Wilson loops: from pseudo-holomorphic surfaces to 2d YM Riccardo - - PowerPoint PPT Presentation

Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Riccardo Ricci

Imperial College London

Cambridge, 12 Nov

Based on 0704.2237, 0707.2699, 0711.3226 and 0905.0665 with N. Drukker, S. Giombi, V. Pestun and D. Trancanelli

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Outline

Wilson loops in N = 4 SYM

Review of 1/2 BPS loops: straight line and circle Zarembo’s construction of supersymmetric Wilson loops

1/16 BPS Wilson loops on S3

General construction Examples

AdS string theory dual: Wilson loops as pseudoholomorphic surfaces Relation to 2d Yang-Mills (YM2)

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Introduction and motivation

In this talk I will discuss a new family of supersymmetric Wilson loop operators in N = 4 SYM theory

Field content: Aµ , ΦI , ΨA I = 1, . . . , 6 , A = 1, . . . , 4 Symmetry group: PSU(2, 2|4) ⊃ SO(5, 1) × SU(4)R + 32 SUSY’s (16Q + 16S)

This gauge theory is believed to be dual to type IIB string theory on AdS5 × S5. This is a strong/weak duality which is in general very hard to test directly. Interesting tests of the conjecture can be obtained by considering certain subsectors of the theory which preserve some fractions of the supersymmetries of the vacuum.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Introduction and motivation

Supersymmetric local operators are well studied both in the gauge theory and in AdS, but less is known about non-local

• perators. One motivation is then the classification of

supersymmetric non-local operators such as Wilson loops. W [C] ∼ Tr R P exp

• C

i A + · · · Wilson loops are described in AdS by strings (and D-branes): probe the duality beyond the strict supergravity limit. Certain supersymmetric Wilson loops may provide exact results interpolating between weak and strong coupling. Remarkable connection with scattering amplitudes

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Part I Field theory analysis

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Wilson loops in N = 4 SYM

In N = 4 SYM it is natural to define W = 1 N Tr R P exp

• ds(iAµ ˙

xµ(s) + |˙ x|ΘI(s)ΦI

• )

Coupling to N = 4 scalars I = 1, . . . , 6

• xµ(s): loop on R4
• ΘI(s): loop on S5,

Θ2 = 1 All such loops are locally supersymmetric δW ∝

• i ˙

xµγµ + |˙ x|ΘIρI

• ǫ0

ǫ0 : 16 comp. constant spinor

Squares to zero (because Θ2 = 1)

So the susy equation δW = 0 has solutions, but in general only point by point along the loop.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

We will be interested in operators which preserve fractions of the full 32 supersymmetries of N = 4 SYM globally. ǫ(x) = ǫ0+xµγµǫ1

• ǫ0 : 16 Q’s

ǫ1 : 16 S’s The susy variation can then be written as δW ∝

• i ˙

xµγµ + |˙ x|ΘIρI ǫ(x) the Wilson loop will be supersymmetric if we can find solutions for constant ǫ0,ǫ1 independent of the point along the loop. Problem: how to choose (xµ(s), ΘI(s)) so that W is supersymmetric?

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

1/2 BPS straight line

x2 x1

xµ = (s, 0, 0, 0) ΘI = (1, . . . , 0) Couples to a single scalar Φ1 It is easy to see that susy variation gives

• γ1 − iρ1

ǫ0 = 0 ,

• γ1 − iρ1

ǫ1 = 0 → Preserves 8 Q’s and 8 S’s, so it is 1/2 BPS It has trivial expectation value to all orders in λ, N W = 1 Checked both in perturbation theory and at strong coupling in AdS.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

1/2 BPS circle

x2 x1

xµ = (cos s, sin s, 0, 0) ΘI = (1, . . . , 0) Again it couples to only one of the scalars Now in the susy variation ǫ0 and ǫ1 are not decoupled ρ1ǫ0 = i γ12ǫ1 → It is also 1/2 BPS, but preserves 16 combinations of Q and S. The VEV is non trivial W = f (λ, N) and remarkably it is exactly computable!

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

1/2 BPS circle and matrix model

The VEV for the 1/2 BPS circle is fully captured by the Hermitean gaussian matrix model:

Erikson, Semenoff, Zarembo ‘00, Drukker-Gross ‘00

W = 1 Z

• Dφ 1

N Tr eφ exp

• −2N

λ Trφ2

• ,

λ = g2

YMN

Crucial observation: the combined gauge-scalar propagator between two points on the loop is a constant

iA + φ

=

• i Aa(x) + Φa(x)
• i Ab(y) + Φb(y)
• = g2

YM

8π2 δab → The sum of all non-interacting graphs (“ladder diagrams”) is equal to the matrix model! Interacting graphs must vanish.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

The matrix model can be exactly solved and yields the prediction W = 1 N L1

N−1

• −g2

YM

4

• e

g2 YM 8

At large N: W = 2 √ λ I1 √ λ

• ∼
• 1 + λ

8 + . . .

