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

Worldsheet string theory in AdS/CFT and lattice Valentina Forini Humboldt University Berlin Junior Research Group “Gauge fields from strings” Women at the intersection of Mathematics and High Energy Physics, Mainz Institute for Theoretical Physics, March 8 2017

Framework String/gauge correspondence, addresses together I understanding gauge theories at all values of the coupling I understanding string theories in non-trivial backgrounds “Quark-antiquark” potential Z Type IIB strings in DXD Ψ e − S string ⇠ e − Area reg � C = Z string � R ⇥ S 1 C AdS/CFT h W [ C ] i = 1 x µ + Φ i ˙ y i ) ds H ( iA µ ˙ super Yang-Mills in 4d N = 4 N Tr P e

Motivation Beautiful progress in obtaining exact results within AdS/CFT f ( g ) = c g + d + e f ( g ) g + . . . Perturbative Perturbative string sigma model string sigma model q Integrability Integrability / Localization / Localization Perturbative Perturbative q g 2 f ( g ) = a g 2 + b g 4 + . . . Y M N R 2 √ g g gauge theory gauge theory λ 4 πα 0 and ≡ g := 4 π = 4 π AdS/CFT AdS/CFT I from integrability (assumed) I from supersymmetric localization (supersymmetric observables)

Motivation Beautiful progress in obtaining exact results within AdS/CFT f ( g ) = c g + d + e f ( g ) g + . . . Perturbative Perturbative string sigma model string sigma model q Integrability Integrability / Localization / Localization Perturbative Perturbative q g 2 f ( g ) = a g 2 + b g 4 + . . . Y M N R 2 √ g g gauge theory gauge theory λ 4 πα 0 and ≡ g := 4 π = 4 π AdS/CFT AdS/CFT I from integrability (assumed) I from supersymmetric localization (supersymmetric observables) In the world-sheet string theory integrability only classically, localization not formulated. 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. Call for genuine 2d QFT to cover the finite-coupling region.

Lattice techniques in AdS/CFT f ( g ) Exciting program on the q 4d susy CFT side, Lattice 4d N=4 SYM subtleties with supersymmetry. [Catterall et al.] g AdS/CFT

Lattice techniques in AdS/CFT Lattice 2d f ( g ) Green-Shwarz string Talks by Shaich, Giedt, Anosh q Lattice for superstring world-sheet Lattice 4d in AdS 5 × S 5 N=4 SYM g AdS/CFT [previous study: Roiban McKeown 2013] 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 AdS 5 × S 5 + RR flux x AdS 5 S 5 [Metsaev Tseytlin 1998] P SU (2 , 2 | 4) Non-linear sigma-model on G/H = SO (1 , 4) × SO (5) Z ∂ a X µ ∂ a X ν G µ ν + ¯ θ Γ ( D + F 5 ) θ ∂ X + ¯ θ ∂θ ¯ ⇥ ⇤ S = g d τ d σ θ ∂θ + . . . Z ⇥ ⇤ Symmetries: I global PSU (2 , 2 | 4) , local bosonic (diffeomorphism) and fermionic ( κ -symmetry) Z I classical integrability P SU (2 , 2 | 4) manifest when written as sigma-model action on G/H = SO (1 , 4) × SO (5) .

Green-Schwarz string in AdS 5 × S 5 + RR flux x AdS 5 S 5 [Metsaev Tseytlin 1998] P SU (2 , 2 | 4) Non-linear sigma-model on G/H = SO (1 , 4) × SO (5) Z ∂ a X µ ∂ a X ν G µ ν + ¯ θ Γ ( D + F 5 ) θ ∂ X + ¯ θ ∂θ ¯ ⇥ ⇤ S = g d τ d σ θ ∂θ + . . . × Highly non-linear, to quantize it use semiclassical methods Γ 0 + Γ 1 g + Γ 2 h i X = X cl + ˜ X Γ = g g 2 + . . . − → R

[ VF Beccaria Dunne Tseytlin, Drukker, Giangreco Ohlson Sax Vescovi .... Green-Schwarz string in AdS 5 × S 5 + RR flux perturbatively Highly non-linear, to quantize it use semiclassical methods √ E 0 + E 1 + E 2 � h i X = X cl + ˜ X E = g g 2 + . . . , g = − → g 4 ⇡ I General analysis of fluctuations in terms of background geometry, e.g. Tr( M ) = a (2) R + b Tr( K 2 ) . [Alvarez-Gaume, Freedman, Mukhi, 81] [Drukker Gross Tseytlin 00] [ VF Giangreco Griguolo Seminara Vescovi 15] I Explicit analytic form of one-loop partition function Z = det O F / √ det O B for a class of effectively one-dimensional problems. Non-trivial differential operators, e.g. elliptic-function potentials: σ + ω 2 + k 2 sn 2 ( σ , k 2 ) . Then use Gelf’and-Yaglom method: O = − ∂ 2 det O u ( L ) O φ ( x ) = λ φ ( x ) , φ (0) = φ ( L ) = 0 = det O free u free ( L ) where u are solutions of auxiliary boundary value problem, u (0) = 0 , u 0 (0) = 1 . Several configurations (GKP string, quark-antiquark potential, generalized cusp) have been “solved” this way at one loop, and agree with predictions.

