Non-Planar AdS/CFT Integrability A Progress Reportbargheer/slides/2019-02-01_berlin-seminar.pdfNon-Planar AdS/CFT Integrability – A Progress Report TILL BARGHEER Leibniz Universität

Non-Planar AdS/CFTIntegrability – A Progress Report

TILL BARGHEER

Leibniz Universität Hannover & DESY Hamburg

1711.05326: TB, J. Caetano, T. Fleury, S. Komatsu, P. Vieira1809.09145: TB, J. Caetano, T. Fleury, S. Komatsu, P. Vieira19xx.xxxxx: TB, F. Coronado, P. Vieira19xx.xxxxx: TB, F. Coronado, V. Gonçalves, P. Vieira

+ further work in progress

HUMBOLDT UNIVERSITY BERLIN SEMINAR

BERLIN, FEBRUARY 2019

http://arxiv.org/abs/1711.05326


http://arxiv.org/abs/19xx.xxxxx

http://arxiv.org/abs/19xx.xxxxx

Invitation

String Theory: String amplitudes are integrals over the moduli space ofRiemann surfaces of various genus.Large-Nc Gauge Theory: Correlation functions are sums over double-lineFeynman (ribbon) graphs of various genus.AdS/CFT: These two quantities/concepts should be the same.Question: How does the continuous worldsheet moduli space integrationemerge from the discrete sum over Feynman graphs? Answering thisquestions is crucial for understanding the nature of holography.

This Talk: Provide one concrete realization. [TB, Caetano, FleuryKomatsu, Vieira ’17][TB, Caetano, Fleury

Komatsu, Vieira ’18]Initial motivation: Compute non-planar corrections to correlators usinghexagon form factors, building on planar methods/results. [Basso,Komatsu

Vieira ’15 ][Fleury ’16Komatsu ]

Along the way understood that the necessary sum over worldsheettessellations quantizes the string moduli space integration.In this respect, a finite-coupling extension of [Gopakumar

’03 ’04 ’04 ’05][Aharony, KomargodskiRazamat ’06 ]

ideas by Gopakumar, Razamat et al. [Aharony, David, GopakumarKomargodski, Razamat ’07 ][Razamat

’08 ]

Till Bargheer — Non-Planar Integrability — HU-Berlin Seminar — 1 February 2019 1 / 33





http://arxiv.org/abs/hep-th/0308184







Invitation

String Theory: String amplitudes are integrals over the moduli space ofRiemann surfaces of various genus.Large-Nc Gauge Theory: Correlation functions are sums over double-lineFeynman (ribbon) graphs of various genus.AdS/CFT: These two quantities/concepts should be the same.Question: How does the continuous worldsheet moduli space integrationemerge from the discrete sum over Feynman graphs? Answering thisquestions is crucial for understanding the nature of holography.

This Talk: Provide one concrete realization. [TB, Caetano, FleuryKomatsu, Vieira ’17][TB, Caetano, Fleury

Komatsu, Vieira ’18]Initial motivation: Compute non-planar corrections to correlators usinghexagon form factors, building on planar methods/results. [Basso,Komatsu

Vieira ’15 ][Fleury ’16Komatsu ]

Along the way understood that the necessary sum over worldsheettessellations quantizes the string moduli space integration.In this respect, a finite-coupling extension of [Gopakumar

’03 ’04 ’04 ’05][Aharony, KomargodskiRazamat ’06 ]

ideas by Gopakumar, Razamat et al. [Aharony, David, GopakumarKomargodski, Razamat ’07 ][Razamat

’08 ]













Illustration

Four strings scattering: Moduli space↔ Strebel graphs. Discretization.


Planar Limit & Genus Expansion

Gauge theory with adjoint matter in the large Nc limit: [’t Hooft1974 ]

I Feynman diagrams are double-line diagrams.I All color lines (propagators and traces) form closed oriented loops.I Can unambiguously assign an oriented disk (face) to each loop.I Obtain a compact oriented surface (operators form boundaries).I The genus of the surface defines the genus of the diagram.