λ ≪ 1 e

√ λ

λ ≫ 1 Large λ limit agrees with string calculation in AdS e−SF1 = e

√ λ

Certain 1/N corrections were also successfully reproduced by D-branes (dual to Wilson loops in large representations).

Drukker, Fiol ’05

The MM/half-BPS loop relation can be proven using localization techniques.

Pestun ’07

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Zarembo’s supersymmetric Wilson loops

Can we find interesting generalizations of the 1/2 BPS Wilson loops? Given a curve xµ(s), how to choose the scalar couplings ΘI(s) so that some supersymmetry is preserved? Two possible directions: i Generalize the straight line loop

Zarembo ’02

ii Generalize the circular loop Given an arbitrary curve xµ(s) ∈ R4 and four scalars Φµ define W = 1 N Tr P exp

• ds(i ˙

xµAµ + ˙ xµΦµ) , Θµ = ˙ xµ/|˙ x| The loop dependence drops out in the supersymmetry variation (γµ − iρµ) ǫ0 = 0 4 indep. projectors → generically only 1 Q is preserved.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

An interesting fact is that the VEV for these loops is always trivial to all orders in λ,N W = 1 as can be checked both in perturbation theory and in AdS.

Dymarsky, Gubser, Guralnik and Maldacena ’06

Generalization of straight line loop as Wstraight line = 1 Unfortunately not much to compute with them! Clearly this family of operators does not contain the 1/2 BPS circle, which is a very interesting observable.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

The loops on S3

In Zarembo’s construction we select four of the scalars Φµ and couple them to the 1-forms dxµ. The construction of susy loops on S3 is similar, but the basic ingredient to define the scalar couplings are now the left invariant one forms of SU(2) = S3. Defining U = iτixi + I x4, these are σL = U†dU = σi ,L iτi 2 , i = 1, 2, 3 In cartesian coordinates, they may be explicitly written as σi ,L = 2σi

µν xµdxν,

xµxµ = 1 where σi

µν are constant matrices: σi jk = ǫijk,

σi

4j = δi j

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

The loops on S3

Given an arbitrary curve xµ(s) on a unit radius S3, we can couple the three one-forms σL

i to three of the scalars. Our

definition for susy Wilson loops on S3 is then W = 1 N Tr P exp i A + 1 2σL

i Φi

• ,

i = 1, 2, 3 It is a locally supersymmetric operator: Θi(s) ds = 1 2σL

i /|˙

x| → Θ2 = 1

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Supersymmetry

Now we show that our definition of Wilson loop on S3 leads to a supersymmetric operator: δW ∝

• i ˙

xµγµ − σi

µν ˙

xµxνρi (ǫ0 + xηγηǫ1) = 0 An important point that allows to solve this equation is that the left invariant one forms are related to the action of the Lorentz group on right (anti-chiral) spinors ǫ−: γµνǫ− = iσi

µντi ǫ−

Similarly we have γµνǫ+ = i ˜ σi

µντi ǫ+ ,

σR = UdU†

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Supersymmetry

A generic curve on S3 is supersymmetric if (τ i + ρi)ǫ−

0 = 0 ,

ǫ−

1 = −ǫ− 0 ,

ǫ+

1 = ǫ+ 0 = 0

• set chiral spinors to zero

for i = 1, 2, 3. This leaves 2 out of the 8 components of ǫ−

0 . Since ǫ− 1 is related to

ǫ−

0 our Wilson loops preserve 2 combinations of ¯

Q and ¯ S The loops are 1/16 BPS

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Examples: Great circle

We have constructed a new infinite family of supersymmetric Wilson loops. For an arbitrary curve on S3, they are 1/16

• BPS. But for special shapes one can have enhanced

supersymmetry. We recover the 1/2 BPS circle: it is a great circle of the S3!

S3

Loop: xµ = (cos s, sin s, 0, 0) Scalar couplings:

σL

1 = σL 2 = 0 , σL

3

2 = 1

Familiar 1/2 BPS circle coupled to a single scalar

As for the 1/2 BPS circle, all loops in this new family have non trivial VEVs. Are they exactly computable?

