analogue of

(4)${F}_{\left[\mu \nu ,\lambda \right]}=0$
(5)${F}^{\mu \nu }{}_{;\nu }=0\phantom{\rule{thinmathspace}{0ex}},$

which isn’t too bad. But maybe you disagree.

Here is a suggestion: You are assigned the special task of providing us with index-free notation of all the formulas that show up in the seminar! ;-)

But seriously, many thanks for your interest and I would very much enjoy if you find the time to follow the seminar here at the Coffee Table and provide us with comments and feedback.

Posted by: Urs Schreiber on March 12, 2004 3:36 PM | Permalink | PGP Sig | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Urs,

Here is a suggestion: You are assigned the special task of providing us with index-free notation of all the formulas that show up in the seminar! ;-)

Not that I have a lot of free time on my hands, but I don’t think it is a bad idea to really try this :)

I agree with your suggestion for ${A}^{*}$, but I’m not sure about $\mathrm{vol}=\sqrt{-h}d\tau$. What you really have is a metric $\text{g}$ on $M$ and you pull this metric back to the string ${\text{g}}^{*}$ and use this metric to construct a volume form on the string. I’ll have to think about a nice coordinate-free/index-free notation for doing this.

If you could give me a more specific section to try to convert, I could work on it. For example, where is a nice exposition of the Nambu-Goto and Polyakov actions? I could start there.

Best regards,
Eric

Posted by: Eric on March 12, 2004 5:03 PM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Thanks for correcting me on that volume form! Yes, for those trying to understand this stuff for the first time now it is of utter importance to understand that in my last comment I have been too cavalier with the notation.

As Eric rightly points out, in the Nambu-Goto action the metric that appears is that induced from the background to the worldvolume, i.e. the pull-back of the background metric. It is the Polyakov action where the metric on the world-volume is a priori independent of the background metric.

For example, where is a nice exposition of the Nambu-Goto and Polyakov actions? I could start there.

See for instance pp. 12-15 of Szabo’s lectures (which I’ll probably stick to most of the time). More details are given on pp. 173 of my sqm.pdf.

Posted by: Urs Schreiber on March 12, 2004 5:51 PM | Permalink | PGP Sig | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Urs,

It is the Polyakov action where the metric on the world-volume is a priori independent of the background metric.

I’m guessing that this is probably an important point, but I don’t yet fully appreciate it. The metrics must be related somehow, right? Sorry, I know this is probably basic (and I probably just read it in one of your spr conversations. Senility *sigh* :))

Eric

Posted by: Eric on March 12, 2004 6:50 PM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Yes, that’s an important point. The general (p-dimensional) Polyakov and Nambu-Goto action are classically equivalent. You can prove this for instance by varying the Polyakov action with respect to the auxiliary worldsheet metric and demanding that the result must vanish. The equation that you get says that the auxiliary metric of the Polyakov action must equal to (for $p=1$ it must only be conformally related to) the induced metric. Inserting this result in the Polyakov action yields the Nambu-Goto action.

In order for this to work for arbitrary $p$-branes one has to take care to include the right ‘cosmological’ term on the brane in the Polyakov action. See page 173 of sqm.pdf.

Posted by: Urs Schreiber on March 13, 2004 12:42 PM | Permalink | PGP Sig | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Urs,

It seems you are having a good time in Ulm. I wish I was there too. An Einstein fest sounds like fun :)

I have to admit I haven’t spent more than 30 minutes thinking about this yet, but it is not immediately obvious how to write the Polyakov action in a coordinate-free/ index-free manner. Come to think of it, I haven’t even made much progress with the Nambu-Goto action, but since that is just the volume form on the brane, I’m not too worried about it. Maybe I should be.

I think this is a nice exercise. Any hints? :)

Anyone else care to take a stab at it? I’m basically looking for the analog of the EM action

(1)$S={\int }_{M}F\wedge \star F$

for the Nambu-Goto and Polyakov actions that does not involve any explicit coordinate indices.

I may be going out on a limb, but the difference between Nambu-Goto and Polyakov reminds me a little bit about the difference between Yang-Mills and BF-theory. In BF theory (I’m sure to get this wrong, but it is something like this), you have an action

(2)$S={\int }_{M}F\wedge B.$

where we want some other input whose equations of motion give $B=\star F$. Does that make any sense? :)

In the Polyakov action, the induced metric seems to be analogous to $B$ (if you know what I mean).

Eric

Posted by: Eric on March 15, 2004 5:39 PM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Eric -

currently I cannot see the connection that you make concerning gauge theory and BF theory.

I’d rather suggest that you look at the Polyakov action as the action of 1+1 dimensional gravity coupled to scalar fields.

I think the index free version of the derivative terms can’t do better than

(1)${g}^{\alpha \beta }{\partial }_{\alpha }X{\partial }_{\alpha }X=\left(\nabla X\right)\cdot \left(\nabla X\right)\phantom{\rule{thinmathspace}{0ex}}.$

For a start, you can ignore the spacetime index $\mu$ on ${X}^{\mu }$ completely and only consider a single scalar field on the worldsheet. I think then it is obvious what to write down.

(BTW, I have found some people who are very interested in our work on discrete differential geometry. See here for the details :-)

Posted by: Urs Schreiber on March 15, 2004 9:45 PM | Permalink | PGP Sig | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Urs,

currently I cannot see the connection that you make concerning gauge theory and BF theory.

I’m still not sure it is relevant, but I was thinking of something along the lines of this:

http://www.lns.cornell.edu/spr/2000-09/msg0028175.html

To paraphrase, you begin with

(1)$S={\int }_{M}\mathrm{tr}\left(B\wedge F+{g}^{2}B\wedge \star B\right),$

which is just the usual BF action when $g\to 0$. Varying with respect to $B$ gives

(2)$F=-2g\star B.$

Varying with respect to $A$ gives

(3)${d}_{A}B=0,$

which combined with the first equation gives

(4)${d}_{A}\star F=0.$

This is what you get from varying the Yang-Mills action

(5)$S={\int }_{M}\mathrm{tr}\left(F\wedge \star F\right)$

with respect to $A$. Quoting Baez,

Sometimes this trick is called “BF-YM theory” - it’s a way of thinking of Yang-Mills as a perturbation of BF theory, which reduces back to BF theory as g -> 0. (It’s hard to see this happening at the classical level, where the Yang-Mills equations don’t depend on g. It’s better to go to the quantum theory - see Witten’s paper on 2d gauge theories for that.)

