Integrability and the Conformal Field Theory of the Higgs branch Bogdan Stefański, jr. City University London 17 November 2015 Based on 1411.3676 , JHEP 1506 (2014) 103 with O. Ohlsson Sax, A. Sfondrini
AdS 3 with 8 + 8 susys and integrability � T 4 � String theory on AdS 3 × S 3 × M 4 where M 4 = S 3 × S 1
AdS 3 with 8 + 8 susys and integrability � T 4 � String theory on AdS 3 × S 3 × M 4 where M 4 = S 3 × S 1 supported by R-R ⊕ NS-NS 3-form flux
AdS 3 with 8 + 8 susys and integrability � T 4 � String theory on AdS 3 × S 3 × M 4 where M 4 = S 3 × S 1 supported by R-R ⊕ NS-NS 3-form flux � All-loop integrable wsheet 2-body S matrix known [Borsato, Ohlsson Sax, Lloyd, Sfondrini, BS, Torrielli]
AdS 3 with 8 + 8 susys and integrability � T 4 � String theory on AdS 3 × S 3 × M 4 where M 4 = S 3 × S 1 supported by R-R ⊕ NS-NS 3-form flux � All-loop integrable wsheet 2-body S matrix known [Borsato, Ohlsson Sax, Lloyd, Sfondrini, BS, Torrielli] � expectation : integrability solves spectral problem (from string side) [Abbott, Aniceto, Babichenko, Bianchi, Beccaria, David, Dekel, Engelund, Hernandez, Hoare, Levkovich-Maslyuk, Macorini, McKeown, Nieto, Pittelli, Prinsloo, Regelskis, Roiban, Sahoo, Stepanchuk, Sundin, Tseytlin, Wolf, Wulff, Zarembo]
AdS 3 with 8 + 8 susys and integrability � T 4 � String theory on AdS 3 × S 3 × M 4 where M 4 = S 3 × S 1 supported by R-R ⊕ NS-NS 3-form flux � All-loop integrable wsheet 2-body S matrix known [Borsato, Ohlsson Sax, Lloyd, Sfondrini, BS, Torrielli] � expectation : integrability solves spectral problem (from string side) [Abbott, Aniceto, Babichenko, Bianchi, Beccaria, David, Dekel, Engelund, Hernandez, Hoare, Levkovich-Maslyuk, Macorini, McKeown, Nieto, Pittelli, Prinsloo, Regelskis, Roiban, Sahoo, Stepanchuk, Sundin, Tseytlin, Wolf, Wulff, Zarembo] � This talk focuses on pure R-R, M 4 = T 4 case Global symmetry is psu ( 1 , 1 | 2 ) 2
CFT 2 integrability: expectations from AdS 3 � λ − → 0 limit gives local spin-chain [Ohlsson Sax, BS, Torrielli ’11]
CFT 2 integrability: expectations from AdS 3 � λ − → 0 limit gives local spin-chain [Ohlsson Sax, BS, Torrielli ’11] � Sites of spin-chain L ⊗ R
CFT 2 integrability: expectations from AdS 3 � λ − → 0 limit gives local spin-chain [Ohlsson Sax, BS, Torrielli ’11] � Sites of spin-chain L ⊗ R � S b L sites : 0 ⊕ 2 ⊕ S f 1 S b / S f 2 -BPS irrep of psu ( 1 , 1 | 2 ) with bos/ferm h.w. state
CFT 2 integrability: expectations from AdS 3 � λ − → 0 limit gives local spin-chain [Ohlsson Sax, BS, Torrielli ’11] � Sites of spin-chain L ⊗ R � S b L sites : 0 ⊕ 2 ⊕ S f 1 S b / S f 2 -BPS irrep of psu ( 1 , 1 | 2 ) with bos/ferm h.w. state � R sites same as L sites
Outline 1 UV gauge theory • D1/D5 system, L UV • Coulomb and Higgs branches in UV and IR 2 CFT H • L IR • ADHM σ -model and small instantons • origin of the Higgs branch, L eff 3 ∆ from L eff and spin-chains 4 Outlook and Conclusions