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Examples: Loops on S2

A very interesting large subfamily for which the VEV might be exactly calculable is given by taking the loop to be inside a maximal S2 ⊂ S3, i.e. an S2 of radius 1, which we may take to be at x4 = 0:

S2

From the definition of the invariant one forms, one can see that for a curve on S2 the left and right forms are related σL

i = −σR i = −ǫijkxj dxk ,

i, j, k = 1, 2, 3 Because of this fact, it turns out that now also positive chirality spinors are allowed, and the supersymmetry is

• doubled. All such loops are therefore 1/8 BPS.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Loops on S2: Latitude

Taking the loop on the equator of the S2 just gives the 1/2 BPS circle, since that is a great circle. But more generally we can take a latitude on S2

θ0

S2

This turns out to be a 1/4 BPS operator

This operator is conjectured to be captured by the same gaussian matrix model upon the replacement

Drukker ’06

λ → λ′ = λ sin2 θ0 The AdS string solution and a D3 brane solution are known and agree with the above conjecture. Drukker, Giombi, Ricci and

Trancanelli ’06

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Loops on S2: Longitudes

Another interesting example is given by taking two arcs of longitudes

δ

S2

Such a loop couples to Φ2 along the first arc and to −Φ2 cos δ + Φ1 sin δ along the second one. It is easy to see that this operator is 1/4 BPS.

This loop is related by a stereographic projection to a supersymmetric cusp on the plane. We can compute the VEV both in perturbation theory and at strong coupling W ≃

• 1 +

λ 8π2 δ(2π − δ) ,

λ ≪ 1 exp

• λ

π2 δ(2π − δ) ,

λ ≫ 1

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Loops on S2

We have seen some explicit examples of loops on a S2 But we can say more: we have an exact answer for all possible loops on S2 (connection with 2d-YM)

S2

In a little bit...

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Part II String theory duals

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Wilson loops in AdS/CFT

Wilson loop string world sheet

boundary AdS5

Given the Wilson loop operator W ∼ P exp

• ds(iAµ ˙

xµ(s) + |˙ x|ΘI(s)ΦI) , ΘIΘI = 1 The couplings ΘI “live” on the S5 of the AdS5 × S5 geometry. Find the minimal surface ending on the loop

• xµ(s), ΘI(s)
• Riccardo Ricci

Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Wilson loops as pseudoholomorphic surfaces

Our family of Wilson loops couples to 3 of the N = 4 scalars Φi: Θi = 1 2σi , ΘiΘi = 1 , i = 1, 2, 3 Therefore the dual string surfaces live on a AdS5 × S2 geometry with metric ds2 = 1 z2 dxµdxµ + z2dyidyi , z−2 ≡ yiyi If ”slice” this space with the constraint xµxµ + z2 = 1 we obtain a AdS4 × S2 space. This is where the dual string lives. The boundary of AdS4 at z = 0 is the S3 where the loop is defined.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

AdS4 × S2 turns out to be an almost complex manifold.

J = B B B B B B B @ z2 B B @ y3 −y2 −y1 −y3 y1 −y2 y2 −y1 −y3 y1 y2 y3 1 C C A z2 B B @ −x4 −x3 x2 x3 −x4 −x1 −x2 x1 −x4 x1 x2 x3 1 C C A z−2 @ x4 −x3 x2 −x1 x3 x4 −x1 −x2 −x2 x1 x4 −x3 1 A z2 @ −y3 y2 y3 −y1 −y2 y1 1 A 1 C C C C C C C A

Remarkably J is very similar to the known almost complex structure of S6. The reason for this “coincidence” is that AdS4 × S2 : xµxµ + z4yiyi = 1 S6 :

7

• k=1

xkxk = 1

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Pseudoholomorphic surfaces

Given the map X M : Σ → M , X M(σ, τ) with M being an almost complex manifold it is natural to consider the “pseudoholomorphicity” condition J M

N∂αX N − √g ǫαβ∂βX M = 0

It reduces to the usual Cauchy-Riemann equations Σ = M = R2 . = C , J = −1 1

• In our case case M = AdS4 × S2 and the dual string turns out

to be pseudoholomorphic with respect to J !

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

The solutions to the pseudoholomorphic equation have the correct behavior at the boundary yi ∼ σi , for z ∼ 0 so that the yi’s asymptote to the scalar couplings in the boundary gauge theory. The surfaces obeying this equation preserve the same supersymmetry as the operators in the gauge theory. The solutions to the pseudoholomorphic equations are automatically solutions to the σ-model equations.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Calibration

From the almost complex structure J M

N one can construct a

2-form JMN = GMPJ P

N, where GMN is the metric.