I think the index free version of the derivative terms can’t do better than

(6)${g}^{\alpha \beta }\left({\partial }_{\alpha }X\right)\cdot \left({\partial }_{B}X\right)=\left(\nabla X\right)\cdot \left(\nabla X\right).$

Hmm… wouldn’t $\nabla X$ be a vector valued 1-form, or something? If so, it is not obvious you wouldn’t get extra terms

(7)$\nabla X={\mathrm{dx}}^{\mu }\otimes {\nabla }_{\mu }X,$

which expands into a mess of connection coefficients. Then again, it is conceivable that metric compatibility will save the day, but I’m too lazy to check right now :)

My inability to write the Polyakov action in a nice coordinate-free/index-free notation is beginning to trouble me. Either there is something wrong with it, or we are lacking the language to express it more clearly. I’m not yet able to see the “meaning” of the expression.

Of course, that is just my own ignorance speaking :)

Best regards,
Eric

Posted by: Eric on March 16, 2004 3:29 AM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Eric -

maybe it is possible to express the worldsheet action using BF theory. I’d have to think about that a little more.

But let me say how I think we should write the Polyakov action in index-free and coordinate-free notation:

First assume there is a single scalar field $X$ on the worldsheet. Let $d$ and $\star$ be the exterior derivative and the Hodge star on the worldsheet with respect to the auxiliary worldsheet metric. Then the Polyakov action is simply proportional to

(1)$S\sim \int \left(dX\right)\wedge \star \left(dX\right)\phantom{\rule{thinmathspace}{0ex}}.$

But really there are $D$ such scalars which carry spacetime indices $\mu$. So really the string is described by

(2)$S\sim \int {g}_{\mu \nu }\left(X\right)\left(d{X}^{\mu }\right)\wedge \star \left(d{X}^{\nu }\right)\phantom{\rule{thinmathspace}{0ex}},$

where ${g}_{\mu \nu }$ is the metric on target space which is a priori independent of the worldsheet metric.

Of course now there are indices again. To remove these we need to introduce new notation. Maybe we should set

(3)$⟨v,w⟩:={w}^{\mu }{v}^{\nu }{g}_{\mu \nu }\phantom{\rule{thinmathspace}{0ex}}.$

Then the full Polyakov action, by a slight abuse of notation, could be written as

(4)$S\sim \int ⟨\left(dX\right),\wedge \star \left(dX\right)⟩\phantom{\rule{thinmathspace}{0ex}},$

Hm, no, this comma followed by a wedge looks a little awkward. Can you think of something better?

Posted by: Urs Schreiber on March 16, 2004 9:28 AM | Permalink | PGP Sig | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Urs,

maybe it is possible to express the worldsheet action using BF theory. I’d have to think about that a little more.

I don’t mean to imply that Polyakov can be derived from BF theory (although that would be neat and a part of me thinks there is some truth to it). Rather, my gut (which has been wrong before) suggests that there is an analogy: Yang-Mills is to Nambu-Goto as BF-YM is to Polyakov :) After this post, I’m thinking the analogy is actually there.

Let ${A}^{p}$ and ${A}^{q}$ be $p$- and $q$- forms, respectively. Same with ${B}^{p}$ and ${B}^{q}$. Define an inner product of ${\Omega }^{p}\otimes {\Omega }^{q}$ via

(1)$\left({A}^{p}\otimes {A}^{q},{B}^{p}\otimes {B}^{q}\right):=\left({A}^{p},{B}^{p}\right)\left({A}^{q},{B}^{q}\right)$

where $\left({A}^{p},{B}^{p}\right)$ is the usual inner product of forms.

Now let

(2)$h={h}_{\kappa \lambda }{\mathrm{dx}}^{\kappa }\otimes {\mathrm{dx}}^{\lambda }$

be the worldsheet metric and

(3)$g={g}_{\mu \nu }{\mathrm{dx}}^{\mu }\otimes {\mathrm{dx}}^{\nu }$

be the target space metric pulled back to the world sheet. Then (on the worldsheet) we have

(4)$\left(g,h\right)={g}_{\kappa \lambda }{h}_{\mu \nu }\left({\mathrm{dx}}^{\kappa },{\mathrm{dx}}^{\mu }\right)\left({\mathrm{dx}}^{\lambda },{\mathrm{dx}}^{\nu }\right)={g}_{\kappa \lambda }{h}_{\mu \nu }{h}^{\kappa \mu }{h}^{\lambda \nu }={h}_{\mu \nu }{g}^{\mu \nu }.$

Unless I’m mistaken, the Polyakov action may then be written in the suggestive form

(5)${S}_{P}={\int }_{M}\left(g,h\right)vol.$

Note that you have $\left(h,h\right)\sim 1$ so that the Nambu-Goto action is

(6)${S}_{\mathrm{NG}}\sim {\int }_{M}\left(h,h\right)vol.$

In this form, the Nambu-Goto action looks very much like a Yang-Mills action

(7)${S}_{\mathrm{YM}}={\int }_{M}\left(F,F\right)vol.$

I can massage the BF action to look something like the Polyakov action. Let $B=\star G$ for some 2-form $G$, then the BF action looks like

(8)${S}_{\mathrm{BF}}={\int }_{M}F\wedge B={\int }_{M}F\wedge \star G={\int }_{M}\left(F,G\right)vol,$

which looks a lot like the Polyakov action above.

How does this look?