1/2 BPS circular Wilson loop [Erickson, Semenoff, Zarembo 00] [Drukker Gross 00] 
 [Pestun 07] p p λ � 3 4 log λ + 1 2 log 2 π + O ( λ � 1 2 2 ) log h W ( λ ) i = log λ I 1 ( λ ) = p The 1-loop disc partition function log Z = log h W i differs: 1 1 [Kruczenski Tirziu 08] 2 log 2 π [Buchbinder Tseytlin 14] 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]

1/2 BPS circular Wilson loop [Drukker 06] 
 [Drukker Giombi Ricci Trancanelli 07] [Pestun 09] p p λ � 3 4 log λ + 1 2 log 2 π + O ( λ � 1 2 2 ) log h W ( λ ) i = log λ I 1 ( λ ) = p The 1-loop disc partition function log Z = log h W i differs: 1 1 [Kruczenski Tirziu 08] 2 log 2 π [Buchbinder Tseytlin 14] 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 θ 0 in S 2 2 S 5 ( λ 0 = λ cos 2 θ 0 ). p log h W ( λ , θ 0 ) i λ (cos θ 0 � 1) � 3 2 log cos θ 0 + O ( λ � 1 2 ) h W ( λ , 0) i =

1/2 BPS circular Wilson loop [Drukker 06] 
 [Drukker Giombi Ricci Trancanelli 07] [Pestun 09] p p λ � 3 4 log λ + 1 2 log 2 π + O ( λ � 1 2 2 ) log h W ( λ ) i = log λ I 1 ( λ ) = p The 1-loop disc partition function log Z = log h W i differs: 1 1 [Kruczenski Tirziu 08] 2 log 2 π [Buchbinder Tseytlin 14] 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 θ 0 in S 2 2 S 5 ( λ 0 = λ cos 2 θ 0 ). h W i p p log h W ( λ , θ 0 ) i log h W ( λ , θ 0 ) i λ (cos θ 0 � 1) � 3 λ (cos θ 0 � 1) � 3 2 log cos θ 0 + log cos θ 0 2 log cos θ 0 + O ( λ � 1 2 + O ( λ � 1 2 ) 2 ) h W ( λ , 0) i = h W ( λ , 0) i = h W i p λ (cos θ 0 � 1) + 3 0 + O ( λ � 1 4 θ 2 2 ) � = � > unphysical cutoff Usual (Gelf’and Yaglom) method fails. > different regulariz. in τ and in σ Perturbative heat-kernel (near AdS 2 expansion) agrees. [Forini Tseytlin Vescovi 17] �

Green-Schwarz string in AdS 5 × S 5 + RR flux perturbatively Green-Schwarz string in AdS 5 × S 5 + RR flux perturbatively Highly non-linear, to quantize it use semiclassical methods Highly non-linear, to quantize it use semiclassical methods Γ 0 + Γ 1 Γ 0 + Γ 1 g + Γ 2 g + Γ 2 h h i i X = X cl + ˜ X = X cl + ˜ X X Γ = g Γ = g g 2 + . . . g 2 + . . . − − → → 2 loops is current limit: “homogenous” configs, “AdS light-cone” gauge-fixing 2 loops is current limit: “homogenous” configs, “AdS light-cone” gauge-fixing [Metsaev, Tseytlin] [Metsaev, Thorn, Tseytlin] [Giombi Ricci Roiban Tseytlin 09] [Bianchi 2 Bres VF Vescovi 14] [Giombi Ricci Roiban Tseytlin 09, Bianchi 2 Bres VF Vescovi 14] 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). Efficient alternative to Feynman diagrams for on-shell objects (worldsheet S-matrix) Efficient alternative to Feynman diagrams for on-shell objects (worldsheet S-matrix) unitarity cuts (on-shell methods) in d=2 unitarity cuts (on-shell methods) in d=2 Indirect evidence of quantum integrability! Indirect evidence of quantum integrability! [Bianchi VF Hoare 2013][Engelund Roiban 2013] [Bianchi Hoare 14] [Bianchi VF Hoare 2013][Engelund Roiban 2013] [Bianchi Hoare 14] Indirect evidence of quantum integrability!

Green-Schwarz string in AdS 5 × S 5 + RR flux perturbatively Unitarity cuts in d = 2 , for worldsheet amplitudes (integrable S-matrix) [Bianchi VF Hoare 13][Engelund, Roiban 13][Bianchi Hoare 14] X ⇣ ⌘ p 1 p 4 R Q M X � A (0) � 2 I bubble , X � A (1) = � l 1 A (0) A (0) l 2 d 2 q 1 Z I bubble ( p ) = N P p 2 p 3 S (2 ⇡ 2 ) 2 ( q 2 − 1 + i ✏ ) (( q − p ) 2 − 1 + i ✏ ) 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