Worldsheet string theory in AdS/CFT and lattice Valentina Forini - - PowerPoint PPT Presentation

Humboldt University Berlin

Valentina Forini

Junior Research Group “Gauge fields from strings”

Worldsheet string theory in AdS/CFT and lattice

Women at the intersection of Mathematics and High Energy Physics, Mainz Institute for Theoretical Physics, March 8 2017

super Yang-Mills in 4d

N = 4

hW[C]i = 1 N TrP e

H (iAµ ˙ xµ+Φi ˙ yi) ds

Type IIB strings in

R ⇥ S1

C “Quark-antiquark” potential

AdS/CFT

Framework

String/gauge correspondence, addresses together

I understanding gauge theories at all values of the coupling I understanding string theories in non-trivial backgrounds

Zstring

• C =

Z DXDΨ e−Sstring ⇠ e−Areareg

I from integrability (assumed) I from supersymmetric localization (supersymmetric observables)

Perturbative string sigma model Perturbative gauge theory Integrability / Localization Perturbative gauge theory Perturbative string sigma model Integrability / Localization

f(g) q

g AdS/CFT

f(g) = c g + d + e g + . . .

f(g) = a g2 + b g4 + . . .

g AdS/CFT

g :=

√ λ 4π = q g2

Y M N

4π

≡

R2 4πα0 and

Motivation

Beautiful progress in obtaining exact results within AdS/CFT

Perturbative string sigma model Perturbative gauge theory Integrability / Localization Perturbative gauge theory Perturbative string sigma model Integrability / Localization

f(g) q

g AdS/CFT

f(g) = c g + d + e g + . . .

f(g) = a g2 + b g4 + . . .

g AdS/CFT

Beautiful progress in obtaining exact results within AdS/CFT In the world-sheet string theory integrability only classically, localization not formulated. Call for genuine 2d QFT to cover the finite-coupling region.

Motivation

I from integrability (assumed) I from supersymmetric localization (supersymmetric observables)

The relevant string sigma-model (Green-Schwarz superstrings in AdS backgrounds with RR-fluxes) is a complicated interacting 2d field theory which has subtleties also perturbatively.

g :=

√ λ 4π = q g2

Y M N

4π

≡

R2 4πα0 and

Exciting program on the 4d susy CFT side, subtleties with supersymmetry.

Lattice 4d N=4 SYM

f(g) q g AdS/CFT

Lattice techniques in AdS/CFT

[Catterall et al.]

Talks by Shaich, Giedt, Anosh Lattice for superstring world-sheet in AdS5 × S5

[previous study: Roiban McKeown 2013]

Lattice 4d N=4 SYM Lattice 2d Green-Shwarz string

f(g) q g AdS/CFT

Lattice techniques in AdS/CFT

I 2d: computationally cheap I no supersymmetry (Green-Schwarz formulation) I all local (diffeo, κ) symmetries are fixed, only scalar fields

(some of which Graßmann-valued) Non-trivial 2d qft with strong coupling analytically known, finite-coupling (numerical) prediction.

The model in perturbation theory

Green-Schwarz string in AdS5 × S5 + RR flux

Non-linear sigma-model on G/H =

P SU(2,2|4) SO(1,4)×SO(5)

S = g Z dτdσ ⇥ ∂aXµ∂aXν Gµν + ¯ θ Γ (D + F5 ) θ ∂X + ¯ θ ∂θ ¯ θ ∂θ + . . . ⇤

AdS5 S5

x

[Metsaev Tseytlin 1998]

Z ⇥ ⇤ Symmetries:

I global PSU(2, 2|4), local bosonic (diffeomorphism) and fermionic (κ-symmetry) I classical integrability

Z manifest when written as sigma-model action on G/H =

P SU(2,2|4) SO(1,4)×SO(5) .

×

Highly non-linear, to quantize it use semiclassical methods X = Xcl + ˜ X − → Γ = g h Γ0 + Γ1 g + Γ2 g2 + . . . i R

Green-Schwarz string in AdS5 × S5 + RR flux

Non-linear sigma-model on G/H =

P SU(2,2|4) SO(1,4)×SO(5)

S = g Z dτdσ ⇥ ∂aXµ∂aXν Gµν + ¯ θ Γ (D + F5 ) θ ∂X + ¯ θ ∂θ ¯ θ ∂θ + . . . ⇤

AdS5 S5

x

[Metsaev Tseytlin 1998]

I General analysis of fluctuations in terms of background geometry,

e.g. Tr(M) = a (2) R + b Tr(K2).

I Explicit analytic form of one-loop partition function Z = det OF /√det OB

for a class of effectively one-dimensional problems. Non-trivial differential operators, e.g. elliptic-function potentials: O = −∂2

σ + ω2 + k2sn2(σ, k2). Then use Gelf’and-Yaglom method:

O φ(x) = λ φ(x) , φ(0) = φ(L) = 0 det O det Ofree = u(L) ufree(L) where u are solutions of auxiliary boundary value problem, u(0) = 0 , u0(0) = 1. Several configurations (GKP string, quark-antiquark potential, generalized cusp) have been “solved” this way at one loop, and agree with predictions.

[VF Beccaria Dunne Tseytlin, Drukker, Giangreco Ohlson Sax Vescovi .... [Drukker Gross Tseytlin 00] [VF Giangreco Griguolo Seminara Vescovi 15] [Alvarez-Gaume, Freedman, Mukhi, 81]

Green-Schwarz string in AdS5 × S5 + RR flux perturbatively

Highly non-linear, to quantize it use semiclassical methods X = Xcl + ˜ X − → E = g h E0 + E1 g + E2 g2 + . . . i , g = √

• 4⇡

1/2 BPS circular Wilson loop loghW (λ)i = log

2 p λI1(

p λ) = p λ 3 4 log λ + 1 2 log 2 π + O(λ 1

2 )

The 1-loop disc partition function log Z = loghWi differs: 1

2 log 1 2π

To avoid measure ambiguities, consider ratio of Zs for surfaces of the same topology: log λ factors and UV divergences (⇠ χ ) should cancel out.