UV gauge theory: D1-D5 system � D1-D5 branes 0 1 2 3 4 5 6 7 8 9 N c × D1 • • N f × D5 • • • • • •
UV gauge theory: D1-D5 system � D1-D5 branes 0 1 2 3 4 5 6 7 8 9 N c × D1 • • N f × D5 • • • • • • � �� � R-symmetry: su ( 2 ) L × su ( 2 ) R
UV gauge theory: D1-D5 system � D1-D5 branes 0 1 2 3 4 5 6 7 8 9 N c × D1 • • N f × D5 • • • • • • � D1: ( 8 , 8 ) susy U ( N c ) vector mplet - dim. red. of N = 4 SYM
UV gauge theory: D1-D5 system � D1-D5 branes 0 1 2 3 4 5 6 7 8 9 N c × D1 • • N f × D5 • • • • • • � D1: ( 8 , 8 ) susy U ( N c ) vector mplet - dim. red. of N = 4 SYM � D5: break susy to ( 4 , 4 )
UV gauge theory: ( 4 , 4 ) susy 2d QCD Open-string low-energy dofs are gluons A µ , quarks λ and (4,4) susy
UV gauge theory: ( 4 , 4 ) susy 2d QCD Open-string low-energy dofs are gluons A µ , quarks λ and (4,4) susy • D1-D1 strings ← → ( 8 , 8 ) U ( N c ) vector-multiplet: φ i , ψ , A µ , D ( 4 , 4 ) vector Φ : t a , χ ( 4 , 4 ) hyper T :
UV gauge theory: ( 4 , 4 ) susy 2d QCD Open-string low-energy dofs are gluons A µ , quarks λ and (4,4) susy • D1-D1 strings ← → ( 8 , 8 ) U ( N c ) vector-multiplet: φ i , ψ , A µ , D ( 4 , 4 ) vector Φ : t a , χ ( 4 , 4 ) hyper T : • D1-D5 strings ← → ( 4 , 4 ) U ( N c ) × U ( N f ) hyper-multiplets: h a , λ ( 4 , 4 ) hyper H :
UV gauge theory: ( 4 , 4 ) susy 2d QCD Open-string low-energy dofs are gluons A µ , quarks λ and (4,4) susy • D1-D1 strings ← → ( 8 , 8 ) U ( N c ) vector-multiplet: φ i , ψ , A µ , D ( 4 , 4 ) vector Φ : t a , χ ( 4 , 4 ) hyper T : • D1-D5 strings ← → ( 4 , 4 ) U ( N c ) × U ( N f ) hyper-multiplets: h a , λ ( 4 , 4 ) hyper H : • D5-D5 strings decouple: suppressed by large V 6789
UV gauge theory: L UV fixed by susy 1 L UV = L Φ ( Φ ) + L T ( T , Φ ) + L H ( H , Φ ) g 2 YM
UV gauge theory: L UV fixed by susy 1 L UV = L Φ ( Φ ) + L T ( T , Φ ) + L H ( H , Φ ) g 2 YM � �� � dim. red. of L N = 4 SYM
UV gauge theory: L UV fixed by susy 1 L UV = L Φ ( Φ ) + L T ( T , Φ ) + L H ( H , Φ ) g 2 YM where F 2 + ( ∇ φ ) 2 + i ¯ ψ ∇ ψ + D 2 + . . . � � L Φ ( Φ ) = tr , ∇ t 2 + i ¯ � � L T ( T , Φ ) = tr χ ∇ χ + . . . , L H ( H , Φ ) = ∇ h 2 + i ¯ λ ∇ λ + h a φ i φ i h a + ¯ λΓ i φ i λ + . . . ,
UV gauge theory: L UV fixed by susy 1 L UV = L Φ ( Φ ) + L T ( T , Φ ) + L H ( H , Φ ) g 2 YM � No H − T couplings
UV gauge theory: L UV fixed by susy 1 L UV = L Φ ( Φ ) + L T ( T , Φ ) + L H ( H , Φ ) g 2 YM � No H − T couplings � L UV has two branches of susy vacua: • Coulomb branch: D1 move away from D5 • Higgs branch: D1 move/dissolve inside D5
UV gauge theory: L UV fixed by susy 1 L UV = L Φ ( Φ ) + L T ( T , Φ ) + L H ( H , Φ ) g 2 YM � No H − T couplings � L UV has two branches of susy vacua: • Coulomb branch: D1 move away from D5 • Higgs branch: D1 move/dissolve inside D5 � g YM dimensionful: theory flows to CFT in IR
