Giant magnons meet giant
gravitons
Diego Hofman
Princeton University
based on work with
Juan Maldacena12 (IAS) and Nick Dorey2 (DAMTP)
Buenos Aires, 10 years of AdS/CFT
version date: December 18, 2007
1
hep-th/0604135, arXiv:0708.2272
2
hep-th/0703104
ptContents
1
Introduction
1.1 String Theory, Gauge Theories AdS/CFT correspondence . . . . . .
1.2 Topics for this talk . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The road to integrability: BMN, Spin Chains and Closed Strings . .
4
4
5
5
2
Giant Magnons in Semiclassical String Theory
2.1 Localized excitations in a string . . . . . . . . . . . . . . . . . . .
2.2 Giant Magnons in the large J limit . . . . . . . . . . . . . . . . .
2.3 LLM coordinates, BPS states and SUSY algebra . . . . . . . . . .
8
8
10
12
3
Giants meet
3.1 Open strings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Strong coupling description . . . . . . . . . . . . . . . . . . . . .
3.3 Weak coupling description . . . . . . . . . . . . . . . . . . . . .
13
13
14
16
4
Exact results
4.1 The symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Y=0 brane . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Yang Baxter Equation . . . . . . . . . . . . . . . . . . .
17
17
18
21
4.2.2 Crossing Equation . . . . . . . . . . . . . . . . . . . . .
The Z=0 brane . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Boundary bound states . . . . . . . . . . . . . . . . . . .
22
23
25
5
Weak coupling results
5.1 Two Loop Hamiltonian . . . . . . . . . . . . . . . . . . . . . . .
5.2 Phases and tests . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
26
27
28
6
Strong coupling results
6.1 Bulk results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
29
30
7
Summary and Conclusions
31
8
Extensions
32
9
Open Problems
32
4.3
Giant magnons meet giant gravitons
4
1.1
String Theory, Gauge Theories AdS/CFT correspondence
1.
Introduction
1.1.
String Theory, Gauge Theories AdS/CFT correspondence
• Gauge Theory/String Theory duality. Can we build a strong coupling description
of QCD out of gravity? This is difficult, but there has been recent progress.
• String Theory on AdS5 × S 5 ←→ N = 4 SYM
• Several limits in which N = 4 is not that different from QCD: Quark-gluon
plasma, transcendentality hypothesis [Lipatov; Beisert, Eden, Staudacher], etc.
• In the very beginning, important objects were BPS objects.
• Weak/Strong duality. Problems and Solutions.
• Can we go beyond BPS objects? Can we go beyond perturbative regime? Integrability might takes us there.
• What is integrability? What do we mean by solving a theory exactly?
Anomalous dimensions ↔ Spectrum → S-matrix → Bethe Ansatz → multiparticle states → Finite Size effects?
Giant magnons meet giant gravitons
1.2.
5
1.2
Topics for this talk
Topics for this talk
• Review of closed string sector of AdS5 × S 5/ Single trace operators of N =
4 SYM.
• Study the open string sector of AdS5 × S 5/ (Excitations of) determinant operators in N = 4 SYM.
• Extend the results from the closed string sector, study the symmetries of the
problem and calculate the S-matrix (up to a phase) for cases corresponding to
different orientation of D3-branes (giant gravitons).
• Describe the spectrum of these theories and find new boundary degrees of freedom.
• Discuss integrability.
• Compare with weak and strong coupling results. Phase.
1.3.
The road to integrability: BMN, Spin Chains and Closed Strings
• Idea that got it all started: consider near BPS operators. [Berenstein, Maldacena,
Nastase]
Giant magnons meet giant gravitons
1.36 The road to integrability: BMN, Spin Chains and Closed Strings
• We consider the BPS state T r[Z J ]. This corresponds to a massless mode of 10d
SUGRA propagating on the sphere. ∆ − J = 0.
• What to do next? Smart idea: Consider T r[W Z J ] = J1+i2,5+i6T r[Z J ]. This
is the state with lowest energy and charge J 0 = 1 over the vacuum. Thus, it is
BPS.
• Now we are getting somewhere. We have an impurity on a chain of letters. Let
us just do what we always do: Let’s construct a Bloch State.