[Drukker Gross Tseytlin 00] [Erickson, Semenoff, Zarembo 00] [Drukker Gross 00]
 [Pestun 07] [Kruczenski Tirziu 08] [Buchbinder Tseytlin 14]

1/2 BPS circular Wilson loop loghW (λ)i = log

2 p λI1(

p λ) = p λ 3 4 log λ + 1 2 log 2 π + O(λ 1

2 )

The 1-loop disc partition function log Z = loghWi differs: 1

2 log 1 2π

To avoid measure ambiguities, consider ratio of Zs for surfaces of the same topology: log λ factors and UV divergences (⇠ χ ) should cancel out.

[Drukker Gross Tseytlin 00]

⇠ E.g. family of 1/4-BPS “latitudes”, parametrized by θ0 in S2 2 S5 (λ0 = λ cos2 θ0). log hW (λ, θ0)i hW (λ, 0)i = p λ (cos θ0 1) 3 2 log cos θ0 + O(λ 1

2 )

[Drukker 06]
 [Drukker Giombi Ricci Trancanelli 07] [Pestun 09] [Kruczenski Tirziu 08] [Buchbinder Tseytlin 14]

1/2 BPS circular Wilson loop log hW (λ, θ0)i hW (λ, 0)i = p λ (cos θ0 1) 3 2 log cos θ0 + O(λ 1

2 )

hW i log hW (λ, θ0)i hW (λ, 0)i = p λ (cos θ0 1) 3 2 log cos θ0+ log cos θ0 2 + O(λ 1

2 )

hW i = p λ (cos θ0 1) + 3 4 θ2

0 + O(λ 1 2 )

• Usual (Gelf’and Yaglom) method fails.

Perturbative heat-kernel (near AdS2 expansion) agrees.

> unphysical cutoff > different regulariz. in τ and in σ

• ⇠

E.g. family of 1/4-BPS “latitudes”, parametrized by θ0 in S2 2 S5 (λ0 = λ cos2 θ0). loghW (λ)i = log

2 p λI1(

p λ) = p λ 3 4 log λ + 1 2 log 2 π + O(λ 1

2 )

The 1-loop disc partition function log Z = loghWi differs: 1

2 log 1 2π

To avoid measure ambiguities, consider ratio of Zs for surfaces of the same topology: log λ factors and UV divergences (⇠ χ ) should cancel out.

[Drukker 06]
 [Drukker Giombi Ricci Trancanelli 07] [Pestun 09] [Forini Tseytlin Vescovi 17] [Drukker Gross Tseytlin 00] [Kruczenski Tirziu 08] [Buchbinder Tseytlin 14]

Green-Schwarz string in AdS5 × S5 + RR flux perturbatively

Highly non-linear, to quantize it use semiclassical methods X = Xcl + ˜ X − → Γ = g h Γ0 + Γ1 g + Γ2 g2 + . . . i 2 loops is current limit: “homogenous” configs, “AdS light-cone” gauge-fixing Efficient alternative to Feynman diagrams for on-shell objects (worldsheet S-matrix) unitarity cuts (on-shell methods) in d=2 Indirect evidence of quantum integrability!

[Giombi Ricci Roiban Tseytlin 09, Bianchi2 Bres VF Vescovi 14] [Bianchi VF Hoare 2013][Engelund Roiban 2013] [Bianchi Hoare 14]

Green-Schwarz string in AdS5 × S5 + RR flux perturbatively

Highly non-linear, to quantize it use semiclassical methods X = Xcl + ˜ X − → Γ = g h Γ0 + Γ1 g + Γ2 g2 + . . . i 2 loops is current limit: “homogenous” configs, “AdS light-cone” gauge-fixing Efficient alternative to Feynman diagrams for on-shell objects (worldsheet S-matrix) unitarity cuts (on-shell methods) in d=2 Indirect evidence of quantum integrability!

[Giombi Ricci Roiban Tseytlin 09] [Bianchi2 Bres VF Vescovi 14]

[Bianchi VF Hoare 2013][Engelund Roiban 2013] [Bianchi Hoare 14]

Indirect evidence of quantum integrability! UV divergences: set to zero power-divergent massless tadpoles (as in dimreg), all remaining log-divergent integrals cancel out in the sum (no need of reg. scheme).

[Metsaev, Tseytlin] [Metsaev, Thorn, Tseytlin]

X

• Ibubble(p) =

Z d2q (2⇡2)2 1 (q2 − 1 + i ✏) ((q − p)2 − 1 + i ✏) X ⇣ ⌘ A(1) = X A(0)2 Ibubble ,

p1 p2 p4 p3 l1 l2 R S M N P Q

A(0) A(0)

[Bianchi VF Hoare 13][Engelund, Roiban 13][Bianchi Hoare 14]

Green-Schwarz string in AdS5 × S5 + RR flux perturbatively

Unitarity cuts in d = 2, for worldsheet amplitudes (integrable S-matrix) Inherently finite, bypasses any regularization issue: may miss rational terms. A large class of 2-d models, relativistic and not (including string worldsheet models in AdS), appears to be cut-constructible.

