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GAUGE/STRING DUALITY AT FINITE TEMPERATURE
Physics beyond the Standard Model Rio de
Janeiro, 12/ 2006 Carlos Alfonso M. B. Bayona
Instituto de Física , Universidade Federal do
Rio de Janeiro . Advisers Nelson R. F Braga ,
Henrique BoschiFilho .
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SUMMARY

• Introduction .
• Gauge/string duality and AdS/CFT correspondence .
  
• Temperature and black hole horizon .
• Some gravitational predictions for strongly
  coupled gauge theories at finite temperature
  using gauge/string duality.
• Conclusions and perspectives .

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1) Introduction

• String theory birth from strong interactions
• Veneziano amplitude (1968)
• It satisfies the duality condition A(s,t)A(t,s)
  .
• Higher spin hadrons and Regge relation
• - String theory predicts Veneziano amplitude and
  Regge relation !
• - String theory also contains gauge fields (spin
  1) and gravitational fields (spin 2) !

...(1)
...(2)
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• Non-abelian gauge theory birth from strong
  interactions
• - Yang-Mills-Shaw lagrangian (1954)
• with ,
• It satisfies isospin invariance (SU(2) ) .
• - Quark model (Gell-Mann and Zweig 1964 ) .
• Renormalizability of non-abelian gauge theories
  (t Hooft 1971) .
• Asymptotic freedom (Gross, Wilczek and Politzer
  1973 ) .
• Quantum chromodynamics (QCD) quarks and
  gluons with SU(3) gauge symmetry . Very
  successful for high energies !!

...(3)
5

• Problems with string theory
• Very difficult to construct !
• Too many dimensions to deal with .
• Problems with QCD
• Non-perturbative at low energies .
• Doesnt explain analitically hadron spectrum of
  masses and confinement of quarks (but note that
  there are very important numerical results on
  these topics using lattice QCD) .

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2) Gauge/string duality and AdS/CFT correspondence
Duality A strongly coupled theory can be dual
to a weakly coupled theory . The t Hooft limit
(1973) - U(N) gauge theory with coupling
constant g . - t Hooft constant - In the very
large N limit only planar diagrams dominate and
the t Hooft constant is the relevant coupling. -
1/N expansion in gauge theory when ? gtgt 1
topological expansion in string theory . - If ?
ltlt 1 , gauge theory is perturbative. - If ? gtgt 1
, gauge theory is non-perturbative but string
theory is perturbative with gs ?/N ltlt 1
...(4)
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Holographic principle relates a theory living
in a n1dimensional space-time with gravity to a
theory living on its n dimensional boundary (t
Hooft, Susskind 1993). Dp-branes p1
dimensional hypersurface (generally flat) where
open string end points can be localized
(Polchinski 1995) . One Dp-brane induces a
U(1) gauge theory with supersymmetry N 4 in
p1 dimensional space-time . Black p-branes
appear in extended black hole solutions of string
theory .
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• Type IIB String solution containing a black
  3-brane ( Horowitz, Strominger 1991)
• with r0 black hole event horizon position ,
  R parameter associated with the charge of the
  black 3-brane.
• AdS/CFT correspondence
• Consider N coincident D3-branes in the limit of
  low energy string theory keeping the t Hooft
  coupling constant (Maldacena limit).
• This system induces a SU(N) gauge theory with N4
  supersymmetry .
• This system is also a black 3-brane solution of
  string theory . Maldacena considered the extremal
  solution when r00 .
• The metric (5) takes the following form

...(5)
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with
...(6)

• This 10 dimensional metric represents a Poincaré
  patch of AdS5 x S5 space-time (AdS
  Anti-de-Sitter , S sphere ) .
• -Maldacena conjecture
• N4 SU(N) gauge theory living in a 4
  dimensional Minkowski space-time is dual to low
  energy string theory living in a 10 dimensional
  AdS5 x S5 space-time .
• The N4 SU(N) gauge theory has conformal symmetry
  , so is frequently denominated CFT (conformal
  field theory) .
• Condition for the existence of the duality t
  Hooft constant ? has to be gtgt 1 (coherent with t
  Hooft result) .

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- Four dimensional Minkowski space-time (after
compactification) is usually considered as the
boundary of five dimensional Anti-de-Sitter
space-time ? AdS/CFT correspondence is
interpreted as a realization of the holographic
principle.

• Dictionary between the 2 theories (Witten ,
  Polyakov et al 1998)
• W generating functional for the fields living
  in M4, f field on the bulk (AdS5) , f0
  field on the boundary (M4) .
• The field f0 is interpreted as a source for the
  fields living in M4 .

...(7)
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3) Temperature and black hole horizon

• Low temperature case
• Euclidean rotation for the time (tit) in the
  metric given by (6).
• Compactification and periodic conditions for t
  with period ß1/T
• (T temperature)
• The resulting metric is known as thermal AdS
  space . String theory in this metric is dual to a
  non-confining gauge theory in M4 but it is
  possible to obtain confinement considering a mass
  gap for the theory (Polchinski 2000) .
• High temperature case
• - There exists two similar gravitational models .

