### 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:

D. Marolf, Resource Letter NSST-1: The Nature and Status of String Theory

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 (**th**eoretical **h**igh **e**nergy **p**hysics) and maybe gr-qc (**g**eneral **r**elativity and **q**uantum **c**osmology) 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

reading, i.e. when I already understand the basics. See

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.)

More advanced introductions are

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

and

L. Dolan TASI Lectures on Perturbative String Theory and Ramond-Ramond Flux (2002)

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.

## 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

pointparticle, 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

exerciseconcerning the material discussed in the first meeting:I had demonstrated how the mass shell constraint

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

How does the mass-shell constraint look now?

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

where $\alpha ,\beta \in \{\mathrm{0,1}\}$ are the indices on the worldsheet, ${h}_{\alpha \beta}=({\partial}_{\alpha}{x}^{\mu})({\partial}_{\beta}{x}^{\nu}){g}_{\mu \nu}$, is the induced metric on the worldsheet and $h=\mathrm{det}{h}_{\alpha \beta}$ is its determinant. ${\u03f5}^{\alpha \beta}$ 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 meetingwill beFriday, 19 Mar 2004, 15:00, in S05 V07 E04.

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