Taming Defects in N=4 Super-Yang-Mills - Yifan Wang Harvard University (Virtual) Strings 2020 Cape Town, South Africa
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Taming Defects in N=4 Super-Yang-Mills Yifan Wang Harvard University (Virtual) Strings 2020 Cape Town, South Africa Based on 2003.11016, 2004.09514 [with Shota Komatsu], 2005.07197 and works in progress
Motivations = 4 SYM is perhaps the most well-studied interacting CFT • Share many features of strongly coupled gauge theories in four dimensions • A variety of methods: e.g. supersymmetric localization, integrability, bootstrap • Close connection to quantum gravity via AdS/CFT • In recent years, lots of progress in determining correlation functions of local operators
Motivations Rich family of defects of different co-dimensions • Sensitive to global structures • New structure constants for the full operator algebra • Refined consistency conditions on a full-fledged CFT • Most works focus on (order-type) Wilson loops. More general disorder-type defects. See also Charlotte’s talk Love defects!
A Program for Defects in = 4 SYM • Classification of (conformal) defects [Gaiotto-Witten, Gukov-Witten Kapustin, Many SUSY examples Kapustin-Witten,…] (e.g. Wilson-’t Hooft loops, Gukov-Witten surface operators, Gaiotto-Witten walls) Holographic descriptions Bootstrap [Karch-Randall, Aharony-Karch, Dewolfe-Freedman-Ooguri, Drukker-Gomis-Matsuura, D’Hoker-Estes-Gutperle,…] Definition • Tools to extract defect observables Analytic methods + Bootstrap philosophy: formulate consistency conditions, [Liendo-Rastelli-van Rees, Liendo-Meneghelli, constraints from unitarity and crossing Billo-Goncalves-Lauria-Meineri,…] • Implications for the fully extended CFT [Aharony-Seiberg- Tachikawa, Gaiotto- Kapustin-Seiberg- See also Clay’s talk Willet…] See Ibou’s talk Higher form symmetries, anomalies through world-volume couplings, complete data to specify extended CFT?
A Program for Defects in = 4 SYM • Classification of (conformal) defect [Gaiotto-Witten, Gukov-Witten Kapustin, Many SUSY examples Kapustin-Witten,…] (e.g. Wilson-’t Hooft loops, Gukov-Witten surface operators, Bootstrap Gaiotto-Witten walls) Definition Holographic descriptions [Karch-Randall, Aharony-Karch, Dewolfe-Freedman-Ooguri, D’Hoker-Estes-Gutperle,…] • Tools to extract defect observables [Liendo-Rastelli-van Rees, Analytic Liendo-Meneghelli, Billo-Goncalves-Lauria-Meineri,…] Bootstrap: Formulate consistency conditions, constraints from unitarity and crossing This talk • Implications for the fully extended CFT [Aharony-Seiberg- Tachikawa, Gaiotto- Kapustin-Seiberg- Willet…] Higher form symmetries, anomalies through world-volume couplings, complete data to specify extended CFT?