Hmm… just before hitting “Post” I had a thought motivated by Baez’ spr post I referenced above. What if we take what I suggested as the “index-free” version of the Polyakov action above and modify it by

(9)$S{\prime }_{P}={\int }_{M}\left[\left(g,h\right)-\frac{1}{2}\left(g,g\right)\right]vol$

then varying ${S}_{P}^{\prime }$ with respect to $g$ results in

(10)$\delta S{\prime }_{P}={\int }_{M}\left[\left(\delta g,h\right)-\left(\delta g,g\right)\right]\mathrm{vol}={\int }_{M}\left(\delta g,h-g\right)vol$

so we get $g=h$. Neat! Maybe THIS is what I really want for the index-free Polyakov action.

Now what do you think? :)

Eric

PS: In general, the term $\frac{1}{2}\left(g,g\right)$ looks like a cosmological constant.

Posted by: Eric on March 16, 2004 3:22 PM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

I think that previous post (aside from errors) is pretty neat so maybe I’ll make the analogy between the index-free Polyakov action and the BF-YM action more explicit. If I begin with the BF-YM action, but with the replacement $B=\star G$ (so maybe I should call it FG theory :)), we get

(1)${S}_{\mathrm{FG}-\mathrm{YM}}={\int }_{M}\left[\left(F,G\right)-\frac{1}{2}\left(G,G\right)\right]vol.$

If we vary with respect to $G$, in complete analogy with the index-free Polyakov action, we get

(2)$\delta {S}_{\mathrm{FG}-\mathrm{YM}}={\int }_{M}\left[\left(F,\delta G\right)-\left(G,\delta G\right)\right]vol={\int }_{M}\left(F-G,\delta G\right)vol$

so that we have $F=G$. Varying with respect to $A$, we get

(3)${d}_{A}^{†}G=0,$

which combined with the first equation gives

(4)${d}_{A}^{†}F=0.$

Voila! Yang-Mills! :)

I think the formal analogy in the way Polyakov relates to Nambu-Goto and FG-YM relates to YM is clear. In fact, I think the relation between the FG-YM action and the Polyakov action is pretty clear (at least formally). Not to mention the relation between the Nambu-Goto and the Yang-Mills actions. Note the key words “I think” :)

Eric

Posted by: Eric on March 16, 2004 4:23 PM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Eric!

Writing

(1)${S}_{\mathrm{P}}={\int }_{M}\left(g,h\right)\mathrm{vol}$

and

(2)${S}_{\mathrm{NG}}={\int }_{M}\left(h,h\right)\mathrm{vol}$

is much better that what I had proposed and I agree that it does look suggestive.

However, we should be carefully distinuishing the roles played by $h$ in these expression. In ${S}_{\mathrm{P}}$ the term $h$ is the auxiliary worldsheet metric and it is understood that $\left(\cdot ,\cdot \right)$ contracts indidices using this auxiliary metric and that $\mathrm{vol}$ is constructed from this auxiliary metric.

But in the expression for ${S}_{\mathrm{NG}}$ as defined above and if we assume $\left(h,h\right)\sim 1$, then $\mathrm{vol}$ must be the volume form of the induced metric, which in the previous equation carried the name $g$! If we still want $\mathrm{vol}$ to be associated with the letter $h$ then now $h$ must be identified with the induced metric, in constrast to what we did above.

So while I think that each of the above expressions given above is a valid notation for the respective actions, the symbolds do not mean the same thing in these expressions. Agreed?

You write:

In general, the term $\frac{1}{2}\left(g,g\right)$ looks like a cosmological constant.

Hm, if this term is supposed to be the contraction of two copies of the induced metric with two copies of the auxiliary metric then I don’t see how it looks like a cosmological constant. A cosmological constant rather gives a term $\sim {\int }_{M}\mathrm{vol}$. But maybe I didn’t fully understand what you have in mind.

Posted by: Urs Schreiber on March 16, 2004 5:12 PM | Permalink | PGP Sig | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Urs! :)

So while I think that each of the above expressions given above is a valid notation for the respective actions, the symbolds do not mean the same thing in these expressions. Agreed?

I dunno. To me, $vol$ MUST be the volume form corresponding to the metric on the (sub)manifold. In the case of the Polyakov action, this metric is apparently called the auxiliary metric $h$ and is assumed to be distinct from the induced metric $g$ pulled back from the target space. I hope I got that right :) However, don’t we get as an equation of motion

(1)$h=g$

? So I don’t think I really understand the difference that you point out. In the end, we get $h=g$, right? Then the $vol$ would also be the same. I’m just confused as usual. Please bear with me :)

Hm, if this term is supposed to be the contraction of two copies of the induced metric with two copies of the auxiliary metric then I don’t see how it looks like a cosmological constant. A cosmological constant rather gives a term $\sim {\int }_{M}vol$. But maybe I didn’t fully understand what you have in mind.

It is more likely that I am confused :) I thought that $\left(g,g\right)~1$, which would mean that

(2)${\int }_{M}\frac{1}{2}\left(g,g\right)vol\sim {\int }_{M}vol.$

I’m obviously still grappling with the difference between the auxiliary and induced metrics.

Eric

Posted by: Eric on March 16, 2004 5:32 PM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Eric -

if we set $h=g$ throughout then there would be no difference between Polyakov and Nambu-Goto and we could just stick to writing either one letter or the other.

The important point is that one of the equations of motions for the Polyakov action says that $h=g$. But in order to have a well-defined action it must be defined even off-shell, i.e. when the equations of motion do not hold.

Note that ${h}_{\alpha \beta }$ is an autonomous metric on the Polyakov worldsheet which is treated exactly like the spactetime metric of GR is. But ${g}_{\alpha \beta }$ is a compound object which involves the scalar fields $X$ on the worldsheet

(1)${g}_{\alpha \beta }=\left({\partial }_{\alpha }{X}^{\mu }\right)\left({\partial }_{\beta }{X}^{\nu }\right){g}_{\mu \nu }\left(X\right)\phantom{\rule{thinmathspace}{0ex}},$

where $\alpha ,\beta \in \left\{0,1\right\}$ vary on the worldsheet while $\mu ,\nu \in \left\{0,1,\cdots ,D-1\right\}$ vary on spacetime and ${g}_{\mu \nu }\left(X\right)$ is the metric on spacetime evaluated at the point $\stackrel{⇀}{X}=\left[{X}^{0},{X}^{1},\cdots ,{X}^{D-1}\right]$.

