ooguri featurette

8/9/2019 Ooguri Featurette

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26 E N G I N E E R I N G & S C I E N C E N O . 3

And in extremely rare cases . . . you get the idea.You can calculate each diagrams individual effect,add them all up, and eventually derive an overalldescription of the particles behavior. In general,the more complicated the diagram, the less likelythe process depicted in that diagram is to happen,so you can cut off the calculation at any level ofcomplexity and get a corresponding level ofaccuracy. Thats how things work when we applythe Standard Model to high-energy collisions, asshown by Professor of Theoretical Physics DavidPolitzer and others, or to the various precisecomputations in quantum electrodynamics thatFeynman studied so successfully, says Ooguri.

Each diagram is represented by a single termin the expansion, or overall calculation, and everyterm contains two key parameters. The first,called g, is the coupling constant, which is ameasure of the strength of the particles interac-tion. Its raised to the power of the number ofvertices, or places where lines meet, in the dia-gram. The second, called N, is raised to the powerof the number of closed loops in the diagram. So,for example, the odds of the Feynman diagram atupper left happening are governed by g8N3.Nis always a positive integer, and in the

Standard Model, Nequals three because quarkscome in three colors. More generally,Nis therank of the matrix in the SU(N) gauge-symmetrygroupdont ask: all you need to know is that theStandard Model is a gauge theory. In gauge

theories, forces are carried by particles, such asgluons and photons; the elusive quantum-gravityparticle is called the graviton.

If youre dealing with electricity, magnetism,or the weak nuclear force, the coupling, g, is verysmallfor electromagnetism at atomic distances,its about 0.1and the high-power terms faderapidly into oblivion. If each vertex costs youg,then the more complicated the diagram becomes,the higher the power ofg you get, and that sup-presses the diagram, Ooguri explains. So ifg

is small, then you need only worry about the rela-tively simple Feynman diagrams.

Unfortunately, the harder you pull quarks apart,the more gluons they will exchange as they tryto keep their grip. The reason more gluons getexchanged is because the coupling constant grows,and the coupling constant grows because thegluons interact. Its a chicken-and-egg problem.The method gets stood on its headthe morecomplex the Feynman diagram, the more likelyit is to occur. You get stuff that looks like fineFrench lace, and the calculation spins wildly outof control. So successive terms get bigger andthe calculation never settles down on an answer.

But Gerardus t Hooft, who shared the 1999Nobel Prize in physics with Martinus Veltmanfor elucidating the quantum structure ofelectroweak interactions, saw a way out. Sincethe calculation depends on Nas well asg, and Nis always greater than one, he figured out a wayto expand the equations in terms of 1/N. You stillhave to consider all the Feynman diagrams, butnow the more complicated the diagram, the bet-teras you divide by higher and higher powersofN, the terms get smaller and smaller.

t Hoofts approach allows you to add upinfinitely many Feynman diagrams by classifyingthem by their topologies rather than their numberof vertices. To see what this means, consider thecase of three lines meeting at two vertices, likethe international do not symbol. This diagram

This three-gluon exchange (left) has two vertices and three

complete loops (right)

The harder you pull quarks apart, the more gluons they will exchange as they

try to keep their grip. The reason more gluons get exchanged is because th

coupling constant grows, and the coupling constant grows because the gluon

interact. Its a chicken-and-egg problem

N

N

g g

N

In the Feynman diagram

above, two quarks emit a

pair of gluons that then

exchange another pair of

gluons among themselves.

The diagram has eight

vertices, labeled g, and

three complete loops,

labeled N, as shown below.

Its contribution to the

overall process is

proportional to g8N3.

N

N

N

g

g

g

g

g

g

g

g


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E N G I N E E R I N G & S C I E N C E N O . X 27 E N G I N E E R I N G & S C I E N C E N O . 3

represents a vacuum exchange of three gluonsin other words, a triple-gluon swap between twoparticles that arent there; in quantum mechanics,empty space is filled with virtual particles thatpop into being from nothingness and promptlydisappear again. The diagrams two vertices giveyoug2, and there are three closed loops for N3.And if you think of the diagram as being madeof flat strips, so that each loop is an edge, you geta disk with two holes in it. So far, so goodbutnow if you take the central strip, give it a half-twist and connect it to the outer edge of the circleinstead of the inner one, the new disk will havea single, continuous edge. (Without going intodetails, the half-twist can happen because Nisrelated to the colors of the quarks.) The twovertices remain, but now theres only one loop,for g2N, as you can prove to yourself by usingthe strips of paper at right. You cant draw thisup-and-over diagram on a sheet of paper, but youcan on the surface of a donut, as we will discover.Mathematicians would say that the two disks havedifferent topologies.

