Thursday, March 24, 2005 ... /////

Maldacena in the Lineland

Joanna Karczmarek is an insider in two-dimensional string theory. She has just described Juan's recent paper with many details that are not even present in his paper:

The paper is relating two families of objects and questions:
• Infinitely long strings in two-dimensional string theory
• Non-singlet wave functions of the corresponding matrix model
Concerning the first point: the dynamics in the Lineland, which is how Brian Greene calls the (1+1)-dimensional backgrounds, may look pretty boring. (We will be discussing the bosonic string theory lineland, which contains a massless "tachyon" as the only local excitation.) When you play chess, you can't even castle. Your destiny is to look into the eyes of your neighbor forever. These eyes (namely the boundaries of line intervals) are just points and they carry no emotions. It's because there are just a few types of extended neighbors in this strange Universe:
• The ZZ brane which is a localized type of a D-brane and plays no role in this article
• The FZZT brane that is spacetime filling, but the tachyon profile goes from the maximum to the minimum in such a way that the brane continuously disappears
• The fundamental string
Concerning the fundamental string, it carries
• D-2 = 2-2 = 0
transverse sets of oscillators which is not too many. Consequently, there is no Hagedorn tower of states, just one ground state which we still call the tachyon but whose mass is actually zero in two dimensions.

The general classical solutions for the motion of the string in the absence of the linear dilaton - solutions both to the wave equation as well as Einstein's equations (which is the same as the Virasoro constraints because the Einstein tensor vanishes in d=2) - happen to be a sum of a zig-zag function of (tau+sigma) and another zig-zag function of (tau-sigma) for the observable PHI(sigma) while X0(sigma,tau) is gauge-fixed to be simply "X0=tau". Here, PHI is the spatial coordinate of the two-dimensional spacetime. A zig-zag function is defined as a piece-wise linear function whose derivative is either +1 (zig) or -1 (zag). You can add the linear dilaton which modifies Einstein's equations a bit - you know that the stress energy tensor contains not only the piece bilinear in the first derivatives, but also a term proportional to the second derivative of the field PHI (the spacetime direction parallel to the dilaton gradient).

At the very end, you also want to add the Liouville wall - the condensate of the tachyonic field that is proportional to "mu.exp(a.PHI)" - whose effect is that it reflects the incoming strings (if it has high energy, you can treat it as a discrete bounce).

The main point of interest is a string that switches the direction from "zig" to "zag" at some point, and this folding point therefore looks like an endpoint. Also, it moves by the speed of light. As far as the information goes, the semi-infinite folded string is described by a position of the folding point - and this point behaves as a relativistic particle that moves by the speed of light (much like the usual endpoint of an open string). Juan also recycles some two-point functions on the disk, and uses the FZZT branes to get some results he need. In this context, he considers an open string attached to the FZZT brane whose worldsheet however reaches the regions outside the line interval between the two endpoints of the open string which is a kind of cute picture.

The matrix model: non-singlets

If you declare the SU(N) symmetry of the matrix model (which is a (0+1)-dimensional quantum field theory) to be a gauge symmetry, it is equivalent to projecting the spectrum onto the subspace of SU(N) singlets. Is it also possible to study other irreducible representations of SU(N), for example the adjoint representation? The answer is Yes, it is. You may also define the corresponding quantum theory in the path integral language. In that case, you must insert the following operator as a factor into the path integral:
where the Wilson loop (either open, or closed, depending on what you calculate) is the usual path-ordered exponential of the integral of the gauge field. Note that the analogous operator
• Tr_{singlet} (Wilson loop)
is simply equal to one, because the Wilson loop acts as an identity operator on the singlet representation, and therefore this insertion may be forgotten. However, if you trace over the adjoint representation, the first trace we mentioned influences the path integral.

With this operator inserted, you want to find the spectrum. Note that the spectrum in the singlet case is described by the system of free fermions. What do you have to do to switch to the adjoint representation? You must add an extra term in the Hamiltonian, an extra index to your wave function, and various additional constraints and generalized symmetry requirements for the wave function.

Joanna claims that Juan admitted that he has neglected one particular requirement that the wave function must still be invariant under the unbroken permutation subgroup of the gauge group. When this requirement is taken into account, it seems that the only wave function that satisfies everything it should satisfy is simply
• Psi = 0.
Unless there exists an elegant loophole that avoids this conclusion, one half of Juan's paper is about a vanishing wave function. It is believed by most people that there is some subtle corrections that will save this half of the paper (as well as its intriguing connections to the other half).

Incidentally, Joanna has also identified an error with an older paper by Gross and Klebanov; they assumed that only very special types of SU(N) representations can occur in the matrix model (essentially those that are the tensor product of a finite Young diagram and its complex conjugate). Finally, she was informed that this error had been known for some time.

One of the punch lines of Juan's paper is that the matrix model where we switch from the singlets to the adjoint representations may be visualized, after some computations, as the same Fermi liquid with an additional particle of a new type inserted - and this new particle is interpreted as the folding point of the string. The inclusion of more complicated representations than the adjoint representation is more subtle, and it involves something that looks like a group of particles.