Toolkit for SYM λ= 2 Ng4 [YW ’20, Komatsu-YW ’20] Probe brane analysis in IIB on AdS5 × S 5 Results (e.g. interface defect) match at common regimes More Handles IIB String theory More planar loops Bootstrap of i ty integrable boundary states Localization of bil general 1/16 BPS gr a at finite λ for interface t e defect networks I n Emergent 2d/1d description Localization N Interface-defect One-point-function # of SUSY determined on the facets
Toolkit for SYM λ= 2 Ng4 To dig deeper … IIB String theory i ty bil r a e g nt I Localization N # of SUSY
Toolkit for SYM λ= 2 Ng4 IIB String theory y ? il i t [Liendo-Rastelli-van Rees, Conformal ab r bootstrap Liendo-Meneghelli, e g Billo-Goncalves-Lauria-Meineri,…] nt I Localization N # of SUSY
Emergent 2d/1d effective theory Defect-Yang-Mills for 1/16 BPS defect observables in = 4 SYM
Two important ingredients: 2d YM
Constrained to 0-instanton
Closed subsector
4 2
4d = 4 SYM on ℝ 2 8π 2d bosonic YM on SR
gYM =− 2 2
Compact 10d notation (M = 1,2,3,4 spacetime directions
g4 R
M = 5,6,7,8,9,0 R-symmetry directions)
1
(2 ) 2 ∫
1 1
2g4 ∫ℝ4
SSYM = − 2 4
d x tr FMN F MN − ΨΓM DM Ψ SYM( ) ≡ − dVS2 tr(⋆ℱ)2
gYM S2
≡ A + iϵijkϕi x kdx j (coupled to 3 of the 6 scalars: Φ7, Φ9, Φ0)
1/8-BPS Wilson loops Ordinary Wilson loops
∮
trrep Pe
{
{
SUSY localization wrt
2
Interacting = − 2(M⊥ − R56) Solvable
(complicated) (quasi-topological)
(1 | 1) ⊂ (2,2 | 4)
[Drukker-Giombi-Ricci-Trancanelli, Pestun, Giombi-Pestun, Giombi-Komatsu…]Two important ingredients: 1d TQM
Closed subsector
1
3d = 4 SCFT on S 3 1d TQM on a great Sφ
( 2)
a φ φ
Twist by u = cos sin
∫
2
Free hypermultiplets (qiI, q̃ jJ
) STQM = ℓ dφQ̃iI(D )ijQ jI
a a
Q(φ) = u qa(φ), Q̃ = u q̃a(φ)
Coupling to vector multiplets Gauged TQM
{
{
Supersymmetric localization wrt 3d
2
3d = − 2(M⊥ − RC) (1 | 1) ⊂ (4 | 4)
Strongly-coupled Solvable
(nontrivial RG) (mini bootstrap)
[Chester-Lee-Pufu-Yacoby, Beem-Peelaers-Rastelli, Dedushenko-Pufu-Yacoby, Dedushenko-Fan-Pufu …]A general picture [YW ’20] • The 3d supercharge 3d is the restriction of to a half- BPS interface? (1 | 1) ⊂ (4 | 4) ⊂ (2,2 | 4) • Study cohomology among general supersymmetric defect observables in the SYM A rich zoo! • localization of the defect-enriched SYM gives rise to defect-enriched 2d YM. Defect-Yang-Mills
1/16 BPS defect networks [YW ’20] In general can have Local op. nontrivial topologies (or twist insertion) (sensitive to global structures): e.g. link of surface operators with line Line defect operators. Op. stuck to interface 2 4d SYM on S 4 2d dYM on S Focus • Boundary/Interface Op. stuck to loop • Boundary/Interface here • Surface operators: transverse • Twist operators, or longitudinal to S 2 or Higgsing the gauge group • Line operators: • Wilson loops, Wilson, ’t Hooft lines Dictionary or (unstable) 2d instantons [Giombi-Pestun] • Local operators • Field strength insertions [Giombi-Pestun] • Local operators on the defect world volume … • … [Giombi-Komatsu]
Application D5-brane interface defects
D5-brane interface for SYM: brane picture • Engineered by intersection of one D5 brane with D3 branes where k D3 branes can end x1 = 0 x⊥ : x8, x9, x0 D5 NS5 x⊥ : x5, x6, x7 S-dual x∥ : x2, x3, x4 x1 x1
