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Title: Apresentao do PowerPoint


1
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 .
2
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 .

3
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)
4
  • 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) .

6
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)
7
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 .
8
  • 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)
9
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) .

10
- 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)
11
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 .

12
  • 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)
13
  • 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)
14
  • 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)
15
  • 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)
17
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)
18
  • 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)
19
  • 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)
20
  • 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 ?

21
  • What can gravity tell about confinement/deconfinem
    ent transition in gauge theory ?
  • Many open problems quarkonium physics ,
    Wilson loops, chiral symmetry restoration , etc .

22
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
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