So a priori ${h}_{\alpha \beta }$ and ${g}_{\alpha \beta }$ are completely independent. When we write down actions (before considering their equations of motion) these actions have to be well defined expressions in terms of these objects.

See, when you write

I thought that $\left(g,g\right)\sim 1$

you are apparantly simply setting $h=g$ and use the relation $\left(h,h\right)\sim 1$ which was postulated before and essentially defines what $\left(\cdot ,\cdot \right)$ is suppoed to mean.

But if we assume $h=g$ everywhere then we are really only dealing with the Nambu-Goto action, because for $h=g$ Polyakov reduces to Nambu-Goto. So there would be no point in having a seperate Polyakov form of the action.

But actually there is good reason to have an a priori independent worldsheet metric. So we may not set $h=g$ throughout.

Posted by: Urs Schreiber on March 16, 2004 7:08 PM | Permalink | PGP Sig | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Urs,

See, when you write

I thought that $\left(g,g\right)\sim 1$

you are apparantly simply setting $h=g$ and use the relation $\left(h,h\right)\sim 1$ which was postulated before and essentially defines what $\left(\cdot ,\cdot \right)$ is suppoed to mean.

Actually, I am even more confused than that :) I didn’t really mean to set $h=g$ in the beginning. I understand that would simply reduce to Nambu-Goto. I was making the mistake that I thought

(1)$\left(g,g\right)={g}_{\mu \nu }{g}^{\mu \nu }={\delta }_{\mu }^{\mu },$

but actually this is not true (I guess). By habit, I was thinking of ${g}^{\mu \nu }$ as the inverse of ${g}_{\mu \nu }$, but actually

(2)${g}^{\mu \nu }={h}^{\mu \kappa }{h}^{\nu \lambda }{g}_{\kappa \lambda }.$

Oops! :)

I am making progress. Thanks!

Also, I enjoy very much your reports of the conference :)

Gotta run!

Eric

Posted by: Eric on March 17, 2004 12:12 AM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Urs,

I’ve been sneaking out and reading Polchinski to try to get a better feel for this stuff. As usual, I can’t seem to content myself with what is written and have to explore alternative ideas :)

In Polchinski, for the point particle he has the action

(1)${S}_{\mathrm{pp}}=-m{\int }_{M}vol,$

where

(2)$vol=\sqrt{-\mathrm{det}\left(\gamma \right)}d\tau =\sqrt{-{\gamma }_{\tau \tau }}d\tau$

and

(3)$\gamma ={\gamma }_{\tau \tau }d\tau \otimes d\tau$

is the auxilliary metric intrinsic to the particle’s worldline. For future reference, I might as well write down

(4)$\delta vol=\left(\frac{{\gamma }_{\tau \tau }^{-1}}{2}vol\right)\delta {\gamma }_{\tau \tau }.$

Then he writes down (what is probably suppose to relate to Polyakov later on)

(5)$S{\prime }_{\mathrm{pp}}=\frac{1}{2}{\int }_{M}\left[\left(\gamma ,h\right)-m\right]vol=\frac{1}{2}{\int }_{M}\left({\gamma }_{\tau \tau }^{-1}{h}_{\tau \tau }-m\right)vol.$

Varying this with respect to ${\gamma }_{\tau \tau }$ results in the equation of motion

(6)${h}_{\tau \tau }=-m{\gamma }_{\tau \tau }.$

Then, plugging this back into the action, we get

(7)${S{\prime }_{\mathrm{pp}}\mid }_{\text{"on shell"}}={S}_{\mathrm{pp}}.$

This is nice except when I compare this $S{\prime }_{\mathrm{pp}}$ with what I was discussing above about BF-YM-type actions, this doesn’t seem to fit. So of course I tried the obvious alternative

(8)$S{″}_{\mathrm{pp}}=-{\int }_{M}\left[\left(\gamma ,h\right)-\frac{m}{2}\left(h,h\right)\right]vol,$

where unlike Polyakov, I treat $\gamma$ AND $h$ as degrees of freedom that generate equations of motion. Of course, varying with respect to $h$ gives directly

(9)$\gamma =mh.$

This is not surprising because this is essentially how $S{″}_{\mathrm{pp}}$ was defined, i.e. so that we would have $\gamma =mh$. The somewhat surprising thing to me was that if we vary $S{″}_{\mathrm{pp}}$ with respect to $\gamma$ WITHOUT plugging in $\gamma =mh$, then I get (with a little more work) the equation of motion

(10)$\gamma =mh$

AGAIN! :) In other words, varying $S{″}_{\mathrm{pp}}$ with respect to $h$ while holding $\gamma$ fixed gives the SAME equation of motion as varying $S{″}_{\mathrm{pp}}$ with respect to $\gamma$ while holding $h$ fixed. It has been so long since I studied Lagrangians that this is probably obvious, but I found it to be pretty neat :)

Of course, plugging in the equation of motion back into the action we get

(11)${S{″}_{\mathrm{pp}}\mid }_{\text{"on shell"}}=\frac{1}{2}{S}_{\mathrm{pp}}.$

My question is, is there anything new or interesting here? I probably just reinvented some ancient wheel, but it was fun :)

Eric

PS: In the above, I was following Polchinski’s convention, which seems to be the opposite of what we had been using. In the above, $\gamma$ is the auxilliary metric instrinsic to the worldsheet/line and $h$ is the induced metric pulled back from target space.

Posted by: Eric on March 18, 2004 5:21 AM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Hi Eric -

I would call your $h$ here auxiliary metric and $\gamma$ the induced metric. But never mind :-) Yes, possibly Polchinski uses other letters than we have, I don’t think there is a generally accepted convention for this stuff.