Beyond perturbation theory

Based on 1601.04670, 1605.01726, 1702.02005, 1703.xxxxx with L. Bianchi, M. S. Bianchi, B. Leder, P. Töpfer, E. Vescovi

The cusp anomaly of N = 4 SYM from string theory

Completely solved via integrability. Expectation value of a light-like cusped Wilson loop hW[Ccusp]i ⇠ e

f(g) φ ln LIR

✏UV

Zcusp = Z [DδX][Dδθ] eSIIB(Xcusp+δX,δθ) = eΓeff ⌘ ef(g) V2 String partition function with “cusp” boundary conditions, evaluated perturbatively A lattice approach prefers expectation values hScuspi = R [DδX][DδΨ] Scusp eScusp R [DδX][DδΨ] eScusp = g d ln Zcusp dg ⌘ g V2 8 f0(g)

AdS/CFT

[Beisert Eden Staudacher 2006]

f(g)|g→0 = 8g2 h 1 − π2 3 g2 + 11 π4 45 g4 − ⇣ 73 315 + 8 ζ3 ⌘ g6 + ... i f(g)|g→∞ = 4g h 1 − 3 ln 2 4π 1 g − K 16π2 1 g2 + ... i

[Bern et al. 2006] [Giombi et al. 2009]

ry of the

• f g =

√ λ 4π .

In Poincaré patch (boundary at z=0) classical solution ( and vary from 0 to ) is a surface bounded by a null cusp, since at the AdS5 boundary it is

τ σ ∞ z = rτ σ x+ = τ x− = − 1 2σ 0 = z2 = −2x+x−. ds2

AdS5 = dz2 + dx+ dx− + dx∗ dx

z2 x± = x3 ± x0 x = x1 ± i x2

[Giombi Ricci Roiban Tseytlin 2009]

Z String partition function with “cusp” boundary conditions, evaluated perturbatively

Φ = i φ

The cusp anomaly of N = 4 SYM from string theory

Completely solved via integrability. Expectation value of a light-like cusped Wilson loop hW[Ccusp]i ⇠ e

f(g) φ ln LIR

✏UV

Zcusp = Z [DδX][Dδθ] eSIIB(Xcusp+δX,δθ) = eΓeff ⌘ ef(g) V2 String partition function with “cusp” boundary conditions, evaluated perturbatively A lattice approach prefers expectation values hScuspi = R [DδX][DδΨ] Scusp eScusp R [DδX][DδΨ] eScusp = g d ln Zcusp dg ⌘ g V2 8 f0(g)

AdS/CFT

[Beisert Eden Staudacher 2006]

f(g)|g→0 = 8g2 h 1 − π2 3 g2 + 11 π4 45 g4 − ⇣ 73 315 + 8 ζ3 ⌘ g6 + ... i f(g)|g→∞ = 4g h 1 − 3 ln 2 4π 1 g − K 16π2 1 g2 + ... i

[Bern et al. 2006] [Giombi et al. 2009]

ry of the

• f g =

√ λ 4π .

[Frolov Tseytlin 02] [Gubser Klebanov Polyakov 02]

Φ = i φ

The cusp anomaly of N = 4 SYM from string theory

Completely solved via integrability. Expectation value of a light-like cusped Wilson loop hW[Ccusp]i ⇠ e

f(g) φ ln LIR

✏UV

Zcusp = Z [DδX][Dδθ] eSIIB(Xcusp+δX,δθ) = eΓeff ⌘ ef(g) V2 String partition function with “cusp” boundary conditions, evaluated perturbatively A lattice approach prefers expectation values hScuspi = R [DδX][DδΨ] Scusp eScusp R [DδX][DδΨ] eScusp = g d ln Zcusp dg ⌘ g V2 8 f0(g)

AdS/CFT

[Beisert Eden Staudacher 2006]

f(g)|g→0 = 8g2 h 1 − π2 3 g2 + 11 π4 45 g4 − ⇣ 73 315 + 8 ζ3 ⌘ g6 + ... i f(g)|g→∞ = 4g h 1 − 3 ln 2 4π 1 g − K 16π2 1 g2 + ... i

[Bern et al. 2006] [Giombi et al. 2009]

ry of the

• f g =

√ λ 4π .

[Frolov Tseytlin 02] [Gubser Klebanov Polyakov 02]

Φ = i φ

Scusp = g Z Lcusp

A lattice approach prefers expectation values hScuspi = R [DδX][DδΨ] Scusp eScusp R [DδX][DδΨ] eScusp = g d ln Zcusp dg ⌘ g V2 8 f0(g)

Simulations in lattice QFT

Spacetime grid with as lattice spacing a, size L = N a, ξ = (an1, an2) ⌘ a n and fields φ ⌘ φn a) natural cutoff for the momenta, π

a < pµ  π a

b) path integral measure [Dφ] = Q

n dφn.

Then R Q

n dφn e−Sdiscr can be studied via Monte Carlo: generate an ensamble

{Φ1, . . . , ΦK} of field configurations, each weighted by P[Φi] = e−SE [Φi]

Z

. Ensemble average hAi = R [DΦ] P[Φ] A[Φ] =

1 K

PK

i=1 A[Φi] + O

• 1

√ K

• Graßmann-odd fields are formally integrated out: P[Φi] = e−SE [Φi]det OF

Z

I action must be quadratic in fermions (linearization via auxiliary fields):

here, interactions at most quartic (AdS light cone gauge)

I determinant must be definite positive

det OF

• !

q det(OF O†

F ) =

Z Dζ D¯ ζ e−

R d2ξ ¯ ζ(OF O†

F )− 1 4 ζ

≡

fields):

Introduce auxiliary fields (complex bosons)

determinant must be positive definite

det OF − → q det(O†

F OF ) ≡

Z Dζ D¯ ζ e−

R d2ξ ¯ ζ(O†

F OF )− 1 2 ζ

Simulations in lattice QFT

Spacetime grid with as lattice spacing a, size L = N a, ξ = (an1, an2) ⌘ a n and fields φ ⌘ φn a) natural cutoff for the momenta, π

a < pµ  π a

b) path integral measure [Dφ] = Q

n dφn.