D5-brane interface for SYM: field theory
• Interpolate between U(N) and U(N + k) SYM at the same coupling
• Singular Nahm pole configuration
⃗
(related to usual N.P. bc by folding trick)
X
X ⃗ = (Φ8, Φ9, Φ0), Y ⃗ = (Φ5, Φ6, Φ7)
X −⃗
( ⋅ ⋅)
+
Aμ =
⋅
,
Aμ−
Y +⃗ = ( Y ⋅ ), ⃗
−
X +⃗ =
⋅
−t⃗
⋅ ⋅ ⋅ x1
k × k block x1
{
Nahm [ti, tj] = ϵijktk given by SU(2) reps
Pole
Config k×k i
t3 = − Diag[k − 1,k − 3,…,1 − k]
2D5-brane interface for SYM: field theory • Hard: disorder-type defect, “non-Lagrangian” • Use Mirror description [Gaiotto,Witten] Superconformal NP BC as IR fixed point Interface theory: 1 2 ... N -1 N N +k N + k -1 ... 2 1 “Bad quiver” • Global symmetry U(N + k) × U(N) (enhancement) Gluing by 3d vectors U(N + k) SYM U(N) SYM with Dirichlet BC with Dirichlet BC 4 • A UV Lagrangian for SYM on S coupled to the D5 interface at equator that preserves ∈ (4 | 4,ℝ) ⊂ (2,2 | 4) For SUSY observables this suffices by non-renormalization [YW ’20, Komatsu-YW ’20]
Defect-Yang-Mills for D5-brane interface • Run localization by to find 2d/1d description ∫S2 −SYM( )−SYM( ′)−STQM( , ′,Q,Q̃) ZdYM = D |HS2 D ′|HS2 DQDQ̃ |S1 e − + TQM YM 1 2 ∫ 2 SYM( ) ≡ − dVS2 tr(⋆ℱ) gYM HS−2 • Ready to use to compute general 1/16 BPS SYM observables in the presence of the D5-brane interface [YW ’20, Komatsu-YW ’20]
Defect One-point function [YW ’20, Komatsu-YW ’20] Basic defect observables: higher point correlators determined by bulk OPE s⊥(x) h ⟨ (x)⟩HS4 = Fixed by conformal s⊥(x) Δ Ward identity J Take J ≡ tr (Φ7 + iΦ8) half-BPS J (−i) ZdYM ∫ Denote gauge fields J −SYM( )−STQM(Q,Q̃, ) ⟨ J⟩dSYM = D DQDQ̃ tr (⋆ℱ) e on the two hemispheres collectively by • Compute by first integrate out 1d fields (Q, Q̃) • Obtain matrix model for ⟨ J⟩dSYM by 2d gauge theory techniques
Full D5 brane matrix model at any (N, λ) deg-J polynomial from G.S. procedure One-point-function of BPS operator k−1 2 N Δ(a) 2 ∫ ∑ ∑ ⟨ J⟩k = [da] fJ(aj) + fJ(is) ∏j=1 2 cosh π (aj + 2 ) N ki k−1 j=1 s=− 2 k−1 Between U(N) and U(N + k) ( ) 2 8π 2 N N − λ ∑i=1 ai2 ∏ ∏ × aj − is e k−1 n s=− 2 Normalize by taking ratio One-point-function of the interface k−1 2 N Δ(a) ( ) 2 ∫ 8π 2 N N − λ ∑i=1 ai2 ∏ ∏ ⟨ ⟩k = [da] a − is e ∏j=1 2 cosh π (aj 2) j N ki + s=− k − 1 j=1 2 [Komatsu-YW ’20]
Large N One-point functions and checks [YW ’20, Komatsu-YW ’20] J i ⟨ J⟩k = J ( + ) g= λ 2 2 J 4π 2πi ( x2 ) [ x )] x π coth πg (x + dx 1 J 1 ∮ g 1 − k ∈ 2ℤ + 1 = 2πi ( x2 ) [ x )] x π tanh πg (x + dx 1 J 1 ∮ g 1 − k ∈ 2ℤ k−1 2 2 2 u+ u − 4g (xs + δJ,2) J Zhukovsky ∑ = xu = Hint for integrability! 2g k−1 s=− 2 Large λ: check with IIB string theory answers [Nagasaki-Yamaguchi] 2 • Probe D5 brane on AdS4 × S with k units of w.v. flux • One-point function from w.v. couplings to closed string KK modes Small λ: check perturbation theory [Buhl-Mortensen-de Leeuw-Ipsen-Kristjansen-Wilhelm,Kristjansen-Müller-Zarembo] ]
Bootstrap approach to Integrable boundary states for interface defects in = 4 SYM
D5 brane and Integrable spin chain Traditional Spin chain approach [Minahan-Zarembo]: = tr(…ZYZ…ZYZ…) “Y magnons” • | ⟩ represented by a matrix-product state • | ⟩ represented by Bethe states, labelled by rapidities (momentum) um of the magnons • One-point function from overlap • Limitation to weak coupling (nearest-neighbor interactions, fixed length) [Buhl-Mortensen, Gimenez Grau, Ipsen, de Leeuw, Linardopoulos, Kristjansen, Pozsgay, Volk, Widen, Wilhelm, Zarembo … ]