Topology, or rubber-sheet geometry, deals withthe invariant properties of objectsthings thatdont change when the object itself is stretched,bent, or otherwise distorted; poking holes ortearing off pieces is not allowed. Thus a donut istopologically equivalent to a coffee mug becauseeach has one loop. If you stood the donut on edgeand very carefully dimpled it with your thumbs,youd create a depression that could hold coffee,albeit briefly. Our twisted do-not symbol isequivalent to a somewhat different mugone

You can make your own one-loop, two-vertex unflat surface.

Cut out the three strips at right. Lay them out in a T,

green side up, and staple them together. Staple the arms

free ends together, forming a ring thats yellow outside and

green inside. Insert the Ts leg through the ring frombelow, bring it out the top, give it a half-twist, and staple it

to the front of the ring. If youve done this correctly, youll

have one all-yellow surface and one all-green surface. To

show that theres only one edge, run a marking pen along

ityou can color all the edges and return to the starting

point without lifting the pen.

A donut is topologically equivalent to a coffee mug because each has one loop.

If you stood the donut on edge and very car efully dimpled it with your

thumbs, youd create a depression that could hold coffee, albeit briefly.


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with a hollow handle thats open to the mugsinterior. In other words, if you filled this cupwith piping-hot coffee, it would go up insidethe handle as well. This could be a popular designin Alaska, but theres a large finger-burning, lap-scalding lawsuit potential in the Lower 48. Andthe twisted do-not donut is equally unsatisfac-toryimagine a chocolate-shelled donut fromwhich a bite has been taken and the donut itselfscraped out, so that only the chocolate remains.Homer Simpson would not be happy.

His daughter Lisa would be ecstatic, however,because thats how you draw a twisted disk on adonut. The intact chocolate shell is the donutssurface, and the bitten-into shell is the drawingon that surface of the twisted do not symbol.In fact, any Feynman diagram can be drawn ona shell made from the right number of donuts.Picture a whole bunch of them, some perhapsstanding on edge, possibly in a big, jumbled pile,all touching one other and completely drenchedin quick-hardening chocolate. After the scraping-out, youd get a hollow shell that looks like one ofHenry Moores sculptures. (Particles that enter orleave the diagram are represented by open-endedtubeshalf-eaten donutssticking out from theshell.) In t Hoofts formulation, if you start withn donuts, any diagram drawn onor bitten outofthat shell comes with a factor of 1/N-2n. Thenumber of donuts is a topological invariant, saysOoguri, and the power ofNkeeps track of it.

Well, then, why not forget about Feynmandiagrams altogether and recast the Standard Modeas a theory of chocolate shells? Ooguri and Vafahave shown this is indeed possiblenot for theStandard Model itself, not yetbut for a largeclass of supersymmetric gauge theories in fourdimensions. (Remember, gauge theories describeforces in terms of particles; supersymmetry issomething required to explain why most particleshave mass.) Ooguri and Vafa adapted the lan-guage of string theory to describe the donut

Above: You can transform

the twisted do-not

symbol into a bitten-out

donut shell by gently

stretching and deforming

it. You start by bringing

the far end of the center

strip around to its near

end, forming a loop that

encircles the donut like a

cigar band. Then stretch

the horizontal and vertical

loops until they cover

most of the surface,

leaving one small hole.


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shells, and it works very well.String theory had been rescued from obscurity

in 1974, when John Schwarz, now the BrownProfessor of Theoretical Physics, and his collabora-tor, the late Joel Scherk of the Ecole NormaleSuprieure in Paris, realized that it could be acandidate for the long-sought Theory of Every-thing. (It had originally been invented for analtogether different purpose that didnt work out,but thats another story.) But while it handled

quantum gravity quite nicely, it predicted auniverse that didnt match ours in one importantrespect. Explains Ooguri, Nature is not symmet-ric under the exchange of left and right. Theworld in the mirror is not the same as our world.This effect, known as parity violation, could notbe reproducedthe string-theory universeremained stubbornly ambidextrous. Undaunted,Schwarz kept plugging away almost single-handedly until 1984, when he and MichaelGreen (then at the University of London, nowat the University of Cambridge) found the fixthat kept the theory internally consistent whileallowing parity to be violated. The field took off,and nowadays you cant pick up a popular-sciencemagazine without reading about superstrings,10- or 11-dimensional universes, M theory,branes, and the like.

