Free Fermions and Thermal AdS/CFT - IISER) Bhopalhome.iiserbhopal.ac.in/~suvankar/downloads/ictp07.pdf · Introduction Gauge Theory on S3 Exact Solution Large N Limit-Saddlepoint

Introduction Gauge Theory on S3 Exact Solution Large N Limit-Saddlepoint Phase diagram Fermionic Description Summary

Free Fermions and Thermal AdS/CFT

Suvankar Dutta

December 5, 2008


Introduction and Motivation

Local diffeomorphism in AdS/CFT

The bulk theory is diffeomorphism invariant.

The precise way in which a local diffeomorphism invarianttheory in one higher dimension is encoded in the dynamicsof the gauge theory.

.

.


Introduction and Motivation

Local diffeomorphism in AdS/CFT

The bulk theory is diffeomorphism invariant.

The precise way in which a local diffeomorphism invarianttheory in one higher dimension is encoded in the dynamicsof the gauge theory.

A QuestionIs there any natural way in the gauge theory tounderstand the diffeomorphism redundancy ingravity ?


Lin Lunin Maldacena (LLM) construction

Geometry of a class of 12 BPS solutions of the bulk theory

is completely determined by specifying a single function′′z(x1, x2)

′′ of two of the bulk coordinates (x1, x2).

z(x1, x2) = 1 x1, x2 ∈ R

= 0 Otherwise

This means the full bulk geometry is determined by the theshape of this 2D droplet.


Identification with Phase space of free fermions

LLM identified this 2D droplet with the phase space of freefermions which captures the dynamics of 1

2 BPS operatorsof boundary gauge theory.

Redundancy in Gauge theory

The phase space picture is also a redundant description ofthe gauge theory → because the shape of the perimeter ofthe phase space completely determines everything.

Hence, the phase space picture of gauge theory seems tobe a correct step to understand the redundancy ofdiffeomorphisim of gravity theory.


LLM construction is a special case

The LLM construction which holds for 12 BPS geometry is a

special case.

Also the free fermion picture is not generally applicable.

Non-supersymmetric case

We find similar free fermionic picture in the context ofnon-supersymmetric thermal AdS/CFT !!


Our Results

We find a similar free fermionic phase space description inthermal AdS/CFT.

Thermal partition function of gauge theory

BBH

Gauge Theory Partition Function on S3

Thermal

AdSSBH

We also find different shape of fermionic phase spacedistribution corresponding to these three different saddlepoints.


The Analysis

Exact N partition function

We write the partition function for weakly coupled gaugetheory, which is exact at any finite N.

Phase TransitionBased on this exact N partition function we find a large Nphase transition of gauge theory similar toDouglus-Kazakov phase transition in 2D YM theory.

Free Fermion PictureApart from this large N phase transition also a freefermionic interpretation comes out from this exact partitionfunction.


Outline

1 Introduction

2 Gauge Theory on S3

3 Exact Solution

4 Large N Limit-Saddlepoint

5 Phase diagram

6 Fermionic Description

7 Summary


Gauge Theory on S3 - a small Review

Gauge theory on S1 × S3

The four dimensional gauge theory lives on the boundaryof AdS5 space which has the topology R × S3.

Finite Temperature Case

The time direction R becomes S1 with radius β = 1T .


The Matrix Model

The gauge theory partition function on S3 at temperature Tcan be reduced to an integral over a unitary N × N matrixU .

The matrix U is the Polyakov loop,

U = exp [iβα]

α is the zero mode of the time component of the gaugefield,

α =1

VS3

∫

S3A0


Integrate out the massive modes

Apart from the zero mode of A0 all other modes of thegauge theory on S3 are massive.

One can integrate them out to obtain an effective action interms of α or U.


Partition Function at λ = 0 [Sundborg, Aharony et. al]

The free gauge theory partition function on S3 × S1 :

Z (β) =

∫

[dU] exp

[

∞∑

n=1

an(T )

nTr(Un)Tr(U†n)

]

The coefficients an(T ) are given by ,

an(T ) = zB(xn) + (−1)n+1zF (xn) , x = e−β

zB(x) and zF (x) are single particle partition functions ofthe bosonic and fermionic modes respectively.


Partition function at λ 6= 0 [Aharony et. al.]