I am still in Ulm, but I have to hurry now to get to the trainstation so that I won’t miss the second meeting of our seminar tomorrow! :-) Meanwhile, since you are now thinking so much about strings, I bet you would enjoy this.

Posted by: Urs Schreiber on March 18, 2004 1:35 PM | Permalink | PGP Sig | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Good morning! :)

The danger of writing up something right before you sleep is that it tends to preoccupy your thoughts all night :)

Anyway, I’m not 100% confident that I performed my calculation correctly and I think the result is pretty neat so I thought I would spell it out here to make it easier to spot any holes. I’m basically proposing what seems to be an alternative (BF-YM-type) action for a point particle (which is supposed to generalize to $p$-branes)

(1)${S}_{0}=-{\int }_{M}\left[\left(\gamma ,h\right)-\frac{m}{2}\left(h,h\right)\right]vol,$

where

(2)$\gamma ={\gamma }_{\tau \tau }d\tau \otimes d\tau$

is the intrinsic metric on the worldsheet and

(3)$h={h}_{\tau \tau }d\tau \otimes d\tau$

is the pullback metric (I’m giving up on the ambiguous terminology auxilliary and induced :)).

I haven’t (yet) figured out how to do the variations in a coordinate-free/index-free manner so for the moment I’ll just follow more standard procedures and compute

(4)$\delta \left(\gamma ,h\right)=\delta \left({\gamma }_{\tau \tau }^{-1}{h}_{\tau \tau }\right)=-{\gamma }_{\tau \tau }^{-2}{h}_{\tau \tau }\delta {\gamma }_{\tau \tau }$
(5)$\delta \left(h,h\right)={\gamma }_{\tau \tau }^{-2}{h}_{\tau \tau }^{2}=-2{\gamma }_{\tau \tau }^{-3}{h}_{\tau \tau }^{2}\delta {\gamma }_{\tau \tau }$
(6)$\delta vol=\frac{{\gamma }_{\tau \tau }^{-1}}{2}vol\delta {\gamma }_{\tau \tau }$

Plugging all this in, I get

(7)$\delta {S}_{0}=-{\int }_{M}\delta {\gamma }_{\tau \tau }\left[-1+m{\gamma }_{\tau \tau }^{-1}{h}_{\tau \tau }\right]\frac{1}{2}{\gamma }_{\tau \tau }^{-2}{h}_{\tau \tau }vol,$

which barring ${\gamma }_{\tau \tau }^{-2}$, ${h}_{\tau \tau }$ or $vol$ being zero gives the equation of motion

(8)${\gamma }_{\tau \tau }=m{h}_{\tau \tau }$

or

(9)$\gamma =mh.$

Plugging this back into the action gives

(10)${{S}_{0}\mid }_{\text{"on shell"}}=-\frac{m}{2}{\int }_{M}vol.$

Ok. Having done this again, I think the chance of an algebra mistake has decreased significantly :)

What appears to be different here than what I see in Polchinski and in Szabo is that I am allowing variation of both the intrinsic and the pullback metrics. Then again, I guess that fact is not too important because we get the same equation of motion even if we think of $h$ as not variable. The thing this makes me think about is the common complaint that string theory is somehow background dependent. Could allowing the pullback metric, i.e. the one pulled back from the target space, also vary somehow be relevent to that question?

You’ve got to tell me if I’m way off base because this seems interesting to me :)

Eric

Posted by: Eric on March 18, 2004 2:48 PM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Oops! I actually get

(1)${{S}_{0}\mid }_{\text{"on shell"}}=-\frac{m}{2}{\int }_{M}\left(h,h\right)vol$

and I stuck in $\left(h,h\right)=1$ but actually we have $\left(\gamma ,\gamma \right)=1$ so that should be $\left(h,h\right)=1/{m}^{2}$ or

(2)${{S}_{0}\mid }_{\text{"on shell"}}=-\frac{1}{2m}{\int }_{M}vol$

so maybe I should replace $m$ with $\alpha =1/m$ in the action. I don’t know if a constant multiple is important or not. In any case, whatever constant you put in there we end up with

(3)${{S}_{0}\mid }_{\text{"on shell"}}\sim {\int }_{M}vol.$

which is the important thing I think.

Eric

Posted by: Eric on March 18, 2004 3:06 PM | Permalink | Reply to this

### Re: Meeting 1: Nambu-Goto, Polyakov and back

Sorry for all the posts, but I just noticed something else that seems interesting. Once I correct for that constant my action is more like

(1)${S}_{0}={\int }_{M}\left[\left(\gamma ,h\right)-\frac{\alpha }{2}\left(h,h\right)\right]vol$

whose equations of motion are

(2)$\gamma =\alpha h⇒h=m\gamma .$

Now when I look back at the equations of motion that Polchinski gets, I see he has

(3)$h=-m\gamma .$

Doesn’t that seem a little odd that the sign is different? I am growing more and more fond of my BF-YM-inspired action so please (anyone) shoot me down quick to ease my inevitable pain :)

I guess the fact that the two actions agree “on shell” means that classically they are equivalent, right? I wonder what happens quantuminally :)

Eric

Posted by: Eric on March 18, 2004 4:08 PM | Permalink | Reply to this

### Meeting 2: Free and yet constrained

This time I started by introducing Nambu-Brackets, which are defined by

(1)$\left\{{X}^{{\mu }_{0}},{X}^{{\mu }_{2}},\cdots ,{X}^{{\mu }_{p}}\right\}:={ϵ}^{{\alpha }_{0}{\alpha }_{1}\cdots {\alpha }_{p}}\left({\partial }_{{\alpha }_{0}}{X}^{{\mu }_{0}}\right)\cdots \left({\partial }_{{\alpha }_{p}}{X}^{{\mu }_{p}}\right)\phantom{\rule{thinmathspace}{0ex}}.$

(Here ${ϵ}^{{\alpha }_{0}{\alpha }_{1}\cdots {\alpha }_{p}}$ is the completely antisymmetric symbol with ${ϵ}^{012\cdots p}=1$.)

