Triality of Two-dimensional (0,2) Theories Jirui Guo - PowerPoint PPT Presentation

Triality of Two-dimensional (0,2) Theories Jirui Guo J.Guo,B.Jia,E.Sharpe,arXiv:1501.00987 April 11, 2015

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Outline 2d N = (0,2) Gauge Theories 1 Triality 2 Proposal Checks Low Energy Description Chiral Rings 3 Chiral Rings in (0,2) Theories Bott-Borel-Weil Theorem Examples 4 Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples 2d N = (0,2) Gauge Theories Two fermionic coordinates: θ + , θ + Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples 2d N = (0,2) Gauge Theories Two fermionic coordinates: θ + , θ + Chiral Multiplet Φ : D + Φ = 0 √ + ∂ + φ 2 θ + ψ + − i θ + θ Φ = φ + Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples 2d N = (0,2) Gauge Theories Two fermionic coordinates: θ + , θ + Chiral Multiplet Φ : D + Φ = 0 √ + ∂ + φ 2 θ + ψ + − i θ + θ Φ = φ + √ Fermi Multiplet Ψ : D + Ψ = 2 E (Φ) √ √ + ∂ + ψ − − + E 2 θ + G − i θ + θ Ψ = ψ − − 2 θ Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples 2d N = (0,2) Gauge Theories Two fermionic coordinates: θ + , θ + Chiral Multiplet Φ : D + Φ = 0 √ + ∂ + φ 2 θ + ψ + − i θ + θ Φ = φ + √ Fermi Multiplet Ψ : D + Ψ = 2 E (Φ) √ √ + ∂ + ψ − − + E 2 θ + G − i θ + θ Ψ = ψ − − 2 θ + λ − + 2 θ + θ + D Vector Multiplet: V = v − 2 i θ + λ − − 2 i θ Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples 2d N = (0,2) Gauge Theories d θ + Ψ i J i (Φ) � � J-term: + = 0 � θ Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples 2d N = (0,2) Gauge Theories d θ + Ψ i J i (Φ) � � J-term: + = 0 � θ d θ + Λ | θ + = 0 ,where t = ir + θ Fayet-Illiopoulos term: t � 2 π 4 Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples 2d N = (0,2) Gauge Theories d θ + Ψ i J i (Φ) � � J-term: + = 0 � θ d θ + Λ | θ + = 0 ,where t = ir + θ Fayet-Illiopoulos term: t � 2 π 4 Potential for the scalars: V = e 2 2 D a D a + | E i ( φ ) | 2 + � � | J i ( φ ) | 2 a a Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description Triality Triality: IR equivalence of three 2d (0,2) gauge theories [Gadde,Gukov,Putrov 1310.0818] Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description Triality Triality: IR equivalence of three 2d (0,2) gauge theories [Gadde,Gukov,Putrov 1310.0818] Matter Content Φ Ψ Γ Ω P − − U( N c ) ✷ ✷ 1 det ✷ − SU( N 1 ) 1 1 1 ✷ ✷ − SU( N 2 ) 1 1 1 ✷ ✷ SU( N 3 ) 1 1 1 1 ✷ SU( 2 ) 1 1 1 1 ✷ N c = ( N 1 + N 2 − N 3 ) / 2 Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description Triality Triality: IR equivalence of three 2d (0,2) gauge theories [Gadde,Gukov,Putrov 1310.0818] Matter Content Φ Ψ Γ Ω P − − U( N c ) ✷ ✷ 1 det ✷ − SU( N 1 ) 1 1 1 ✷ ✷ − SU( N 2 ) 1 1 1 ✷ ✷ SU( N 3 ) 1 1 1 1 ✷ SU( 2 ) 1 1 1 1 ✷ N c = ( N 1 + N 2 − N 3 ) / 2 d θ + Tr (ΓΦ P ) � � J-term: L J = + = 0 � θ Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description Triality Triality: IR equivalence of three 2d (0,2) gauge theories [Gadde,Gukov,Putrov 1310.0818] Matter Content Φ Ψ Γ Ω P − − U( N c ) ✷ ✷ 1 det ✷ − SU( N 1 ) 1 1 1 ✷ ✷ − SU( N 2 ) 1 1 1 ✷ ✷ SU( N 3 ) 1 1 1 1 ✷ SU( 2 ) 1 1 1 1 ✷ N c = ( N 1 + N 2 − N 3 ) / 2 d θ + Tr (ΓΦ P ) � � J-term: L J = + = 0 � θ Triality under permutation of N 1 , N 2 , N 3 Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description Quiver Diagram Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description Checks Non-abelian Flavor Anomalies: k SU ( N 1 ) = − 1 4 ( − N 1 + N 2 + N 3 ) k SU ( N 2 ) = − 1 4 (+ N 1 − N 2 + N 3 ) k SU ( N 3 ) = − 1 4 (+ N 1 + N 2 − N 3 ) Invariant under cyclic permutations of N 1 , N 2 , N 3 Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description Checks Central Charge: ( − N 1 + N 2 + N 3 )( N 1 − N 2 + N 3 )( N 1 + N 2 − N 3 ) c R = 3 N 1 + N 2 + N 3 4 c L = c R − 1 4 ( N 2 1 + N 2 2 + N 2 3 − 2 N 1 N 2 − 2 N 2 N 3 − 2 N 3 N 1 ) + 2 Both are invariant under the permutations of N 1 , N 2 , N 3 Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description Checks Central Charge: ( − N 1 + N 2 + N 3 )( N 1 − N 2 + N 3 )( N 1 + N 2 − N 3 ) c R = 3 N 1 + N 2 + N 3 4 c L = c R − 1 4 ( N 2 1 + N 2 2 + N 2 3 − 2 N 1 N 2 − 2 N 2 N 3 − 2 N 3 N 1 ) + 2 Both are invariant under the permutations of N 1 , N 2 , N 3 Elliptic Genus is also invariant under the permutations. Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description General Quiver Gauge Theories The triality transformation can be applied to any gauge node in a general quiver gauge theory [Gadde,Gukov,Putrov 1310.0818] Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description Geometry of the Triality S ⊕ N 3 ⊕ Q ∗⊕ N 1 → G ( n 3 , N 2 ) S ⊕ N 1 ⊕ Q ∗⊕ N 3 → G ( n 1 , N 2 ) = ✉ + - T N 1 , N 2 , N 3 T N 3 , N 1 , N 2 S ∗⊕ N 3 ⊕ Q ∗⊕ N 2 → G ( n 3 , N 1 ) S ∗⊕ N 1 ⊕ Q ∗⊕ N 2 → G ( n 1 , N 3 ) T N 2 , N 3 , N 1 - + + - � � ✉ ✉ S ⊕ N 2 ⊕ Q ⊕ N 3 → G ( n 2 , N 1 ) S ∗⊕ N 2 ⊕ Q ⊕ N 1 → G ( n 2 , N 3 ) N = N 1 + N 2 + N 3 , n i = N − N i 2 Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description IR Fixed Point At the fixed point, the simple global symmetry G is enhanced to the affine symmetry G 2 | k G | Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description IR Fixed Point At the fixed point, the simple global symmetry G is enhanced to the affine symmetry G 2 | k G | Chirality of the affine algebra is determined by the sign of k G Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description IR Fixed Point At the fixed point, the simple global symmetry G is enhanced to the affine symmetry G 2 | k G | Chirality of the affine algebra is determined by the sign of k G Left-moving affine algebra is: R = � 3 i = 1 SU ( N i ) n i × U ( 1 ) NN i Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description IR Fixed Point At the fixed point, the simple global symmetry G is enhanced to the affine symmetry G 2 | k G | Chirality of the affine algebra is determined by the sign of k G Left-moving affine algebra is: R = � 3 i = 1 SU ( N i ) n i × U ( 1 ) NN i � n i ( N 2 i − 1 ) � Sugawara central charge for R is c R = � 3 + 1 i = 1 n i + N i Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description IR Fixed Point At the fixed point, the simple global symmetry G is enhanced to the affine symmetry G 2 | k G | Chirality of the affine algebra is determined by the sign of k G Left-moving affine algebra is: R = � 3 i = 1 SU ( N i ) n i × U ( 1 ) NN i � n i ( N 2 i − 1 ) � Sugawara central charge for R is c R = � 3 + 1 i = 1 n i + N i � H λ λ H λ The full spectrum H = � R , λ runs over all L integrable representations of R Jirui Guo Triality of Two-dimensional (0,2) Theories

2d N = (0,2) Gauge Theories Triality Chiral Rings Examples Proposal Checks Low Energy Description The partition function Z ( τ, ξ i ; τ, η ) = � λ χ λ ( τ, ξ i ) K λ ( τ, η ) Jirui Guo Triality of Two-dimensional (0,2) Theories