Strongly Coupled Gauge Strongly Coupled Gauge Theories and Strings Theories and Strings Igor Klebanov Igor Klebanov Department of Physics Department of Physics Princeton University Princeton University Talk at the Galileo Galilei Galilei Institute Institute Talk at the Galileo Inaugural Conference Inaugural Conference Firenze, September 21, 2005 , September 21, 2005 Firenze
The Beginnings The Beginnings String Theory was born out of attempts to understand String Theory was born out of attempts to understand the Strong Interactions! the Strong Interactions! Empirical evidence for a string-like structure of hadrons Empirical evidence for a string-like structure of hadrons comes from arranging mesons and baryons into Regge Regge comes from arranging mesons and baryons into trajectories. trajectories. Dolen-Horn-Schmid Dolen-Horn-Schmid duality conjecture: in meson duality conjecture: in meson scattering, sum over s-channel exchanges scattering, sum over s-channel exchanges s s equals sum over t-channel exchanges t equals sum over t-channel exchanges t
• The I= 1 leading • The I= 1 leading meson Regge Regge meson trajectory trajectory ( ρ , a ) ( ρ , … ) a 2 … 2 • The I= 0 leading • The I= 0 leading meson Regge Regge meson trajectory trajectory ω , f ...) ( ω , ...) f 2 ( 2
• Veneziano • Veneziano proposed a manifestly proposed a manifestly dual amplitude for elastic pion pion dual amplitude for elastic scattering: scattering: with linear Regge Regge trajectory trajectory with linear • Nambu • Nambu, Nielsen and , Nielsen and Susskind Susskind independently proposed its open independently proposed its open string interpretation string interpretation
• • The string world sheet dynamics is governed by the The string world sheet dynamics is governed by the Nambu-Goto area action area action Nambu-Goto • • The string tension is related to the Regge Regge slope through slope through The string tension is related to the • • The quantum consistency of the Veneziano Veneziano model requires model requires The quantum consistency of the that the Regge Regge intercept is intercept is that the so that the spin 1 state is massless massless but the spin 0 is a but the spin 0 is a so that the spin 1 state is tachyon. tachyon. • • Calculation of the string zero-point energy gives Calculation of the string zero-point energy gives • • Hence the model has to be defined in 26 space-time Hence the model has to be defined in 26 space-time dimensions. dimensions.
Spinning Open String Picture of Mesons Spinning Open String Picture of Mesons • The linear relation between angular • The linear relation between angular momentum and mass-squared is provided momentum and mass-squared is provided by a semi-classical spinning relativistic by a semi-classical spinning relativistic string with massless massless quark and anti-quark quark and anti-quark string with at its endpoints. at its endpoints.
Crossroads in the 1970’ ’s s Crossroads in the 1970 • • Attempts to quantize such a string model in 3+ 1 Attempts to quantize such a string model in 3+ 1 dimensions lead to tachyons, problems with unitarity unitarity. . dimensions lead to tachyons, problems with • • Consistent supersymmetric supersymmetric string theories were string theories were Consistent discovered in 9+ 1 dimensions, but their relation to discovered in 9+ 1 dimensions, but their relation to strong interaction was initially completely unclear. strong interaction was initially completely unclear. • • Most importantly, the Asymptotic Freedom of strong Most importantly, the Asymptotic Freedom of strong interactions was discovered by Gross, Wilczek Wilczek; ; Politzer Politzer in in interactions was discovered by Gross, 1973. This singled out the Quantum Chromodynamics 1973. This singled out the Quantum Chromodynamics (QCD) as the exact theory of strong interactions. (QCD) as the exact theory of strong interactions. • • Most physicists gave up on strings as a description of Most physicists gave up on strings as a description of strong interactions. Instead, string theory emerged as strong interactions. Instead, string theory emerged as the leading hope for unifying quantum gravity with other the leading hope for unifying quantum gravity with other forces (the graviton appears in the closed string forces (the graviton appears in the closed string spectrum). Scherk spectrum). Scherk, Schwarz; , Schwarz; Yoneya Yoneya
QCD gives strings a chance! QCD gives strings a chance! • At short distances, • At short distances, must smaller than 1 must smaller than 1 fermi, the quark- , the quark- fermi antiquark potential is potential is antiquark Coulombic, due to the , due to the Coulombic Asymptotic Freedom. Asymptotic Freedom. • At large distances the • At large distances the potential should be potential should be linear (Wilson) due to linear (Wilson) due to formation of confining formation of confining flux tubes. flux tubes.