Op ∼
X
eilp(· · · ZZZW ZZZ · · · )
(1)
l
• There’s a catch! An honest string state should contain, at least, two excitations
with opposite momenta to be non vanishing because of the trace. These states
are not BPS. They are the next best thing to BPS states: Impurities running in
the chain. This problem can now be mapped to a Spin Chain with ∆ → H
where our fundamental blocks are the magnons we just described. [ Minahan
and Zarembo; Beisert, ...].
• To one loop this is the Heisenberg Spin Chain. We obtain different spin chains
to higher order. All these Spin Chains are integrable!
Giant magnons meet giant gravitons
1.37 The road to integrability: BMN, Spin Chains and Closed Strings
• It turns out that in the J → ∞ limit, the energy of these states can be calculated
in perturbation theory with the effective coupling λ0 = Jλ2 . Using symmetry
arguments it is possible to obtain the exact formula:
r
E−J =
1+
λ
2 p
sin
π2
2
(2)
[Beisert]
This result uses the su(2|2) × su(2|2) symmetry algebra extended by central
charges that make the chain dynamic. Closed string states are neutral under these
charges. Symmetries do not constrain f (λ) → f (λ) = πλ2 experimental result!
• Because the system is integrable we are tempted to forget about the trace an
consider single excitation states (magnons) at infinite J . Then we can scatter
these and calculate an S-matrix. Are magnons BPS?
• What happens in the string theory side of the story? Classically, the model is
integrable [Bena, Polchinski, Roiban]
• In the Plane Wave (BMN) limit, an exact quantization at J, λ → ∞ and pJ =
Giant magnons meet giant gravitons
2.1
8
fixed is possible [Berenstein, Maldacena, Nastase]. λ0 =
r
E−J =
λn2
1+ 2
J
p=
2πn
J
λ
J2
Localized excitations in a string
expansion.
(3)
• In the full geometry the dual states to the spin chain magnons are called giant
magnons [DH, Maldacena].
2.
Giant Magnons in Semiclassical String Theory
2.1.
Localized excitations in a string
Strings in Flat Space
• Strings of finite worldsheet length.
Giant magnons meet giant gravitons
2.1
9
X
(a)
+
−p
p
(c)
(b)
X−
X1
(d)
• Strings of infinite worldsheet length.
X
+
p
(a)
(b)
X1
(c)
X
−
Localized excitations in a string
Giant magnons meet giant gravitons
2.2.
10
2.2
Giant Magnons in the large J limit
Giant Magnons in the large J limit
• Solution obtained by minimizing the string action.
sin p2
cos θ0
i=
i
h
h
cos θ =
x−cos p2 t
x−sin θ0 t
cosh cos θ0
cosh sin p
(4)
2
x − cos p2 t
x − sin θ0t
p
tan(ϕ − t) = cot θ0 tanh
= tan tanh
cos θ0
2
sin p2
(5)
Giant magnons meet giant gravitons
2.2
11
Giant Magnons in the large J limit
• Coordinates and momentum identified by
√
λ
dJ
=
dx
2π
vgauge
√
or
dl =
√
dl d(p)
p
λ
=
=
=
cos ,
dt
dp
2π
2
vstring =
λ
dx
2π
for p > 0
dx
∆ϕ
= sin θ0 = cos
dt
2
(6)
(7)
(8)
• Note that we allowed for an open string solution. This is equivalent to relaxing
the zero mode of the Virasoro constraints (or the trace in the gauge theory). We
can do this for an infinite string.
Giant magnons meet giant gravitons
2.3.
12
2.3
LLM coordinates, BPS states and SUSY algebra
LLM coordinates, BPS states and SUSY algebra
ip
e
p
1
x2
x1
(b)
(a)
• This coordinates are just a projection of the sphere onto the plane.
• The SUSY algebra in 10d SUGRA contains gauge transformation of the B field
under which the string is charged [Schwarz]. Therefore, open stretched strings
can carry winding charges associated. It is these charges that make the state
BPS. Analogous to strings stretched between branes.
