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.

Wednesday, April 27, 2005

Freund-Rubin Superpotentials Revisited

Neil Lambert gave a seminar today on his paper from earlier this year entitled "Flux and Freund-Rubin Superpotentials in M-Theory", which is a continuation of work begun here. As usual my desire for an efficient arrival, i.e. get there at start time +/-delta where delta is minimized by my journey resulted in a small additive delta, and so I missed the very beginning. I'm not sure this is crucial as most of the material was unfamiliar to me.

Anyway Neil was asking 2 key questions as I got there, about the vacua in the landscape or haystack:

-Is the standard model there?
-How special or generic is it, if it is indeed there?

At this point NL emphasised that Douglas et al. who are working on the landscape simply aim to count the number of vacua there are and not the probability density of each vacuum existing (which may or may not be a delta function). NL summarised that "the counting statistics of the vacua is maths, the probability density is the physics." In the coffee break afterwards he pointed out the potential usefulness of the counting statistics, in that one may be able to find for some cases (i.e. specific vacua criteria) an absolute count that is less than one, in which case there is no such vacuum. This tells us about the probability density of the vacua and in such cases we learn some physics. It seems, at least in London, that the landscape is not as quite the rage here as it is in North America at the moment, so this elaboration was needed.

In the introduction NL stated that in the talk we would be concentrating on M-theory vacua M_4 x X, and that for N=1 Minkowski vacua X is the group G_2. He then told us that over twenty years ago Freund and Rubin introduced a class of M-theory (D=11 supergravity back then) "compactifications" with spacetimes AdS_4xX. To obtain N=1 supersymmetry in this case X is weak G_2. Weak G_2 was defined as a pair of conditions on the three-form \phi and its dual *\phi contsructed in the theory:

d\phi=4\lamdba *\phi, d*\phi=0

At a later point NL mentioned that while G_2 is restored under the condition \lambda=0, it is not true to think of weak G_2 being close to G_2 for small values of \lamda. In the paper it is pointed out that this is expected as \lambda can be made small by increasing the volume of the manifold, but this clearly doesn't change the properties of the manifold in a non-trivial way. NL said that his paper was essentially redoing the paper of Beasley and Witten but with weak G_2 rather than G_2.

The main body of the talk closely followed the layout of the paper, but without giving the detail of the calculations, and was a lively talk, however since so much of this was alien to me I don't feel comfortable trying to regurgitate it here. So I will just give three of NL's summary points:

1. the consistent construction of the the potential and superpotential of the Freund-Rubin compactications in the presence of topological fluxes.
2. that when the fluxes were turned on the F-R terms were driven to zero, resulting in a non-supersymmetric minimum
3. there are no supersymmetric vacua other than F-R or pure G_2

A major concern of NL's was that the Kaluza-Klein modes arising from the compactification are of the same order as the cosmological constant, and he was particularly interested in looking for ways to lift a decently small, positive cosmological constant from the theory using a "KKLT mechanism". For NL this means not fine-tuning the KK terms to get the desired cosmological constant but rather seeing if it was even a possibility. One scheme he considered involved the wrapping of a 9-cycle at which point he stopped and admitted, with spirit, that he didn't think there was a single person who beleived in a 9-brane. This was perhaps the second-most humerous comment of NL's upbeat seminar, the first being upon a specialisation to the bosonic case when he said "since I'm a supersymmetry guy, I never talk about fermions." Quite right too - in truth, I only wish that I could.

After one question, Nicolas announced the usual "coffee and cookies" in the coffee room, which was greeted by a very silent joy and gratitude.

Tuesday, April 26, 2005

Duff Talk Missed

Due to the relatively early hour of Duff's talk, 1300hrs, and the need to crusade across London from East to West to get there, well, in short, I missed his layman's guide to M-theory. Any second-hand news of this talk will be reported here at a later date, and sources will be given.

Monday, April 25, 2005

Donaldson on Curvature and Physics

I trotted out of Drury Lane and off to Imperial College's Clore Lecture Theatre this afternoon to see Field's medallist Simon Donaldson talk on "Curvature and Geometry in Physics". The talk is the first of a series this week to mark the launch of IC's very own Institute for Mathematical Sciences, of which Donaldson will be the first president. Tomorrow Michael Duff, soon to be Principal of the Faculty of Physical Sciences at IC, will give a talk entitled "A Layman's Guide to M-Theory" - hopefully we will hear his favourite meaning for the M. But the rest of the week's talks are less relevant to theoretical physics, it's almost as if there are other mathematical sciences beyond high energy physics :)