Strings can be thought of as flexible Os. Astime passes, a string sweeps out a world sheet,as shown at left. If the string is moving, thesheeta cylinder, reallyleans in the directionof motion. If the string emits another string,the cylinder forks. As more strings interact, theircollective world sheet becomes a network of fuseddonut shells.

But its not enough for the world sheet to looklike a shell. It has to taste like chocolate, or inthis case it has to reproduce the adding-up of theFeynman diagrams. Ooguri and Vafa have shownthat one particular variant of string theory doesjust that. Says Ooguri, When we did the com-putations, the world sheet started generating someexotic domains because of its internal dynamics.It tore open here and there to create a new phasein which space-time decayed into nothing. Suchbehavior tends to be the death of theories, as themath generally breaks down, but Ooguri and Vafawere thunderstruck to discover that the stringsstayed in the sheets normal regions, flowingaround the exotic domains like water around rocks

in midstream. That is, the strings developed gapsas needed to avoid entering these uncharted zones,and then magically closed up again when thedanger had passed. It turns out that this corre-sponds exactly to a Feynman-diagram computa-tion. The exotic domains create holes in the worldsheet, and if you throw them out, you recover thecomputation from gauge theory. This provides away to generate open strings out of closed strings,and once you have open strings, you almost have agauge theory.

Why not forget about Feynman diagrams altogether and recast the Standard

Model as a theory of chocolate shells?

Top left: As in a Feynman

diagram, time moves from

left to right. Here a string

(red) emits another string,

causing the world sheet

(gray) to fork.

Left: If a string comes into

existence briefly and then

vanishes, its world sheet is

a sphere. Ooguris and

Vafas exotic domains tear

the spheres surface open,

and by stretching three

openings in just the right

way, you can get the flat

disk with two holes. The

bottom two spheres have

been rotated to show how

one hole engulfs nearly an

entire hemisphere before

the flattening.


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The work also has mathematical applications,particularly in three-dimensional knot theory.A knot can be thought of as a length of rope withits two ends attached to each other. The simplestknot is a circle or ellipse, the so-called unknot; innontrivial knots the line is wrapped around itself.So in the next-simplest knot, you cross the lineover and under itselfonce, as if you werepreparing to tie yourshoelaces, before youjoin the ends. This iscalled a trefoil knot.And truly complicatedknots have loopsstuffed through otherloops and lines twistedaround themselves likethe Gawd-awful tangle that that 150-foot, bright-orange outdoor extension cord in your garage is in

One of knot theorys fundamental problems isto determine whether one knot is equivalent toanotherwhether the one can be transformed intothe other without forcing the line to pass throughitself like a magicians linking rings. Mathemati-cians eventually hope to be able to classify allknots in this manner. A related question is thatof deciding whether a given knot is trivial, thatis, if it can be disentangled into a circle. Say youhave a flat loop of ropea very long, thin oval.If you treat the tips of the oval like the ends of an

ordinary piece of cordage, you can tie the doubled-up rope into additional knots. The result suredoesnt look trivial, but it isyou can get backto the original oval without cutting and splicinganything.

In their quest to classify knots, mathematicianshave come up with several invariants, or math-ematical expressions that remain unchanged asyou pry the knots loops apart. If the invariantsfor two knots are different, then, clearly, so arethe knots. But nobody has yet come up with a

The Number of the Knot

Ooguri and Vafa were working in four dimen-sions; the current universe-explaining superstringtheory operates in 10. (The other six are curledup on themselves, so we dont experience them.)Ahead lies the job of twiddling with those othersix dimensions until the Standard Model comestumbling out. The clincher will come when thecalculations predict the masses of the proton,neutron, and so on that are actually observed,and to the same level of precision.

Theres already a string theory that approxi-mates the strong interaction pretty well, but itsnot exact, says Ooguri. In this regard, the stringtheorists are in the same boat as everybody else.Because the Feynman diagrams are so intractable,the other folks have resorted to something calledlattice gauge theory, in which space-time isdivided into a finite set of points, called a lattice.Then a computer calculates all the fields at eachlattice point. Says Ooguri, The technique hasgotten to the point where we can compute particlemasses fairly well. But it is not very illuminating.

