Defects preserving N = 2 supersymmetry between two free CFTs - PowerPoint PPT Presentation

Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos Advisor: Prof. Ilka Brunner Faculty of Physics Ludwig Maximilians Universit¨ at M¨ unchen 20/12/2018 Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

Motivation To look into which conformal interfaces preserve N = 2 worldsheet SUSY between two theories CFT 1 and CFT 2 , each compactified on a D -dimensional torus. → Use ”Folding Trick”. Equivalently, to formulate the gluing conditions and construct the boundary states/D-branes that leave N = 2 SUSY unbroken. → Use ”Boundary CFT”. Why??? Applications to: ◮ CFT (Gepner models) ◮ String Theory (D-branes in Type IIB/IIA) ◮ Mathematics (K ¨ ahler Geometry) ◮ Statistical Mechanics Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

The Boundary CFT approach How to impose boundary conditions in String Theory? Open string: Neumann, Dirichlet at the endpoints Closed string: Every point equivalent... Idea: Need a more generic way to describe boundary conditions. Solution: Boundary CFT → Boundary conditions encoded in boundary states, which are coherent states. Let: � S n z − n − h , ˜ � ˜ z − n − h S (¯ S n ¯ S ( z ) = z ) = n ∈ Z n ∈ Z be the generators of the symmetry algebra A defined on the upper-half complex plane and ρ the automorphisms of A . Relate S , ˜ S at the bound- ary, i.e. S ( z ) = ρ (˜ S (¯ z )) , z ∈ R . Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

Boundary states Using this relation and mapping the upper-half plane to the (unit) circle we obtain: ( S n − ( − 1 ) h ρ (˜ S − n ) || B �� = 0 , for all n ∈ Z . This is called gluing condition and || B �� are the boundary states. Boundary states are a linear combination of Ishibashi states, i.e. � || B �� = N i | i �� , i where N i non-negative integers (consistency). In every case, the conformal symmetry should be preserved. This is translated to ( L n − ˜ L − n ) || B �� = 0 . !!! More symmetries → More constraints Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

Free bosonic field theory ( c = 1 , h = 1 ) Primary fields: S ≡ ∂ z X L ( z ) = − i z ) = − i ˜ � a n z − n − 1 , � z − n − 1 S ≡ ∂ ¯ z X R (¯ ˜ a n ¯ 2 2 n ∈ Z n ∈ Z Symmetry algebra: A ≡ u ( 1 ) , ρ = ± id Generators: S n ≡ a n , ˜ S n ≡ ˜ a n ˜ Zero modes: a 0 ≡ p L = k , a 0 ≡ p R = k → p L = p R = k ( k : center-of-mass momentum) Gluing condition: ( a n ± ˜ a − n ) || B �� = 0 ◮ + → Neumann ( n = 0 → k = 0 ) ◮ − → Dirichlet Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

Circle Compactification Purpose: Too many dimensions (26 in bosonic string, 10 in superstring). → Wrap some of them around small compact spaces. Simplest case: Circle S 1 ∼ = R / 2 π R Z with radius R . Identify: X ∼ X + 2 π Rw , where w is the winding number (no analogue in particles). k k Zero modes become: p L = 2 R + wR , p R = 2 R − wR , with k , w ∈ Z !!! → p L � = p R Gluing condition: ( a n ± ˜ a − n ) || B �� = 0 Boundary states: √ � � 1 1 e iw ˜ � φ 0 exp � || 0 , w �� N − na − n ˜ | 0 , w � N , = R a − n 1 2 4 w ∈ Z n > 0 �� � 1 1 1 R φ 0 exp � e − i k || k , 0 �� D √ na − n ˜ | k , 0 � D = a − n 1 R 2 4 k ∈ Z n > 0 Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

Torus compactification D-Torus: T D ∼ = R D / 2 π Λ D . Identify: D X I ∼ X I + 2 π i , w i ∈ Z . � w i e I i = 1 Reduce to D = 2 → T 2 ∼ = S 1 R 1 × S 1 R 2 This is a CFT with c = 2 . → Two real bosons, each compactified on a circle. Zero modes: L = k µ R = k µ p µ + w µ R µ , p µ − w µ R µ , µ = 1 , 2 2 R µ 2 R µ Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

Boundary states for rotated branes Gluing condition: � ˜ � 1 �� a 1 � a 1 �� � 0 n − n + O || B �� = 0 , O = a 2 a 2 ˜ − 1 0 n − n Rotate: D1-brane rotated by an angle θ . Gluing condition becomes: � ˜ � cos 2 θ �� a 1 � a 1 �� � sin 2 θ n − n + O || B �� = 0 , O = ∈ O ( 2 ) a 2 a 2 ˜ sin 2 θ − cos 2 θ n − n Rational rotation ( 0 < θ < π 2 ): tan θ = NR 2 , N , M ∈ N & coprime MR 1 Boundary state: � � � NM 1 e ik α − iw β exp � � na µ a ν || B �� = − − n ˜ − n O µν | kN , wM ; − kM , wN � sin 2 θ k , w ∈ Z n > 0 Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

