Anomalous Dimensions @ Strong Coupling Luca Mazzucato Simons Center - PowerPoint PPT Presentation

Anomalous Dimensions @ Strong Coupling Luca Mazzucato Simons Center for Geometry and Physics, Stony Brook based on [Brenno Carlini Vallilo, LM, arXiv:1102.1219 [hep-th] ] for a pedagogical review, [LM, arXiv:1104.2604][hep-th] ] XI Workshop on Nonperturbative QCD, Paris Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Outline Motivations: conformal dimensions AdS/CFT recap @ Strong coupling with string theory Konishi multiplet on the string worldsheet Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Outline Motivations: conformal dimensions AdS/CFT recap @ Strong coupling with string theory Konishi multiplet on the string worldsheet Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Outline Motivations: conformal dimensions AdS/CFT recap @ Strong coupling with string theory Konishi multiplet on the string worldsheet Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Outline Motivations: conformal dimensions AdS/CFT recap @ Strong coupling with string theory Konishi multiplet on the string worldsheet Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Motivations What are anomalous dimensions in QFT? Take a local gauge invariant op O ( x ) = Tr φ i ( x ) φ i ( x ) In a CFT e.g. N = 4 SYM: global symmetry ⊃ dilations D [ D , O ( x )] = i ∆ O ( x ) ∆ ≡ conformal dimension ↔ 2 pt functions 1 �O ( x ) O ( y ) � ∼ | x − y | 2 ∆ Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Motivations What are anomalous dimensions in QFT? Take a local gauge invariant op O ( x ) = Tr φ i ( x ) φ i ( x ) In a CFT e.g. N = 4 SYM: global symmetry ⊃ dilations D [ D , O ( x )] = i ∆ O ( x ) ∆ ≡ conformal dimension ↔ 2 pt functions 1 �O ( x ) O ( y ) � ∼ | x − y | 2 ∆ Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Motivations What are anomalous dimensions in QFT? Take a local gauge invariant op O ( x ) = Tr φ i ( x ) φ i ( x ) In a CFT e.g. N = 4 SYM: global symmetry ⊃ dilations D [ D , O ( x )] = i ∆ O ( x ) ∆ ≡ conformal dimension ↔ 2 pt functions 1 �O ( x ) O ( y ) � ∼ | x − y | 2 ∆ Luca Mazzucato Anomalous Dimensions @ Strong Coupling

What do we know about ∆ ? hereafter: Planar limit N → ∞ g YM = 0 : classically ∆ = ∆ 0 ≡ "engineering dimension" E.g. Tr φ i φ i has ∆ 0 = 2. Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Turn on ’t Hooft coupling λ = g 2 YM N ⇒ generate anomalous dimensions ≡ ∆ − ∆ 0 � = 0 Luca Mazzucato Anomalous Dimensions @ Strong Coupling

What do we know about ∆ ? hereafter: Planar limit N → ∞ λ = 0 : classically ∆ = ∆ 0 ≡ "engineering dimension" λ << 1 : weak coupling planar loop expansion ∆ − ∆ 0 = c 1 λ + c 2 λ 2 + . . . Luca Mazzucato Anomalous Dimensions @ Strong Coupling

What do we know about ∆ ? hereafter: Planar limit N → ∞ λ = 0 : classically ∆ = ∆ 0 ≡ "engineering dimension" λ << 1 : weak coupling planar loop expansion ∆ − ∆ 0 = c 1 λ + c 2 λ 2 + . . . Luca Mazzucato Anomalous Dimensions @ Strong Coupling

: strong coupling! λ >> 1 Luca Mazzucato Anomalous Dimensions @ Strong Coupling

What do we know about ∆ ? hereafter: Planar limit N → ∞ λ = 0 : classically ∆ = ∆ 0 ≡ "engineering dimension" λ << 1 : weak coupling ∆ − ∆ 0 = c 1 λ + c 2 λ 2 + . . . λ >> 1 : strong coupling protected (BPS) : ∆ − ∆ 0 = 0 √ 4 ∆ − ∆ 0 ∼ short non-BPS : λ + . . . √ long non-BPS : ∆ − ∆ 0 ∼ λ + . . . Luca Mazzucato Anomalous Dimensions @ Strong Coupling

What do we know about ∆ ? hereafter: Planar limit N → ∞ λ = 0 : classically ∆ = ∆ 0 ≡ "engineering dimension" λ << 1 : weak coupling ∆ − ∆ 0 = c 1 λ + c 2 λ 2 + . . . λ >> 1 : strong coupling protected (BPS) : ∆ − ∆ 0 = 0 √ 4 ∆ − ∆ 0 ∼ short non-BPS : λ + . . . √ long non-BPS : ∆ − ∆ 0 ∼ λ + . . . Luca Mazzucato Anomalous Dimensions @ Strong Coupling

What do we know about ∆ ? hereafter: Planar limit N → ∞ λ = 0 : classically ∆ = ∆ 0 ≡ "engineering dimension" λ << 1 : weak coupling ∆ − ∆ 0 = c 1 λ + c 2 λ 2 + . . . λ >> 1 : strong coupling protected (BPS) : ∆ − ∆ 0 = 0 √ 4 ∆ − ∆ 0 ∼ short non-BPS : λ + . . . √ long non-BPS : ∆ − ∆ 0 ∼ λ + . . . Luca Mazzucato Anomalous Dimensions @ Strong Coupling