Then R Q

n dφn e−Sdiscr can be studied via Monte Carlo: generate an ensamble

{Φ1, . . . , ΦK} of field configurations, each weighted by P[Φi] = e−SE [Φi]

Z

. Ensemble average hAi = R [DΦ] P[Φ] A[Φ] =

1 K

PK

i=1 A[Φi] + O

• 1

√ K

• Graßmann-odd fields are formally integrated out: P[Φi] = e−SE [Φi]det OF

Z

I action must be quadratic in fermions (linearization via auxiliary fields):

here, interactions at most quartic (AdS light cone gauge)

I determinant must be definite positive

det OF

• !

q det(OF O†

F ) =

Z Dζ D¯ ζ e−

R d2ξ ¯ ζ(OF O†

F )− 1 4 ζ

≡

fields):

Introduce auxiliary fields (complex bosons)

determinant must be positive definite

det OF − → q det(O†

F OF ) ≡

Z Dζ D¯ ζ e−

R d2ξ ¯ ζ(O†

F OF )− 1 2 ζ

Simulations in lattice QFT

Spacetime grid with as lattice spacing a, size L = N a, ξ = (an1, an2) ⌘ a n and fields φ ⌘ φn a) natural cutoff for the momenta, π

a < pµ  π a

b) path integral measure [Dφ] = Q

n dφn.

Then R Q

n dφn e−Sdiscr can be studied via Monte Carlo: generate an ensamble

{Φ1, . . . , ΦK} of field configurations, each weighted by P[Φi] = e−SE [Φi]

Z

. Ensemble average hAi = R [DΦ] P[Φ] A[Φ] =

1 K

PK

i=1 A[Φi] + O

• 1

√ K

• Graßmann-odd fields are formally integrated out: P[Φi] = e−SE [Φi]det OF

Z

I action must be quadratic in fermions (linearization via auxiliary fields):

here, interactions at most quartic (AdS light cone gauge)

I determinant must be definite positive

det OF

• !

q det(OF O†

F ) =

Z Dζ D¯ ζ e−

R d2ξ ¯ ζ(OF O†

F )− 1 4 ζ

≡

fields):

Introduce auxiliary fields (complex bosons)

determinant must be positive definite

Pf OF − → (det O†

F OF )

1 4 ≡

Z Dζ D¯ ζ e−

R d2ξ ¯ ζ (O†

F OF )− 1 4 ζ

potential ambiguity!

Scusp = g Z dtdsLcusp Lcusp = |∂tx + 1

2x|2 + 1 z4 |∂sx −1 2x|2 +

⇣ ∂tzM + 1

2zM + i z2 zNηi

• ρMNi

j ηj⌘2

+ 1

z4

• ∂szM − 1

2zM2

+i

• θi∂tθi + ηi∂tηi + θi∂tθi + eηi∂tηi

− 1

z2

• ηiηi

2 (2.1) +2i h

1 z3 zMηi

ρM

ij

• ∂sθj − 1

2θj −i zηj

∂sx −1

2x

• + 1

z3 zMηi(ρ† M)ij

∂sθj − 1

2θj + i zηj

• ∂sx − 1

2x

∗ i

I 8 bosonic coordinates: x, x∗, zM (M = 1, · · · , 6), z =

p zMzM;

I 8 fermionic variables, θi = (θi)†, ηi = (ηi)†, i = 1, 2, 3, 4

transforming in the fundamental of SU(4)

I ρM are off-diagonal blocks of SO(6) Dirac matrices γM ≡

✓ ρ†

M

ρM ◆

exp n − g Z dtds h − 1

z2

• ηiηi

2 + ⇣

i z2 zNηiρMN i jηj⌘2i

} (2.5) ∼ Z DφDφM exp n − g Z dtds [ 1

2φ2 + √ 2 z φ η2 + 1 2(φM)2 − i √ 2 z2 φM i z2 zNηiρMN i jηj

]

• .

Green-Schwarz string in the null cusp background

Manifest global symmetry is SO(6) × SO(2). Quartic fermionic interactions that can be linearized

[Giombi Ricci Roiban Tseytlin 2009]

Scusp = g Z dtdsLcusp Lcusp = |∂tx + 1

2x|2 + 1 z4 |∂sx −1 2x|2 +

⇣ ∂tzM + 1

2zM + i z2 zNηi

• ρMNi

j ηj⌘2

+ 1

z4

• ∂szM − 1

2zM2

+i

• θi∂tθi + ηi∂tηi + θi∂tθi + eηi∂tηi

− 1

z2

• ηiηi

2 (2.1) +2i h

1 z3 zMηi

ρM

ij

• ∂sθj − 1

2θj −i zηj

∂sx −1

2x

• + 1

z3 zMηi(ρ† M)ij

∂sθj − 1

2θj + i zηj

• ∂sx − 1

2x

∗ i

I 8 bosonic coordinates: x, x∗, zM (M = 1, · · · , 6), z =

p zMzM;

I 8 fermionic variables, θi = (θi)†, ηi = (ηi)†, i = 1, 2, 3, 4

transforming in the fundamental of SU(4)

I ρM are off-diagonal blocks of SO(6) Dirac matrices γM ≡

✓ ρ†

M

ρM ◆

exp n − g Z dtds h − 1

z2

• ηiηi

2 + ⇣

i z2 zNηiρMN i jηj⌘2i

} (2.5) ∼ Z DφDφM exp n − g Z dtds [ 1

2φ2 + √ 2 z φ η2 + 1 2(φM)2 − i √ 2 z2 φM i z2 zNηiρMN i jηj

]

• .