It is a general property that pseudoholomorphic surfaces are “calibrated” by J A(Σ) =

• Σ

J This quantity is generically divergent: A(Σ) = AReg(Σ) + Div. and the regularized Wilson loop VEV is then given by W = exp

• −

√ λ 2π AReg(Σ)

• .

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

In our case the 2-form may be written as J = J0 + dΩ and the exact one-form dΩ precisely removes the divergence. The strong-coupling prediction is then W = exp

• −

√ λ 2π

• Σ

J0

• with

Areg(Σ) =

• Σ

J0 = 1 2

• Σ

d2σ√g

• ∂αθi ∂αθi + ∇2z

z

• Riccardo Ricci

Wilson loops: from pseudo-holomorphic surfaces to 2d YM

A feature of the 2-form J is that it is not closed: dJ = 0 This implies that a surface calibrated by J is not necessarily

• minimal. This is related to the fact that the loops can wrap the S2
• f AdS4 × S2 geometry in two different ways:

S2

Θi(σ, τ)Θi(σ, τ) = 1 One of the two possible wrappings will have minimal area. The non-minimal area solution contribution to the Wilson loop VEV is subleading: W ∼ e

√ λ A1 + e− √ λ A2

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Part III Connection with 2d Yang-Mills

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Loops on S2 and the 2d Yang-Mills Conjecture

A particularly interesting subclass is obtained by restricting the Wilson loops in gauge theory to live on an S2 sphere: x4 = 0; xixi = 1, i = 1, 2, 3 The Wilson loop operator becomes W = P exp

• ds(i Ai ˙

xi + ( x × ˙

• x) ·

Φ) The corresponding string solutions are 1/8-BPS and live in AdS3 × S2 (instead of AdS4 × S2).

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

A tantalizing conjecture is that all loops on S2 are related to the analogous observables in 2d bosonic YM on S2 W N=4 = W YM2 where W N=4 ∼ Tr P exp

• C

(i A4d + scalar-couplings) W YM2 ∼ Tr P exp

• C

i A2d and the couplings of the two theories are related as follows g2

2d = −g2 YM

A

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

The basic insight comes by looking at the combined gauge-scalar propagator in perturbation theory ∆ab

µν(x, y) =

iA + 1/2 σ . φ

W = 1 − 1 2N Tr (T aT b)

• dxµ dyν ∆ab

µν(x, y) + . . .

Explicitly the effective propagator reads ∆ab

µν(x, y) = g2 YMδab

4π2 1 2δµν − (x − y)µ(x − y)ν (x − y)2

• Riccardo Ricci

Wilson loops: from pseudo-holomorphic surfaces to 2d YM

2d Yang-Mills Conjecture

Despite its origin from 4d fields, ∆ab

µν(x, y) turns out to be a

propagator for a gauge field on S2 (in a certain generalized Feynman gauge), provided the 2d and 4d couplings are related as g2

2d = −g2 YM

A This may lead to conjecture that for these loops the perturbative sum in N = 4 and YM2 is the same (non trivial to prove because of interacting graphs).

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Exact solution from YM2

YM2 is an exactly solvable theory in particular one has exact results for arbitrary Wilson loops on S2. Migdal ’75, Rusakov ’90 These exact results are non-perturbative and they can be written as a sum over instanton sectors. To compare to the perturbative sum in N = 4 SYM, we extract the contribution

• f the “zero-instanton” sector. Bassetto-Griguolo ‘98

W (A1, A2)pert. YM2 = 1 N L1

N−1

• −g 2

YM

A1A2 A2

• exp

g 2

YM

2 A1A2 A2

• A1

A2

S2

Usual gaussian matrix model result up to the simple rescaling g 2

YM → g 2 YM 4A1A2 A2

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

So we can rephrase the conjecture as follows W (A1, A2)N=4 = W (A1, A2)pert. YM2 = 1 N Treφ

• M.M.

with 1 N Treφ

• M.M.

= 1 Z

• Dφ 1

N Treφ exp

• −2N

λ′ Trφ2

• λ′ = 4A1A2

A2 λ

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Explicit checks

It is simple to verify that to leading order in λ the above matrix model reproduces the expected result W = 1 + λ 4π A1A2 A + O(λ2) Matching the higher orders is much harder, but all known examples agree, both in perturbation and in string theory.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Explicit confirmations at higher orders:

Bassetto et al. ’08, Young ’08, Bassetto et al. ’09

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Invariance under area preserving diffeomorphisms

The YM2 conjecture implies that the sector of N = 4 SYM under consideration should be invariant under area-preserving

• diffeomorphisms. This is an infinite dimensional symmetry.