Correlators of single-trace operators:Count powers of Nc and g2

YM for propagators (∼g2YM), vertices (∼1/g2

YM),and faces (∼Nc), define λ = g2

YMNc, use Euler formula:

〈O1 . . .On〉 =1

Nn−2c

∞

∑g=0

1

N2gc

G(g)n (λ) Oi = Tr(Φ1Φ2 . . . )

∼ 1N2

c+

1N4

c+

1N6

c+ . . .


Proposal

Concrete and explicit realization [TB, Caetano, FleuryKomatsu, Vieira ’17][TB, Caetano, Fleury

Komatsu, Vieira ’18]of the general genus expansion:

〈Q1 . . . Qn〉 =∏n

i=1√

ki

Nn−2c

S ◦ ∑Γ∈Γ

1

N2g(Γ)c

×

×

∏b∈b(Γ4)

d`bb

∫Mb

dψbW(ψb)

2n+4g(Γ)−4

∏a=1

Ha .

Remarkable fact:For N = 4 SYM, all ingredients of the formula are well-defined andexplicitly known as functions of the ’t Hooft coupling λ.

This talk:I Explain all ingredients of the formula.I Demonstrate match with known data.I Show (moderate) predictions.




Proposal: Ingredients I

〈Q1 . . . Qn〉 =∏n

i=1√

ki

Nn−2c

S ◦ ∑Γ∈Γ

1

N2g(Γ)c

×

×

∏b∈b(Γ4)

d`bb

∫Mb

dψbW(ψb)

2n+4g(Γ)−4

∏a=1

Ha .

Half-BPS operators: Qi = Q(αi, xi, ki) = Tr[(αi ·Φ(xi))

ki]

, α2i = 0 .

Internal polarizations αi, positions xi, weights (charges) ki.Set of all Wick contractions Γ ∈ Γ of the free theory, genus g(Γ).Promote each Γ to a triangulation Γ4 of the surface in two steps:I Collect homotopically equivalent lines in “bridges”→ skeleton graph.

The number of lines in a bridge b is the bridge length (width) `b.I All faces are triangles or higher polygons.

Subdivide all faces into triangles by inserting zero-length `b = 0 bridges.Set of all bridges: b(Γ4).

On each bridge: Propagator factor d`bb .


Proposal: Ingredients II

〈Q1 . . . Qn〉 =∏n

i=1√

ki

Nn−2c

S ◦ ∑Γ∈Γ

1

N2g(Γ)c

×

×

∏b∈b(Γ4)

d`bb

∫Mb

dψbW(ψb)

2n+4g(Γ)−4

∏a=1

Ha .

On each bridge b ∈ b(Γ4):Sum/integrate over states ψb ∈Mb of the mirror theory on the bridge b,with a kinematical weight factorW(ψb) that depends on the cross ratiosdefined by the surrounding four operators.

By Euler, the triangulation Γ4 contains 2n + 4g(Γ)− 4 faces.For each face a, insert one hexagon form factorHa: Accounts forinteractions among three operators and three mirror states ψb.

Finally, S : Stratification. Sum over graphs quantizes the integration overthe string moduli space. S accounts for contributions at the boundaries.


The Sum over Mirror States

On each bridge lives a mirror theory, which is obtained from the physicalworldsheet theory by an analytic continuation, a double-Wick (90 degree)rotation (σ, τ)→ (iτ, iσ) that exchanges space and time:

τ

R

σL

eHL

−→

σ R

τL

eHR

In all computations, the volume R can be treated as infinite.⇒Mirror states are free multi-magnon Bethe states, characterized by

rapidities ui, bound state indices ai, and flavor indices (Ai, Ai).

The mirror integration therefore expands to∫Mb

dψb =∞

∑m=0

m

∏i=1

∞

∑ai=1

∑Ai ,Ai

∫ ∞

ui=−∞dui µai (ui) e−Eai (ui) `b .

µai : measure factor, E: mirror energy, `b: length of bridge b (discrete “time”).