Integrating out the massive modes as before and obtain amore general and complicated effective action for U.

The structure of the effective action is now

Z (β, λ) =

∫

[dU] exp Seff (U)

The general form of the effective action is

Seff (U) =∑

{ni}

a{ni}(λ, T , N)1

Nk

k∏

i=1

TrUni ,∑

i

ni = 0


A phenomenological model[Alvarez-Gaume Gomez Liu Wadia]

Integrate out all the TrUn with n 6= ±1

Obtain an effective model in terms of x = 1N2 TrUTrU†

Z =

∫

[dU]eN2Seff (x)

Seff (U) = a1(λ, T )TrUTrU† +b1(λ, T )

N2 (TrUTrU†)2

+c1(λ, T )

N4 (TrUTrU†)3 · ··

S(x) → convex and S′(x) → concave.


The (a , b) Model

This the simplest class of this effective model.

Keep only the first two coefficients in the Seff given

Z (a1, b1) =

∫

[dU] exp[

a1TrUTrU† +b1

N2

(

TrUTrU†)2

]

a1 and b1 are functions of temperature T and λ.


The Solution: Eigenvalue Analysis

Eigenvalue density

Introduce the eigenvalue density

σ(θ) =1N

N∑

i=1

δ(θ − θi)

θi ’s are eigenvalues of U

U = diag(eiθi ).

The partition function can be expressed in terms of afunctional S[σ(θ)]

Z (β) =

∫

[Dσ]eN2S[σ(θ)],


Zero coupling Result

The functional S[σ(θ)] involves all the coefficients an’s andit is very complicated to analyze.

To see the phase transition and the behaviour near thetransition point, it is sufficient to consider the simple modelwhere an = 0, n > 1.

Let us summarize the saddle points and the phasestructure for this special case.


Saddle points at zero coupling

Saddle point I

For a1 < 1 (T < TH ), the minimum action is for theeigenvalue density

σ(θ) =1

2π.

The free energy is zero for this configuration (to order N 2).

.

.



Saddle point I

For a1 < 1 (T < TH ), the minimum action is for theeigenvalue density

σ(θ) =1

2π.

The free energy is zero for this configuration (to order N 2).

Saddle point II

For a1 = 1 (T = TH ): ∃ continuous family of minimumaction configurations for which the eigenvalue distribution :

σ(θ) =1

2π(1 + 2ξ cos(θ)) 0 ≤ 2ξ ≤ 1

All these configurations also have free energy zero.



Saddle point III

For a1 > 1 (T > TH ) the eigenvalue distribution function:

σ(θ) =1

π sin2(

θ02

)

√

sin2(

θ0

2

)

− sin2(

θ

2

)

cos(

θ

2

)

sin2(

θ0

2

)

= 1 −√

1 − 1a1(T )

≡ 12ξ

The free energy for this configuration is

F = −N2T[

ξ − 12

ln(2ξ) − 12

]

≤ 0.


Zero coupling phase diagram

There is a first order phase transition at a1 = 1, whichcorresponds to a temperature T = TH .

For T < TH the gauge theory is in the confined phase.

T > TH the theory is in the deconfined phase .

Does not exhibit the complete phase structure

This zero coupling phase diagram does not show all thesaddle points which appear in the strong coupling phasediagram.

To see the complete phase diagram one has to turn on asmall ’t Hooft coupling.


Weak coupling results

To recover the above strong coupling phase structure let usconsider a weakly coupled unitary matrix model: (a,b)model.

One can study the eigenvalue analysis as before and findthe saddle points.

We will indicate the weak coupling result.


Phase diagram at weak coupling

H

T<T0T=T0 T>T0, T<T1

T=T1 T>T1, T<TH T=T


Exact Solution at Finite N

Gauge theory Partition function S3

Start with the matrix model,

Z =

∫

[dU] exp

[

∞∑

n=1

an(T )

nTrUn TrU†n

]

.

Expand the exponential to obtain for the integrand

exp

[

∞∑

n=1

an(T )

nTrUn TrU†n

]

=∑

~k

1z~k

∏

j

akj

j Υ~k(U)Υ~k

(U †)

z~k =∏

jkj !jkj and Υ~k

(U) =∞∏

j=(TrU j)kj .


