r. ramachandran- superstrings: theory of everything

8/3/2019 R. Ramachandran- Superstrings: Theory of Everything

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2 1 N O V 1 3 8 5IC/85/211

INTERNAL REPORT(Limited distribution)

International Atomic Energy Agencyand

fons Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SUPERSTRING S - THEORY OF EVERYTHING? *

R. RamachandranInternational Centre for Theoretical Phys ics, Trieste, Italy

anrlIndian Institute of Technology, FCanpur 208016,India **

ABSTRACT

A "brief overview af the essential ingredientsof the theory of superstrings is presented.

MIRAMARE - TRIESTESeptember 1985

Physics of the 'basic interactions now appears to be in a -veryinteresting stage of developmen t. The effort to link gravity with otherinteractions - electromagnetic , weak (or radioactive forces) and chromodynamics(responsible for strong or nuclear forces) - has had a spectacular succe ss.We are now talking about a theory of fundamental interactions which could inprinciple envelop everything. Superstring theory is a theory of gravitationthat appears to have the necessary ingredients to serve, on the one hand,as theproper quantum version of the classical gravity and on the other hand,as thebasic framework for all matter together with thei r strong, electromagnetic andweak interaction. It makes use of several notions that have been around, suchas local gauge symmetry, local supersymmetry and Higgs-lik e mechanism forspontaneous symmetry breaking. It is formulated in space-time of dimensionD > h and the superfluous dimensions ar e supposed to have playe d a crucial rolein the very early stages of the evolution of the universe (until about 10" seconds

33after the big bang) when temperatures ranged 10 K. Subsequently and formost phenomena of both the macroscopic and microscopic wor ld, these extradimensions remain compactified at length scale? of order 10" metres andwhat has been achieved is that there is room In the theory for such acompactification with no undesirable side effects and the resultant formalismis phenomenologically c onsistent and interesting. We may also point out atechnical aspect that has had a profound ef fect: symmetries present in theclassical act ion, it is known,not to persist on quantization flue to the presen ceof what are known as anomali es. In order to have a consistent renorrnalizabletheory anomalies have to be avoided. The spectacular development ushered inby Hike Green and John Schwarz was the discovery that the superstring th eoriesare anomaly free if the gauge interaction and symmetry associated with themis either 30(32 ) or F.n x En itivariance. Thus, in these theories, we appearo oto insist on a fairly unique set of internal symmetries of the interactionsand there is hope that further analysis will yield a rather unique solutionfor the low energy symmetries of eleetroweak and strong interactions that areembedded in them. Further there is a prospect that these theories may turnout to be finite to all orders in perturbation theory.

Historically strings were invoked in the early seventies to explainthe excitation spectrum of strongly Interaction particles (hadrons). Mesonsand baryons were found to have regularly spaced (in mass ) resonan ces. They becomethe .infinite tower of quantum states of a relativistic string and the equalspacing between the levels related to the only dimensional parameter in the

* To be submitted for publication.** Permanent addrnsc.


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proble m, namely the string tension (T = l/2Tia'), which has the dimensionsof (Energy) orin natural units L~ . The main interest in them was theproperty ofduality of the scattering amplitude of the string interaction. Bythis one understood that the same scattering am plitude could heconsidered asboth aninfinite sequence of resonances in the direct channel (a + b - 6. -+c + d)and equivalently as analytic continuation of aninfinite sequence of resonances(a + c + T * b + d) inthe crossed channel. There was a lot of phenomenological

Jsupport for the principle ofduality and the string models that imply a tower offermionic states (as amodel for baryon resonance sequence) were also discovered.(Incidentally supersymmetry- symmetry linking fermions and bosons asdegeneratestates - had its origin inthese Ramond-Heveu-Schwarz models.) However, theinterest inthe string models for hadrons diminished principally because oftwo unresolved "shortcom ings" of the theory (apart from the emergence of QuantumChromodynamics as the prototype field theory describing hadrons). First, thereva s a problem with Lorentz covarianee ofthe theory. A consistent quantumtheory ofstrings could beformulated only in 26space-time dimensions. (Thisnumber reduces to 10for the spinning strings.) Second, itrequired thePomeron (or a family ofstates that can have vacuum quantum numbers relatedto the excitations ofclosed string configurations) trajectory intercept a (o)to be2 instead of 1 as was expected from the asymptotic high energy 'bounds forscattering amplitude s. Both these shortcomings are turned toadvantages bya simple reinterpretation.