These are in a sense a generalization of Poisson brackets, to which they reduce for $p=1$. Using these brackets the determinant of an induced metric can conveniently be rewritten as

(2)$\mathrm{det}\left({h}_{\alpha \beta }\right)=\mathrm{det}\left(\left({\partial }_{\alpha }{X}^{\mu }\right)\left({\partial }_{\beta }{X}^{\nu }\right){g}_{\mu \nu }\right)=\frac{1}{\left(p+1\right)!}\left\{{X}^{{\mu }_{0}},\cdots ,{X}^{{\mu }_{p}}\right\}\left\{{X}_{{\mu }_{0}},\cdots ,{X}_{{\mu }_{p}}\right\}\phantom{\rule{thinmathspace}{0ex}}.$

Using this notation it is relatively easy to show that the general bosonic Nambu-Goto action for the $p$-brane

(3)${S}_{\mathrm{NG}}=-T\int \sqrt{h}\phantom{\rule{thinmathspace}{0ex}}{d}^{p+1}\sigma$

gives rise to the constraints

(4)${P}_{\mu }{\partial }_{i}{X}^{\mu }=0$
(5)${P}^{\mu }{P}_{\mu }+{T}^{2}\stackrel{˜}{g}=0\phantom{\rule{thinmathspace}{0ex}}.$

Here ${P}_{\mu }$ is the canonical momentum conjugate to ${X}^{\mu }$, ${\partial }_{i}$ is a spatial derivative along the brane and $\stackrel{˜}{g}$ is the determinant of the spatial part of the induced metric on the brane.

The first set of constraints are the spatial reparameterization constraints and the last one is known as the Hamiltonian constraint.

This clearly generalizes the constraint ${P}^{\mu }{P}_{\mu }=-{m}^{2}$ of the point particle: Every piece of membrane moves like a point particle with mass proportional to its volume ($\stackrel{˜}{g}$).

For the string with $p=1$ this reduces to the two constraints

(6)${P}_{\mu }{X}^{\prime \mu }=0$
(7)${P}_{\mu }{P}^{\mu }+{T}^{2}{X}^{\prime \mu }{X}_{\mu }^{\prime }=0\phantom{\rule{thinmathspace}{0ex}},$

where ${X}^{\prime }\left(\sigma \right)={\partial }_{\sigma }X\left(\sigma \right)$ is the spatial derivative of $X$ along the string.

The string ($p=1$-brane) is special in many respects. Here the special property is that the above constraints can be reassambled for $p=1$ in the symmetric form

(8)$⇔\left(P±{\mathrm{TX}}^{\prime }{\right)}^{2}=0\phantom{\rule{thinmathspace}{0ex}}.$

These are known as the Virasoro constraints.

In order to better understand them it is very helpful to make a Fourier decomposition.

To that end assume for simplicity that we are considering a flat Minkowski background spacetime (${g}_{\mu \nu }={\eta }_{\mu \nu }$) and define the objects

(9)${𝒫}_{±}^{\mu }\left(\sigma \right)=\frac{1}{\sqrt{2T}}\left({P}^{\mu }±T{X}^{\prime \mu }\right)\phantom{\rule{thinmathspace}{0ex}}.$

Using canonical quantization with commutator

(10)$\left[{X}^{\mu }\left(\sigma \right),{P}_{\nu }\left(\kappa \right)\right]=i{\delta }_{\nu }^{\mu }\delta \left(\sigma -\kappa \right)$

these have the commutators

(11)$\left[{𝒫}_{±}^{\mu }\left(\sigma \right),{𝒫}_{±}^{\nu }\left(\kappa \right)\right]=±{\eta }^{\mu \nu }{\delta }^{\prime }\left(\sigma -\kappa \right)\phantom{\rule{thinmathspace}{0ex}}.$
(12)$\left[{𝒫}_{±}^{\mu }\left(\sigma \right),{𝒫}_{\mp }^{\nu }\left(\kappa \right)\right]=0\phantom{\rule{thinmathspace}{0ex}}.$

Using this one can check that the Fourier modes defined by

(13)${\alpha }_{m}^{\mu }:=\frac{1}{\sqrt{2\pi }}\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{𝒫}_{-}^{\mu }\left(\sigma \right){e}^{-\mathrm{in}\sigma }$
(14)${\stackrel{˜}{\alpha }}_{m}^{\mu }:=\frac{1}{\sqrt{2\pi }}\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{𝒫}_{+}^{\mu }\left(\sigma \right){e}^{+\mathrm{in}\sigma }$

satisfy the oscillator algebra

(15)$\left[{\alpha }_{m}^{\mu },{\alpha }_{n}^{\nu }\right]=m{\delta }_{m,-n}{\eta }^{\mu \nu }$
(16)$\left[{\stackrel{˜}{\alpha }}_{m}^{\mu },{\stackrel{˜}{\alpha }}_{n}^{\nu }\right]=m{\delta }_{m,-n}{\eta }^{\mu \nu }$
(17)$\left[{\alpha }_{m}^{\mu },{\stackrel{˜}{\alpha }}_{n}^{\nu }\right]=0\phantom{\rule{thinmathspace}{0ex}}.$

Up to an inessential factor this are many copies of the well known relation of the creator ${a}^{†}$ and annihilator $a$ of the harmonic oscillator

(18)$\left[a,{a}^{†}\right]=1\phantom{\rule{thinmathspace}{0ex}}.$

This suggest that we construct the Hilbert space of string states from a Fock vacuum $\mid 0⟩$, which by definition is annihilated by all the ${\alpha }_{m>0}$ and ${\stackrel{˜}{\alpha }}_{m>0}$ with:

(19)${\alpha }_{m>0}^{\mu }\mid 0⟩=0$
(20)${\stackrel{˜}{\alpha }}_{m>0}^{\mu }\mid 0⟩=0\phantom{\rule{thinmathspace}{0ex}}.$

An arbitrary state in the Fock Hilbert space is then constructed by acting with creators ${\alpha }_{m<0}$ and ${\stackrel{˜}{\alpha }}_{m<0}$ on this vacuum state.