Flux Tubes in QCD Flux Tubes in QCD • These objects may • These objects may be approximately be approximately described by the described by the Nambu strings strings Nambu (animation from lattice work by (animation from lattice work by D. Leinweber Leinweber et al, Univ. of et al, Univ. of D. Adelaide) Adelaide) • The tubes are widely • The tubes are widely used, for example, in used, for example, in jet hadronization hadronization jet algorithms (the Lund algorithms (the Lund String Model) where String Model) where they snap through they snap through quark-antiquark antiquark quark- creation. creation.
Large N Gauge Theories Large N Gauge Theories • Connection of gauge theory with string • Connection of gauge theory with string theory is strengthened in ` t Hooft Hooft’ ’s s theory is strengthened in ` t generalization from 3 colors (SU(3) gauge generalization from 3 colors (SU(3) gauge group) to N colors (SU(N) gauge group). group) to N colors (SU(N) gauge group). • Make N large, while keeping the ` t • Make N large, while keeping the ` t Hooft Hooft coupling fixed. coupling fixed. • The probability of snapping a flux tube by • The probability of snapping a flux tube by quark-antiquark antiquark creation (meson decay) is creation (meson decay) is quark- 1/N. The string coupling is 1/N. 1/N. The string coupling is 1/N.
• In the large N limit the gauge • In the large N limit the gauge theory simplifies (only planar theory simplifies (only planar diagrams contribute). Adding a diagrams contribute). Adding a non-planar line (green) causes a non-planar line (green) causes a suppression. suppression. • But it is still very difficult! • But it is still very difficult! • Between mid-70 • Between mid-70’ ’s and mid-90 s and mid-90’ ’s s many theorists gave up hope of many theorists gave up hope of finding an exact gauge/string finding an exact gauge/string duality. duality. • An important exception is • An important exception is Polyakov, who already in 1981 , who already in 1981 Polyakov proposed that the string theory proposed that the string theory dual to a 4-d gauge theory should dual to a 4-d gauge theory should have a 5-th hidden dimension, and have a 5-th hidden dimension, and later argued that the 5-d space later argued that the 5-d space must be “ “warped warped” ”. . must be
Breaking the Ice Breaking the Ice • • Dirichlet branes branes ( (Polchinski Polchinski) led string theory back to ) led string theory back to Dirichlet gauge theory in the mid-90’ ’s. s. gauge theory in the mid-90 • • N = 4 A stack of N Dirichlet Dirichlet 3-branes realizes 3-branes realizes N = 4 A stack of N supersymmetric SU(N) gauge theory in 4 dimensions. SU(N) gauge theory in 4 dimensions. supersymmetric It also creates a curved background of 10-d theory It also creates a curved background of 10-d theory of closed superstrings (artwork by of closed superstrings (artwork by E.Imeroni E.Imeroni) ) which for small r approaches which for small r approaches • • Successful matching of graviton absorption by D3- Successful matching of graviton absorption by D3- branes, related to 2-point function of stress-energy branes, related to 2-point function of stress-energy tensor in the SYM theory, with a gravity calculation in tensor in the SYM theory, with a gravity calculation in the 3-brane metric (IK; Gubser Gubser, IK, , IK, Tseytlin Tseytlin) was a ) was a the 3-brane metric (IK; precursor of the AdS AdS/CFT correspondence. /CFT correspondence. precursor of the
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