Saturday, April 30, 2005

Giuseppe describes Maldacena's Long String

This is a delayed write-up of a similarly postponed group meeting. It seemed appropriate. The meeting was scheduled for monday but started last wednesday at 1500hrs. The informative email from Sylvain told us that Giuseppe d'Appollonio would talk about Maldacena's paper on the long string in two-dimensional string theory and non-singlets in the matrix model, hep-th/0503112. For the definition of long strings as classical string trajectories in a non compact space, we were referred to hep-th/0001053. Naturally I managed to get to grips with all of the first four pages of the main paper, and I printed off the second. I had also read Lubos Motl's description of a talk on this very paper (I would encourage you to read this article for your general education, and then come back and read this article to satisfy your curiosity to see just how little I manage to get out of a group meeting, in fact just read Lubos' blog). This, I felt, put me in a strong position, to just maybe get something out of a group meeting for once. However, I know nothing of the matrix model.

Fortunately, and kindly, Giuseppe began with a brief overview of singlets in the matrix model, beginning by telling us that the matrix model is a way of describing two-dimensional gravity, and that regularisation of the theory comes from the triangulation of the two-dimensional surface. If we take M to be an N by N matrix then we have can write an action of the form

\Int dM exp{-\beta \Int[ Trace {(dM/dt)^2} + V(M) ]}

Where "Trace {(dM/dt)^2}" is a kinetic term and "V(M)" is a generic potential term defined on the matrix. \beta was added after the rest when a short discussion of the continuum, or double scaling, limit occurred, see later. GdA asked us to consider the case when M is a matrix of constants, correpsonding to a flat geometry, so that we could get a beginner's feel for the model. We note that the kinetic term vanishes in the action in this case.

GdA next wrote an explicit formula for the potential:

V(M)=\sum_N[ { g_N \over N } Trace {M^N} ]

GdA described quadratic potential as a harmonic oscillator and drew two parallel lines of interaction on the board, the cubic potential and drew three interactions, the quartic and drew four interactions. For triangulation, he said in passing, a cubic potential was used. Next the double scaling limit was defined as the continuum limit coming from sending N to infinity along with appropriate limits on \beta, such that the string coupling constant g_s, (not shown here, but perhaps hidden in \beta, or maybe a notational error on my part: perhaps g_N should be g_s, any comments are welcomed) which is proportional to 1/N, remains finite. I should emphasise the potential for my notational errors in this write-up, so be wary.

GdA then lead us through a diagonalisation of the M, commencing with a Lagrangian, L:

L = 1/2 Trace {dM/dt}^2 - V{M}

M(t)=U'(t)L(t)U(t) where U' is the complex-conjugate of U, a unitary matrix; and L is the diagonal matrix of eigenvalues of M. We note that V(M) is unchanged, but the kinetic term is rewritten:

Trace (dM/dt)^2 = Trace (dL/dt)^2 + Trace [L, (dU/dt)U']^2

To see this judicious use is made of (dU/dt)U'=-U(dU'/dt), which comes from d/dt{UU'}=0. The Hamiltonian may be written as

H=-1\over D(L)\sum d^2/dL_i^2 D(L) +\sum V(L_i) +\sum_{i lt j} [\Pi_{ij}^2 + \Pi'_{ij}^2]\over [L_i - L_j]^2

Where D(L)=\Pi_{i lt j}(L_i-L_j), and "i lt j" means "i less than j". The case where the difficult last term vanishes is considered, i.e. when \Pi_{ij}\Psi = 0, so that \Psi transforms trivially under SU(n). Under this condition this is the Hamiltonian for the matrix model in the singlet sector. D(L)\Psi gives an antisymmetry condition in the eigenvalues so we believe this is really a fermionic theory in an upside down harmonic oscillator potential. States are filled up on only one side of this potential. So that's my beginner's knowledge of the matrix model, in a nutshell.

GdA moved onto the long string paper, which is a proposal for the non-singlet sector in the matrix model. Maldacena considers an action,

S{\phi}=\int\sqrt{-g}d^2x[g^{ab}\partial_a\phi\partial_b\phi + QR^(2)\phi + 4\pi\mu exp{2b\phi}] - \int d\tau d\sigma [\partial_+X^0\partial_-X^0]

(Yes, I know, I need to start using mathplayer or come up with some trick for displaying formulae.) Where we have the usual k.e. term, a potential poportional to the dilaton and a Liouville potential term. The ansatz X^0=\tau has been used. GdA told us that Maldacena first considers the case where \mu=0, only the linear dilaton potential remains. The solution is described by a path in the worldsheet (\tau,\phi) that commences at -\tau at minus \phi infinity with a gentle positive gradient. This increases quickly at positive \phi and crosses the \tau=0 in this region; the rest of the solution is the reflection in the \phi axis, giving an overall path like the contour of the end of a cigar. Despite this being a worldsheet picture, the target space, time-evolution picture matches the intuition that the string is stretched all the way in from infinity. This is called the long string, as it extends back to minus infinity twice, so is very long; in fact it is so long it can even be seen in London all the way from Princeton where it originates :) One considers the tip of the string as the centre of mass, \phi_m. This can be thought of as a massive particle moving under a force that pulls the centre of mass back towards negative infinity. In terms of string tension, T, the energy, E, is E=2T(\phi_m - \phi_c). The "2" comes because the string doubles back on itself and the tension acts on the centre of mass twice in the same direction. N.B. this is the energy required to stretch the string a length 2(\phi_m-\phi_c). \phi_c is a cutoff in the energy, and is therefore a little contentious, but we can understand that if we didn't make a finite cutoff then the change in energy in our calculation would be divergent, due to the stretch being from minus infinity.

Now comes a neat suggestion, Maldacena introduces an FZZT brane that is extended back to minus infinity in \phi. The long string is then considered as an open string attached to the FZZT and stretched to its limit in positive \phi. The cutoff at \phi_c is now seen to cut the FZZT brane out of the picture and also subtracts the divergent energy associated to the excitations of the brane, which extends back to infinity.

This took us up to page 9 of the paper. The remainder was covered much more quickly and no meaningful commentary on it can be made here by me, alas. So in all, I gained 5 pages on my preparation through the meeting, this was very satisfying, and perhaps a record.

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