The Hexagon Form FactorsHexagon = Amplitude that measures the overlapbetween three mirror and three physical off-shellBethe states. Worldsheet branching operator thatcreates an excess angle of π. [Basso,Komatsu

Vieira ’15 ]

Explicitly: H(χA1 χA1 χA2 χA2 . . . χAn χAn)

= (−1)F(

∏i<j

hij

)〈χA1 χA2 . . . χAn |S|χAn . . . χA2 χA1〉

I χA, χA: Left/Right su(2|2) fundamental magnonsI F: Fermion number operatorI S: Beisert S-matrix

I hij =x−i − x−jx−i − x+j

x+j − 1/x−ix+2 − 1/x+1

1σij

,x±(u) = x(u± i

2 ) , ug = x + 1

xσij: BES dressing phase

Example:Two magnons( , )

⊗ =

SS



Frames & Weight FactorsHexagon depends on positions xi and polarizations αi of the three half-BPSoperators Oi = Tr[(αi ·Φ(xi))

k]. These preserve a diagonal su(2|2) thatdefines the state basis and S-matrix of excitations on the hexagon.Two neighboring hexagons share two operators, but the third/fourthoperator may not be identical. ⇒ The two hexagon frames are misaligned.

In order to consistently sum over mirror states, need to align the two framesby a PSU(2, 2|4) transformation g that maps O3 onto O2:

1 2

3 4H1H2 eiφL

e−D log |z|O1 O3 O4

O2

0 1 ∞

(z, z) g = e−D log |z|eiφL··eJ log |α|eiθR ,

e2iφ = z/z ,e2iθ = α/α .

HexagonH1 = H is canonical, andH2 = g−1Hg.

Sum over states in mirror channel: [Fleury ’16Komatsu ]

∑ψ

µ(ψ)〈H2|ψ〉〈ψ|H1〉 = ∑ψ

µ(ψ)〈g−1H|ψ〉〈ψ|g|ψ〉〈ψ|H〉

Weight factor: W(ψ) = 〈ψ|g|ψ〉 = e−2i pψ log |z|eJψ ϕeiφLψ eiθRψ , i p = (D− J)/2.

→ Contains all non-trivial dependence on cross ratios z, z and α, α.Till Bargheer — Non-Planar Integrability — HU-Berlin Seminar — 1 February 2019 9 / 33


Some Remarks

〈Q1 . . . Qn〉 =∏n

i=1√

ki

Nn−2c

S ◦ ∑Γ∈Γ

1

N2g(Γ)c

×

×

∏b∈b(Γ4)

d`bb

∫Mb

dψbW(ψb)

2n+4g(Γ)−4

∏a=1

Ha .

Separates sum over graphs and topologies from λ dependence:At given genus, can construct sum over graphs once and for all.

Should in principle hold at any value of the coupling λ.

Still a sum over infinitely many mirror contributions that cannot beevaluated in general. But may admit high-loop or even exact expansions inspecific limits.

Stratification operator S looks innocent, but is in fact non-trivial.


Known Non-Planar DataHalf-BPS operators: First non-trivial correlator: Four-point function.

Qki ≡ Tr

[(αi ·Φ(xi))

k] , Φ = (φ1, . . . , φ6) , α2i = 0 .

Specialize to equal weights k1 . . . k4 = k,and to specific polarizations αi with α1 · α4 = α2 · α3 = 0.Possible propagator structures:

X ≡ α1 · α2 α3 · α4

x212x2

34=

1 2

3 4, Y ≡

1 2

3 4,

(Z ≡

1 2

3 4

).

Correlators for general Nc: [ ArutyunovSokatchev ’03][ Arutyunov,Penati ’03

Santambrogio,Sokatchev]

Gk ≡ 〈Qk1Qk

2Qk3Qk

4〉loops = Rk−2

∑m=0Fk,m XmYk−2−m

Supersymmetry factor: R = zzX2 − (z + z)XY + Y2

Main data: Coefficients Fk,m = Fk,m(λ, Nc; z, z)




The Data: CoefficientsF (1),U

k,m (z, z) =

− 2k2

N2c

{1 +

1N2

c

[[ 176 r4 − 7

4 r2 + 1132

]k4 +

[ 92 r2 − 13

8

]k3 +

[ 16 r2 + 15

8

]k2 − 1

2 k]}

F(1) ,

F (2),Uk,m (z, z) =

4k2

N2c

[{1 +

1N2

c

[[ 176 r4 − 7

4 r2 + 1132

]k4 +

[ 92 r2 − 13

8

]k3 +

[ 16 r2 + 15

8

]k2 − 1

2 k]}

F(2)