Michael Duff introduced Lord May of Oxford, head of the Royal Society, who said hello and then told us that "Simon Donaldson needs no introduction", but, of course, we had already had two. With a slight buzz of interference the microphone system was switched to SD who began by telling us that he was there to fight the pure mathematician's corner, but had in deference to the name of the new institute included a science, namely physics, in his title. This was to be expected. SD used a series of very clear slides, making three or less points per slide, and began the talk by asking the audience to consider the manner in which a line field gives rise to integral curves and whether or not there was an analog for plane fields giving rise to integral surfaces. The answer in general was no. Then to captivate the mind of the physicists in the audience SD exhibited the example of a trolley moving on a flat surface, and then transported it to a plane field. In the flat case the trolley moved around a quadrilateral returns to its starting point, but in the plane field example the trolley is generally displaced along the normal to the plane field at the start/end point. The curvature was defined at this point as a skew-symmetric map from the plane field, P, to the normal to the plane field P': PxP->P'. And from this starting point four relations to curvature were summarised as

1. Deviation from the flat model
2. Intrinsic/Extrinsic
3. Parallel transport
4. Integrability conditions for an overdetermined set of variables

No. 2 was elaborated on and the difference between the two types of curvature was described as whether or not a fly (presumably a mathematician constrained to the manifold [the mathematican having by supposition the dimension of the manifold, without any harm coming to him or her] and not really a fly) living on the surface would be able to deduce if it was living on a plane or not. If the 'fly' could tell, e.g. if it lived on a sphere, then the curvature is intrinsic. If the 'fly' could not tell, e.g. if it lived on the curved side of a cylinder, then the curvature is extrinsic. This was good to know.

SD then made some comments about general relativity because it is the most common place curvature turns up in physics, arising from the pseudo-Riemannian metric, g, on the space-time manifold. But SD decided to focus on curvature in electromagnetism instead. The geometric interpretation of EM comes through complex line bundles, L, over flat space-time, R(3,1); so that L~CxR(3,1), but not as he emphasised canonically. There is an association between the scalar and vector potentials of EM and the field of subspaces inside L (connection) and furthermore that the Maxwell field strength, F, is asssociated to the curvature of L. SD continued that the complex line bundles were only apparent in quantum mechanics, where the wavefunction was really a section of L, and the Hilbert space of wavefunctions is really a set of sections of L. Gamma(L) was defined, here for use later, to be the set of sections of L. This part of the talk was rounded off by a comment that in string theory there is a more complicated bundle.

The focus was then shifted to curvature and geometry, commencing with topology and the Poincare conjecture in three dimensions and SD's quiet endorsement of Grigori Perelman's work. The Ricci flow of the metric with respect to a parameter, theta, was given in a short equation as partial{g}/partial{theta}=-Ric(g_theta) (here Ric means the Ricci curvature defined from the metric) and the question of whether one could deform an arbitrary three-dimensional metric to Ric=constant of the three sphere was described.

Next, complex algebraic geometry (CAG) was looked at. SD commenced by saying that CAG involved making an association between a set and a ring. For example the set of points {x,y} making a unit circle is associated to the ring {Set of all polynomials on x, y with complex coefficients}/{}. Projective geometry was briefly described as the association between a vector z~Kz where K is a non-zero real(?) coefficient (as an example of the results of a projection was the equivalency of the parabola, ellipse and the hyperbola, and this motivated the definition). An example set was given using the projection as X={z: f(z)=0}/{z~Kz}, where f is a homogeneous polynomial. SD then declared that in projective geometry one has sections of line bundles as oppose to functions, such as L -> X and that a ring can be constructed as R= Direct Sum from k=0 to infinity of Gamma(kth order tensor product of L). At this point the example of toric varieties was discussed and entirely lost on me but it looked interesting nonetheless, and can be added to my list of interesting things to look-up one day, which is frankly far too long. When I came to, the basic principle of the talk was being given inside a red box, so it must be important. It is that there is an association between positive curvature (high g) and Gamma(L) being large (many sections).

Some more was said. Then a recap was given, which consisted of a picture with a line bundle, L, over a set then from the left hand side of L emerged an arrow to algebraic geometry and the ring defined as the direct product sum of Gamma(L)'s, and from the right came an arrow to differential geometry, and a line of arrows from curvature of L = F, to g (suppose positive definite), to curvature of g (Ricci curvature). Finally the question why was asked and the answer was given in the example of Calabi-Yau metrics, where the set was given as X={a^5+b^5+c^5+d^5+e^5=0}/{Some equivalency relation, not given}.

Afterwards a reception was held on the eight floor and some very nice food was enjoyed by Vid and myself and I had a glass of wine. We didn't talk about curvature.

New Blog Promise and a False Start

Ok, so this has become one of the worst kept blogs online. But, hopefully, no more! The new semester is just beginning here at King's so I hereby vow to attempt to publish the happenings of the seminars and group meetings each week. However today's group meeting has been postponed till wednesday. However, Donaldson is talking at Imperial later today, so all being well, I will record the goings on here later.