UV gauge theory: L UV fixed by susy 1 L UV = L Φ ( Φ ) + L T ( T , Φ ) + L H ( H , Φ ) g 2 YM � No H − T couplings � L UV has two branches of susy vacua: • Coulomb branch: D1 move away from D5 • Higgs branch: D1 move/dissolve inside D5 � g YM dimensionful: theory flows to CFT in IR � IR CFT = CFT C ⊕ CFT H [Witten ’95, ’97]
UV gauge theory: L UV fixed by susy 1 L UV = L Φ ( Φ ) + L T ( T , Φ ) + L H ( H , Φ ) g 2 YM � No H − T couplings � L UV has two branches of susy vacua: • Coulomb branch: D1 move away from D5 • Higgs branch: D1 move/dissolve inside D5 � g YM dimensionful: theory flows to CFT in IR � IR CFT = CFT C ⊕ CFT H [Witten ’95, ’97] � CFT H dual to AdS 3 [Maldacena ’97]
CFT H : L IR � In IR g YM → ∞ so L Φ irrelevant and can be dropped [Witten ’97] L IR ( Φ , T , H ) = L T ( Φ , T ) + L H ( Φ , H )
CFT H : L IR � In IR g YM → ∞ so L Φ irrelevant and can be dropped [Witten ’97] L IR ( Φ , T , H ) = L T ( Φ , T ) + L H ( Φ , H ) � L IR marginal if Φ has geometric dimensions [ φ i ] = 1 , [ A µ ] = 1 , [ Ψ ] = 3 / 2 , [ D ] = 2 while H and T have canonical scaling dimensions [ h a ] = [ t a ] = 0 , [ χ ] = [ λ ] = 1 / 2
CFT H : L IR � In IR g YM → ∞ so L Φ irrelevant and can be dropped [Witten ’97] L IR ( Φ , T , H ) = L T ( Φ , T ) + L H ( Φ , H ) � L IR marginal if Φ has geometric dimensions [ φ i ] = 1 , [ A µ ] = 1 , [ Ψ ] = 3 / 2 , [ D ] = 2 while H and T have canonical scaling dimensions [ h a ] = [ t a ] = 0 , [ χ ] = [ λ ] = 1 / 2 � Φ enter quadratically as auxiliary fields in L IR
CFT H : L ADHM � Φ auxiliary: eliminate using eoms [Witten 97] [Berkooz, Verlinde ’99] [Aharony, Berkooz ’99] L IR ( Φ , T , H ) − → L ADHM ( H , T )
CFT H : L ADHM � Φ auxiliary: eliminate using eoms [Witten 97] [Berkooz, Verlinde ’99] [Aharony, Berkooz ’99] L IR ( Φ , T , H ) − → L ADHM ( H , T ) � L ADHM is ( 4 , 4 ) σ -model with target space M N c , N f the moduli space of N c instantons in su ( N f ) gauge theory
CFT H : L ADHM � Φ auxiliary: eliminate using eoms [Witten 97] [Berkooz, Verlinde ’99] [Aharony, Berkooz ’99] L IR ( Φ , T , H ) − → L ADHM ( H , T ) � L ADHM is ( 4 , 4 ) σ -model with target space M N c , N f the moduli space of N c instantons in su ( N f ) gauge theory � L ADHM gives conventional picture of Higgs branch: D- and F-flatness conditions equivalent to ADHM construction
CFT H : L ADHM � Φ auxiliary: eliminate using eoms [Witten 97] [Berkooz, Verlinde ’99] [Aharony, Berkooz ’99] L IR ( Φ , T , H ) − → L ADHM ( H , T ) � L ADHM is ( 4 , 4 ) σ -model with target space M N c , N f the moduli space of N c instantons in su ( N f ) gauge theory � L ADHM gives conventional picture of Higgs branch: D- and F-flatness conditions equivalent to ADHM construction � L ADHM has small instanton singularity: Metric on M N c , N f singular when instanton size goes to zero
CFT H : states near origin � CFT H states localised near origin of Higgs branch (near small instanton singularity) not captured by L ADHM σ -model
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