@ Strong coupling? Use holography! [Maldacena ’97] gauge gravity IIB string theory on AdS 5 × S 5 N = 4 SYM = √ radius 2 /α ′ λ = λ >> 1 = small curvature strongly coupled CFT = perturbative string theory CFT conformal dim. ∆ = E energy in AdS State of the Art: [Gromov et al. ’09] conjectured a set of Y-system eqs. whose numerical solution gives ∆( λ ) at any value of the coupling. Problem: very hard coupled integral eqs.! Luca Mazzucato Anomalous Dimensions @ Strong Coupling

@ Strong coupling? Use holography! [Maldacena ’97] gauge gravity IIB string theory on AdS 5 × S 5 N = 4 SYM = √ radius 2 /α ′ λ = λ >> 1 = small curvature strongly coupled CFT = perturbative string theory CFT conformal dim. ∆ = E energy in AdS State of the Art: [Gromov et al. ’09] conjectured a set of Y-system eqs. whose numerical solution gives ∆( λ ) at any value of the coupling. Problem: very hard coupled integral eqs.! Luca Mazzucato Anomalous Dimensions @ Strong Coupling

@ Strong coupling? Use holography! [Maldacena ’97] gauge gravity IIB string theory on AdS 5 × S 5 N = 4 SYM = √ radius 2 /α ′ λ = λ >> 1 = small curvature strongly coupled CFT = perturbative string theory CFT conformal dim. ∆ = E energy in AdS State of the Art: [Gromov et al. ’09] conjectured a set of Y-system eqs. whose numerical solution gives ∆( λ ) at any value of the coupling. Problem: very hard coupled integral eqs.! Luca Mazzucato Anomalous Dimensions @ Strong Coupling

What do we know about ∆ ? hereafter: Planar limit N → ∞ λ = 0 : classically ∆ = ∆ 0 ≡ "engineering dimension" λ << 1: weak coupling ∆ − ∆ 0 = c 1 λ + c 2 λ 2 + . . . λ >> 1: strong coupling AdS/CFT dictionary BPS : ∆ − ∆ 0 = 0 massless vertex op = SUGRA √ 4 short non-BPS : ∆ − ∆ 0 ∼ λ + . . . massive vertex op = quantum strings √ long non-BPS : ∆ − ∆ 0 ∼ λ + . . . semi-classical strings [Beisert et al. ’10] Luca Mazzucato Anomalous Dimensions @ Strong Coupling

What do we know about ∆ ? hereafter: Planar limit N → ∞ λ = 0 : classically ∆ = ∆ 0 ≡ "engineering dimension" λ << 1: weak coupling ∆ − ∆ 0 = c 1 λ + c 2 λ 2 + . . . λ >> 1: strong coupling AdS/CFT dictionary BPS : ∆ − ∆ 0 = 0 massless vertex op = SUGRA √ 4 short non-BPS : ∆ − ∆ 0 ∼ λ + . . . massive vertex op = quantum strings √ long non-BPS : ∆ − ∆ 0 ∼ λ + . . . semi-classical strings [Beisert et al. ’10] Luca Mazzucato Anomalous Dimensions @ Strong Coupling

What do we know about ∆ ? hereafter: Planar limit N → ∞ λ = 0 : classically ∆ = ∆ 0 ≡ "engineering dimension" λ << 1: weak coupling ∆ − ∆ 0 = c 1 λ + c 2 λ 2 + . . . λ >> 1: strong coupling AdS/CFT dictionary BPS : ∆ − ∆ 0 = 0 massless vertex op = SUGRA √ 4 short non-BPS : ∆ − ∆ 0 ∼ λ + . . . massive vertex op = quantum strings √ long non-BPS : ∆ − ∆ 0 ∼ λ + . . . semi-classical strings [Beisert et al. ’10] Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Anomalous dimensions @ strong coupling ∆ = E ⇒ Compute the spectrum of perturbative string states in AdS in the near flat space limit √ √ λ + c 1 + c 2 4 √ E ∼ c 0 + O ( 1 / λ ) 4 λ � n /α ′ Flat space is mass ∼ Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Anomalous dimensions @ strong coupling ∆ = E ⇒ Compute the spectrum of perturbative string states in AdS in the near flat space limit √ √ λ + c 1 + c 2 4 √ E ∼ c 0 + O ( 1 / λ ) 4 λ � n /α ′ Flat space is mass ∼ Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Anomalous dimensions @ strong coupling: Konishi multiplet Simple string: sitting still at the center of AdS (with same quantum #s as SYM op. in the Konishi multiplet) Type IIB string σ model in AdS 5 × S 5 Compute worldsheet physical state condition (Virasoro) at 1-loop T ∼ E ( E − 4 ) √ + quantum corrections = 0 λ Solve for E = E ( λ ) AdS/CFT dictionary: ∆ = E [LM+Vallilo ’11] Luca Mazzucato Anomalous Dimensions @ Strong Coupling

Anomalous dimensions @ strong coupling: Konishi multiplet Simple string: sitting still at the center of AdS (with same quantum #s as SYM op. in the Konishi multiplet) Type IIB string σ model in AdS 5 × S 5 Compute worldsheet physical state condition (Virasoro) at 1-loop T ∼ E ( E − 4 ) √ + quantum corrections = 0 λ Solve for E = E ( λ ) AdS/CFT dictionary: ∆ = E [LM+Vallilo ’11] Luca Mazzucato Anomalous Dimensions @ Strong Coupling