Green-Schwarz string in the null cusp background

Manifest global symmetry is SO(6) × SO(2). Quartic fermionic interactions that can be linearized

[Giombi Ricci Roiban Tseytlin 2009]

≡ OF = B B B @ i∂t −iρM ∂s + m

2

zM

z3

i∂t −iρ†

M

• ∂s + m

2

zM

z3

i zM

z3 ρM

∂s − m

2

• 2zM

z4 ρM

∂sx − mx

2

• i∂t − AT

i zM

z3 ρ† M

• ∂s − m

2

• i∂t + A

−2zM

z4 ρ† M

• ∂sx∗ − mx

2 ∗

1 C C C A

Green-Schwarz string in the null cusp background

After linearization the Lagrangian reads (m ∼ P+) Lcusp = |∂tx+ m 2 x|

2

+ 1 z4

• ∂sx− m

2 x|

2

+ (∂tzM + m 2 zM)2 + 1 z4 (∂szM − m 2 zM)2 + 1 2 φ2 + 1 2 (φM)2 + ψT OF ψ ,

B @

• −
• A =

1 √ 2z2 φMρMNzN − 1 √ 2z φ + i zN z2 ρMN ∂tzM

2 2

I +7 bosonic auxiliary fields φ, φM (M = 1, · · · , 6)) I formal variable ψ ≡ (θi, θi, ηi, ηi)

Green-Schwarz string in the null cusp background

After linearization the Lagrangian reads (m ∼ P+) Lcusp = |∂tx+ m 2 x|

2

+ 1 z4

• ∂sx− m

2 x|

2

+ (∂tzM + m 2 zM)2 + 1 z4 (∂szM − m 2 zM)2 + 1 2 φ2 + 1 2 (φM)2 + ψT OF ψ ,

ˆ OF = B B B @ W+ −˚ p01 (˚ p1 − i m

2 )ρM zM z3

−˚ p01 −W †

+

ρ†

M(˚

p1 − i m

2 ) zM z3

−(˚ p1 + i m

2 )ρM zM z3

2zM

z4 ρM

∂sx − mx

2

• + W−

−˚ p01 − AT −ρ†

M(˚

p1 + i m

2 ) zM z3

−˚ p01 + A −2zM

z4 ρ† M

• ∂sx∗ − mx

2 ∗

− W †

−

1 C C C A

@ (2) where W± = r

2

• ˆ

p2

0 ± i ˆ

p2

1

• ρMuM, |r| = 1, and ˆ

pµ ≡ 2

a sin pµa 2 . It is such that

A naive discretization pµ →

• pµ ≡ 1

a sin(a pµ) leads to fermion doublers.

• Add to the action a “Wilson term” W

Lattice perturbation theory reproduces its continuum counterpart for a → 0

I

It preserves the SO(6) global symmetry, breaks the SO(2).

I

The simulation: parameter space

I In the continuum model there are two parameters, g =

√ λ 4π and m ∼ P+.

In perturbation theory divergences cancel, dimensionless quantities are pure functions of the (bare) coupling F = F(g)

I Our discretization cancels (1-loop) divergences, and reproduces the 1-loop

cusp anomaly. Assume it is true nonperturbatively, for lattice regularization. Only additional scale: lattice spacing a. Three dimensionless (input) parameters: g , N ≡ L a , M ≡ m a Therefore FLAT = FLAT(g, N, M)

Line of constant physics

The continuum limit must be taken through a series of simulations in a controlled way: lattice spacing a → 0 while physical (renormalized) quantities should be kept constant. Line of constant physics: curves in the bare parameter space, where dimensionless physical quantities are kept fixed as a changes. In the continuum, “effective” masses undergo a finite renormalization m2

x(g) = m2

2 ⇣ 1 − 1 8 g + O(g−2) ⌘ (?) The dimensionless physical quantity to keep constant when a → 0 is L2 m2

x = const ,

leading to (L m)2 ≡ (NM)2 = const , if (?) is still true on the lattice and g is not (infinitely) renormalized.

[Basso 2010] [Giombi Ricci Roiban Tseytlin 2010]

Continuum limit a → 0

We assume that, on the lattice, no further scale but a is present. A generic observable FLAT = FLAT(g, N, M) = F(g) + O ⇣ 1 N ⌘ + O ⇣ e−MN⌘ where g = √ λ 4π , N = L a , M = a m . Recipe:

I fix g I fix MN, large enough so to to keep small finite volume effects I evaluate FLAT for N = 6, 8, 10, 12, 16, · · · I obtain F(g) extrapolating to N → ∞.

finite lattice spacing (~a) effects finite volume (~ m L) effects

Measurement I: hx, x⇤i correlator

From the correlator of the x fields Cx(t; 0) = X

s1, s2

hx(t, s1)x⇤(0, s2)i = X

n

|cn|2etEx(0; n)

t1

⇠ et mxLAT extract the x-mass mxLAT = lim

t!1 meff x

⌘ lim

t!1

1 a log Cx(t; 0) Cx(t + a; 0) No infinite renormalization occurring, no need of tuning m to adjust for it. This corroborates our choice of line of constant physics.

0.5 1 1.5 2 2.5 3 1 2 3 4 5 6 ·10−3 tmxLAT Cx(t) 0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 tmxLAT (meff

x )2/m2

X

t mxLAT

X

t mxLAT

Cx(t; 0)

(meff

x )2

m2

Measurement I: hx, x⇤i correlator

From the correlator of the x fields Cx(t; 0) = X

s1, s2

hx(t, s1)x⇤(0, s2)i = X

n

|cn|2etEx(0; n)

t1

⇠ et mxLAT extract the x-mass mxLAT = lim

t!1 meff x

⌘ lim

t!1

1 a log Cx(t; 0) Cx(t + a; 0) No infinite renormalization occurring, no need of tuning m to adjust for it. This corroborates our choice of line of constant physics.

0.5 1 1.5 2 2.5 3 1 2 3 4 5 6 ·10−3 tmxLAT Cx(t)

X

t mxLAT

X

t mxLAT

Cx(t; 0)

(meff

x )2

m2

0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1 gc m2

x/m2

Lm = 4 Lm = 6 gc = 0.04g

m2

x

m2

gc

Courtesy of B. Basso

Consistent with large g prediction, no clear signal of bending down. No infinite renormalization occurring, no need of tuning m to adjust for it. This corroborates our choice of line of constant physics. Consistent with large g prediction, no clear signal of bending down. No infinite renormalization occurring. This corroborates our choice of line of constant physics.