• This charges are proportional to the string length. Therefore:
2
k 1 + ik 2 =
√
R
λ i p2
p
ip
(e
−
1)
=
i
e
sin
2πα0
π
2
(9)
Giant magnons meet giant gravitons
3.1
13
r
E − J = k0 =
p
1 + |k1 + ik2|2 =
1+
Open strings
λ
2 p
sin
π2
2
(10)
• This algebra looks like the usual 2+1 Poincare algebra. Notice, however, that
boosts are not a symmetry but an outer automorphism of the algebra.
• Finally: Closed string states are neutral. (therefore, not BPS in general).
3.
Giants meet
3.1.
Open strings
• Consider a D3-brane wrapping a maximal S 3 inside S 5. We can attach strings
to this D-brane.
• Gauge theory operators
B
...jN −1 A i1 i2
OY = ji11ij22...i
Yj1 Yj2 . . . YjiNN−1−1 (ZZZ . . . ZZZ)A
N −1 B
(11)
Giant magnons meet giant gravitons
14
3.2
Strong coupling description
• One loop results consistent with integrable open spin chain with dirichlet boundary conditions [Berenstein, Vazquez]
• Two loop results in conflict with integrability. [Agarwal]
• At strong coupling, the classical sigma model is integrable [Mann, Vazquez]
3.2.
Strong coupling description
• We consider strings with angular momentum mostly in the Z plane.
• There are two inequivalent configurations for a giant graviton attached to these
strings.
• The string interpolates between the giant and the BMN vacuum.
Giant magnons meet giant gravitons
3.2
15
Strong coupling description
• Finite J strings do not reach the rim of the circle.
• Novel degree of freedom? Energy:
p
B = 1 + 4g 2 ,
• Zero mode quantization must yield B (n) =
g2 =
√
λ
16π 2
n2 + 4g 2.
(12)
Giant magnons meet giant gravitons
3.3.
16
3.3
Weak coupling description
Weak coupling description
• Y brane
B
...jN −1 A i1 i2
iN −1
(. . . ZZZχZZZ . . .)A
OY (χ) = ij11ij22...i
Y
Y
.
.
.
Y
j
j
j
B
1
2
N −1
N −1
(13)
• Z brane
B
...jN −1 A i1 i2
iN −1
(χZZ . . . ZZZχ0ZZZ . . . ZZχ00)A
OZ (χ, χ0, χ00) = ji11ij22...i
Z
Z
.
.
.
Z
j
j
j
B
1
2
N −1
N −1
(14)
• Can we fill the vacuum?
B
...jN −1 A i1 i2
iN −1
O0 = ji11ij22...i
.
.
.
Z
Z
Z
jN −1 (Y Y Y . . . Y Y Y )A
j1 j2
N −1 B
(15)
Giant magnons meet giant gravitons
4.1
17
4.
Exact results
4.1.
The symmetries
The symmetries
• AdS5 × S 5 / N = 4 SYM has a P SU (2, 2|4) symmetry group.
• Choice of BMN vacuum breaks this to SU (2|2)2 × R
• Maximal central extension of this algebra is equivalent to consider gauge dependent operators in SY M [Beisert] or strings with winding charge in String
Theory [DH, Maldacena]
{Qαa, Qβb}
αβ k
= ab
2
{Saα , Sbβ }
,
k∗
= αβ
2
ab
(16)
• Magnons transform in the fundamental (BPS) representation of each SU (2|2)
Qαa|φbi = aδab |ψ α i ,
Qαa|ψ β i = bαβ ab|φbi
Saα |φbi = cαβ ab|ψ β i ,
Saα |ψ β i = dδαβ |φai
√
√
√
√
g
giη
g
x+
with a = gη b = η f 1 − x−
c = f x+ d = iη (x+ − x−)
and ad − cb = 1
k
2
= ab
k∗
2
= cd
x+
x−
= eip
Giant magnons meet giant gravitons
18
4.2
The Y=0 brane
• If we demand that the S-matrix (2 → 2 operator) commutes with symmetry
generators we can determine it up to a phase [Beisert]. Notice that quantum
numbers change after scattering in a non local way.
• The S-matrix satisfies the Yang Baxter equation.
