Fermion Droplets, Bubbling Geometries and Bubbling Wilson Loops

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Fermion Droplets, Bubbling Geometries and
Bubbling Wilson Loops

• Gordon W. Semenoff
• University of British Columbia

NBI, August, 2006
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In the context of the AdS/CFT correspondence, the
Wilson loop of N4 supersymmetric Yang-Mills
theory is the gauge theory operator most
closely related to the fundamental string.
We shall consider some highly symmetric Wilson
loops, ½ BPS operators which have some special
properties - conjectures that they are exactly
computable with an interesting
relationship to random matrix models -
conjectures that some correlation functions with
other operators are exactly calculable
- comparison with supergravity and confirmation
of AdS/CFT results - an interesting
relationship to bubbling geometries
K.Okuyama and G.S. hep-th/0604209
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N4 supersymmetric Yang-Mills theory
SU(N) gauge group, SO(2,4)XSO(6) symmetry
Wilson loop operator
Compute ltWgt by summing Feynman diagrams
expansion in or
Has good UV properties
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AdS/CFT correspondence in the dual string
theory we would compute an open string amplitude
C
boundary of AdS
IIB superstring
bulk of AdS
String theory is classical when World-sheet
sigma model is weakly coupled when
lt0
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Straight-line Wilson loop is a ½-BPS operator.
It is thought to be protected from quantum
corrections so that
C
This has been confirmed to order three loops in
perturbation theory. The circle Wilson loop is
obtained from the straight line by a (singular)
conformal transformation and
is also a ½-BPS operator.
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½-BPS Circle Wilson loop as a Matrix integral
Propagator between arcs of same circle ½
Diagrams with internal vertices seem to cancel
at least to order three loops.
Summing ladder Feynman diagrams reduces to
solving combinatorics of matrix-valued
propagators, can be summarized in the matrix
model
The large limit., agrees
with AdS/CFT computation.
J. Erickson, G.S. and K.Zarembo, hep-th/0003055
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Correlator of Circle Wilson loop with chiral
primary operator
R
x
Wilson loop
When seen at a large distance, a Wilson loop with
a compact contour should look like local operators
For a primary operator,
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Weak coupling
Strong coupling
Berenstein, Corrado, Fischler, Maldacena
hep-th/9809188
Sum of planar ladder diagrams
Corrections to planar ladders vanish in leading
order.
G.S. and K.Zarambo hep-th/0106015
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Correlator of Circle Wilson loop with chiral
primary operator
When seen at a large distance, a Wilson loop with
a compact contour should look like local operators
For a primary operator,
For the circle Wilson loop, and chiral primary
operators, the
coefficients can also be computed using a
2-matrix model
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The set of all ½ - BPS (circle) Wilson loops R
an irreducible representation of the SU(N)
gauge group.
The symmetry group is SL(2,R)XSU(2)XSO(5),
subgroup of SO(2,4)XSO(6) Compute using the
matrix integral
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The set of all ½ - BPS Wilson loops R an
irreducible representation of the SU(N) gauge
group.
The symmetry group is SL(2,R)XSU(2)XSO(5),
subgroup of SO(2,4)XSO(6) For large
representations, the string theory dual of this
operator is a D3-brane or D5-brane with boundary
pinched to the contour C residing at the
boundary of AdS. S.Yamaguchi hep-th/0603208
J.Gomis and F.Passerini, hep-th/0604007
R
R
D5-brane wrapping
D3-brane wrapping N.Drukker and B.Fiol,
hep-th/0501129 Large wrapping fundamental rep.
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Bubbling Geometries ½ - BPS Chiral primary
operators
Two- and three-point functions do not depend on
the Yang-Mills coupling constant
J number of boxes in Young diagram corresponding
to R
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The operators are in 1-1
correspondence with states of the matrix harmonic
oscillator
action
Hamiltonian
Physical state condition
Physical states are created by gauge
invariant operators
Phase space action
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Physical states depend on eigenvalues only.
Eigenvalues are fermions
Slater determinant
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particle
hole
One-to-one correspondence between states and
Young tableaux.
Trace basis Representation basis
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In the (semiclassical) large N limit, filled
fermionic levels form a droplet in phase space
p
x
Constant density
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Geometry
Solve IIB supergravity with the assumption that
the solution has SO(4)XSO(4)XR symmetry. H.Lin,
O.Lunin, J.Maldacena hep-th/040917
Spacetime is
Boundary of is the complex plane and is
divided into two regions where one of the
3-spheres shrinks to zero size black and white
areas. Black areas are droplets.
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A giant graviton is a D3-brane wrapping a
3-sphere either extended into AdS-direction (
row tableau particle) or
S5-direction ( column tableau
hole)
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Geometry dual to big Wilson loops
Solve IIB supergravity with the assumption that
the solution has SL(2,R)XSO(3)XSO(5) symmetry
S.Yamaguchi, hep-th/0601089
Boundary of is the line and is divided
into two regions where one of the spheres shrinks
to zero size black and white areas.
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The expectation of a circle Wilson loop is found
by computing a Gaussian Hermitian one matrix
model
Ground state of N fermions in a harmonic
oscillator potential.
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Single-particle wave-function as a coherent state
Eigenvalue integral in the matrix model is the
same as an expectation value of exp(x) between
ground states of N-fermions in a harmonic
potential well.
We can use coherent states to compute these
matrix elements.
Normal matrix model !
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Solution of Gaussian normal matrix model at large
N electrostatic analogy
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Large winding Wilson loop, KN
electric field
Extra eigenvalue is separated from the droplet.
Drucker and Fiol This coincides with the result
of Born-Infeld action for a fundamental string
blown up to a D3-brane extended in the AdS
direction (large lambda)
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Anti-symmetric representation
N-K
K
N
N2
Found from Born-Infeld action of D5-brane
(Yamaguchi)
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Symmetric Representation
All indices are equal
Same as large winding loop matrix model gives
Same as D3-brane
All indices different
similar to anti-symmetric rep loop
D5-brane with an entropy factor
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Symmetric Representation
In the large N limit, which configuration is
preferred depends on K and lambda
Small
Large
N-K
K
first order phase transition
antisymmetric rep. D5-brane
large winding loop D3-brane
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Correlator of Circle Wilson loop with chiral
primary operator
When seen at a large distance, a Wilson loop with
a compact contour should look like local operators
For a primary operator,
For the circle Wilson loop, and chiral primary
operators, the coefficients Can also be computed
using a 2-matrix model
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Conclusion, the Wilson loop operator, truncated
to the space of ½-BPS chiral primary operators
is, in terms of the effective matrix quantum
mechanics,
Selection rule
F
R
Is non-zero only if R has one hook.