1/100 1/50 1/30 1/20 1/10 1/5 7.4 7.45 7.5 7.55 7.6 7.65 7.7 7.75 1/g c/2 Lm = 4 Lm = 6 g >= 50 g >= 10

Measurement II: (derivative of the) cusp anomaly

In measuring hScuspi ⌘ g V2 m2

8

f0(g) quadratic divergences appear. At large g, hSLATi ⌘ g N2 M2

4

4 + c

2 (2N2)

where c = nbos = 8 + 7 = 15. This is because hSi = ∂ ln Z

∂ ln g and Z ⇠ Πnbos(det g O) 1

2 .

Therefore a factor proportional to g (2N2)

2

for each bosonic field species. In lattice codes, coupling omitted from the (pseudo)fermionic part of the action.

Simulation: the cusp action

In measuring hScuspi ⌘ g V2 m2

8

f0(g) quadratic divergences appear. They appear also at finite g, hSLATi ⌘ g N2 M2

4

f0(g)LAT + c(g)

2 (2N2)

In continuum perturbation theory dim. reg. set them to zero. In continuum perturbation theory dim. reg. set them to zero. Here, expected mixing of the Lagrangian with lower dimension operator O(φ(s))r = X

α:[Oα]D

Zα Oα(φ(x)) , Zα ⇠ Λ(D[Oα])⇠ a(D[Oα])

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 1 1.5 2 2.5 3 1/N (< SLAT > −7.5(2N 2))/S0 g = 5, Lm = 4 g = 10, Lm = 4 g = 20, Lm = 4 g = 30, Lm = 4 g = 50, Lm = 4 g = 100, Lm = 4 g = 30, Lm = 6

h i S0 = g N2 M2

1/100 1/50 1/30 1/20 1/10 1/5 7.4 7.45 7.5 7.55 7.6 7.65 7.7 7.75 1/g c/2 Lm = 4 Lm = 6 g >= 50 g >= 10

Measurement II: (derivative of the) cusp anomaly

1 g with

Divergences appear also at finite g,

O(φ)r = X

α:[Oα]D

Zα Oα(φ) , Zα ⇠ Λ(D[Oα])⇠ a(D[Oα])

Measurement II: (derivative of the) cusp anomaly

To compare, assume g = α gc: then from f0(g) = f0(gc)c is gc = 0.04g.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.2 0.4 0.6 0.8 1 gc f’(gc)/4 BES, gc=0.04 g PT, gc=0.04 g Lm = 4 Lm = 6

The phase

After linearization LF = ψT OF ψ, integrating fermions leads to a complex Pfaffian Pf OF = |(det OF )

1 2 | eiθ.

The phase is encoded in the linearization: we deal with a fermionic hermitian bilinear b ⇠ η2 whose corresponding quartic interaction e−Lferm

4

= e− b2

4 a =

Z dx e−a x2+i b x comes in the exponential as with the “wrong” sign. The phase can be treated via reweighting: incorporate the non positive part of the Boltzmann weight into the observable hOireweight = hO eiθiθ=0 heiθiθ=0 It gives meaningful results as long as the phase does not average to zero.

The phase

In the interesting (g = 1) region the phase has a flat distribution. Alternative algorithms: active field of study, no general proof of convergence.

g=30 g=5 g=1

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 Re(eiφ)

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 5 10 15 20 Re(eiφ) −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 50 100 150 200 250 300 350 Re(eiφ)

Re(eiθ)

number of configs

Re(eiθ) Re(eiθ)

number of configs number of configs

Alternative linearization

We identified a problem in the ”wrong sign” of the quartic fermionic interaction. Consider a simple SO(4) invariant four-fermion interaction L4F = 1 2 ✏abcd a(x) b(x) c(x) d(x) ≡ Σab e Σab where Σab = a b , e Σab = 1

2 ✏abcd c d.

Introducing the (anti)self-dual fermion bilinears Σab

± = 1

2 ⇣ Σab ± 1 2 ✏abcd Σcd ⌘

• ne can rewrite

L4F = ± 2 ⇣ Σab

±

⌘2 just exploiting the Graßmann character of the underlying fermions.

[Catterall 2015]

Alternative linearization

In our case LF 4 = 1 z2 (⌘2)2+ 1 z2 (i ⌘i(⇢MN)i

jnN⌘j)2

• ne analogously defines - notice (⇢M)im(⇢M)kn = 2✏imkn - the bilinears

Σij = ⌘i⌘j e Σji = (⇢N)iknN(⇢L)jlnL⌘k⌘l and again introduces Σ±

j i = Σj i ± e

Σj

i to rewrite

LF 4 = 1 z2 (⌘2)2⌥ 2 z2 (⌘2)2⌥ 1 z2 Σ±

j i Σ±i j

Now choose the good sign (). The new set of Yukawa terms (now 1 + 16 real auxiliary fields) LF 4 ! 12 z ⌘2 + 62 + 2 z Σ±i

jj i + i jj i

ensures the full Lagrangian to be hermitian, and a full (including auxiliary fields) non negative det OF .

Alternative linearization

The Pfaffian is now real (Pf OF )2 = det OF 0 but not definite positive: Pf OF = ±(det OF )

1 2 .

For g  5 equal number of + and . Phase problem traded for a purely sign problem? Gain in computational costs: for large values of N (finer lattices) the algorithm for evaluating complex determinants is very inefficient. Now just a sign flip. hOireweight = hO eiθiθ=0 heiθiθ=0

• !

hOireweight = hO wi hwi where w = ±1, and pdet OF = (det O†

F OF )

1 4 .

Allows removing a systematic error (omission of reweighting factor for large N). NOTICE: there’s a region of the coupling which is free from sign problem and that clearly sees nonperturbative physics.

and pdet OF and pdet OF

det OF

W

ensures to be real and non-negative. with Γ†

5Γ5 = 1 , Γ† 5 = −Γ5

A Γ5-hermiticity and antisymmetry O†

F = Γ5 OF Γ5 ,

OT

F = −OF

− Pfaffian is real, (PfOF )2 = det OF ≥ 0, but not positive definite, PfOF = ± det OF .