• Scattering a singlet we can obtain an equation for this phase [Janik,Beisert]
4.2.
The Y=0 brane
• The subgroup that leaves Y invariant is SU (1|2)2. This algebra cannot be centrally extended → the complex charge k is not conserved.


 + −
− + 
−̇
χ χ −χ χ
φ

ϕϕ

  φ+̇ 

• Magnon multiplet transforms as = 
=
ϕχ− + χ−ϕ   ψ − 
ϕχ+ + χ+ϕ
ψ+
where = (ϕ, χ± )
• This representation breaks into a singlet and a fundamental for states that do not
carry central charges.
• The reflection matrix R is a 1 → 1 operator.
Giant magnons meet giant gravitons
4.2
19
The Y=0 brane

r− 0 0 0
 0 r+ 0 0 
• Commutation with the symmetry generators impliy R = 
0 0 r 0
0 0 0 r
ar − a0r+ = 0
c
b0
−
r
=
r
=
r
0
c
b
br− − b0r = 0
−→
with
cr − c0r− = 0
d0
a
+
r
=
r
=
r
+
0
a0
d
dr − d r = 0
• Quantum numbers change because of reflection. Energy must be conserved.


−eip 0 0 0
 0 1 0 0
• The reflection matrix is RL = R0L(p) 
0 0 1 0
0 0 0 1

Giant magnons meet giant gravitons
20
4.2
The Y=0 brane
Giant magnons meet giant gravitons
4.2.1.
21
4.2
The Y=0 brane
Yang Baxter Equation
• There are three intermediate representations. Each one carries a state that scatters diagonally → parity invariance of the bulk implies YBE.
• The Yang Baxter Equation is obeyed for the full SU (1|2)2 theory! → integrability
Giant magnons meet giant gravitons
4.2.2.
4.2
22
The Y=0 brane
Crossing Equation
• We build an identity state: particle/hole pair with no energy. Scattering should
−
− −
α β
be trivial. 1(p,p̄) = fp eip/2 (φ+
p φp̄ − φp φp̄ ) + αβ ψp ψp̄ .
• The crossing equation is obtained from
that the scattering phase is 1. demanding
β
α
1(p,p̄) = hbS0(p, −p̄)R0R (p)R0R (p̄) f−peip/2(φ+−p̄φ−−p − φ−−p̄φ−−p) + αβ ψ−p̄
ψ−p
= hbS0(p, −p̄)R0R (p)R0R (p̄)1(−p̄,−p) with hb ≡
1
+x−
x−
1
+x+
x+
Giant magnons meet giant gravitons
4.3.
4.3
23
The Z=0 brane
The Z=0 brane
• The boundary now preserves the same symmetries as the vacuum. We have the
full SU (2|2)2 symmetry.
• The boundary transform in the fundamental representation of the group with
quantum numbers
√
aB = √ gηB
gf
bB = η B B
√
giη
cB = xB fBB
√
gx
dB = iηBB
• This implies xB =
√
1 + 4g 2.
i
2g
1+
√
1 + 4g 2 and = ad + bc =
g
i
xB −
1
xB
=
• The reflection matrix is a Bp → Bp operator. It is again fixed by the symmetries to be:
Giant magnons meet giant gravitons
{a b}
24
[a b]
β
R|φaB φbpi = A|φB φ−pi + B|φB φ−pi + 21 Cabαβ |ψbα ψ−p
i
{α β}
[α β]
R|ψBα ψpβ i = D|ψB ψ−pi + E|ψB ψ−pi + 12 F αβ ab|φaB φb−pi
α
i
R|φaB ψpα i = G|ψBα φa−pi + H|φaB ψ−p
α a
a α
α a
R|ψB φp i = K|ψB φ−pi + L|φB ψ−pi
+
+
+xB )
A = R0 xx−(x
(x− −xB )
+ −
+
+ 2
− 2
+ −
B )[−2(x ) +2(x ) +x x ]
B = R0 2x x xB +(x −x
(x− )2 (x− −xB )
− +x+ )(x− x −x+ x −x− x+ )
(x
B
B
C = R0 2ηηf B
xB x− (x+ )2 (x− −xB )
D = R0
+ 2
− 2
+ −
−
+ +x− (x+ )2 )]−x (x+ x− )2 (x −x− )
B
B
E = R0 2[(x ) −(x ) ][−x x +x(xB−(xx+)−x
2 x (x− −x )
B
B
2f (x+ −x− )(x+ +x− )(xB x+ −xB x− +x+ x− )
F = R0 ηηB
(x− )3 (x− −xB )
+ −x− )(x+ +x− )
(x
G = R0 ηηB (x−−xB )x−
+ 2
−
H = R0 (xx−(x) −−x−xBBx)
+
− )2 ]
K = R0 [xxB−x(x−+(x
−xB )
η (x+ +x− )xB
L = R0 ηB x−(x−−xB )
• This form can be tested perturbatively.