Frobenius formulaUse Frobenius formula,

Υ~k(U) =∑

R∈U(N)

χR(C(~k)) TrRU.

χR(C(~k )) : character of the conjugacy class C(~k) of thepermutation group Sl , (l =

∑

jkj ) in the representation R ofU(N). ~k = {k1, k2, .....}.

Orthonormality

Orthogonality relation between the characters of U(N),∫

[dU] TrR(U) TrR′(U†) = δRR′ .

Carry out the integral over U.


The Exact N partition function : λ = 0

Finally we obtain

Z (β) =∑

~k

Q

j akjj

z~k

∑

R

[

χR(C(~k))]2

.

The Special Case

In a special case where an = 0 for n > 1 the partitionfunction becomes

Z (β) =

∞∑

k=0

∑

R

1k !

[dR(Sk )]2 ak1.

χR(C(~k )) → dR(Sk ) → dimension of the representation.


The (a,b) Model: Effective Action

This method can easily be generalized to the (a,b) model.

The effective action for (a,b) model is given by,

Seff = a1(λ, T )TrUTrU† +b1(λ, T )

N2

(

TrUTrU†)2

.

Exact N partition function for (a,b) Model

The exact N partition function for (a,b) model is given by,

Z (a1, b1) =∞∑

k=0

k/2∑

l=0

ak−2l1 bl

1k !

N2l l!(k−2l)!

∑

Rd2

R(Sk )

k ! .


Rearrange the sum over representation

One can rearrange the sum over representations of U(N)in terms of the number of boxes of the correspondingYoung tableaux.

ni → Number of boxes in the ‘i’ th row of the Youngtableaux.


Partition function at λ = 0

Therefore the partition function reads as

Z (β) =

∞∑

k=0

∞∑

{ni}=0

1k !

[dR(Sk )]2 ak1 δ(ΣN

i=1ni − k).

Dimension FormulaThe dimension dR(Sk ) is given by the formula

dR(Sk ) =k !

h1!h2!...hN !

∏

i<j

(hi − hj),

where,hi = ni + N − i ,

withh1 > h2 > ... > hN ≥ 0.


The large N Limit and Saddle Point Analysis

Taking the large N limit

Partition function as stat. mech system

In N → ∞ the limit, the partition function can be viewed asa statistical mechanical system.

The group characters → entropy contribution .a1 (b1) → Boltzmann suppression factors.

Phase TransitionInterplay between these two factors.

The balance between them leads to a dominantrepresentation at any particular value of the temperature.

At large N, as we vary temperature the dominant saddlepoints changes non-analytically => phase transition.


Continuum Limit

DefinitionsIn N → ∞ limit, let us define

n(x) =ni

N, h(x) =

hi

N, x =

iN

x ∈ [0, 1] .

h(x) and n(x) are related by: h(x) = n(x) + 1 − x .

The function n(x) or h(x) captures the profile of the largeN Young tableaux.

Constraint on h

h(x) > h(y) for y > x .


Introduce Young tableaux density

Introduce the density of boxes in the Young tableaux u(h)defined by

u(h) = −∂x(h)

∂h.

NormalizationBy definition, it obeys the normalization

∫ hU

hL

dh u(h) = 1, hL = h(1) and hU = h(0).

ConstraintFrom the monotonicity of h(x), it follows that u(h) obeysthe constraint

u(h) ≤ 1.


The effective action: λ = 0

In terms of the saddle point density the partition functionbecomes : Z (β) =

∫

[dh] exp[−N2Seff ].

−Seff =

∫ hU

hL

dh−∫ hU

hL

dh′u(h)u(h′) ln |h − h′|

− 2∫ hU

hL

dh u(h) h ln(h) + k ′ + 1 + k ′ ln(a1k ′).

Total No. of boxes

Where k ′ is related to total number of boxes ’k’ in a Youngtableaux

k = N2∫ hU

hL

dh h u(h) = N2k ′.


The Saddle point Equation: λ = 0

Varying Seff with respect to u(h), we obtain the saddlepoint equation (similar to [Kazakov-Staudacher-Wynter]),

−∫ hU

hL

dh′ u(h′)

h − h′= ln h − 1

2ln

[

a1k ′]

= ln[

hξ

]

where ξ2 ≡ a1k ′.