+

{t4+

1N2

c

[([ 72 r2 − 1

8

]k2 + 5

8 k− 14

)s+ − r

([ 176 r2 − 7

8

]k3 + 3k2 − 13

12 k)

s−

+([ 29

24 r4 − 1116 r2 + 15

128

]k4 +

[ 178 r2 − 21

32

]k3 −

[ 2324 r2 − 39

32

]k2 − 9

8 k + 12

)t]}(

F(1))2

− 1N2

c

[r{[ 7

6 r2 − 18

]k3 + 3

2 k2 + 103 k}

F(2)C,− +

{[ 54 r2 − 19

48

]k3 +

[ 32 r2 + 7

8

]k2 + 1

3 k}

F(2)C,+

]+

14

{1 +

(k− 1)(k3 + 3k2 − 46k + 36)12N2

c

}(sδm,0 + δm,k−2

)(F(1))2

+

{1 +

(k− 2)4

12N2c

}(δm,0F(2)

z−1 + δm,k−2F(2)1−z)]

,

where r = (m + 1)/k− 1/2. Fk,m: Coefficient of XmYk−2−m.Till Bargheer — Non-Planar Integrability — HU-Berlin Seminar — 1 February 2019 12 / 33

First Test: Genus One, Large ChargesFocus on leading order in large charges (large k)→ several simplifications:Data, sum over graphs, and loop expansion (mirror states) all simplify.

Data: F (1),Uk,m (z, z) = − 2k2

N2c

{1 +

1N2

c

[[ 176 r4 − 7

4 r2 + 1132

]k4 +O(k3)

]}t F(1) ,

F (2),Uk,m (z, z) =

4k2

N2c

[{1 +

1N2

c

[[ 176 r4 − 7

4 r2 + 1132

]k4 +O(k3)

]}t F(2)

+

{1 +

1N2

c

[[ 296 r4 − 11

4 r2 + 1532

]k4 +O(k3)

]}t2

4(

F(1))2]

,

where r = (m + 1)/k− 1/2. Fk,m: Coefficient of XmYk−2−m.

Step 1: Sum over propagator graphs: Split in two steps:I Sum over torus “skeleton graphs” with non-parallel edges (≡ bridges).I Sum over distributions of parallel propagators on bridges.

= −→A B

DC


First Test: Large Charges: GraphsLarge k: Combinatorics of distributing propagators on bridges:Two operators typically connected by j > 1 bridges.Sum over distributions of m propagators on j bridges→ mj−1/(j− 1)!⇒ Only graphs with a maximum number of bridges contribute.Genus-one four-point graphs with the maximal number of 8 bridges:

B G L M P QSum over labelings:

Case Inequivalent Labelings (clockwise) Combinatorial Factor

B (1, 2, 4, 3), (2, 1, 3, 4), (3, 4, 2, 1), (4, 3, 1, 2) m3(k−m)/6B (1, 3, 4, 2), (3, 1, 2, 4), (2, 4, 3, 1), (4, 2, 1, 3) m(k−m)3/6G (1, 2, 4, 3), (3, 4, 2, 1) m4/24G (1, 3, 4, 2), (2, 4, 3, 1) (k−m)4/24L (1, 2, 4, 3), (3, 4, 2, 1), (2, 1, 3, 4), (4, 3, 1, 2) m2/2 · (k−m)2/2M (1, 2, 4, 3), (2, 1, 3, 4), (1, 3, 4, 2), (3, 1, 2, 4) m2(k−m)2/2P (1, 2, 4, 3) m2(k−m)2/2Q (1, 2, 4, 3) m2(k−m)2


First Test: Large Charges: Hexagons

Graphs:

B G L M P Q

All graphs consist of only octagons!Split each octagon into two hexagons with azero-length bridge.

1 2

3 4H1H2

Example:

(a)(b)

(c)

(d)

G

−→


First Test: Large Charges: Mirror Particles

Loop Counting:Expand mirror propagation µ(u) e−`E(u) and hexagonsH in coupling g.→ n particles on bridge of size `: O(g2(n`+n2))All graphs consist of octagons framed by parametrically large bridges.→ Only excitations on zero-length bridges inside octagons survive.