But we have to be careful because the 0-modes ${\alpha }_{0}$ and ${\stackrel{˜}{\alpha }}_{0}$ are not oscillators but proportional to the ‘center of mass’ momentum of the string (in the given parameterization):

(21)${\alpha }_{0}^{\mu }={\stackrel{˜}{\alpha }}_{0}^{\mu }=\frac{1}{\sqrt{4\pi T}}\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{P}^{\mu }\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}.$

As usual the (generalized) ‘eigenstates’ of the momentum operator are plane waves and hence the Hilbert space of the string is really the direct sum of the above oscillator excitations for a given center of mass momentum $p$. For the Fock vacuum at com-momentum $p$ we write $\mid p,0⟩$.

And this is where the content of this second meeting rather abrubtly ended. To be continued on Wednesday, 24th or March.

Posted by: Urs Schreiber on March 22, 2004 5:09 PM | Permalink | PGP Sig | Reply to this

### Re: Meeting 2: Free and yet constrained

Hi Urs,

Just another note on notation…

Given ANY p 0-forms (i.e. scalar functions on a manifold) f,g,…,h, you can define the Nambu brackets free of indices via

(1)$\left\{f,g,...,h\right\}=\epsilon \left(\mathrm{df},\mathrm{dg},...,\mathrm{dh}\right),$

where $\epsilon$ is a $p$-vector, i.e. the “Nambu tensor”.

:)

Eric

Posted by: Eric on March 22, 2004 6:48 PM | Permalink | Reply to this

### Meeting 3: The importance of being invariant

Today we talked a bit about the meaning of the classical Virasoro constraints and their Poisson algebra. Then we set out to completely ‘solve’ the classical closed bosonic string by constructing a complete set of classical invariants i.e. of observables that Poisson-commute with all the constraints - namely the classical DDF observables. Essentially all of what I wrote on the blackboard is what I previously typed into section 2.3.1 of this pdf. Please see there for more details.

In the process of these derivations a couple of important concepts came up, such as reparameterizations, the notion of conformal weight and the idea of reparameterization invariant observables. Having understood the DDF invariants classically should allow us next time to understand the massless spectrum of the closed bosonic quantum string as well as the need for it to propagate in the critical number of exactly 26 spacetime dimensions.

But maybe I won’t be able to refrain from first showing how from the DDF invariants one can construct the so-called Pohlmeyer invariants. This is the content of section 2.3.2 of the above mentioned pdf. Besides being very simple and instructive (and still related to current research) this would give a nice first opportunity to say something about Wilson lines and in particular strings as Wilson lines, which I plan to say more about in the near future.

We will meet next time on

Tuesday, 30. April, 15:00 c.t.

After that the schedule may become problematic: From April 4th-8th most of us will be in Bad Honnef, after that we have Easter (phew, what a horrible website… ;-), the week after that I’ll be at the AEI in Potsdam, and from Apr 16-19 I’ll be in New York, visiting Ioannis Giannakis at Rockefeller University. After that the semester starts again and we’ll have to figure out how to proceed anyway. The most recent information will always be found here in the latest comment.

Finally, here is a little exercise:

We have seen that the classical DDF observables ${A}_{m}^{\mu }$, ${\stackrel{˜}{A}}_{m}^{\mu }$ are morally similar to the ordinary worldsheet oscillators ${\alpha }_{m}^{\mu }$ and ${\stackrel{˜}{\alpha }}_{m}^{\mu }$ but with appropriate corrections inserted to make them invariant under one copy of the Virasoro algebra.

Compute the Poisson algebra of the DDF observables ${A}_{m}^{\mu }$, i.e. compute

(1)$\left[{A}_{m}^{\mu },{A}_{n}^{\nu }{\right]}_{\mathrm{PB}}=\cdots \phantom{\rule{thinmathspace}{0ex}}.$

Hint: Consider first the case where $\mu =i$ and $\nu =j$ are indices transversal to the lightlike vector $k$ (which enters the definition of the ${A}_{m}^{\mu }$), i.e.

(2)$\left[{A}_{m}^{i},{A}_{n}^{j}{\right]}_{\mathrm{PB}}=\cdots$

Then consider another lightlike vector $l$ with $l\cdot k\ne 0$ and compute

(3)$\left[l\cdot {A}_{m},{A}_{n}^{i}{\right]}_{\mathrm{PB}}=\cdots$

and

(4)$\left[l\cdot {A}_{m},l\cdot {A}_{n}\right]=\cdots$

Do you recognize the algebra of the $l\cdot {A}_{m}$?

Why don’t we consider the algebra of the $k\cdot {A}_{m}$?

Posted by: Urs Schreiber on March 24, 2004 5:45 PM | Permalink | PGP Sig | Reply to this

### Meeting 4: Subtle is the spectrum..

Today we finally did OCQ (old canonical quantization) and determined the spectrum of open and closed bosonic strings as well as the conditions for the Hilbert space of physical states to have non-negative metric and a maximum of null states.

What I said was a mixture of the following sources:

- - Polchinski, pp. 123-125

- - Szabo’s lecture pp. 19-27

- - Green,Schwarz& Witten, pp. 112-113 .

For reasons mentioned before we won’t meet again before at least 20th of April. In any case I will announce the new date here in the comment section.

Posted by: Urs Schreiber on March 30, 2004 5:46 PM | Permalink | PGP Sig | Reply to this

### Cosmological Conformality

Hello string people :)

I had a chance to spend a few minutes here and there during a mini-vacation to study some strings and I had a couple of questions.