One important note

ξ involves k ′.

Which in turns depends on the density u(h).

We will therefore have to solve the equationself-consistently.


The Solutions

Two kinds of saddle points

In presence of constraint on u(h) there are two differentpossible solutions → depending on the value of theparameter ξ.

We will sketch the procedure and show the final result inthis talk.


The Saddle points

Solution Class 1

0 ≤ u(h) < 1; h ∈ [q, p].

.

.


The Saddle points

Solution Class 1

0 ≤ u(h) < 1; h ∈ [q, p].

Solution Class 2

u(h) = 1 h ∈ [0, q]

= u(h) h ∈ [q, p]

with 0 ≤ u(h) < 1.

As we vary ξ, one will have to switch from one of thebranches to the other.


Solution Class 1 Solution Class 2

n (x=0)

n (x=1)

1

N

Generic plots of Young tableaux


The sketch of the analysis

ResolventIntroducing the resolvent H(h) defined by

H(h) =

∫ hU

hL

dh′ u(h′)

h − h′.

Find the expression for H(h) for two different solutionclasses.

Young tableaux density is given by,u(h) = − 1

2πi [H(h + iε) − H(h − iε)] for h ∈ [q, p].

The support of u(h) as well as k ′ is determined byexpanding H(h) for large h and matching withH(h) ∼ 1

h + (k ′ + 12) 1

h2 (as h → ∞).


The Saddle densities

Solution Class 1:λ = 0The Young tableaux density in this solution class is givenby,

u(h) =1π

cos−1

[

h − 12 ξ

+

(

ξ − 12

)2

2 ξ h

]

=2π

cos−1[

h + ξ − 1/22√

ξh

]

. h ∈ [q, p]


The supports : p and q

The support of u(h) as well as k ′ is given by,

√q =

√

ξ − 1√2

,

√p =

√

ξ +1√2

k ′ =√

q p + 14 => a1 = 4 ξ2

4 ξ−1

q is a real and positive quantity => this branch of solutionexists for ξ ≥ 1

2 .

Conclusion : this class of solutions only exist for

a1 ≥ 1 or T > TH where a1(TH) = 1


Plot of u(h) Vs. h : Solution class 1


The Saddle densities

Solution Class 2: λ = 0The Young tableaux density in this solution class

u(h) =1π

cos−1[

h − 12ξ

]

, h ∈ [q, p]

The supports: p and q

The support of u(h) is given by,

q = 1 − 2ξ ,

p = 1 + 2ξ

• k ′ = ξ2

ξ ≤ 12 .


Two possible solutions in solution class 2

From the definition a1k ′ = ξ2 we obtain

Either ξ = 0 (2A)

or , a1 = 1 (2B).

Solution 2A : ξ = 0

Uniform distribution: u(h) = 1 h ∈ [0, 1].

This is a saddle point for any value of a1.

This saddle point corresponds to the trivial representationi.e. ni = 0.

Solution 2B : a1 = 1

A family of saddle points labeled by ξ .

Exists only at a1 = 1 i.e at T = TH .


Plot of u(h) vs. h: Solution Class 2


The Free Energies

In large N limit the free energy is given by,

F = −T ln Z = N2TS0eff .

S0eff is the value of effective action at the (dominant) saddle

point.

Zero coupling Free energy: Solution Class 1

F = −N2T[

ξ − 12

ln(2ξ) − 12

]

≤ 0.

Zero coupling Free energy: Solution Class 2

F = 0 + O(1

N2 )


Phase Diagram

Zero coupling phase diagram

T < TH

For low enough temperature (T < TH ) when a1 < 1 thereexists only one saddle point i.e. ξ = 0.

This saddle point corresponds to trivial representation.

Zero free energy.

T = TH

a1 = 1.

ξ = 0 configuration is a limiting member of this family.

A finite fraction of rows of Young tableaux are empty.

Zero free energy.


T > TH

As we further increase the temperature T > TH , thena1 > 1.

ξ has a solution greater than 12 .

An exchange of dominance of the saddle point at T = TH .

All the rows of Young tableaux are filled.

a1 > 1 phase has negative free energy of order O(N2).

Confinement-deconfinement phase transition


Extension to non-zero coupling

To see the complete phase structure we have to turn onsmall ’t Hooft coupling.

