Sunday, December 02, 2007

Projection


Between Princeton and New York, the past few days have been too exciting to describe. Saturday at 11:00 AM saw the premiere of Grand Central Terminal's Kaleidoscopic Light Show. As music played, stars and fireworks were projected onto the enormous vault and columns. More about this place soon.

I spoke with Peter Woit when he showed up in Princeton Wednesday. Gauge Theory and Representation Theory are small fields, and this post will show why. He was curious how many would show up, whether they would fill the 100 seats. Next week I'll be far away at a meeting with 11,000 scientists. Woit was curious where that was.

As Woit expected, the biggest audience was Wednesday morning for Edward Witten and his collaborator Sergei Gukov. The latter's talk was more interesting, for it told what Witten is working on. Woit gives a far longer description in Not Even Wrong, which may not be long enough to please all mathematicians. An even shorter summary is recounted here, though this gets mathematical.

G is the compact Lie Group. The goal is to understand G_R representations in terms of D-branes. G_R the real form of G_C where G_C is the complexification of G. This leads to a 4-dimensional topological gauge theory. M_4 is a 4-dimensional manifold, a product of a 3-manifold and an interval I.

M_4 = $ \omega x I$

To connect this to reality, a boundary condition is to preserve topological Supersymnetry. One should note that SUSY itself is a highly speculative idea. The many particles predicted have never been detected.

If we remove one dimension, the manifold can be pictured as the ceiling of Grand Central Terminal with the additional dimension projected onto that manifold.

This all leads to a 3-dimensional Quantum Field Theory on W, where W is a Chernin-Simons theory with gauge group G. "Surface Operators" are operators in a 4-dimensional theory supported on a 2-dimensional surface D (like Grand Central's ceiling) which is a subset of M_4.

It is considered "natural" to take D = $\gamma x I$, where $\gamma$ is a member of W and I is an interval.

Next we take W = R x C, where C is a Riemann surface and R is time and $\gamma$ = R_x.

The Hamiltonian approach leads to a Hilbert space H. C is replaced by a punctured disk, leading to a representation space H.

In 4-dimensional gauge theory, M_4 = $\Sigma$ x C where $\Sigma$ = R x I and R is time. We have seen before that R = t in Planck units. In MKS units R = ct where t is time and c is the speed of light.

A 4-dimensional gauge theory on M = $\Sigma$ x C is equivalent to a 2-dimensional topological model.

$\Sigma$ leads to M_H(G,C)

For applications to gauge and representation theory, C = D* a punctured disk. Boundary conditions are specified only at the puncture. Some lively audience questions asked what happens at the puncture, whether it represents a singularity.

Next we have solutions to Hitchins equations:

M_H = T*(E/T) = N = $\Theta_reg$
$\Theta_reg$ takes the form W_i = $\alpha$, W_j = $\beta$, W_k = $\gamma$ where $\alpha$, $\beta$, and $\gamma$ are members of L, the compact Lie algebra.

Finally a Hilbert space is proposed H = hom(Bcc, B') which could be a space of open string states between two branes Bcc and B' on M_H = T*(G/T). B' is a brane supported on G/$\pi$ and Bc is the canonical isotropic brane.

As all can see, this is not just complicated but highly speculative. "Branes" are a theoretical surfaces existing in higher dimensions that intersect with ours. Strings enter the picture only as one possible way to connect the branes. While the maths are interesting, none of this leads to a single testable prediction. Projecting these speculations onto the ceiling of reality will be quite difficult.

For years the string enterprise dominated theoretical physics, pushing other promising ideas and people out. Thanks to the hammering of critics like Peter Woit, strings are rapidly falling out of fashion.Though Edward Witten was once considered a priest of the enterprise, his latest work moves away from strings. The way is open for more useful theories that make testable predictions.

Bored yet? More interesting Space news is in the new Carnival of Space!

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