First is just a statement, I’ll probably start bugging you at sci.physics.strings as soon as I can get our news server to make it available for subscription :)

Second, I have seen it emphasized how the Polyakov action is conformally invariant so that string theory is a conformal field theory. I’m sure I’m glossing over important details, but Urs has emphasized that in order for Polyakov to really be equivalent to Nambu-Goto, then you must add a cosmological constant term. However, unless I’m mistaken, adding a cosmological constant makes the action no longer conformally invariant so that Polyakov + cosmological constant doesn’t seem to be a conformal field theory to me. Is that correct? How important is it for string theory that it be a conformal field theory?

I guess a related question is that I’ve seen it also emphasized that string theory contains general relativity. Is general relativity a conformal field theory? If not, how is that supposed to work? :)

That will probably do it for now.

Best regards,
Eric

Posted by: Eric on April 1, 2004 2:40 PM | Permalink | Reply to this

### Re: Cosmological Conformality

Hi Eric -

recall that the ‘cosmological constant’ type of term in the Polykov-like action for a $p$-brane was proportional top $p-1$. So it vanishes precisely for the string!

Also note that when you do the computation which demonstrates the classical equivalence of the Polyakov-like action of a $p$-brane with the NG-like action of the same $p$-brane, you’ll find that for $p\ne 1$ the induced metric and the auxiliary metric must be precisely the same. But for $p=1$ you find that they may differ by a conformal factor!

(I did emphasisze this in the seminar, but maybe not so much here at the SCT.)

How important is it for string theory that it be a conformal field theory?

Conformal on the worldsheet, by the way, not in spacetime. GR is not conformal, but the strings which are in graviton states are conformal on their worldsheet.

Worldsheet conformal invariance is of paramount importance for the whole theory. For one, it dictates the form of the target space theory. (One can in principle do non-critical string theory which gives up conformal invariance on the worldsheet, though.)

Posted by: Urs Schreiber on April 2, 2004 12:17 PM | Permalink | PGP Sig | Reply to this

### Next meeting: Cosmological Billiards

The next meeting of our string theory seminar will be

Friday, 23. April 2004

10:00 c.t.

in room S05 V06 E22 .

I originally indended to give an introduction to two dimensional conformal field theory. But due to recent developments I now want to give an introduction to cosmological billiards instead. The CFT discussion will be made good for later.

In case anyone wants to have a look at some literature, here is a brief list of relevant papers (but my talk will be fully introductory, no familarity with either cosmology or billiards or in fact much else will be assumed):

T. Damour, M. Henneaux & H. Nicolai Cosmological Billiards (2002)

T. Damour, M. henneaux & H. Nicolai ${E}_{10}$ and the ‘small tension expansion’ of M Theory (2002)

J. Brown, O. Ganor & C. Helfgott M-theory and ${E}_{10}$: Billiards, Branes, and Imaginary Roots (2004)

R. Gebert & H. Nicolai ${E}_{10}$ for beginners (1994)

Anyone interested might also want to have a look at (and maybe participate in) a discussion thread on sci.physics.strings called Supergravity Cosmological Billiards and the BIG group.

Posted by: Urs Schreiber on April 21, 2004 5:05 PM | Permalink | PGP Sig | Reply to this

### Next Meeting: 2D conformal field theory

Our next meeting will be

Friday, 14th May, 10:15 S05 V06 E22 .

I’ll introduce some basic concepts of conformal field theory (CFT), discuss the Polyakov action from that point of view and show how the oscillator computations that we have discussed before are performed in terms of the more sophisticated CFT language.

The meeting after that will be

Friday, 4th June, 10:15 S05 V06 E22 .

Posted by: Urs Schreiber on April 28, 2004 4:15 PM | Permalink | PGP Sig | Reply to this

### Re: Next Meeting: 2D conformal field theory

Tomorrow morning I’ll have 60-90 minutes to explain from scratch some notions of CFT, what they are good for in string theory and in particular how to derive the Virasoro anomaly using CFT techniques. Some students in the audience won’t have heard much of QFT, so I’ll need to account for that. Pretty difficult task, eh? :-)

I thought it would be best to hand out some notes and then roughly go through the main ideas in these notes at the blackboard, with lots of specific pointers to the literature. I am under pretty tight time pressure, so the best I could come up with before going to bed tonight is this:

I know that there is lots of room for improvement and additions, but it’s a first step that should serve the purpose of showing how the Nambu-Goto/oscillator derivations that I have used so far have their complement in terms of Polyakov/CFT language.

Next time, on 4th of June, I’ll talk about BRST quantization.

Posted by: Ur s on May 13, 2004 7:13 PM | Permalink | PGP Sig | Reply to this

### Meeting 6: BRST quantization

The next meeting will be

Friday, 4th of June, 10:15 in S05 V06 E22 .

I’ll try to explain the BRST quantization of the bosonic string in Minkowski background.

Those interested might enjoy looking at some of the references listed by Eric Forgy in the SFT thread and the discussion which can be found there, especially maybe the sketch of the main idea, which I had given here.

I seem to recall some nice discussion of BRST formalism applied to some simple systems by Warren Siegel somewhere, probaly in FIELDS (nothing under the sun which is not mentioned in this book), but currently I can’t find it.

I’ll probably pretty much stick to section 4.2 of Polchinski’s book.

Posted by: Urs Schreiber on May 14, 2004 1:52 PM | Permalink | PGP Sig | Reply to this

### Re: Meeting 6: BRST quantization

Today I have talked about BRST techiques in general and how they are used in string theory and string field theory, in particular. I mostly followed appendix A in my Notes on OSFT which contains a simple description of exterior derivatives on gauge groups, on gauge fixing of path integrals and how both are related via the BRST operator.

I then motivated the construction of the cubic OSFT action as described for instance on the first couple of pages of hep-th/0102085 and briefly talked about the effective action obtained from level truncation and the meaning of tachyon condensation in OSFT.

The next meeting will be on 16th of July and I hope then to be able to say more interesting things about string field theory and in particular how deformations of worldsheet CFTs can be understood from the SFT point of view.

Posted by: Urs Schreiber on June 4, 2004 11:44 AM | Permalink | PGP Sig | Reply to this

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