We want to do much better. By transformingthe calculations into string-theory problems, thetechniques Vafa and I, and other collaborators,have worked out over the last 10 years give usa way to compute various quantities exactly fora large class of gauge theories. These are calcula-tions we couldnt even approach before, and thatsvery exciting.

To date, nobody has found a general analytical

method capable of handling the strong interaction.In fact, its such a tough nut to crack that the ClayMathematics Institute has named it one of sevenMillennium Problems, and has offered a millionbucks to the person or persons who succeed. Andwhile the money would be nice, if we get a han-dle on this, Ooguri says, well surely learn tonsof new things about gauge theory. Thats our aim.

Above: If you make a

world sheet from enough

donuts, you can reproduce

any Feynman diagram, no

matter how complicated.

The Feynman diagram at

left has 30 vertices and a

tangle of gluons. The red

lines are half-twisted paths

that rise up out of the

page, while the green lines

are half-twisted ones that

hang under the page. This

diagram can be drawn on

the five-donut surface at

right. (The forked bridge,

when squashed flat,

becomes three donuts.)


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An unknot (red) extended

into two dimensions

creates a cylindrical

surface. The unknotsminimum, or soap-bubble,

surface in the three-

dimensional space

containing the cylinder is

the slanted ellipse shown

in pink.

along with string theory. Says Ooguri, We thenlooked for minimum surfacessurfaces of mini-mum area, like a soap film on a wire loopthatare bounded by the 3-D knot. Thats pretty mind-

bending, but its easierto follow in fewerdimensions. Forexample, take a cylin-der and slice it on thebias to make an oval.This oval is an unknotIf you extend theunknot into twodimensions, you getthe cylindrical surface.

And the unknots minimum surface in three-dimensional space is the diagonal disk that lieswithin the cylinder and whose edge is the oval.Moving up the food chain, a trefoil knot can havea fluted minimum surface with a donutlike holein the center.

The surfaces come with various topologies,

says Ooguri, so we count up the number ofsurfaces in each topological class. There areinfinitely many topological classesbasicallythe number of donuts againso we have infinitelymany integers. (Of course, a lot of those integerscan be zero.) And the way you count them hasclose ties to other branches of mathematics, soI hope that insights from those branches will givefresh perspectives to problems in three-dimen-sional topology. DS

Mathematicians like integers. They think

integers are more noble than real numbers. So

when we found integers in an unexpected place, it

got their attention.

formulation for a complete invarianta formu-lation that says that two knots must be the sameif their invariants are the same.

One nearly complete class of knot invariantsis called the Jones polynomials, discovered byUC Berkeleys Vaughn Jones. This work won himthe Fields Medal, often called the Nobel Prize ofmathematics, in 1990. Says Ooguri, Joness workinitiated a proliferation of knot invariants in the1980s. Unfortunately, these invariants have notprovided much insight into knot theory itself. Inparticular, the relationships between these invari-ants and the intrinsic geometric properties of theknots remain obscure.

But, he adds, whilewe were trying to fig-ure out the equivalencebetween gauge theoryand string theory, andthe physical conse-quences of that equiva-lence, we came upwith a surprising pre-diction: for every knot,you can extract an infinite set of integers from theJones invariant and its generalizations, and theseintegers have clear geometric meaning. Mathema-ticians like integers. They think integers are morenoble than real numbers. So when we foundintegers in an unexpected place, it got theirattention. In fact, some aspects of Ooguri

and Vafas conjecture have already been provenmathematically.The conjecture arose by analogy to t Hoofts

method for adding up Feynman diagrams drawnon chocolate surfaces. A knot is a one-dimensionalobject, but its embedded in three-dimensionalspace. So Ooguri and Vafa added two dimensionsto the knot to make it 3-D, and then placed this3-D knot in six-dimensional spacesix-dimen-sional because they were trying to work out whathappens in those six extra dimensions that come

Surface tension drives

soap films to span the

minimum possible

area, so heres the

minimum surface

of a trefoil knot,

wire-loop and

soap-film style. The

saddle-shaped surface

curves gracefully to

connect adjoining turns of

the wire, leaving a void in

the middle analogous to a

donut hole on a closed

surface.


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