In simpler models with four-fermion interactions, similar manipulations ensure a definite positive Pfaffian. There real, antisymmetric operator with doubly degenerate eigenvalues: quartets (ia, ia, −ia, −ia) , a ∈ R.

[Catterall 2016, Catterall and Schaich 2016]

Alternative linearization

The Pfaffian is now real (Pf OF )2 = det OF 0 but not definite positive: Pf OF = ±(det OF )

1 2 .

For g  5 equal number of + and . Phase problem traded for a purely sign problem? Gain in computational costs: for large values of N (finer lattices) the algorithm for evaluating complex determinants is very inefficient. Now just a sign flip. hOireweight = hO eiθiθ=0 heiθiθ=0

• !

hOireweight = hO wi hwi where w = ±1, and pdet OF = (det O†

F OF )

1 4 .

Allows removing a systematic error (omission of reweighting factor for large N). NOTICE: there’s a region of the coupling which is free from sign problem and that clearly sees nonperturbative physics.

and pdet OF and pdet OF

A Γ5-hermiticity and antisymmetry O†

F = Γ5 OF Γ5 ,

OT

F = −OF

In simpler models with four-fermion interactions, similar manipulations ensure a definite positive Pfaffian. There real, antisymmetric operator with doubly degenerate eigenvalues: quartets (ia, ia, −ia, −ia) , a ∈ R.

[Catterall 2016, Catterall and Schaich 2016]

det OF

W

ensures to be real and non-negative. with Γ†

5Γ5 = 1 , Γ† 5 = −Γ5

− Pfaffian is real, (PfOF )2 = det OF ≥ 0, but not positive definite, PfOF = ± det OF .

Spectrum of OF

−4 −3 −2 −1 1 2 3 4 −4 −2 2 4 λ λ L = 8, g = 10, r = 1

From Γ5-hermiticity and antisymmetry, P(λ) = det(OF − λ1) = det(Γ5 (OF − λ1) Γ5) = det(O†

F + λ1) = det(OF + λ∗1)∗ = P(−λ∗)∗

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5 λ λ L = 8, g = 25, r = 0

Spectrum characterized by quartets {λ, −λ∗, −λ, λ∗}. det OF = Y

i

|λi|2 |λi|2 − → Pf(OF ) = ± Y

i

|λi|2 Choosing a starting configuration with positive Pfaffian, no sign change possible.

Spectrum of OF

From Γ5-hermiticity and antisymmetry, P(λ) = det(OF − λ1) = det(Γ5 (OF − λ1) Γ5) = det(O†

F + λ1) = det(OF + λ∗1)∗ = P(−λ∗)∗

−4 −3 −2 −1 1 2 3 4 −4 −2 2 4 λ λ L = 8, g = 10, r = 1 −1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5 λ λ L = 8, g = 25, r = 0

For λ = ± λ∗, no four-fold property: due to zero crossings, Pfaffian may change sign.

Alternative linearization: measures

Eigenvalue distribution of fermionic operators well separated from zero, no sign problem for g ≥ 10, where nonperturbative physics is captured. 0.02 0.04 0.06 0.08 0.1 0.12 2 4 6 8 10 12 14 1/N < λmin > /σmin g = 10, r = 1 g = 30, r = 1 g = 25, r = 0 g = 50, r = 0

On the CFT side

The control is in the perturbative region (matching with NNLO). Strong sign problem at strong coupling (λ 1). David Schaich at Lattice 2016

Solving a non-trivial 4d QFT is hard reduce the problem via AdS/CFT: solve (finding a good regulator for) a non-trivial 2d QFT. For Green-Schwarz string worldsheet in AdS backgrounds, it is possible to improve perturbative techniques e.g. via cross-fertilization of QFT methods. The model is amenable to study using lattice QFT techniques (Wilson-like discretizations, standard simulation algorithms). Interesting beyond string community. Non-perturbative definition of string theory? Not quite yet. Still, suitable framework for first principle statements (proofs of AdS/CFT), and potentially efficient tool in numerical holography.

Concluding remarks

I All correlators, different backgrounds (e.g. for ABJM cusp). I Further observables? . . . I . . .

Future

Thanks for your attention.

A remark on numerics

The most difficult part of the algorithm is the inversion of the fermionic matrix |Pf OF | ≡ (det O†

F OF )

1 4 ≡

Z dζd¯ ζ e−

R d2ξ ¯ ζ (O†

F OF )− 1 4 ζ .

The RHMC (Rational Hybrid Montecarlo) uses a rational approximation ¯ ζ (O†

F OF )− 1

4 ζ = α0 ¯

ζ ζ +

P

X

i=1

¯ ζ αi O†

F OF + βi

ζ with αi and βi tuned by the range of eigenvalues of OF . Defining si ≡

1 O†

F OF +βi ζ, one solves

(O†

F OF + βi) si = ζ ,

i = 1, . . . , P. with a (multi-shift conjugate) solver for which number of iterations ∼ λ−1

min

In our case the spectrum of OF has very small eigenvalues. And:

g T/a × L/a Lm am τ S

int

τ mx

int

statistics [MDU] 5 16 × 8 4 0.50000 0.8 2.2 900 20 × 10 4 0.40000 0.9 2.6 900 24 × 12 4 0.33333 0.7 4.6 900,1000 32 × 16 4 0.25000 0.7 4.4 850,1000 48 × 24 4 0.16667 1.1 3.0 92,265 10 16 × 8 4 0.50000 0.9 2.1 1000 20 × 10 4 0.40000 0.9 2.1 1000 24 × 12 4 0.33333 1.0 2.5 1000,1000 32 × 16 4 0.25000 1.0 2.7 900,1000 48 × 24 4 0.16667 1.1 3.9 594,564 20 16 × 8 4 0.50000 5.4 1.9 1000 20 × 10 4 0.40000 9.9 1.8 1000 24 × 12 4 0.33333 4.4 2.0 850 32 × 16 4 0.25000 7.4 2.3 850,1000 48 × 24 4 0.16667 8.4 3.6 264,580 30 20 × 10 6 0.60000 1.3 2.9 950 24 × 12 6 0.50000 1.3 2.4 950 32 × 16 6 0.37500 1.7 2.3 975 48 × 24 6 0.25000 1.5 2.3 533,652 16 × 8 4 0.50000 1.4 1.9 1000 20 × 10 4 0.40000 1.2 2.7 950 24 × 12 4 0.33333 1.2 2.1 900 32 × 16 4 0.25000 1.3 1.8 900,1000 48 × 24 4 0.16667 1.3 4.3 150 50 16 × 8 4 0.50000 1.1 1.8 1000 20 × 10 4 0.40000 1.2 1.8 1000 24 × 12 4 0.33333 0.8 2.0 1000 32 × 16 4 0.25000 1.3 2.0 900,1000 48 × 24 4 0.16667 1.2 2.3 412 100 16 × 8 4 0.50000 1.4 2.7 1000 20 × 10 4 0.40000 1.4 4.2 1000 24 × 12 4 0.33333 1.3 1.8 1000 32 × 16 4 0.25000 1.3 2.0 950,1000 48 × 24 4 0.16667 1.4 2.4 541 Table 1: Parameters of the simulations: the coupling g, the temporal (T) and spatial (L) extent of the lattice in units of the lattice spacing a, the line of constant physics fixed by Lm and the mass parameter M = am. The size of the statistics after thermalization is given in the last column in terms of Molecular Dynamic Units (MDU), which equals an HMC trajectory

• f length one. In the case of multiple replica the statistics for each replica is given separately.

The auto-correlation times τ of our main observables mx and S are also given in the same units.

Parameters of the simulations

[McKeown Roiban, arXiv: 1308.4875]

Previous study

Boundary conditions

We use periodic BC for all the fields (antiperiodic temporal BC for fermions). In the infinite volume limit BC should not play a substantial role (unless what is studied is topological). Finite volume effects ⇠ e−m L ⌘ e−M N. Most run are done at M N = 4 (e−4 ' 0.02), some at M N = 6 (e−6 ' 0.002). Appear to play a role only in evaluating the coefficient of divergences. Simulations with Dirichlet BC (which we are going to do) are not expected to change the outcome significantly.

1/100 1/50 1/30 1/20 1/10 1/5 7.4 7.45 7.5 7.55 7.6 7.65 7.7 7.75 1/g c/2 Lm = 4 Lm = 6 g >= 50 g >= 10

g T/a × L/a Lm am τ S

int

τ mx

int

statistics [MDU] 5 16 × 8 4 0.50000 0.8 2.2 900 20 × 10 4 0.40000 0.9 2.6 900 24 × 12 4 0.33333 0.7 4.6 900,1000 32 × 16 4 0.25000 0.7 4.4 850,1000 48 × 24 4 0.16667 1.1 3.0 92,265 10 16 × 8 4 0.50000 0.9 2.1 1000 20 × 10 4 0.40000 0.9 2.1 1000 24 × 12 4 0.33333 1.0 2.5 1000,1000 32 × 16 4 0.25000 1.0 2.7 900,1000 48 × 24 4 0.16667 1.1 3.9 594,564 20 16 × 8 4 0.50000 5.4 1.9 1000 20 × 10 4 0.40000 9.9 1.8 1000 24 × 12 4 0.33333 4.4 2.0 850 32 × 16 4 0.25000 7.4 2.3 850,1000 48 × 24 4 0.16667 8.4 3.6 264,580 30 20 × 10 6 0.60000 1.3 2.9 950 24 × 12 6 0.50000 1.3 2.4 950 32 × 16 6 0.37500 1.7 2.3 975 48 × 24 6 0.25000 1.5 2.3 533,652 16 × 8 4 0.50000 1.4 1.9 1000 20 × 10 4 0.40000 1.2 2.7 950 24 × 12 4 0.33333 1.2 2.1 900 32 × 16 4 0.25000 1.3 1.8 900,1000 48 × 24 4 0.16667 1.3 4.3 150 50 16 × 8 4 0.50000 1.1 1.8 1000 20 × 10 4 0.40000 1.2 1.8 1000 24 × 12 4 0.33333 0.8 2.0 1000 32 × 16 4 0.25000 1.3 2.0 900,1000 48 × 24 4 0.16667 1.2 2.3 412 100 16 × 8 4 0.50000 1.4 2.7 1000 20 × 10 4 0.40000 1.4 4.2 1000 24 × 12 4 0.33333 1.3 1.8 1000 32 × 16 4 0.25000 1.3 2.0 950,1000 48 × 24 4 0.16667 1.4 2.4 541 Table 1: Parameters of the simulations: the coupling g, the temporal (T) and spatial (L) extent of the lattice in units of the lattice spacing a, the line of constant physics fixed by Lm and the mass parameter M = am. The size of the statistics after thermalization is given in the last column in terms of Molecular Dynamic Units (MDU), which equals an HMC trajectory

• f length one. In the case of multiple replica the statistics for each replica is given separately.

The auto-correlation times τ of our main observables mx and S are also given in the same units.

Parameters of the simulations

Measurement II: (derivative of the) cusp anomaly

We proceed subtracting the continuum extrapolation of c

2 multiplied by N2:

divergences appear to be completely subtracted, confirming their quadratic nature. Errors are small, and do not diverge for N → ∞. Flatness of data points indicates very small lattice artifacts. We can thus extrapolate at infinite N to show the continuum limit.

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 2 3 4 5 6 g = 5 g = 10 g = 20 g = 30 g = 50 g = 100 1/N (< SLAT > −c/2(2N 2))/S0 + ln(g)