The trick is tochange the length scale of the theory from the hadronicvalue of 10 m. toPlanck value 10~ m. With this vastly increasedstring tension, the theory describes gravity and now it isquite natural thatit incorporates a massless spin-2 graviton, signalled by 0^(0) = 2 referredto above. The infinite tower ofstates (level spacing lo W Q eV, Planckenergy!) are presumably necessary to make the quantum theory of gravityrenormalizable and internally consistent and in a sense explains why earlierattempts atquantizing gra vity which did not recognize th is rich structureproved futile. Thus , if we are able to make some sense out ofthe extra spatialdimensions, wewill have made significant progress in understanding g ravity.What isremarkable is that the attempts tounderstand the extra dimensions -that the space is socurled up inthese dimensions that they become visibleonly at extremely short distances i> 10 ~ m - has yielded as a byproduct arealistic formalism that can explain the observe dbasie gauge interactions(electroveak and ehromodynamic orsuitable unified gauge interac tion)

together with local supereymmetry (which was found essential for some technicalreasons tokeep apart different energy scales) asthe 'low energy' effect ormassless sector of such superstring theories.

At the classical level , the string ischar acte rized 'by x ( C T , T ) ,U = 0,1,2,... ,D-1; cr a real parameter along the length of the string, chosento be the closed interval [O,TT] and T islike proper time, anevolutionparameter. The classical action isproportional tothe two-dimensiona l surfacearea embedded in the space- time, much like the invariant length of the worldline inthe case of a point particle

A 1 A

The action isinvariant under arbitrary reparametrization since itrepresentsthe invariant area of the world sheet. This can beexplicitly displayed inth eequivalent form

where oufi are two-dimensional vector indice s, = (t.a) 9 = -,r ft 3T do,t l i e i: tensor and g = det(g

a n dUnder the reparametrization

and

X S -- o 'If we eliminate g by solving its algebraic field equation and substitute in" atsthe action, we will get the earlier expression called Nambu-Goto action. Howeve r,in th e latter form, it displays what astring really is. We may regard xu (E, ) asD scalar fields in the two-dimensional space ^ and the reparnjnetri Ratio n1 i ssimply th e ^enertilcoordinate transformatio n. Thus th e action i s that of two-dimensional gravity. It is natural that this large symmetry will imply arich set of constraints for the string theory. In the first stage it ispossible to choose an ort.honormal set of coordinates by imposing

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. * *r w i i's ft


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a *-._ - oSO" * "

ft - - D

and the classical equations of motion are

with the boundary condition

at

The solution gives various modes of string vibrations and their ampli tudessatisfy an infinite set of constraints. On quantizing the theory these'amplitudes' become operator valued and the system is simply an infinite setof harmonic oscillators. Thus the "basic ingredients of the theory are modesof string vibrations and represent an infinite tower of states generated bythe action of infinite number of harmonic oscillation creation operators.Their masses are given by (just as in quantum electrodynami cs, the photonhas only transverse polariz ations , the physical states of string correspondto only transverse vibrations).

R , T at:1*1where D is the number of space-time dimension. .The Lorentz invariance canbe satisfied only if D = 26. Further the ground state in this tower hasimaginary mass (such a state is called tachyon and can cause havoc to causalityif it interacts vith the normal particles) and should therefore be eliminated.Before we turn to this task , let us make a digression to make these essentiallytechnical features (D = 26 and presence of tachyons ) plausible.

The ground state of an infinite set of harmonic oscillator will carrythe corresponding zero point fluctuations leading to a massi

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This divergent su mmation can "be regularized by associ ating Hiemann zeta functio na when Re a > 0. The factort.{a), which has a representation %(a) = ^T^ n~" n=l^> * n may be regarded as the analytic continuation and identified as

n=lwhich is finite and known to be -1/1^. Thus

making the ground state a tachyon. The first excited state a. |0 > can nowbe seen to be massless (vector meson with only transverse co mponents) prov ided

= oThis calls for D = 6.

The elimination of the tachyon is achieves by going supersymmetric -i.e. by introducing in addition to space-time coordinates x > a set of anti-commuti ng c-number Grs.ssmu.rinvariables 9 . We then have a rela tion shipbetween bosonic and fermionic states. In the string context this implies inaddition to bosonic oscillators (a , a ), fermionic oscillators (d . d orI It n n n nb^ , b ) whose creation and annihilation operators anticommut e. Theresulting mass spectrum has two sectors. For the fermionic sector , we have-- 2That this describes fermions can be understood by noting that the zero modeoperators dj! satisfy the anticommuation relation

making them proportional to Dirac gamma matrices and ban therefore only spinorialrepresentation. (The zero mode bosonic oscillator aji is like the moment umfor the string state.) For the bosonic sector the mass operator is given by

The index r for the anticommuting oscillators runs over half odd integers .


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Lorentz algebra is now satisfied only when D = 10. (Arguments basedon the zero point energy, as made for the bosonic strings can also be givensee L. Brink .) In the ten-dimensio nal space-time , we can impose bothMajorana (real spinor) and Weyl (chir al) condition and the number of independentcomponents of Majorana-Weyl sp inors is reduced (from 32) to just 8, same asthe number of transverse polarizations for a vector representat ion of masslessin ten dimension . This is the reason that we could use the same label I bothon commuting (a n> a^) and antIcommuting (d , a* or b , b +) harmonicoscillat ors. The ground state in the fermionic sector has an eightfolddegeneracy and is massle ss.

The bosonic sector ground state on the contrary appears to be a scalartachyon. Thia state is eliminated by truncating the bosonic sector to evenG-parity subsector; G is defined by

and is a symmetry of the theory. It is therefore consistent to have a theoryrestricted to even G-sector. The lowest energy state in the bosonic sector is^1/2 ^ ' an( ^ ^ o r m s ^ massless vector mesons.

Ground state of the theory so obtained is a supermultiplet and it turnsout that for every mass leve l, there are equal bosonic and fermionic states , anexplicit signal of supersymmetry.

Interactions in string theory can be pictured as simply splitting andJoining of strings. This brings in another class of states related to closedstrings obtained by joining the two ends of an open string. If the internalsymmetry is realized through some charges that reside at the end of the strings,then the closed string configurations correspond to neutral states and wereearlier associated with Pomeron in the old hadronic strings. Here the closedstrings are the states at the base of which is the graviton supermu ltiplet.The mass spectrum Is given by

with

and a constraint N|physical stat ed = N|physical state >, where N and Nare the number operators of the left moving and right moving vibrations inclosed string and the constraint in a consequence of the periodicity inherentin a closed string. (We have now introduce d, in place of d and b , $ avariables which are 8 component spinors in S0(8). ) The ground state Is asupergravity multiplet consisting of a graviton g (degeneracy 35 ), gravitinoi|i (5 6 states), and antisymmetric second rani: tensor field B (28 states),a Majorana-Weyl fermion 41 (8 states) and a scalar (1 state).

The quantum theory of strings still suffers from two deseases - Itpossesses chiral and conformal anomalies and can potentially cause breakdownof energy momentum conservation and axial current conservation. As mentionedearlier, anomalies cancel if the gauge symmetry group Is S0(32) or Eo x E R .In the ease of S0(32), we may attach to the end of an open string charges thattransform as fundamental representatio ns of the gauge group , with the resultthe open string states belong to the adjoint representation of E0(32)(dimension U96) - The closed string will be 3 SO(32) singlet. Phenomen ological ly,more interesting Is the case for En x En and is referred to in the literature ,as heterotic (heterosis is a greek word meaning increased vigour by cross-breeding in animals and plants) string. This is obtained by combining theright movers in ten-dimensional superstrlng and left movers in the 26-dimensionalbosonic string. This bizarre combination lets one associate a Yang-Mill ssymmetry with a closed string! This is achieved by compaetifying l6 of the6 dimensions of the right movers into a lfi-dimensional torus and the iso-metries on this torus are linked to the 16-dimensional Ab elian (tartan) sub-group of the group G (rank of both S0(32 ) and Eg x Eg is 16 and matchesthe number of dimensi ons, being compactifled). The additionalk8(i gaug e TTieyons needed to compl ete the adjoin t rt:j>T"cr3entat ion of 0as the mas sless solitori", whicjh result when the closed strings windsaround the torous T . We tave thus a totally geometrical realization ofnot onl y gravity but all internal cymmetry as well . An f ar a:1, the interactionsare concerned aince Iie1.erotic string theory eonni iits of only oriented closedstrings, only one type of interaction is possible:

N -

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ooHence there is only one diagram for each order of perturbation. Because of theintrinsic relationship between the charges of the internal symmetry and thegravitat ion, the relevant coupling parameters are related

where g in Yang-Mills gauge interaction coupling parameter (of dimensionL" in ten- dimen sion al space -time and L in four-dimem-.n'onal space),< is gravitatio nal parameter (dimension L in D ~ 10 and L in D = k)and a' is string tension (i* [10 GeV] ) . This relation is characteristicof the Kaluza-Klein type of gravity theori es. The original five-dimensionalKaluza-Klein unified gravity theory incorporated the electromagnet!Sm as theU(l) gauge group defined as the isometry group on the fifth compactifieddimension. The electromagnetic coupling constant e is related to thegravitational coupling constant r and the compactiflcation scale R through

which is in essence the same relation as above.In order to relate this ten-dimensional theory to the four-dimensional

space-ti me, we now require compactIfication of the remaining six superfluousdimensions; the compactifi cation scale perhaps should lie in the region10 -10 GeV. This process is expected to also simultaneously break theinternal symmetry down to the observed low energy symmetries. In a sense thecompactification and the choice of six-dimensional space is like Higgs mechanismof spontaneous "breakdown of the higher symmetry. At the first stage one hopesthat this process vill yield one of the grand unified theories (such as SU(5),SO(lO) or E/-) together with the super symmetry, Solutions for suchcompactifications have "been discovered in what are called Cala bi-Yau manifolds .(These man ifol ds h;ive :;ll(0 !inloi;omy, i.e. n njanor or a vector trririniio ;.*"!]parallel on rt closed path returns to a new npi nor vector related to the

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oriR.inal by some KU(3) transformati on. They possess no isometrics whichmeans that there is no intrinsic symmetries in the manifold. This breaks

i rEg x Eg to Eg x En, Eg being the GUT group and finds the number ofgenerations of chiral families (that transform as 2T. in E,-) as a topologi calnumber associated with the solution. It is necessary to look for geometricstructures that can accommod atevario us necessary Higgs boson s so that ultimatelyonly the really low energy symmetries survive. Since at no stage there isany room for introducing arbitrary parameter s or mass scales, it will be aremarkable f eature of the string theory that all 'basic' mass ratios and theunrenormalized coupling parameter s will be topological invariants associatedwith the solution. As an example, the number of families is given "by halfthe Wuler characteristic of the compact manifold (Euler characteristic is atopological invariant. For example in two dimensions it i 0 given byX = 2 - SH - B, where H are number of handles and B are number of holes).Solution that imply four generations have been around and ones with threegenerations are also now spoken of. It is speculated that furtherphenomenologi cal analysis wil l pick a fairly uniqu e solut ion, wh.ich indeedwill prove to be a theory of everything.

Finally, while the group Eg is the parent of all low energy symmetr ies,is there a role for the other group Eg? There are two tasks that this canperform. First the supersymmetry has to be broken since there is no evidencefor the presence of bosonic or fermionic partne rs for the various basicingredients in the low energy world. It is speculated - and there is roomfor this speculation in the theory - that the gaugino condensates are made out

iof En fields and this could cause soft supersymmetry breakin g. Last ly,the missing mass of the universe - there is evidence that stars are su bjectedto more gravitational force than what could be expected from the amount ofmatter estimated to "be present in the various galaxies and intergalactic spaces

tcould belong to the 'shadow' universe made up of matter arising from Egpart of the complete Eg X Eg stringy univer se. This part of the world willinteract with our visible universe only through gravitational mea ns, since thelatter is made up of singlets (neutral) under E group.

There have been several other technical miracles in the superstringtheory. There are horrible divergences present in the one loop amplitudefor bosonic string s, but dramatic cancellatio ns take place for superstrings andwhatever divergences remain are tamed when specific symmetries are employed.While the main reason for this cancellation is supersy mmetry, there still

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remains aspects (niodular invaria nce, to use the jargon) that are yet to beclearly understood. Compactification of the erstwhile gravity theoriesformulated in dimensions larger than 1+, have often been accompanied bygeneration of a large cosmological constant which has teen regarded ascatastrop hic. String compactification avoids it. Considerable amount ofactivity in particle physics is presently devoted to understanding thesesubtle aspects of this exciting discovery. Hopefull y, we will soon havedefinite experimental signatures associated with this beautiful theoreticalformulation.

ACKNOWLEDGMENTS

The author would like to thank Professor Abdu s Salam, theInternational Atomic Energy Agency and UNESCO for hospitality at theInternational Centre for Theoretical Pfcyajcs, Trieste.

REFERENCES

For a general review:J.H. Sehva rz, Phys. Rep. 89 (1982) 2S3;M.B. Gre en, Surveys in High Energy Physics 3_ (1983) 127 ;L. Brink, CFKM preprint Th.l40O6 (198M.

For old strings:Y. Nambu, Proc. Int. Conf. on Symmetries and Quark Models , Wayne State

University (Gordon and Breach, 19&9) p.26 9;T. Goto , Prog. Theor. Ph ys. 16 (1971) 1560;M. Jacob, Ed. Dual Models (Phys, Rep. reprint collection , North Holland ,297I1);J. Scherk, Rev. Mod, Phys. kj_ (1975) 123;P. Rumontl, Phya. Rev. D3, (1971) S^IS;A. Neveu and J.H. Schvar z, Nucl. Phys. B31. [197 1) 86; Ph ys. Rev. lA_ (1971)

1109;P. God dard, .T. Goldston e, C. Re bM and C. B. Tho rn, Nucl, Ph ys, Bjifj (1973)

109.Two-dimensional gra vity, conformal invariance etc:

J. Scher k and J. Sehvar z, Wucl. P hys, Bf L (l97li) .1.18;D. Friedan, Proc. 1982 Les Houehes Summer S chool , Eds. J. ?uber and R. Stora

North Holland, 1901*);S. Jain, R, Shankar and 5, Wa dia, TIFR, preprint, to be published;A. Polyakov, Phys. Lett. B103 (1981) 207 and 211.

Loop amplitudes , anomalies cancellations etc:L.J. Clavelli and J.A. Shapiro, Nucl. Phys. B57 (1973) 90;L.Alvarez-Gaume and E. Witt en, Nucl. Phys. B2'jh (I9O3) 26$;M.B. Green and J.H. Schvarz, Phys. Lett. l^lB (1985) ?1; CALTECH, preprint

68-1182 (1985).Keterotic string:

D. Gross, J. Harvey, E. Martineo and R. Rohm, Nucl. Phys, B256 (1985) 253;Princeton, preprint, June 1985);

P.G.O. Freund, Phys. Lett. T"1B (I9B5) 387;M. Dine, R. Rohm, M. Seiberg and E. Witten , Phys. Lett. I56B (1985) 55;M. Din e, V. Kaplonovsky, C. Ha ppi, H. Mongano and IJ. Seibe rg, Princet on,

preprint (198^);P. Candela.T, C,. Horowitz, A. Strominger and E. Wi tten, Santa Barbara, preprint

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r. ramachandran- superstrings: theory of everything

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