A precision test ofAdS/CFT with flavor
Talk by Andreas Karch, (UW Seattle) at “Quantum Matter, Spacetime andInformation” conference
Kyoto, June 16 2016

work in collaboration with Brandon Robinson and Christoph Uhlemann

How do we know AdS/CFT is correct?

It is difficult to prove AdS/CFT.
Equality between what?

How do we know AdS/CFT is correct?
N=4 SYM with gauge group SU(N)=Type IIB string theory on AdS5 x S5

How do we know AdS/CFT is correct?
N=4 SYM with gauge group SU(N)This is a gauge theory. In principle
defined on lattice. Well posed problem.

=
Type IIB string theory on AdS5 x S5

How do we know AdS/CFT is correct?
N=4 SYM with gauge group SU(N)=We basically only know this as a perturbative expansion.
These days we say its non-perturbatively defined via AdS/CFT.
Therefore true by assumption?

Type IIB string theory on AdS5 x S5

How do we know AdS/CFT is correct?
N=4 SYM with gauge group SU(N)=Practical question: does IIB SUGRA + classical strings
describe the strong coupling, large N limit of N=4?

Type IIB string theory on AdS5 x S5

Does IIB SUGRA describe strongly
coupled N=4?
Established beyond reasonable doubt.
Early evidence:

BPS quantities. Take thesame value at all couplings.

BPS: Dimensions of operators.

dimension

BPS operators = chiral primaries = SUGRA fields

2
XI XJ in 20’
coupling

BPS: Dimensions of operators.
Δ~λ1/4Generic Operators?

dimension

XI XI in 1

2
XI XJ in 20’
coupling

Non-BPS evidence also exists!
About 10 years after the AdS/CFT proposal

Integrability

Beisert, Eden, Staudacher

Inspired guess work

Ansatz for dimensionof certain operatos asfunction of coupling.

BES conjecture matches both weak and strong expansion.

Additional evidence:• Qualitative Predictions sensible.

o Thermodynamics
o Entanglement Structureo Correlation functions
o Real time dynamics

Additional evidence:• Numerical checks (in low dimensions).
(Hanada, Hyakutake, Ishiki,
Nishimura, published in SCIENCE)

Monte-Carlo simulationof D0 brane quantum mechanics.

Rigorous checks from localization.
The probably most compelling checks performed
to date probably come using the technique of(supersymmetric) localization.

Rigorous checks from localization.
The probably most compelling checks performed
to date probably come using the technique of(supersymmetric) localization.

NOTLocalization:
Starting point: Nil-potent symmetry generator:

2𝑄 =0
Easily found in theories with extended supersymmetry
Plays role of exterior derivativeoperator on field space:

2𝑑 =0Localization:
In particular, “Stokes theorem” for Path integrals now reads:

𝒟𝜙 𝑄 …

=0• Integral of total derivative vanishes• Need to be able to drop boundary terms.
• … includes e(-action) term that suppresses “boundaries of field space”

LocalizationWant:(meaning, Q is a symmetry)

Define:with:Localization.
independent of t.
(also true if we insert any operators A with QA=0)
At t → infinity the path integral is dominated by saddle
Path integral localizes to zeroes of QV.

Often path integral reduces to sum/ordinary integral

Localization and AdS/CFT
For generic quantities, localization is just another limit.

tlocalization

Perturbation
Theorygravity

couplingLocalization and AdS/CFT
But free energy of N=4 SYM independent of t !!
Also works for expectation value of SUSY Wilson loops.

tlocalization

Perturbation
Theorygravity

coupling

Free energy of N=4 on S4Can calculate free energy of N=4 SYM on S4 at any
coupling and compare to supergravity.

𝐹 = − log 𝑍
But: scheme dependent!

𝑆 = ….+𝑅2Finite counterterms.

••••

no dynamical field
local in “sources” (here metric)
only affect contact terms
coefficient ambiguousFree energy of N=2* (massive adjoint
hypermultiplet) on S4
Need second mass scale (in addition to radius).
𝐹 = 𝐹(𝑚 ∗ 𝐿)
Finite number of scheme dependent terms.
Perfect agreement.
Alternatively: Wilson loops in N=4

Free energy of N=2*

orange:numerical sugra solution

black:
𝑣 𝜇 = −2𝜇 − 𝜇 log(1 − 𝜇2 )? We should be able to do better ?

analytic answer from localization.

Does IIB SUGRA describe strongly
coupled N=4?

Established beyond reasonable doubt.

Testing flavoredholography

Flavored Holography(Katz, AK)(picture from CLMRW-review, 2011)

Add fundamental matter quarks via probe branes!

N,

not N2 ; quenchedtension negligible;
no backreaction
26

Numerous applications• No QCD without quarks• Wilson lines• Simplest model of dissipation• Holographic lattices• charged matter for CM applications• non-equilibrium steady states• interacting topological states• single EPR pairs• ….