Excited Octagons:n particles on a zero-length bridge→O(g2n2

)→ Octagons with 1/2/3/4 particles start at 1/4/9/16 loops.

Octagon 1–2–4–3 with 1 particle: [Fleury ’16Komatsu ][TB,Caetano,Fleury

Komatsu,Vieira ’18]

M(z, α) =[z + z−

(α + α

)αα + zz2αα

]·(

g2F(1)(z)− 2g4F(2)(z) + 3g6F(3)(z) + . . .)

For Z = 0: R-charge cross ratiosα = zz X/Y and α = 1.

1 2

3 4




First Test: Large Charges: Match & Prediction

We are Done:Sum over graph topologies and labelings (with bridge sum factors),Sum over one-particle excitations of all octagons.⇒ Result matches data and produces prediction for higher loops!

Summing all octagons gives:

FUk,m(z, z)

∣∣torus = −

2k6

N4c

{g2[ 17

6 r4 − 74 r2 + 11

32]t F(1) X match

− 2g4[[ 17

6 r4 − 74 r2 + 11

32]t F(2) +

[ 296 r4 − 11

4 r2 + 1532] t2

4(

F(1))2]

X match

+ 3g6[[ 17r4

6 −7r2

4 + 1132]t F(3) +

[ 29r4

18 −11r2

12 + 532]t2 F(2)F(1) +

[ (1−4r2)2

96](

F(1))3]

+O(g8) +O(1/k)}

. prediction!

In fact, the octagon can be evaluated to much higherloop orders, is a polynomial in ladder integrals. [Coronado

2018 ][Coronado2018 ]

⇒ Immediate high-loop prediction for the four-point function.




More Tests: Finite ChargesFinite k: Need to include all four-point graphs on the torus.

Complete list of “maximal” graphs:

All other graphs can be obtained from maximal graphs by deleting bridges.Number of genus-one graphs by number of bridges:

#bridges: 12 11 10 9 8 7 6 5 ≤4 any#graphs: 7 28 117 254 323 222 79 11 0 1041

Even for our Z = 0 correlators, graphs with Z propagators need to beincluded: Mirror contributions depend on R-charges and can effectivelycancel Z-propagators contained in free-theory graphs.


Finite-Charge Tests

Small and finite k:Few propagators→ Fewer bridges→ Graphs with fewer edges⇒ Graphs composed of not only octagons, but bigger polygons

Example: Graphs for k = 3:

Hexagonalization:Each 2n-gon: Split into n− 2 hexagons by n− 3 zero-length briges.

Loop Expansion: Much more complicated!All kinds of excitation patterns already at low loop orders

I Single particles on several adjacent zero-length (or ` = 1) bridgesI Strings of excitations wrapping around operators


Finite k: One Loop: Sum over ZLB-Strings

Restrict to one loop: Only single particles on one or more adjacentzero-length bridges contribute.⇒ Excitations confined to single polygons bounded by propagators.

For each polygon: Sum over all possible one-loop strings:

12

34

5

6(1 loop) = + +

+ + +

One-strings: understood X [Fleury ’16Komatsu ]

Two-strings: understood X [Fleury ’17Komatsu ]

Longer strings: need to compute!




One-String and Two-String

One-String: Can be written as

M(1)(z, α) = m(z) + m(z−1) ,

with building block

1 2

3 4

[Fleury ’16Komatsu ]

m(z) = m(z, α) = g2 (z + z)− (α + α)

2F(1)(z, z)

Two-string: Despite complicated computation, simplifies to [Fleury ’17Komatsu ]

M(2)(z1, z2, α1, α2)

= m(

z1 − 1z1z2

)+ m

(1− z1 + z1z2

z2

)+ m

(z1(1− z2)

)−m(z1)−m(z−1

2 ) ,

with the same building block m(z)!

1

2

3 4

5




Finite Charges: Larger StringsLarger strings: Computation will be even more complicated!But: Can in fact bootstrap all of them by using flip invariance!

12

34

5

6(1 loop) = + +

+ + +

= + +

+ + +

Apply recursively:I 3-string '

1-strings & 2-stringsI . . . iterate . . .I n-string '

1-strings & 2-strings

⇒ Can write all polygonsin terms of only1-strings & 2-strings.