4.3
The Z=0 brane
Giant magnons meet giant gravitons
4.3.1.
4.3
25
The Z=0 brane
Boundary bound states
• This matrix contain poles at x− = xB . This is similar to what happens in the
bulk S-matrix.
• In the bulk, the analytic porperties of S-matrix can be related to the spectrum via
Landau diagrams. They can provide non trivial tests of the overall phase [Dorey,
DH, Maldacena]. Not all poles are bound states.
• This particular pole is associated with a bound state. Other singularities might
also be present (crossing demands this).
• The energy of the intermediate state is B =
√
1 + 4g 2.
• Notice that, by changing the initial representation of the boundary degree of
Giant magnons meet giant gravitons
5.1
26
Two Loop Hamiltonian
freedom (cf. Bootstrap) we obtain more bound states with energies B (n) =
√
n2 + 4g 2.
• YBE and a crossing equation can be obtained in the same fashion as in the
previous case.
5.
Weak coupling results
5.1.
Two Loop Hamiltonian
• We can use the dilatation operator of SY M [Beisert, Kristjansen, Staudacher]
to obtain a two loop Hamiltonian for the spin chain in the SU (2) subsector.
2
4
D = −2 gN : Tr[Y, Z][Y̆ , Z̆] : −2 Ng 2 : Tr[[Y, Z], Z̆][[Y̆ , Z̆], Z] :
4
4
−2 Ng 2 : Tr[[Y, Z], Y̆ ][[Y̆ , Z̆], Y ] : +4 gN : Tr[Y, Z][Y̆ , Z̆] :
P
• The P
bulk hamiltonian is well known to be Hbulk = i (2g 2 − 8g 4 )(I − Pi,i+1 ) +
2g 4 (I − Pi,i+2). By truncating this operator we can obtain the boundary
contribution Hnaive = (2g 2 − 8g 4 )q1XB + 2g 4 q2XB . This is related to the fact that
operators can’t have a field of the same flavor of the determinant next to it to
leading order in N1 .
Giant magnons meet giant gravitons
27
5.2
Phases and tests
XBN −1| |{z}
X0 |{z}
X1 |{z}
X2 . . .
0
1
2
• There is, however, one more contribution coming from the dilatation operator
acting on the determinant. Each term is suppressed by N1 but there are N − 1
such terms. This corrects the previous hamiltonian by Hdet = 4g 4 q1B [DH,
Maldacena]
5.2.
Phases and tests
• Using these results (and one loop results for the SU(3) sector) we can check the
exact reflection matrix. This agrees with previous result at one loop by [Berenstein, Vazquez]
• We check the exsitence of a boundary degree of freedom for SU (2|2)2.
• Also, we can calculate the overall phase at two loops in perturbation theory.
• For the SU (1|2)2 theory this is R20L = −e−2ip
ip
1−2e
• For the SU (2|2)2 theory the result is more complicated R0 = − 1−2e
−ip +
−ip
ip
3
ip
ip
i2p
+e )
2g 2 e (e −1) (e(eip+1)(1−4e
. At two loops this is equivaent to
−2)2
Giant magnons meet giant gravitons
R20L
=
R0
A2
=−
5.3
28
Integrability
1
1
+
−
x− 2 (x− −xB ) (x + xB ) (x− +xB ) (x + xB )
x+
(x+ −xB ) (x+ − 1 ) (x+ +xB ) (x− − 1 )
xB
xB
• This expression could be correct to higher loops (the equivalent calculation in
the bulk is correct to four loops).