(a,b) Model

We will consider the (a,b) model to explore the weak couplingphase diagram.

Seff = a1(λ, T )TrUTrU† +b1(λ, T )

N2

(

TrUTrU†)2

.

All the expressions for u(h) and the supports p and q for boththe solution classes are same as that of zero coupling.

We will only discuss the phase diagram and the correspondingpattern of Young tableaux for different saddle points.


Weak coupling phase diagram

T < T0

For temperature T < T0 there exists only one saddle point,ξ = 0. This saddle point corresponds to trivialrepresentation.

T<T0

Thermal AdS

u

h


T = T0 and T > T0

T>T0

T=T0

SBH/BBH

SBH BBH

SBH

BBH


T = Tc

CT=TSBH

BBH

SBH

BBH

BH-String transition : identified by Alverez Gaume-C.Gomez-H. Liu-Spenta Wadia


T > Tc

CT>T

SBH

BBH

SBH

BBH


T = TH

At T = TH saddle point corresponds to SBH and saddlepoint corresponds to global AdS merge.

T > TH BBH is the only saddle point.

H

Thermal AdS/SBH

BBH

T=T


Free Fermionic Phase space Description

Young tableauxVs. eigenvalue distribution

There exists a simple relation between (h, u(h)) and(θ, σ(θ)).

Low temperature case

The saddle point for which u(h) and h relation is one toone, we have a mapping,

u =θ

π,

h2π

= σ(θ)


u(h) =1π

cos−1[

h − 12ξ

]

, h ∈ [q, p]

h = 1 + 2ξ cos[πu]

σ(θ) =1

2π(1 + 2ξ cos θ)


Young tableaux Vs. eigenvalue distribution

High temperature case

Where as, for the other saddle point, the mapping is nontrivial, but simple.

u =θ

π,

h+ − h−

2π= σ(θ)

sin2 θ0

2=

12ξ


h2 − [1 + 2 ξ cos[πu(h)]] h +

(

ξ − 12

)2

= 0

h+ − h− = 2√

2ξ

√

1 − 2ξ sin2(πu

2

)

cos(πu

2

)

σ(θ) =1

π sin2(

θ02

)

√

sin2(

θ0

2

)

− sin2(

θ

2

)

cos(

θ

2

)

.


Interpretation

The above relations have a natural interpretation in termsof free fermionic picture.

The eigenvalues θ’s of the matrix U behave likecoordinates of fermions.

On the other hand the representation of U(N) also have aninterpretation in the language of free fermions with ni ’s arelike momenta. [Douglas]

This suggests that eigenvalue density is like a positiondistribution.

Young tableauxdensity is like momentum distribution.


The phase space distribution

Define a phase-space distribution which gives rise to theseindividual distributions.

Phase space density

At saddle point define a phase space density

ρ(h, θ) =1

2π(h, θ) ∈ R

= 0; otherwise

The position distribution and momentum distributions:

σ(θ) =

∫ ∞

0ρ(h, θ)dh

u(h) =

∫ π

−πρ(h, θ)dθ


Phase space distributions for saddle points

We therefore see that the large N saddle points of thegauge theory effective action, which correspond to theThermal AdS, the small black hole and the big black holecan all be thought of in terms of a particular configurationin a free fermionic phase space.

There is a particular shape associated to each of them.


Free fermi phase space distribution for thermal AdS


Free fermi phase space distribution for SBH


Free fermi phase space distribution at GWW point


Free fermi phase space distribution for BBH


Summary

We studied thermal gauge theory by evaluating its partitionfunction at finite N.

Taking large N limit of the full partition function gives us anew perspective on some already known facts about thephase diagram of gauge theory.

We saw there is a close relation between Young tableauxdensity and eigenvalue density.

This identification has a natural meaning in terms of Freefermion phase space distribution.

We found different phase space distributions for differentsaddle points of the partition function.

This formulation may be helpful to reconstruct the localtheory in bulk with all its redundancies.

Documents

Free Fermions and Thermal AdS/CFT - IISER) Bhopalhome.iiserbhopal.ac.in/~suvankar/downloads/ictp07.pdf · Introduction Gauge Theory on S3 Exact Solution Large N Limit-Saddlepoint