⇒ All n-strings can be written as linear combinationsof one-string building blocks m(z).


Finite k: General Polygons at One LoopPolygon with 2n edges:Sum over all strings inside the polygon greatly simplifies to:

P (1)2n = ∑

{j,k} non-consecutive

m

(zjk ≡

x2j,k+1x2

j+1,k

x2jkx2

j+1,k+1

)

→ Sum over m(z) evaluated in each subsquare:

1 loop =m

+ m + m + m

+ m + m + m + m +m

Recall the one-loop building block:

m(z) = g2 (z + z)− (α + α)

2F(1)(z, z)


Finite k, One Loop: Result

Done! Sum over all graphs, expand all polygons to their one-loop values.

Numbers of labeledgraphs with assignedbridge sizes:

k: 2 3 4 5

g = 0: 3 8 15 24g = 1: 0 32 441 2760

Data:

F (1),Uk,m (z, z) =

− 2k2

N2c

{1 +

1N2

c

[[ 176 r4 − 7

4 r2 + 1132]k4 +

[ 92 r2 − 13

8]k3 +

[ 16 r2 + 15

8]k2 − 1

2 k]}

F(1) ,

where r = (m + 1)/k− 1/2. Fk,m: Coefficient of XmYk−2−m.

Result: For k = 2, 3, 4, 5, . . . :Matches the U(Nc) data Fk,m, up to a copy of the planar term!

Fk,m : Result = (torus dataXXX

) +1

N2c(planar data

? ? ?)

⇒ Stratification.


Resolution of Mismatch: Stratification

We have based the computation on a sum over genus-one graphs of the freetheory that cover all cycles of the torus.We therefore miss contributions from purely virtual handles. In thelanguage of hexagons, these come from graphs where a handle of the torusis traversed only by zero-length bridges (no propagators).Resolution: Include graphs that are by themselves planar, but drawn on thetorus, and fully tessellate the torus by zero-length bridges (as before).

New Problem: This adds the missing contributions (mirror states traversinga handle that contains no propagators). But it also adds many genuinelyplanar (and therefore unwanted) contributions.Get rid of these unwanted contributions by subtracting the same graphs,but now drawn on a degenerate torus where the empty handle has beenpinched, such that the torus becomes a sphere with two marked points.

This goes under the name of stratification (S in our formula).


Resolution of Mismatch: Stratification

We have based the computation on a sum over genus-one graphs of the freetheory that cover all cycles of the torus.We therefore miss contributions from purely virtual handles. In thelanguage of hexagons, these come from graphs where a handle of the torusis traversed only by zero-length bridges (no propagators).Resolution: Include graphs that are by themselves planar, but drawn on thetorus, and fully tessellate the torus by zero-length bridges (as before).

New Problem: This adds the missing contributions (mirror states traversinga handle that contains no propagators). But it also adds many genuinelyplanar (and therefore unwanted) contributions.Get rid of these unwanted contributions by subtracting the same graphs,but now drawn on a degenerate torus where the empty handle has beenpinched, such that the torus becomes a sphere with two marked points.

This goes under the name of stratification (S in our formula).


Stratification: Examples

→

(2) (2′)

→

(4) (4′)


Stratification & Moduli Space

Stratification is also natural from the string theory point of view:The sum over graphs discretizes the integration over the moduli space ofworldsheet Riemann surfaces.

The moduli space includes boundaries. In continuous integrations, theseboundaries are measure-zero sets and hence do not contribute. But in adiscretized sum, it matters which terms are included or dropped.