• Here we see an interesting pole structure that could be analized by Bootstrap
methods.
5.3.
Integrability
• Once we have the S-matrices we can construct multiparticle states using the
Bethe Ansatz technique.
• This was carried out in [Agarwal] and a problem was found. Extra restriction
had to be imposed.
P∞
• When we consider
the improved Hamiltonian H = (2g 2 − 8g 4 ) i=1 (I −
P∞
4
2
4 XB
4 XB
Pi,i+1) + 2g
i=1 (I − Pi,i+2 ) + (2g − 4g )q1 + 2g q2 , this construction
can be carried out explicitly [DH, Maldacena]. This matches nicely with the
strong coupling results of [Mann, Vazquez].
Giant magnons meet giant gravitons
6.1
29
6.
Strong coupling results
6.1.
Bulk results
Bulk results
• Classical correspondence between Sine-Gordon and strings on R × S 2.
• Sine-Gordon solitons ←→ String Theory Magnons.
√
• Nontrivial energy relation =
λ 1
.
π Esg
• Lorentz symmetry in Sine-Gordon responsible for one parameter solutions in
string theory. Again, this is not a symmetry of the theory.
• Scattering solutions in sine-gordon map to scattering of magnons.
• x and t are the same −→ time delays are the same (S-matrices are not though).
• WKB approximation gives phase as
∂δ12 (1 ,2 )
∂1
= ∆T12 [Jackiw, Woo]. This
agrees (modulo length redefinition) with the AFS formula, derived in an indirect
way.
∆T12 =
2
log vcm
γ1v1
(17)
Giant magnons meet giant gravitons
30
6.2
Boundaries
√
√ h
p1 −p2 i
1 − cos 2
p1
p2
p2
λ
λ
− cos + cos
−
p
sin
(18)
δ=
log
1
2
π
2
2
π
2
1 − cos p1+p
2
• Length redefinition. Far away from the excitation E and J have constant densities. But inside the excitation, our gauge says the have just constant E . To
compare with gauge theory we should go to coordinates where J is constant.
The mismatch will be given in the phase as eip∆Length . (Compare with BA equations).
Z
∆l =
6.2.
dJ
dx
=
dx
Z
dx
dE
dE dJ
2π
−(
−
) = √ ∆x − dx
dx
dx
λ
Boundaries
• Boundaries can be incorporated by the methof of images.
(19)
Giant magnons meet giant gravitons
31
6.2
Boundaries
• Different giant magnons orientations correspond to different boundary conditions. In our cases
Z = 0 ↔ Dirichlet φ = π2 → Boundary bound state.
Y = 0 ↔ Dirichlet φ = 0 or Neumann, depending on the polarization. Results
are the same, as matrix structure is subleading at strong coupling.
• Zero
mode quantization
yields the right energy for boundary bound states B =
p
√
1
(2n)2 + 16g 2 = n2 + 4g 2
2
• The crossing equation is satisfied in the classical limit δB,R (p) + δB,R (p̄) =
−δ(p, −p̄) + O(1)
7.
Summary and Conclusions
• We have extended the techniques from the bulk theory to boundary theories and
were able to obtain exact results for the reflection matrices. These results were
checked perturbatively.
• Crossing equations were developed for the overall phase factor.
• Two different theories appear, depending on the relative orientations of giant
gravitons and the BMN vacuum.
Giant magnons meet giant gravitons
32
6.2
Boundaries
• An interesting boundary bound state structure was found for one of these cases.
• Issues with integrability were resolved at two loops.
• We are closer to solving exactly (different sectors of) String Theory on AdS5 ×
S 5 ←→ N = 4 SYM.
• Do not underestimate what integrable theories can teach us!
8.
Extensions
• Theories with flavors and defects.
• Wilson loops (in progress).
• Non maximal giants. Integrability?
9.
Open Problems
• Fill the vacuum and connect reflection matrices. Lessons to be learned here.
• Finite Size. Wrapping interactions
• Deformations and other theories. Confinement? QCD!?