D5 brane and Integrable field theory Boundary state 5 D5-brane interface defect on IIB worldsheet in AdS5 × S Large N (lightcone G.S.) • Single trace operators (non-BPS) - closed string state | Ψ⟩ • One-point function from overlap ⟨ ⟩ = ⟨ | Ψ⟩ Weak coupling evidences for defect observable: 1) Rapidities pair up (u1, − u1, u2, − u2, …, uM/2, − uM/2), i.e. no particle production 2) Overlaps involve ratio of Gaudin determinants [Jiang-Komatsu-Vescovi] is Integrable [Komatsu-YW ’20]
General Strategy for non-perturbative boundary state Apply bootstrap philosophy to integrable boundary state [Ghoshal-Zamolodchikov, Jiang-Komatsu-Vescovi] • Integrability implies general overlaps determined by two particle overlaps e.g. ⟨ | Y(u)Y(ū)⟩ • Explore symmetry constraints under SU(2 | 1) × SU(2 | 1) ⊂ SU(2 | 2) × SU(2 | 2)c.e. • Impose consistency conditions Bulk S-matrix Boundary Young-Baxter Boundary unitarity [Komatsu-YW ’20] Watson (crossing) See also [Gombor-Bajnok]
Large J R-charge non-BPS one-point function SU(2) sector: (k) cu single trace operators of length L ⟨ u(x)⟩ k = built out of Z = Φ7 + iΦ8 and 2J Lx⊥Δ excitations Y = Φ5 + iΦ9 i k−1 (a) ⟨ k | u⟩ Q( 2 )Q(0) M (∏ ) det G− (k) 2 det G+ ∑ cu ≡ = k ik σB(um) k−1 a=− ⟨u | u⟩ Q( 2 ) m=1 2 Extend to SO(6) in [Gombor-Bajnok] ( 2 ) 2 M ik ∏ k−1 Q( ) M Q(u) = (u − um) 2 J+M (a) m=1 ∑ ∏ k ≡ (xa) σ̃ k (um) k−1 Q (ia)Q (ia) m=1 − + Important: excited boundary stares a=− 2 (bound states with bdy) • Zero magnon M = 0 case gives large J BPS one point function • Weak coupling expansion agrees with spin chain method More checks in [Kristjansen-Müller-Zarembo] • Finite J (non-pert.) from wrapping corrections (using TBA and analytic continuation)
Summary A toolkit for defects in = 4 SYM • Localization: 1/16 BPS, arbitrary N, arbitrary λ • Integrability: non-BPS, large N, arbitrary λ • Application: Interface-defect one-point-function
More applications and future directions • Correlation functions of both bulk and defect operators [YW ’20] 4 2 • Unorientable spacetimes, e.g. SYM ℝℙ governed by YM on ℝℙ [YW ‘20] • Surface defects (linked with line defects) • Discrete theta angles and global structures of gauge theories: from 4d SYM to 2d dYM 5 • Classification of integrable boundary states in IIB string on AdS5 × S e.g. D7, D-instanton, more exotic ones? • Topological defects (bi-branes) on w.s. sigma model are known to generate boundary states [Fuchs-Schweigert-Waldorf, Kojita-Maccaferri-Masuda-Schnabl] Relation to SYM interfaces? • T T̄ deformation: 2d YM has solvable T T̄ [Sentilli-Tierz] learn about T T̄ deformations 4d = 4 SYM [Caetano-Rastelli-Peelaers] • SL(2,ℤ) properties: Recent progress in SL(2,ℤ) invariant four point functions [Chester-Green-Pufu-YW-Wen] including defects which transform nontrivially under SL(2,ℤ)? • Extend the defect analysis to other AdS/CFT pairs? e.g. deriving AdS3 /CFT2 beyond local correlators by incorporating defects [See Rajesh’s talk]
Thank you!
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