Free fermionic field theory ( c = 1 / 2 , h = 1 / 2 ) Need also fermions to talk about SUSY! ˜ r ˜ r ψ r z − r − 1 z − r − 1 Primary fields: Ψ( z ) = � 2 , Ψ(¯ z ) = � ψ r ¯ 2 Gluing condition: � � ψ r − i ηρ ( ˜ ψ − r ) || B �� = 0 , where η = ± 1 (+: Neveu-Schwarz, -: Ramond). For a two-fermion theory: � ˜ �� ψ 1 � ψ 1 �� r − r + i O F || B �� = 0 , O F ∈ O ( 2 ) ψ 2 ˜ ψ 2 r − r Boundary state:   � ψ µ − r ˜  | 0 � NS ψ ν || B �� NS = exp  − i − r ( O F ) µν r ∈ N − 1 2 ! Omit the discussion for the Ramond sector. Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

N = 1 -supersymmetric free field theory ( c = 3 / 2 ) Combine the previous results: → Two bosons and two fermions on the upper-half complex plane. Symmetry algebra: N = 1 superconformal algebra Generators: T ( h = 2 ), G ( h = 3 2 ) Gluing conditions: ( L n − ˜ L − n ) || B �� = 0 ( G r − i η ˜ G − r ) || B �� = 0 Boundary state: || B �� = || B �� bos ⊗ || B �� ferm Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

N = 2 -supersymmetric free field theory ( c = 3 ) ! Works only for even spacetime dimensions. Question: Which N = 1 boundary states preserve N = 2 SUSY as well? Preliminaries: Combine two real bosons into a complex boson (complexi- fication). Same for the fermions ( i = 1 , 2 ): 1 1 ( X 2 i − 1 + iX 2 i ) , X − i = ( X 2 i − 1 − iX 2 i ) X + i √ √ = 2 2 1 1 (Ψ 2 i − 1 + i Ψ 2 i ) , Ψ − i = (Ψ 2 i − 1 − i Ψ 2 i ) Ψ + i = √ √ 2 2 Primary fields (left sector): − i n z − n − 1 , ∂ X − i = − i � � ∂ X + i a + i a − i n z − n − 1 = 2 2 n > 0 n > 0 2 , Ψ − i = � r z − r − 1 � r z − r − 1 Ψ + i ψ + i ψ − i = 2 r > 0 r > 0 Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

N = 2 gluing conditions Symmetry algebra: N = 2 superconformal algebra Generators: T , G + , G − , J Gluing conditions (B-type): T , G + = ˜ G + , G − = ˜ T = ˜ G − , J = ˜ J Equivalently: ( L n − ˜ L − n ) || B �� = 0 r − i η ˜ ( G + G + − r ) || B �� = 0 r − i η ˜ ( G − G − − r ) || B �� = 0 ( J n + ˜ J − n ) || B �� = 0 A-type ⇔ B-type (mirror symmetry) Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

B-type gluing conditions B-type gluing conditions: �� a + 1 � ˜ a + 1 � �� n − n + Ω || B �� = 0 a + 2 a + 2 ˜ n − n � ˜ �� a − 1 a − 1 � �� n + Ω † − n || B �� = 0 a − 2 a − 2 ˜ n − n �� ψ + 1 � ˜ ψ + 1 � �� r − r + Ω F || B �� = 0 ˜ ψ + 2 ψ + 2 r − r �� ψ − 1 � ˜ ψ − 1 � �� + Ω † r − r || B �� = 0 ˜ ψ − 2 ψ − 2 F r − r Here: Ω , Ω F ∈ U ( 2 ) ֒ → O ( 4 ) . Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

B-type boundary states (NS-sector) Bosonic part: � � 1 � � a − j a + j − n Ω † n ( a + i − n Ω ij + a − i | k + , k − , w + , w − � , || B �� bos = − − n ˜ − n ˜ N i exp ij ) i n > 0 where | k + , k − , w + , w − � = | kN + 1 , kN − 1 , wM + 1 , wM − 1 ; − kM + 1 , − kM − 1 , wN + 1 , wN − 1 � Fermionic part:     � � − r ˜ ψ − j − r ˜ ψ + j − r (Ω F ) † ( ψ + i − r (Ω F ) ij + ψ − i || B �� NS =  − i  | 0 � NS N i exp ij ) i r ∈ N − 1 2 Full boundary state: || B �� full = || B �� bos ⊗ || B �� NS Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24

The unfolding procedure Conformal interfaces can be described in two equivalent ways: As boundary conditions in the tensor-product theory CFT 1 ⊗ CFT ∗ 2 . 1 As operators mapping the states of CFT 2 to those of CFT 1 . 2 Figure 1: The folding trick Unfolding: Hermitean conjugation and exchange of left with right movers in CFT 2 . Defects preserving N = 2 supersymmetry between two free CFTs Vangelis Giantsos / 24