Extra subtleties.
Many questions beyond the probe limit:• Asymptotic freedom lost
• are there still branes in backreacted geometry?
• can the probe be completely geometrized?
• …..
Could this all be wrong???

Extra subtleties.
Many questions beyond the probe limit:• Asymptotic freedom lost
• are there still branes in backreacted geometry?
• can the probe be completely geometrized?
• …..
In any case: not nearly as well tested as N=4/AdS

Goal:
Calculate the free energy of a massive
fundamental representation
N=2 supersymmetric hypermultiplet coupled to
N=4 SYM on S4 using localizationand compare, in the large N strong coupling
limit, to the probe brane answer.

Supersymmetry on curved space.
Challenge 1:
Generically SUSY completely broken byconnection terms in action.
For superconformal theories SUSY obviously
preserved for spaces that are conformally flat:

AdS4

S4

Conformal field theory on conformallyflat spaces.

Generically mass terms break SUSY.

Non-supersymmetric embeddings
These non-supersymmetricembeddings have been
constructed before.
(AK, O’Bannon, Yaffe, ...)

Exhibit interesting topology
changing phase transition withuniversal, calculable
exponents:

Not suited for our purpose.

Restoring SUSY.
To restore (at least some) SUSY we need to add
new terms to the action. Compensating terms.
Ex: “topological twisting”(Witten)

Compensating term = background R-charge gauge field
equal to spin connection.
Keeps some SUSY alive on any curved space (creates a scalar supercharge).

Restoring SUSY.
For special spaces, simpler compensating terms suffice:(Pestun)superpotential mass accompanied with purely scalar mass.

AdS4

S4

real mass
imaginary mass → unitarity lost

Can be understood as due to auxiliary terms in
non-dynamical supergravity background.
(Festucchia, Seiberg)

Holographic compensating terms.
To find holographic duals to SUSY theories,
compensating terms are crucial.
N=2* on Minkowski: 2 scalars in 5d gauged sugra turned on
N=2* on sphere: 3 scalars in 5d gauged sugra turned on

Mass terms for flavor branes.

𝜃(𝑧)Superpotential mass = slipping mode

𝜃(𝑧)Mass terms for flavor branes.

compensating scalar mass = internal gauge field

𝐴=𝑓 𝑧 𝜔particular sphericalharmonic on S3(Kruczenski, Mateos, Myers, Winters)

Mass terms for flavor branes.
SUSY embedding completely characterizedby two scalar functions:

𝜃(𝑧), f(z)(f purely imaginary for flavors on S4)Supersymmetry to the rescueκ- symmetry for D-branes

This equation yields:• Projection condition on SUSY preserved• 1st order equation on background fields

The devil is in the details:

ProjectorBackground Killingspinor

With just a little bit of algebra…..
k=2

c fixed in terms of m by regularity condition

Phase Diagram of D7 embeddings on S4
extend to ρ=0
EquatorPhase transition
at m=1terminate at finite ρPoleSmall mass embeddings (m<1):xSmall mass embeddings (m<1)

xBrane slides off.
Reaches finite angle at center

Large mass embeddings

xBrane smoothly caps off
at finite value of ρCritical embedding at m=1x
For critical embedding branecaps of exactly at center, ρ=0

Phase Diagram of D7 embeddings on S4extend to ρ=0Equator

Phase transition
at m=1terminate at finite ρPoleGeometry of the Phase Transition

Free energy and critical exponents

Free energy and critical exponents

Free energy and critical exponents

Free energy and critical exponents:

Localization for flavored SYMStart with N=4. Vanishing locus of QV:
single (position independent) scalar.

Path integral → Matrix model
Action from 1-loop determinant around vanishing locus:

Localization at large N
At large N Matrix Model solved by saddle point approximation.
Can find eigenvalue distribution. N=4: Wigner semi-circle

endpoint ~ λ1/2

Localization with flavors
Massive flavors enter via 1-loop factor.
Modify action of Matrix model(Pestun)Nightmare even at large N! Potential involves Barnes G.
(Russo, Zarembo)

The Matrix Model in the Probe Limit
For a finite numberof fundamental rep
hypers, semi-circle unchanged.

Calculate flavor contribution
to F with this eigenvalue density.
Free energy = “integrals of Barnes G”

… and large λargument of G = eigenvalue ± m ~ λ1/2
can use asymptotic form of G: logs

Phase transition at large λ
m>1

m<1

Showdown….

Showdown….

Showdown….

IDENTICAL TO PROBE BRANE ANSWER !!!

Free energy and critical exponents:

Conclusions:

FLAVORED HOLOGRAPHY LIVES.
Interesting things to do for the future:• Understand phase structure of AdS4 flavors• Find analytic solution for N=2*. Maybe possible for AdS4
• Finite N, finite λ• Holographic backgrounds for topologically twisted theories• Holography at finite t. Non-BPS quantities?
• other probe branes: D3/D5