Moduli space discretizations have been considered before in the context ofmatrix models, and the right treatment of boundary contributions in theknown cases is in line with the above prescription (stratification). [Chekhov

1995 ]



Stratification: Degeneration Type I

(a)

(b) (c)


Stratification: Degeneration Type II

(a)

(b) (c)


Stratification: Final Formula

At higher genus, simple degenerations subtract terms multiple times→ need to be compensated by adding double degenerations etc.→ alternating sum.Also need to account for disconnected degenerations. Final result:

S ◦ ∑Γ∈Γ≡

∞

∑g=0

2g+n−2

∑c=1

∑τ∈τg,c,n

(−1)∑i mi/2 ∑Γ∈Στ

.

c: Number of components of the surfaceτ: Genus-g topology with c components and n punctures:

τg,c,n ={{(g1, n1, m1) . . . (gc, nc, mc)}

∣∣∑i

ni = n, ∑i(gi +

mi2 )− c + 1 = g

}where (gi, ni, mi) labels the genus, the number of punctures, and thenumber of marked points on component i.Στ : Set of all graphs Γ (connected and disconnected) that are compatiblewith the topology τ and that are embedded in the surface defined by τ in allinequivalent possible ways (Γ may cover all or only some components ofthe surface).


Dehn Twists and Modular Group

We implicitly identified graphs thatonly differ by “twists” of a handle: '

This makes sense at weak coupling: Identity at the level of Feynman graphs.

Also makes sense from string moduli spaceperspective: Dehn twists are modulartransformations that leave the complexstructure invariant.

Modding out by Dehn twists has non-trivial implications for the summationover mirror states, especially for stratification terms:

Dehn twists along cycles not covered by thepropagator graph act trivially in the absenceof mirror particles:

Once the cycle is dressed with zero-length bridges and mirror particles,Dehn twists will non-trivially map sets of mirror magnons onto each other.→ Need to mod out by this non-trivial action!


Stratification: EvaluationStratification graphs: Cycles that are traversed only by zero-length bridges.→ Infinitely many mirror contributions already at one loop!

Example:1 2

3 4+

1 2

3 4+

1 2

3 4+

1 2

3 4+ . . .

Presently: Cannot evaluate strings of magnons that wrap a cycle,or cross any edge more than once.

However, reasonable to assume that almost all such contributions will eithercancel agains stratification subtractions, or be projected out by Dehn twists.

Use (partly heuristic) simple rules: Drop configurations with closed loops;identify one-loop strings that are “superficially” related by Dehn twists.

All remaining contributions can be honestly computed.

Including stratification indeed gives the missing (planar)/N2c term!

⇒ Now have a perfect match for k = 2, 3, 4, 5!


Summary & OutlookSummary: Method to compute higher-genus terms in 1/Nc expansion.

I Sum over free graphs, decompose into planar hexagons.I Infinite sum over mirror excitations.I Discretizes string moduli space integration.I Non-trivial match with various one/two-loop correlators.I New, bottom-up approach to string perturbation theory?

Outlook: There are many things to do!I Study more examples: Higher loops / genus, more general operators.I Understand details/implications of stratification beyond one loopI Better understand summation/integration of mirror particles!I Find a limit that can be resummed (λ and/or 1/Nc).I Most promising: Large-charge limit. No stratification.I Other non-planar observables: Anomalous dimensions? Double-traces?I Can we do better? Combine hexagons with quantum spectral curve?

Thank you!


Summary & OutlookSummary: Method to compute higher-genus terms in 1/Nc expansion.

I Sum over free graphs, decompose into planar hexagons.I Infinite sum over mirror excitations.I Discretizes string moduli space integration.I Non-trivial match with various one/two-loop correlators.I New, bottom-up approach to string perturbation theory?

Outlook: There are many things to do!I Study more examples: Higher loops / genus, more general operators.I Understand details/implications of stratification beyond one loopI Better understand summation/integration of mirror particles!I Find a limit that can be resummed (λ and/or 1/Nc).I Most promising: Large-charge limit. No stratification.I Other non-planar observables: Anomalous dimensions? Double-traces?I Can we do better? Combine hexagons with quantum spectral curve?

Thank you!


Graph Statistics

Numbers of maximal graphs for various g and numbers of insertions:

genus : 0 1 2 3 4

n = 2 : 1 1 4 82 7325n = 3 : 1 3 38 661n = 4 : 2 16 760 122307n = 5 : 4 132 18993n = 6 : 14 1571 487293n = 7 : 66 20465n = 8 : 409 278905n = 9 : 3078

n = 10 : 26044


Documents

Non-Planar AdS/CFT Integrability A Progress Reportbargheer/slides/2019-02-01_berlin-seminar.pdfNon-Planar AdS/CFT Integrability – A Progress Report TILL BARGHEER Leibniz Universität