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• Non-extremal D3 brane solution (charged black
  hole solution)
• Consider again the metric (5) but with r0
  different from zero .
• In the Maldacena limit we obtain
• The temperature is inserted by going to Euclidean
  time and choosing a periodicity for Euclidean
  time.
• - The period ß1/T in this case is not arbitrary.
  Its value is determined by the condition that
  there is no singularity at z z0. This condition
  leads to the following relation

with
...(8)
...(9)
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• Bekenstein Hawking (1974) the black hole
  entropy is proportional to the horizon area .
  Using (9) we can get the entropy per unit volume
  ( Gubser, Klebanov , Peet 1996 )
• - It is useful to compare it with the weakly
  coupled gauge theory result
• - The figure below shows the behavior expected
  for the entropy when the t Hooft coupling varies
  .

...(10)
...(11)
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• Schwarzchild AdS black hole (not charged)
• In the limit of large mass this solution is
  almost the same as (8).
• Scharzchild AdS is stable only for high
  temperatures (Hawking and Page 1983 , Witten 1998
  ) .
• A similar analysis was done recently for the
  non-extremal D3-brane space (Herzog 2006 ).

...(12)
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• 4) Some gravitational predictions for strongly
  coupled gauge theories at finite temperature
  using gauge/string duality.
• The hydrodynamic limit of gauge theories
• Long distance, low frequency behavior of gauge
  theories described by hydrodynamics .
• Hydrodynamics implies very precise constraints on
  the forms of the correlation functions of
  conserved currents and components of the
  stress-energy tensor.
• For example we have
  ...(13)
• for a conserved current in the low energy
  momentum regime (diffusion equation) .
• This equation implies one pole in the correlation
  function for j0 .

16

• Similarly we expect
  ...(14)
• for the transverse components of the momentum
  density which implies a pole for their
  correlation functions.
• - The gravitational model is given by (8) in
  Minkowskian signature .
• Gauge/string duality calculate correlator
  functions for the SYM R-currents on the boundary
  (M4) from classical gauge field action of
  correspondent fields living on the bulk (AdS5) .
• In the hydrodynamic limit the results are the
  following (G. Policastro, D. T. Son, and A. O.
  Starinets 2002)
• For the j0 current we find a pole in the
  correlation function
  
• with

...(15)
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This is a non-trivial prediction for the
diffusion constant in a strongly coupled N 4
Super Yang- Mills theory at finite temperature.
Similarly we find a pole for the momentum
density T0i correlator function


with
...(16) From (10) , (14) and (16) we obtain
...(17) This is a
prediction for the shear viscosity of a
strongly coupled N 4 Super Yang- Mills theory
at finite temperature. - It is useful to compare
eq (15) and (17) with the weakly coupled N 4
Super Yang- Mills theory
,
...(18)
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• Schwinger-Keldysh propagators
• - We define sources and fields in the circuit
• 
  (vanishing sources for the
  other
• 
  parts of the circuit )
• The Schwinger-Keldysh propagators for these
  fields are
• 
  
• If we define the retarded propagator

....(19)
....(20)
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• We can find the following relations
• Considering the Kruskal extension for the
  non-extremal D3 brane metric (8) with Minkowskian
  signature we find 2 boundaries for this space
  (Maldacena 2001, Herzog and Son 2002).
• The field f(x, r) approaches f1 and f2 on the
  boundaries .
• Using gauge/string duality we can obtain the
  propagators given by (19) and (20) and reproduce
  eq (21) !!!!
• But , for this mechanism to work, we have to
  choose s ß/2 .

....(21)
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• 6) Conclusions and perspectives
• There are very strong evidences of the validity
  of the gauge/string duality even at finite
  temperature .
• It is important to understand how supersymmetry
  is broken by the temperature .
• How does the gravity interpretation of the
  temperature change if we modify the
  asymptotically AdS space-time (aAdS Slice,
  deformed aAdS, etc)
• - It is not known how to incorporate a complex
  time . (Real time seems to forbid imaginary time
  and viceversa ).
• - Why s ß/2 ? Is it possible to graph a
  general Schwinger-Keldysh circuit on AdS ?

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• What can gravity tell about confinement/deconfinem
  ent transition in gauge theory ?
• Many open problems quarkonium physics ,
  Wilson loops, chiral symmetry restoration , etc .
  

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References 1 J. Maldacena, The Large N limit
of Superconformal Field Theories and
Supergravity, hep-th/9711200, Adv. Theor. Math.
Phys 2 (1998) 231 . 2 E. Witten, Anti de
Sitter space and holography, Adv. Theor. Math.
Phys. 2, (1998) 253 . 3 O. Aharony, S.S.
Gubser, J. Maldacena, H. Ooguri and Y. Oz, Large
N field theories, string theory and gravity,
Phys. Rept. 323 (2000) 183 hep-th/9905111. 4
E. Witten, Anti-de Sitter space, thermal phase
transition, and confinement in gauge theories,
Adv. Theor. Math. Phys. 2 (1998) 505,
hep-th/9803131. 5 G. Policastro, D. T. Son and
A. O. Starinets, From AdS/CFT correspondence to
hydrodynamics, JHEP 0209, 043 (2002)
arXivhep-th/0205052. 6 J. Maldacena TASI
2003 lectures on AdS/CFT, hep-th/0309246 . 7
C.P.Herzog, D.T.Son , Schwinger-Keldysh
propagators from AdS/CFT correspondence ,
hep-th/ 0212072 . 8 C.A.Bayona, N.R.F.Braga ,
Anti-de-Sitter boundary in Poincaré coordinates
, hep-th/ 0512182