It would be nice to explicitly show that the 1/8-BPS loops on S2 enjoy this non-trivial invariance. Unfortunately it is hard to prove this conjecture directly from the gauge theory side. Do we see this infinite dimensional symmetry in the dual string picture?

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Area preserving diffeos at strong coupling

Let us consider an infinitesimal deformation of the loop

XM(τ,σ) XM(τ,σ) + δXM(τ,σ) C C + δC

σ=0

τ

We are interested in the difference of the on-shell action for these infinitesimally close solutions S[X + δX] − S[X] as a function of the loop deformation δC

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

The variation of the Polyakov action S = 1 2

• d2σ∂αX M∂αX MGMN

is a boundary term δS = −

• ∂Σ

dτδX M∂σX MGMN =

• ∂Σ

dτδX M∂τX MJMN Explicitly we have δS = −

• ∂Σ

dτ

• δxi∂τ(xjyk) + δyi∂τxjxk

ǫijk We only need to know the behaviour of the solution close to the boundary.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

The net result is rather simple δS = −CδAloop It is the variation of the area Aloop enclosed by the loop. Therefore we have shown that the on-shell action is invariant under loop transformations which preserve the area Aloop. Crucial to this result was the use of the pseudo-holomorphic equation for 1/8 BPS loops.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Because of invariance under area-preserving diffeomorphisms we can compute the action of any 1/8 BPS string solution by simply comparing to a known solution with same loop area. We can deduce the general result (A1 = Aloop) Sreg = − √ λ 2π

• A1A2

in agreement with the YM2 prediction.

A1 A2

S2

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Correlators

It is also possible to consider correlators of Wilson loop operators, W (C1)W (C2) In the string side the correlator corresponds to a worldsheet with topology of a cylinder and which ends on the loops C1 and C2. A concrete and simple example is the correlator between two latitude loops

S2

x3=cosθ1 x3=cosθ2

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

If we use an ansatz with circular symmetry in the string equations

• ne quickly ends up with the condition

x3 = const. But x3 should vary continuously between the two loops. As a consequence no cylindrical solution exists! What else?

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

The only possible topology is a “disconnected” configuration, i.e. a cylinder made up by two disks joined by a degenerate cylinder of zero area. Physically the tiny tube corresponds to the exchange of supergravity modes between the two disks

C1 C2

The strong coupling prediction for the correlator is therefore W (C1)W (C2) ∼ exp  √ λ sin θ1

• 1stdisk

+ √ λ sin θ2

• 2nddisk

 

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

This degeneration to disconnected configurations is a general feature for all loops on S2. Can we explain this behavior from the 2d-YM side (Matrix Model)?

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Correlators from a double Matrix-Model

In the case of a correlator of two loops, the YM2 conjecture leads to the following 2MM model computation 1 Z

• Dφ1Dφ2 treφ1 treφ2e

−

A 2g2 YM tr

“

1 A1 φ2 1+ 1 A12 (φ1−φ2)2+ 1 A2 φ2 2

”

where A1 + A12 + A2 = A

S2

A1 A12 A2

A correlator of k loops corresponds to a k-matrix model.

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

At large N the matrix model yields W (C1)W (C2) =

∞

• n=1

nρnIn

• λ′

1

• In
• λ′

2

• with

λ′

1 = λ4A1(A − A1)

A2 , λ′

2 = λ4A2(A − A2)

A2 , ρ = ρ(A1, A2, A12) By taking the strong coupling limit λ ≫ 1 we obtain W (C1)W (C2) ∼ exp(

• λ′

1 +

• λ′

2) × subleading

The exponential saddle point agrees with the disconnected disks picture emerging from string theory (for the correlator of two latitudes: λ′

i = λ sin2 θi)!

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Conclusions

New family of Wilson loops on preserving between 1/16 and 1/2 of the supercharges The well known 1/2 BPS circle is included All these operators have non trivial VEV. The strings dual to these operators are pseudoholomorphic surfaces The dynamics of N = 4 loops on S2 seems to be captured by YM2 Remarkable dynamical reduction : 4D → 2D

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM

Open problems

i Can we give a direct proof of the YM2 conjecture? Pestun ’09 ii Test the YM2 conjecture by studying sub-leading corrections in string theory (one loop fluctuactions). iii Can we exactly compute the VEV for generic loops on S3?

Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM