From Solitons to Gauge/String Duality

Igor Klebanov

Department of Physics and Center for Theoretical Science

Talk at Kerman’s 80th Birthday Celebration MIT, April 24, 2009 • In January 1982, Arthur Kerman helped launch my research career by agreeing to supervise my Bachelor Thesis and suggesting an exciting topic: fractional charge on 1+1 dimensional solitons (domain walls). • The thesis, `Charge Fluctuations in Fractionally Charged Excitations’ was completed exactly 27 years ago. • After further work in collaboration with Roman Jackiw and Gordon Semenoff, we wrote a related paper (my first!)

• As shown in 1974 by Jackiw and Rebbi, one-dimensional topological solitons carry ½ units of fermion number. However, asked Kerman and Weisskopf, is this fractional charge a sharp quantum observable, or does it fluctuate, as in a 2-centered molecule? • In the paper we considered a 1-d Yukawa model, with a scalar field describing a kink and an anti-kink separated by a finite distance L

• The total charge is integer, but the charge localized near the kink is ½. Does it fluctuate to the anti-kink and back? It does, but we showed that the fluctuations decrease exponentially with distance, as e-L • Thus, for widely separated solitons, the fractional charge becomes a sharp quantum observable, as claimed by Jackiw and Rebbi. • This type of fractionation has been observed in poly-acetylene (Heeger et al). More recently, a similar effect is serving as the basis for the theory of topological insulators, which is causing a great deal of excitement. • 27 years ago work on this problem introduced me to cutting edge research. Arthur kindly allowed me to approximate the kinks by step functions, which made all the wave functions simpler. String Theory as a Branch of Nuclear Physics • One often hears of string theory as the leading hope for unifying all known interactions-- strong (nuclear), electromagnetic, weak (-decay) and gravitational-- into a consistent quantum theory. Some have dubbed it `The Theory of Everything.’ • Actually, string theory has had a more humble beginning. It was invented in the late 60’s to model `just’ the strong (nuclear) interactions. • The strong interactions are short range (~ 1 fm = 10 -15 m), but much stronger than the electromagnetic. • The strong interaction analogue 2 s = gYM /(4 ) of the electromagnetic `fine structure constant’ (=1/137) is about 100 times bigger. • As more hadrons were discovered, they were grouped into multiplets. For example, the lightest spin-0 mesons form an `octet.’ This gave impetus to the Quark Model. Gell-Mann, Zweig • Lots of heavier mesons, with higher spin, have also been discovered. • Early empirical evidence for the string-like structure of hadrons comes from arranging mesons and baryons into `Regge trajectories’ on plots of angular momentum vs. mass-squared. • A leading `Regge trajectory’ of mesons is shown Ua2 … Open String Picture of Mesons

• In the string model, excited mesons are identified with excitations (rotational and vibrational) of a relativistic string of energy density ~ 1 GeV/fm, which is around 1.6 kilo-Joules/cm. • The rest energy of a  meson is 0.78 GeV. • The linear relation between angular momentum and mass-squared is provided by a spinning relativistic string. Later it was understood that a quark and anti-quark are located at the string endpoints. • String theory replaces point particles by strings and develops their quantum mechanical description.

• Such strings naturally interact via splitting and joining. Supersymmetry • Attempts to construct a quantum mechanical string model in 3 spatial and 1 time dimensions ran into problems, e.g. tachyons (particles with imaginary mass). • Consistent string theories were discovered in 9 spatial and 1 time dimensions. Such theories have a remarkable new symmetry, called Supersymmetry (SUSY), which pairs up bosonic particles (integer spin) with fermionic particles (half-odd-integer spin). Such pairs are called superpartners. • We hope that Supersymmetry (SUSY) is approximately realized in Nature. Superpartners of known particles have not been observed yet, but physicists hope to discover them in the next round of high-energy experiments. • Yet, the relation of 10-dimensional superstring theories to strong interaction was completely obscure in the 70’s. • For a number of reasons, most physicists gave up on strings as a description of strong interactions. Instead, string theory emerged as the leading hope for unifying quantum gravity with other forces (the graviton predicted by Einstein’s General Relativity is the lightest vibrational mode of the closed string). Scherk, Schwarz; Yoneya QCD • In 1973 a point particle theory of strong interactions, inspired by the quark model, was proposed: the Quantum Chromodynamics. • The hadrons are made of spin 1/2 constituents called quarks and spin 1 ones called gluons. Quarks come in 6 known flavors (up, down, strange, charm, bottom, top), and each flavor comes in 3 different color states (red, green, blue). • The adjoint gluons come in 32 -1=8 different color states: Asymptotic Freedom • A crucial property of QCD, discovered in 1973 by Gross and Wilczek at Princeton, and Politzer at Harvard.

• As the energy increases, the interactions between the quarks and gluons, though still quite strong, gradually weaken. This provided a theoretical explanation of such effects observed in the late 60’s at energies around 10 GeV. QCD Gives Strings A Chance

• At short distances, much smaller than 1 fm, the quark-antiquark potential is nearly Coulombic. • At larger distances the potential should be linear (Wilson) due to formation of confining flux tubes. Their dynamics is approximately described by the Nambu-Goto area action. So, strings have been observed, at least in numerical simulations of gauge theory. Large N Gauge Theories • Connection of gauge theory with string theory is strengthened in `t Hooft’s generalization from 3 colors (SU(3) gauge group) to N colors (SU(N) gauge group). • Make N large, while keeping the `t Hooft coupling fixed.

• The probability of snapping a flux tube by quark-antiquark creation (meson decay) is 1/N. The string coupling is 1/N. D-Branes vs. Geometry • Dirichlet branes (Polchinski) led string theory back to gauge theory in the mid-90’s. • A stack of N Dirichlet 3-branes realizes N=4 supersymmetric SU(N) gauge theory in 4 dimensions. It also creates a curved background of 10-d theory of closed superstrings (artwork by E.Imeroni)

which for small r approaches whose radius is related to the coupling by Conformal Invariance

• In the N=4 SYM theory the Asymptotic Freedom is canceled by the extra fields; the gauge coupling is independent of energy. The theory is invariant under scale transformations xP -> a xP  • It is also invariant under space-time inversions. • The conformal group in 3+1 dimensional field theory is SO(2,4). The AdS/CFT Duality Maldacena;; Gubser,, IK,, Polyakov;y ; Witten • Relates conformal gauge theory in 4 dimensions to string theory on 5-d Anti-de Sitter space times a 5-d compact space. For the N=4 SYM theory this compact space is a 5-d sphere.

• The geometrical symmetry of the AdS5 space realizes the conformal symmetry of the gauge theory.

• The AdSd (or hyperbolic) space is

with metric • When a gauge theory is strongly coupled, the radius of curvature of the dual AdS5 and of the 5-d compact space becomes large:

• String theory in such a weakly curved background can be studied in the effective (super)-gravity approximation, which allows for a host of explicit calculations. Corrections to it proceed in powers of

• Feynman graphs instead develop a weak coupling expansion in powers of O At weak coupling the dual string theory becomes difficult. • For an introduction to gauge/gravity duality, see the article by I.K. and J. Maldacena in the January 2009 issue of Physics Today.

• Gauge invariant operators in the CFT4 are in one-to-one correspondence with fields

(or extended objects) in AdS5 • Operator dimensions are an important set of quantities • The operator dimension is related to mass of the corresponding field in AdS space: Thermal gauge theory

• Thermal CFT is described by a black hole

in AdS5

• The CFT temperature is identified with the

Hawking T of the horizon located at zh • Any event horizon contains Bekenstein- Hawking entropy • A brief calculation gives the entropy density Gubser, IK, Peet • This is interpreted as the strong coupling limit of

• For small `t Hooft coupling, Feynman graph calculations in the N=4 SYM theory give

• We deduce from AdS/CFT duality that

• The entropy density is multiplied only by ¾ as the coupling changes from zero to infinity. Gubser, IK, Tseytlin • A similar effect is observed in lattice simulations of non- supersymmetric gauge theories for N=3: the arrows show free field values. Karsch (hep-lat/0106019).

• N-dependence in the pure glue theory enters largely through the overall normalization. Bringoltz and Teper (hep-lat/0506034) The quark anti-quark potential

• The z-direction of AdS is dual to the energy scale of the gauge theory: small z is the UV; large z is the IR. • Because of the 5-th dimension z, the string picture applies even to theories that are conformal. The quark and anti- quark are placed at the boundary of Anti-de Sitter space (z=0), but the string connecting them bends into the interior (z>0). Due to the scaling symmetry of the AdS space, this gives Coulomb potential (Maldacena; Rey, Yee) String Theoretic Approaches to Confinement • It is possible to generalize the AdS/CFT correspondence in such a way that the quark- antiquark potential is linear at large distance but nearly Coulombic at small distance. • The 5-d metric should have a warped form (Polyakov):

• The space ends at a maximum value of z where the warp factor is finite. Then the confining string tension is A Warped Calabi-Yau Space • To construct a confining gauge theory, consider N D3-branes and M D5-branes wrapped over the S2 at the tip of the conifold. • The 10-d geometry dual to the gauge theory on these branes is the warped deformed conifold (IK, Strassler)

• is the metric of the deformed conifold, a simple Calabi-Yau space defined by the following constraint on 4 complex variables: • The upper graph, from the recent Princeton Senior Thesis of V. Cvicek shows the string theory result for the quark anti-quark potential in the warped deformed conifold theory. • It is qualitatively similar to that found in numerical simulations of QCD. The lower graph shows lattice QCD results by G. Bali et al with r0 ~ 0.5 fm. • Perhaps the dual gravity provides a `hyperbolic cow’ approximation to QCD. M2-Branes and Chern-Simons • A similar approach leads to `solving’ certain strongly interacting theories in 2+1 dimensions. These theories arise on coincident membranes in 11-dimensional M-theory. Due to recent progress we understand that they are certain supersymmetric versions of Chern-Simons gauge theory with group U(N) x U(N)

Bagger, Lambert; Gustavsson; Aharony, Bergman, Jafferis, Maldacena • When N is large, the dual description of the k=1 Chern-Simons matter gauge theory is 7 provided by the weakly curved AdS4 x S background in 11-dimensional M-theory which is described by Einstein gravity coupled to other fields. • This leads to many non-trivial predictions, e.g. the number of degrees of freedom 3/2 scales as N I.K., A. Tseytlin • Such models may provide `hyperbolic cow’ approximations to strongly interacting many-body systems on a plane. Conclusions • Throughout its history, string theory has been intertwined with the theory of strong interactions. • The AdS/CFT correspondence makes this connection precise. It makes a multitude of dynamical statements about strongly coupled conformal gauge theories, both at zero and finite temperature. • Extensions of AdS/CFT provide a new geometrical understanding of color confinement and chiral symmetry breaking. • Applications of the AdS/CFT correspondence to heavy ion collisions, and even to condensed matter physics, are being widely explored. Happy 80th Birthday, Arthur!

And Many Happy Returns…