The String Coffee Table

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

$$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

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 $h=\det h_{\alpha \beta }$ 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).

Meeting 2: Free and yet constrained

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

(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

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

gives rise to the constraints

Here $P_{\mu }$ is the canonical momentum conjugate to $X^{\mu }$, ${\partial }_i$ is a spatial derivative along the brane and $\tilde{g}$ 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 ($\tilde{g}$).

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

where $X^{\prime }(\sigma )={\partial }_{\sigma }X(\sigma )$ 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

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

Using canonical quantization with commutator

these have the commutators

Using this one can check that the Fourier modes defined by

satisfy the oscillator algebra

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

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

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

But we have to be careful because the 0-modes ${\alpha }_0$ and ${\tilde{\alpha }}_0$ are not oscillators but proportional to the ‘center of mass’ momentum of the string (in the given parameterization):

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 $\mid p\mathrm{,0}\rangle$.

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

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 $A_m^{\mu }$, ${\tilde{A}}_m^{\mu }$ are morally similar to the ordinary worldsheet oscillators ${\alpha }_m^{\mu }$ and ${\tilde{\alpha }}_m^{\mu }$ 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

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.

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

and

Do you recognize the algebra of the $l\cdot A_m$?

Why don’t we consider the algebra of the $k\cdot A_m$?

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.

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.

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 .

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.